Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent Optical Imaging to Quantitative Phase Imaging

*Joseph Rosen, Angika Bulbul, Nathaniel Hai and Mani R. Rai*

## **Abstract**

Coded aperture correlation holography (COACH) combines incoherent digital holography with coded aperture imaging. COACH is also a method to record incoherent digital holograms of three-dimensional object scenes. Still, COACH can be used for several other incoherent and coherent optical applications. In this chapter, we survey the prime landmarks on the topic of COACH from two major perspectives: architectures and applications of the various systems. We explore the main configurations of hologram recorders in the COACH systems. For each design, we describe some of the recent implementations of these recorders in optical imaging. We conclude the chapter with general ideas on this technology.

**Keywords:** incoherent holography, digital holography, Fresnel incoherent correlation holography, digital holographic microscopy, phase-shifting interferometry

## **1. Introduction**

Imaging by optical waves has been known in the technology world for centuries [1]. For most of this time, imaging has been direct in the sense that images recorded on the eye retina, photographic film, or electronic sensor have been replicas of the observed scenes. However, the computing revolution of the second half of the twentieth century has opened many possibilities for indirect rather than direct imaging. In indirect imaging, a modified version of the observed scene is transferred from the image sensor to the computer to process and reconstruct the image of the original scene. One of the indirect imaging methods is coded aperture imaging, proposed in the sixties for X-ray imaging [2–4] and later adapted to the visible light using coded phase-masks [5] instead of an array of randomly distributed pinholes used in X-ray imaging [4].

Digital holography [6–8] can also be classified as indirect imaging, although it is special in the sense that the pattern recorded by the image sensor is an interference pattern between two light beams. At least one of the beams originates from the object. However, in the case of incoherent digital holography by self-interference, both interfering beams originate from the object [7]. In 2016, the two different concepts of coded phase-aperture imaging and incoherent digital holography were combined into a new indirect imaging method, dubbed coded aperture correlation holography (COACH) [9]. COACH merges the merits of these two different imaging modalities and enables three-dimensional (3D) imaging with interesting and unexpected features. More specifically, COACH is an electro-optical technique to record digital holograms of two- and three-dimensional scenes, where at least part of the light from the object passes through a coded phase-mask. COACH was initially proposed as an additional method to record incoherent digital holograms without scanning and evolved in several different directions. The COACH concept was inspired by several previous methods and systems [2–5, 10–12] and has already stimulated several studies since then [13–33]; some of them are mentioned in the following. This chapter provides an overview of research activities in the technology of COACH done by several researchers in the field.

About a year after the invention of COACH, a simpler version of it was proposed. This version operates without two-wave interference and can demonstrate some applications, such as 3D imaging. The modified version was called interferenceless COACH (I-COACH) [15]. Usually, the I-COACH is preferred whenever an application can be performed by both COACH and I-COACH with the same quality. This rule of thumb is reasonable because the calibration of a single wave system, such as I-COACH, is simpler, and its noise immunity is higher than that of an incoherent interferometer, such as COACH. However, not all the applications successfully implemented by COACH can be executed by I-COACH, and examples are given in the following.

COACH and I-COACH were initially invented for 3D imaging of incoherently illuminated scenes. Recently, the concept of coherent COACH with and without two-wave interference has been examined [34–36]. A central application of coherent digital holography is quantitative phase imaging (QPI) [37], and hence in the following, we review different ways to implement QPI using COACH [35, 36].

This review consists of six main sections. The development of incoherent COACH and I-COACH architectures, with different modalities and characteristics, are reviewed in the following two sections. We describe the coherently illuminated I-COACH and COACH techniques in the fourth and fifth sections, respectively. The concluding section summarizes the review.

## **2. Incoherent COACH**

Incoherent COACH belongs to the family of self-interference digital holography systems [38]. The general optical configuration of these systems is shown in **Figure 1**. The flow of information starts from the light emitted from each object point in the upper part of **Figure 1**. The light propagates toward a beam splitting unit and is split into two waves. Each wave is modulated differently by a modulation component. The two waves originate from the same object point and hence are mutually coherent, although the light emitted, or reflected, from the object is spatially incoherent. Therefore because of the mutual coherence, the two waves with different wavefronts interfere at the sensor plane. The image sensor accumulates the entire interference patterns of all the input points to an incoherent hologram. A single hologram, or

*Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

**Figure 1.**

*Recording and reconstruction of holograms in a general self-interference digital holography system.*

several acquired holograms, are introduced into a digital computer, where the various operations of the digital processor are schematically shown in the lower part of **Figure 1**. In the case of several holograms, they are superposed into a single digital hologram. Finally, the image of the object is reconstructed from the processed hologram by an appropriate numerical algorithm.

COACH was proposed as a generalized case of Fresnel incoherent correlation holography (FINCH) [10–12], a well-known technique of recording holograms, which also belongs to the self-interference systems. In FINCH, a quadratic phase-mask modulates at least one of the two waves. In COACH, on the other hand, the quadratic phase-mask of FINCH is substituted by a diffractive chaotic phase-aperture. The initial goal of COACH was like FINCH, that is, to acquire a hologram of the 3D observed scene illuminated by quasi-monochromatic spatially incoherent light. COACH's optical scheme is depicted in **Figure 2**. The light from an object is split into two beams, and only one of the object beams is modulated by the chaotic mask termed coded phase-mask (CPM). The modulated beam is coherently interfered with the unmodulated object beam due to their common origin. Because COACH is on-axis system, it needs a phase-shifting procedure and complex hologram synthesis [39]. That means that three holograms are recorded, each of which with the CPM multiplied by a different phase-constant. The three holograms are superposed digitally in the computer such that the result is a complex-valued hologram. This digital hologram is reconstructed into a single image without the twin image and the bias term.

Unlike other well-known incoherent hologram recorders, such as FINCH [10–12, 40] and Michelson-interferometer-based incoherent holographic systems [41–44], COACH does not have a defined image plane where the wavefront can numerically propagate from the hologram to the reconstruction plane. Hence, COACH has different recording and reconstruction procedures. In other words, COACH consists of a two-step recording procedure: a one-time calibration and then imaging. In the

#### **Figure 2.**

*Schematic diagram of coded aperture correlation holography (COACH). SLM - spatial light modulator.*

initial stage of the calibration, one illuminates a moving pinhole along the optical axis, and the image sensor records a point spread hologram (PSH) for every axial location of the pinhole. The set of PSHs is accumulated in a library for later use in the imaging stage. Following the calibration process, an object hologram is recorded under the same restrictions and with the coded apertures as the PSH acquisition. The 3D image of the observed scene is reconstructed by a two-dimensional (2D) cross-correlation between the object hologram and the corresponding elements of the PSH library.

Although FINCH influenced the COACH structure, COACH has different features than FINCH. The image reconstruction has been modified to 2D cross-correlations with guidestar responses instead of the Fresnel back-propagation of FINCH [10–12, 40]. Compared to FINCH, COACH has better axial resolution but worse lateral resolution [9, 45]. However, the main difference is that COACH can do the same holographic 3D imaging without two-wave interference [15]. Nevertheless, several applications can only be performed by a version of the original COACH with two-wave interference. One of such applications is a one-channel-at-a-time incoherent synthetic aperture imager [46], summarized next.

### **2.1 One-channel-at-time incoherent synthetic aperture**

An interesting application for COACH is incoherent imaging with synthetic aperture (SA). SA is a familiar super-resolution method and a conventional technique in astronomy to accomplish image resolution beyond the diffraction limit dictated by the physical aperture [47] of the telescope. Since its invention a century ago [48], incoherent SA imaging was usually realized by at least two optical channels operating simultaneously. The wave interference between two incoming light beams, both originated from the same object, was recorded over time from several viewpoints within the SA region. Then, the interference intensity patterns were processed to produce an image of the object with a resolution equivalent to complete SA [22, 48]. A singlechannel SA is possible for cases of imaging systems with coherent light [49], but astronomical imaging is usually done with incoherent light sources. A solution to this double-channel problem of SA incoherent imaging is the lately proposed incoherent single-channel SA technique termed one-channel-at-time incoherent synthetic aperture imager (OCTISAI) [46].

As in many other COACH systems, the CPM of OCTISAI is synthesized using a modified version of the Gerchberg-Saxton algorithm (GSA) [50]. Then, the CPM is

## *Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

divided into *N* (in the following example *N* = 64) equal parts for the SA implementation. The optical setup of the OCTISAI experiment is shown in **Figure 3** and described next. The system is first calibrated by collimating the light diffracted from a pinhole, where the collimating lens *L*<sup>1</sup> mimics the far-field imaging condition. A polarizer *P*<sup>1</sup> polarizes the collimated light to be oriented at 450 regarding the active orientation of a spatial light modulator (SLM). The SLM is used as the display on which the CPMs of OCTISAI and all other systems in this chapter are displayed. Only a partial area of the SLM is used at a time, and all other parts are activated in a raster scan mode. Because of the polarization angle, the light is split into two orthogonal linear polarizations beyond the SLM. The CPM modulates one polarized wave, and the other wave passes the SLM without any change. Beyond the polarizer *P*2, also oriented at 450 to the SLM's active axis, both beams have the same orientation enabling to record a pattern of interference between the two beams. The interference pattern between the modulated and unmodulated beams is captured by the image sensor. Three phase-shifted PSHs for the input point object (pinhole) are recorded for every partial aperture at each position in the SA region. Then, three phase-shifted object holograms are captured for the input object with the same phase-apertures as before in the calibration. Next, using the digital computation capabilities, the entire PSH parts are stitched together into one synthetic PSH. The parts of the object hologram are also processed into one synthetic object hologram by a similar procedure. The final image with the enhanced resolution is obtained by a 2D cross-correlation between the two synthetic holograms.

The complete experiment of OCTISAI is extensively described in [46], and here we briefly describe only the main results. In the experiment, a collection of PSHs was produced using three CPMs, each having a phase-shift exp(*iθj*), where *θ*1,2,3 = 0<sup>o</sup> , 120<sup>o</sup> , and 240o . A pinhole of 25 μm diameter was positioned in the input. After the PSH creation, group 3, element 1 of the United States Air Force (USAF) negative resolution chart, replaced the pinhole. We recorded the three object holograms with the same three CPMs used for the PSHs. The synthetic object holograms and PSHs were produced by stitching respective partial holograms and superimposing corresponding

#### **Figure 3.**

*The tabletop experimental setup for one-channel-at-time incoherent synthetic aperture imager (OCTISAI) inside the blue rectangle,* BS*<sup>1</sup> and* BS*<sup>2</sup> – beamsplitters, CMOS camera - Complementary metal-oxide-semiconductor camera,* L*01,* L*02, and* L*<sup>1</sup> - refractive lenses, LED1 and LED2 - identical light-emitting diodes,* P*<sup>1</sup> and* P*<sup>2</sup> – polarizers, SLM spatial light modulator, and USAF - United States Air Force resolution target. Adapted from [46].*

**Figure 4.**

*(a1, a2) COACH reconstructed images and (a3, a4) direct images of limited aperture (a1, a3) and full aperture (a2, a4), magnitude and phase of (b1, b2) PSH and (b3, b4) object holograms of the complete SA, (c1-c6) reconstructed images after stitching of (c1) 8 central horizontal sub-holograms, (c2) eight central vertical subholograms, (c3) 2 2, (c4) 4 4, (c5) 6 6 central sub-holograms, and (c6) full 64 (8 8) sub-holograms. Adapted from [46].*

synthetic three-intensity responses. Finally, the object hologram was cross-correlated with the phase-only filtered version of the synthetic PSH. The outcome of this crosscorrelation is the final reconstructed image. The COACH images related to the partial and complete apertures are shown in **Figures 4(a1)** and **(a2)**, respectively. For comparison, **Figures 4(a3)** and **(a4)** show the corresponding images of direct imaging with a setup of a single lens and similar numerical apertures. The stitched holograms after the superposition are shown in **Figure 4(b)**. **Figure 4(c)** presents the reconstructed images for OCTISAI with various area sizes of the SA holograms. **Figures 4(c1)** and **(c2)** are produced using the central eight, horizontally [**4(c1)**] and vertically, [**4(c2)**] stitched partial holograms, respectively. **Figures 4(c3)-(c6)** show the reconstruction results with 2 2, 4 4, and 6 6 central sub-holograms, and the entire 64 sub-holograms. The resolution enhancement by raising the number of stitched partial holograms is demonstrated. Comparing **Figure 4(c6)** with **Figures 4 (a1)** and **(a3)**, one can conclude that OCTISAI's images have higher resolution than the images taken with a limited aperture in both techniques of COACH and direct imaging.

## **3. Interferenceless incoherent COACH**

As mentioned above, interferenceless coded aperture correlation holography (I-COACH) was published in 2017 [15] as a simpler configuration of the earlier proposed COACH [9]. Both systems spatially modify incoherent light by chaotic phase-masks. However, unlike COACH, I-COACH records holograms without twobeam interference. I-COACH is an incoherent 3D imaging method in which the image is digitally obtained by numerical 2D cross-correlation between the hologram of the object and the library of PSHs. The PSHs are recorded once in the calibration mode of the system, before the imaging stage, as shown in **Figure 5**. The same chaotic CPMs modulate the light waves in both the calibration and imaging stages. The modulated light is recorded by a digital camera after propagating in the free space. I-COACH system without two-beam interference can produce similar results as COACH because the intensity point-response of I-COACH on the sensor plane is highly sensitive to the *Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

**Figure 5.**

*Schematic diagram of interferenceless coded aperture correlation holography (I-COACH). The upper scheme refers to the calibration mode, whereas the lower describes the imaging mode.*

axial location of the input point. Mathematically, the high sensitivity means that the cross-correlation between two intensity responses for two points located at two different axial positions is much smaller than the autocorrelation of each response [15]. Thus, the entire object points can be reconstructed in the 3D image space using 2D cross-correlations between a multi-point object and the library PSHs. The early configuration of I-COACH [15] has been developed into different systems with various architectures and with a variety of algorithms [16–33], each with strengths and weaknesses. The typical design of I-COACH is shown in **Figure 5**, where the same physical setup is schematically depicted in two modes of operation. The upper scheme shows the calibration process, in which the system collects a library of PSHs acquired for an object point positioned at different axial locations. When the library is completed, the same setup works in the imaging mode shown in the lower part of **Figure 5**. An incoherently illuminated 3D object replaces the single point in the system's input. The object intensity response recorded by the sensor is 2D cross-correlated with each PSH of the library. The assembly of cross-correlation results is the desired reconstructed 3D image. This general scheme describes most I-COACH types and has been the basis for developments that have evolved since 2017 [16–31]; one example of an I-COACH application is depth-of-field engineering, briefly described next.

## **3.1 Depth-of-field engineering**

Long depth-of-field (DOF) in imaging systems has been important for many applications [51]. Generally, the DOF is dictated by the numerical aperture of the optical system. Reducing the numerical aperture extends the DOF, but it also unfavorably decreases the lateral image resolution of the system. Several methods have been advanced to extend the DOF of the optical system [51–59] with a minimal resolution decrease. Still, the complicated experimental and computational requirements have stimulated a search for simpler methods. This subsection reviews a new technique proposed first in [60] to engineer the DOF of imaging systems. DOF

engineering is done by integrating radial quartic phase-functions (RQPFs) [61, 62] into the incoherent I-COACH shown schematically in **Figure 6**. The phase-mask displayed on the SLM of **Figure 6** is a fusion of three separate phase-masks. The first is the chaotic CPM generated by GSA with the constraints of sparse dots on the camera plane [63] and a constant magnitude on the SLM plane. The second element is a positive diffractive lens used to fulfill the 2D Fourier relations of the GSA between the planes of the SLM and camera. The focal length of the diffractive lens *f* is determined such that each object is imaged on the camera. In other words, for the distance object-SLM (*dOS*) and SLM-camera (*dSC*), the three lengths satisfy the imaging equation 1*=f* ¼ ð Þþ 1*=dOS* ð Þ 1*=dSC* . The distance object-SLM is chosen as the distance from the center of the object space to the SLM. The third mask is the above-mentioned RQPF implemented to extend the DOF as desired. The RQPF with the phase-function exp *i*2*π*ð Þ *r=p* <sup>4</sup> h i stretches the DOF of the sparse dots created by the CPM, where *<sup>p</sup>* is the modulation parameter controlling the length of the DOF, and *r* is the radial coordinate on the SLM plane. Near the back focal point of the diffractive lens on the camera plane, the RQPF generates sword beams with an almost constant intensity along a controlled propagation distance and a relatively narrow beam-like shape in any transverse plane [61, 62]. The 3D location and the length of the DOF can be determined by changing the parameters of the RQPF and the focal length of the lens. Multiplexing various threesomes of phase-masks (diffractive lens, CPM, and RQPF) with different modulation parameters can create various focusing curves. For instance, an imaging system that can image objects in two non-connected sub-volumes in the object space. In this example, the entire objects inside these sub-volumes remain in focus, while the images outside these sub-volumes are blurred and seen out-of-focus. The unusual DOF enables to image targets in specific sub-volumes simultaneously (or successively), whereas objects in other sub-volumes are blurred. Moreover, the engineered DOF allows to transversely shift an image from one volume relative to another image from another volume. Mutual transverse shifts of sub-volumes can avoid overlap between images when one object is behind or in front of another object.

Back to **Figure 6**, the incoherent light source critically illuminates the observed 3D scene using a lens *L*0. In this scheme, an object volume is defined as the volume along *z* for which the DOF is extended, such that each object inside the volume produces an in-focus image in the output. Off-axis sub-volumes in **Figure 6** indicate images of onaxis objects that are reconstructed out of the *z*-axis in the output due to an additional linear phase-mask attached to the other three-phase-masks (diffractive lens, CPM,

#### **Figure 6.**

*Optical scheme of the depth-of-field engineering system. DL - diffractive lens, CPM - coded phase-mask, RQPF radial quartic phase-function. Adapted from [60].*

*Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

#### **Figure 7.**

*(a, b) Direct images of the objects with two different lenses and (c) Reconstructed images from a single hologram using depth-of-field engineering. Adapted from [60].*

and RQPF). The light emitted from the object scene is modulated by the combination of the phase-masks displayed on the SLM. For any point inside the object volume, the intensity recorded by the camera is distributed in the form of chaotically sparse dots. Like other I-COACH schemes, this dot pattern is used as the PSH, which reconstructs the image of any object by cross-correlation with the object hologram. In previous demonstrations of I-COACH [15, 16], the PSH has been recorded experimentally by illuminating a pinhole positioned at the system input. However, in [60], the PSH is digitally computed based on the known experimental parameters. The object reconstruction is done by a nonlinear cross-correlation [21] between the computed PSH and the object hologram.

Next, we show results of only a single object volume. In other words, the following I-COACH has extended DOF compared to direct imaging with the same numerical aperture. More complicated examples of DOF engineering can be found in Ref. [60]. The proposed technique is verified by an experimental setup like the scheme in **Figure 6**. Unlike typical I-COACH systems, two targets are positioned in two separate channels of the experimental setup such that they are located at each end of the object volume [60]. Two LEDs with refractive lenses separately illuminated the object in each channel. Both objects are from the USAF transmission resolution chart. In one of the channels, the object is element 6 of Group 2, and in the other channel, element 1 of Group 3 is used as the object. The targets are located at distances of 24 and 26 cm from the SLM, respectively. Light from the two targets was combined by a beamsplitter and projected on the SLM. The gap between the SLM and the camera was 22 cm. Direct images of the objects [shown in **Figures 7(a)** and **(b)**] on the camera plane were achieved by displaying only a single diffractive lens on the SLM with the focal length that satisfies the imaging equation, each for a different object in its own depth. For the I-COACH system, the PSH was computed using the optimal CPM that yielded ten randomly distributed dots on the camera plane. The reconstructed images of I-COACH are shown in **Figure 7(c)**. In the case of direct imaging, it is clear from **Figures 7(a)** and **(b)** that the axial gap between the targets was too large to focus both targets at the same time. On the other hand, in the case of the I-COACH with the engineered DOF of 3 cm, the reconstructed images of **Figure 7(c)** show that both targets are in focus without any resolution decrease.

## **4. Interferenceless coherent COACH**

Optical recording of digital holograms with coherent light traditionally involves interference between object and reference waves, complicating the image acquisition [39]. With the coherent I-COACH, the concept of the coded aperture is adapted from the area of incoherent holography to record digital holograms of three-dimensional coherently illuminated scenes without two-wave interference or any kind of scanning. In addition to the obvious advantages of combining interferenceless holographic systems with coherent light, the proposed method enables relatively rapid image acquisition made possible by its inherent high signal-to-noise ratio (SNR). In [34], the I-COACH method was implemented for generating coherent holograms without interference between reference and object waves. The technique, called interferenceless coherent coded aperture correlation holography (IC-COACH), creates a bi-polar digital hologram of a 3D scene from two camera shots where the scene is illuminated by coherent laser light. The 3D image of the observed scene is reconstructed from the hologram by a deconvolution-like process.

To understand the evolution from incoherent to coherent I-COACH, we briefly summarize the principles of incoherent I-COACH first. Generally, an incoherent I-COACH hologram denotes a 2D function containing an image of a 3D scene, such that the image can be digitally reconstructed from the 2D function. Mathematically, the 2D digital hologram of a 3D object is given by,

$$H\_{OBI}(\overline{r}) = \int I\_{OBI}(\overline{r}; z) \ast p(\overline{r}; z) dz,\tag{1}$$

where ∗ is 2D convolution at each *z* plane, *r* ¼ ð Þ *x*, *y* are the transverse coordinates, and *p*ð Þ *r*; *z* is the PSH of the recording system, which can be a general complex [15] or bi-polar real [16] function. The library of PSHs is a priori acquired in a calibration process with a guidestar, in which each *p r*; *zj* � � from the PSH library is computed as a response to an object point at *zj*. Once the library is ready, and an object hologram is recorded, 2D cross-correlations between the object hologram and each PSH from the library reconstruct each *zj* plane of the 3D image. This computation process is based on the linearity of incoherent optical systems with 2D intensity signals expressed by the following familiar convolution,

$$I\_{Out}(\overline{r}) = I\_{In}(\overline{r}) \* |h(\overline{r})|^2,\tag{2}$$

where *h*ð Þ*r* is the coherent point spread function of the optical system. *IIn*ð Þ*r* and *IOut*ð Þ*r* are the system input and output intensities, respectively. In contrast to incoherent, coherent optical systems are linear in corresponding to 2D complex amplitudes, and they obey the relation,

$$I\_{Out}(\overline{r}) = \left| A\_{In}(\overline{r}) \* h(\overline{r}) \right|^2,\tag{3}$$

where *AIn*ð Þ*r* is the input 2D complex amplitude fulfilling the equation *IIn*ð Þ¼ *r* j j *AIn*ð Þ*r* 2 *:* Because of the nonlinearity of Eq. (3), the implementation of the I-COACH concept in the coherent system is possible only for special cases. Hence, the coherent processor should be adapted in such a way that can satisfy the relation,

$$\left|A\_{\ln}(\overline{r})\*h(\overline{r})\right|^2 \approx \left|A\_{\ln}(\overline{r})\right|^2 \* q(\overline{r}),\tag{4}$$

where *AIn*ð Þ*r* represents a broad set of input objects, and we assume that *h*ð Þ*r* and *q*ð Þ*r* are nontrivial functions. Eq. (4) is satisfied if *h*ð Þ*r* is a set of points distributed over the *Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

camera plane such that the gap between any two points is wider than the size of *AIn*ð Þ*r* [34]. The pattern of the random points on the camera plane is considered the system's PSH. Hence, the CPM replicates the object to a set of the same images chaotically distributed over part of the camera plane. Such CPMs are created by a modified version of GSA [50], in which iterative transformations between the CPM's plane and the spectral plane are done with suitable constraints at each plane. The constraint at the CPM's plane is a constant magnitude distribution because the CPM is displayed on a phase-only SLM. In the spectral plane, which is also the camera plane, the intensity is constrained to be in a shape of randomly distributed dots over all or part of the plane.

The optical configuration of the IC-COACH system of [34] shown in **Figure 8** is based on the classical 4-f spatial filtering system, with the SLM positioned at the Fourier domain and the camera at the image plane. In this setup, the spatial spectrum of the object is modulated by the CPM displayed on the SLM. The CPM was produced by the GSA to duplicate the input object over an ensemble of points randomly distributed at the camera plane. Two different chaotic CPMs are sequentially displayed on the SLM to create two different random sets of replications of the object. These two sets are subtracted from each other to produce a bi-polar object hologram. The ability of IC-COACH to image multi-plane objects is accomplished by multiplexing on the SLM, a few independent CPMs, each of which yields an in-focus different set of dots on a different transverse plane. Each ensemble of out-of-focus dots becomes focused on the camera plane for a point object positioned at the corresponding transverse plane. Lastly, the desired transverse image of the observed 3D scene is reconstructed by cross-correlation between the object hologram and the corresponding PSH. **Figure 9** shows the reconstructed images for two different planes and two different gaps between the object planes, forming two different multi-plane scenes.

#### **Figure 8.**

*Experimental setup of interferenceless coherent coded aperture correlation holography (IC-COACH) with two independent illumination channels. BS1,2,3: Beamsplitters, M1,2: mirrors, and SLM: spatial light modulator. Adapted from [34].*

#### **Figure 9.**

*Image reconstructions of different object planes obtained by a correlation with the corresponding PSH in IC-COACH. Adapted from [34].*

Experimental demonstration for imaging diffusely reflective objects also appears in [34], making the IC-COACH system suitable for processing speckle images obtained by coherent illumination.

## **5. Coherent COACH with two-wave interference**

IC-COACH described in the previous section is an adaptation of the incoherent I-COACH to the case of coherent illumination. This system is capable of imaging 3D scenes holographically, but it cannot do phase imaging of any kind. To enable QPI of transparent objects, COACH has been integrated with a Mach-Zehnder interferometer [35, 36]. QPI, in general, is done by capturing the wavefront passing through thin transparent objects and converting it to an optical thickness map of the examined objects. This method is useful for many applications, including label-free biological cell imaging [64, 65] and nondestructive quality tests [66, 67].

Like the previous demonstrations of I-COACH [34, 63], the image of the observed object is projected to randomly and sparsely distributed replications over the camera plane. As before, the replications are obtained by a pseudorandom CPM synthesized by modified GSA [50]. The CPM is displayed on a phase SLM in the configuration of the coherent sparse COACH (CS-COACH) shown in **Figure 10**. The image sensor records the interference pattern between the waves of the image replications and of a reference tilted plane wave as follows:

*Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

#### **Figure 10.**

*Optical configuration of coherent sparse COACH. MO: Microscope objective, BS1,2,3: Beamsplitters, M1: Mirror, and SLM: Spatial light modulator. Adapted from [36].*

$$I(\mathbf{x}, \mathbf{y}) = \left| O(\mathbf{x}, \mathbf{y}) \exp\left[j\phi(\mathbf{x}, \mathbf{y})\right] \* \sum\_{i=1}^{N} \delta(\mathbf{x} - \mathbf{x}\_{i}, \mathbf{y} - \mathbf{y}\_{i}) + \mathbf{R} \cdot \exp\left[j\frac{2\pi}{\lambda} \left(\mathbf{x}\sin\theta\_{\mathbf{x}} + \mathbf{y}\sin\theta\_{\mathbf{y}}\right)\right]\right|^{2},\tag{5}$$

where *O*(*x, y*) is the object amplitude, and *ϕ*(*x, y*) is its phase, *R* is the reference wave amplitude, *λ* is the illumination wavelength, *N* is the number of image replications, (*xi*, *yi*) are the displacement values of the *i*-th replica from the camera origin, and (*θx, θy*) are the angles between the object and reference waves in the *x-z* and *y-z* planes, respectively. It should be noted that off-axis holography is used to acquire holograms by a single camera shot. A digital filtering process in the spatial frequency domain eliminates the bias term and the twin image from the recorded intensity pattern. The processed object hologram is:

$$H\_{\rm OB/}(\mathbf{x}, \boldsymbol{y}) = \boldsymbol{R}^\* \cdot \mathcal{O}(\mathbf{x}, \boldsymbol{y}) \exp\left[j\phi(\mathbf{x}, \boldsymbol{y})\right] \ast \sum\_{i=1}^N \delta(\mathbf{x} - \mathbf{x}\_i, \boldsymbol{y} - \boldsymbol{y}\_i). \tag{6}$$

This hologram includes several randomly distributed replications of the object over the image plane. Like the procedure explained in [34, 63], the reconstruction of the object's complex amplitude is performed by 2D cross-correlation between the object hologram *HOBJ* and the PSHs. **Figures 11(a)** and **(b)** show the phase-image of polystyrene microspheres (FocalCheck, 6 μm diameter) with the proposed CS-COACH method. For comparison purposes, the phase-images extracted from a regular Mach-Zehnder interferometer using conventional off-axis holography are shown in **Figures 11(c)** and **(d)**. It is apparent that the image of CS-COACH has higher SNR than the conventional technique. This advantage is attributed to the averaging procedure over several replications accompanied by the reconstruction using crosscorrelation. Noise reduction is one of the several advantages of CS-COACH in comparison to open-aperture equivalent systems. Another advantage presented in Ref. [36] is extending the field-of-view (FOV) of the imaging system. Extended FOV realized with the same focal length of the microscope objective and without sacrificing the image resolution is an important advantage in microscopy.

#### **Figure 11.**

*Reconstructed phase images of polystyrene microspheres and the phase cross-sections using (a)-(b) CS-COACH and (c)-(d) conventional off-axis holography. Units of the left panel color bars are radian. Adapted from [36].*

## **6. Discussion and summary**

For all its forms, COACH is a rapidly evolving technology because of the desire to enhance the resulting images and due to the new applications supported by the method. Any technology of imaging is expected to be as quickly as possible with the least camera shots. While the early version of I-COACH [15] operated with three camera shots taken under three independent CPMs, the number of shots and CPMs was decreased to two in [16]. By multiplexing two CPMs in space instead of time as before [15, 16], a single-camera shot was applied in Ref. [18]. I-COACH [19] and CS-COACH [36] with extended FOV were demonstrated by calibrating the systems with extended PSHs beyond the conventional FOV. The numerical reconstruction procedure was changed in [21] by substituting the ordinary linear cross-correlation with new nonlinear cross-correlation optimized to yield a correlation distribution with the lowest entropy. A different nonlinear cross-correlation with other cost-function in the optimization process was employed in [26, 30]. Some of the noise on the resulting images in the early versions [15, 16] appeared because of the low-intensity level per pixel of the PSH on the sensor plane. This difficulty was treated in [63] by imposing a PSH with the structure of sparse dots of light intensity distributed chaotically inside a limited region. The same problem was differently solved in [30] with PSHs of a ring shape. The electro-optical calibration in the upper part of **Figure 5** was changed by a

## *Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

pure digital technique of synthesizing the library of PSHs in the computer [68]. Lateral resolution can be considered one of the holy grails of optical imaging. Improving the lateral resolution by I-COACH has been treated in [23, 45, 69, 70] by different approaches. Usually, I-COACH's lateral and axial resolutions are the same as those of lens-based imaging systems with the same numerical aperture. The methods of [23, 45, 69, 70] improve the lateral resolution beyond the diffraction limit enforced by the finite numerical aperture of optical systems. In [23], resolution-enhanced images of the observed objects are reconstructed by a nonlinear cross-correlation between object holograms and PSHs. In [69, 70], a CPM displayed on the SLM was introduced between the object and the input aperture of a regular lens-based imager. Thus, the effective numerical aperture was increased beyond the characteristic numerical aperture of the imaging system. The effective numerical aperture and the improved resolution limits can be tuned by altering the scattering degree of CPMs [69, 70]. Other applications of COACH and I-COACH and their context in a frame of systems with dynamic diffractive phase-apertures are reviewed in [17, 71, 72].

To conclude this review, we note that COACH for all its modes is based on the extension of the resources available for imaging in a few ways. First, the real-valued aperture function of ordinary direct imaging is replaced with the complex-valued aperture function of COACH. Second, the COACH aperture is modified over time in the multiple-shot versions. Finally, an additional stage of digital processing is integrated with the optical system. These additional resources add to the COACH system new capabilities and unique features. Even though I-COACH is a simpler form of COACH and thus is preferred for many 3D imaging projects, there are some unusual applications in which COACH with two-beam interference is required. Incoherent synthetic aperture imagers [20, 22], the hybrid FINCH-COACH system [45], and quantitative phase-imagers [35, 36] are characteristic examples of systems that twobeam interference is necessary for their operations. However, other applications can be implemented successfully on I-COACH; some are presented herein others might be proposed in the future.

## **Author details**

Joseph Rosen\*, Angika Bulbul, Nathaniel Hai and Mani R. Rai School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel

\*Address all correspondence to: rosenj@bgu.ac.il

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Mait JN, Euliss GW, Athale RA. Computational imaging. Advances in Optics and Photonics. 2018;**10**:409-483

[2] Ables JG. Fourier transform photography: A new method for X-ray astronomy. Proceedings of the Astronomical Society of Australia. 1968; **1**:172-173

[3] Dicke RH. Scatter-hole cameras for X-rays and gamma rays. Astrophysics Journal. 1968;**153**:L101

[4] Fenimore EE, Cannon TM. Coded aperture imaging with uniformly redundant arrays. Applied Optics. 1978; **17**:337-347

[5] Chi W, George N. Optical imaging with phase-coded aperture. Optics Express. 2011;**19**:4294-4300

[6] Goodman JW, Lawrence RW. Digital image formation from electronically detected holograms. Applied Physics Letter. 1967;**11**:77-79

[7] Liu J-P, Tahara T, Hayasaki Y, Poon T-C. Incoherent digital holography: A review. Applied Sciences. 2018;**8**:143

[8] Javidi B, Carnicer A, et al. Roadmap on digital holography. Optics Express. 2021;**29**:35078-35118

[9] Vijayakumar A, Kashter Y, Kelner R, Rosen J. Coded aperture correlation holography—a new type of incoherent digital holograms. Optics Express. 2016; **24**:12430-12441

[10] Rosen J, Brooker G. Digital spatially incoherent Fresnel holography. Optics Letters. 2007;**32**:912-914

[11] Brooker G, Siegel N, Wang V, Rosen J. Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy. Optics Express. 2011;**19**:5047-5062

[12] Bouchal P, Kapitán J, Chmelík R, Bouchal Z. Point spread function and two-point resolution in Fresnel incoherent correlation holography. Optics Express. 2011;**19**:15603-15620

[13] Vijayakumar A, Rosen J. Spectrum and space resolved 4D imaging by coded aperture correlation holography (COACH) with diffractive objective lens. Optics Letters. 2017;**42**: 947-950

[14] Vijayakumar A, Kashter Y, Kelner R, Rosen J. Coded aperture correlation holography system with improved performance. Applied Optics. 2017;**56**: F67-F77

[15] Vijayakumar A, Rosen J. Interferenceless coded aperture correlation holography—a new technique for recording incoherent digital holograms without two-wave interference. Optics Express. 2017;**25**: 13883-13896

[16] Kumar M, Vijayakumar A, Rosen J. Incoherent digital holograms acquired by interferenceless coded aperture correlation holography system without refractive lenses. Scientific Reports. 2017;**7**:11555

[17] Rosen J, Anand V, Rai MR, Mukherjee S, Bulbul A. Review of 3D imaging by coded aperture correlation holography (COACH). Applied Sciences. 2019;**9**:605

[18] Rai MR, Vijayakumar A, Rosen J. Single camera shot interferenceless coded aperture correlation holography. Optics Letters. 2017;**42**:3992-3995

*Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

[19] Rai MR, Vijayakumar A, Rosen J. Extending the field of view by a scattering window in I-COACH system. Optics Letters. 2018;**43**:1043-1046

[20] Bulbul A, Vijayakumar A, Rosen J. Partial aperture imaging by systems with annular phase coded masks. Optics Express. 2017;**25**:33315-33329

[21] Rai MR, Vijayakumar A, Rosen J. Nonlinear adaptive three-dimensional imaging with interferenceless coded aperture correlation holography (I-COACH). Optics Express. 2018;**26**: 18143-18154

[22] Bulbul A, Vijayakumar A, Rosen J. Superresolution far-field imaging by coded phase reflectors distributed only along the boundary of synthetic apertures. Optica. 2018;**5**:1607-1616

[23] Rai MR, Vijayakumar A, Ogura Y, Rosen J. Resolution enhancement in nonlinear interferenceless COACH with a point response of subdiffraction limit patterns. Optics Express. 2019;**27**: 391-403

[24] Ji T, Zhang L, Li W, Sun X, Wang J, Liu J, et al. Research progress of incoherent coded aperture correlation holography. Laser and Optoelectronics Progress. 2019;**56**:080005 (in Chinese)

[25] Anand V, Ng SH, Maksimovic J, et al. Single shot multispectral multidimensional imaging using chaotic waves. Scientific Reports. 2020;**10**:13902

[26] Liu C, Man T, Wan Y. Optimized reconstruction with noise suppression for interferenceless coded aperture correlation holography. Applied Optics. 2020;**59**:1769-1774

[27] Jiang Z, Yang S, Huang H, He X, Kong Y, Gao A, et al. Programmable liquid crystal display based noise

reduced dynamic synthetic coded aperture imaging camera (NoRDS-CAIC). Optics Express. 2020;**28**: 5221-5238

[28] Dubey N, Rosen J, Gannot I. Highresolution imaging with an annular aperture of coded phase masks for endoscopic applications. Optics Express. 2020;**28**:15122-15137

[29] Anand V, Ng SH, Katkus T, Juodkazis S. Spatio-spectral-temporal imaging of fast transient phenomena using a random array of pinholes. Advanced Photonics Research. 2021;**2**: 2000032

[30] Wan Y, Liu C, Ma T, Qin Y, lv S. Incoherent coded aperture correlation holographic imaging with fast adaptive and noise-suppressed reconstruction. Optics Express. 2021;**29**: 8064-8075

[31] Anand V, Ng SH, Katkus T, Juodkazis S. White light threedimensional imaging using a quasirandom lens. Optics Express. 2021;**29**: 15551-15563

[32] Dubey N, Kumar R, Rosen J. COACH-based Shack-Hartmann wavefront sensor with an array of phase coded masks. Optics Express. 2021;**29**: 31859-31874

[33] Yu X, Wang K, Xiao J, Li X, Sun Y, Chen H. Recording point spread functions by wavefront modulation for interferenceless coded aperture correlation holography. Optics Letters. 2022;**47**:409-412

[34] Hai N, Rosen J. Interferenceless and motionless method for recording digital holograms of coherently illuminated 3-D objects by coded aperture correlation holography system. Optics Express. 2019;**27**:24324-24339

[35] Hai N, Rosen J. Doubling the acquisition rate by spatial multiplexing of holograms in coherent sparse coded aperture correlation holography. Optics Letters. 2020;**45**:3439-3442

[36] Hai N, Rosen J. Coded aperture correlation holographic microscope for single-shot quantitative phase and amplitude imaging with extended field of view. Optics Express. 2020;**28**:27372-27386

[37] Balasubramani V et al. Roadmap on digital holography-based quantitative phase imaging. Journal of Imaging. 2021; **7**:252

[38] Rosen J, Vijayakumar A, Kumar M, Rai MR, Kelner R, Kashter Y, et al. Recent advances in self-interference incoherent digital holography. Advances in Optics and Photonics. 2019;**11**:1-66

[39] Yamaguchi I, Zhang T. Phaseshifting digital holography. Optics Letters. 1997;**22**:1268-1270

[40] Rosen J et al. Roadmap on recent progress in FINCH technology. Journal of Imaging. 2021;**7**:197

[41] Kim MK. Adaptive optics by incoherent digital holography. Optics Letters. 2012;**37**:2694-2696

[42] Watanabe K, Nomura T. Recording spatially incoherent Fourier hologram using dual channel rotational shearing interferometer. Applied Optics. 2015;**54**: A18-A22

[43] Nobukawa T, Muroi T, Katano Y, Kinoshita N, Ishii N. Single-shot phaseshifting incoherent digital holography with multiplexed checkerboard phase gratings. Optics Letters. 2018;**43**: 1698-1701

[44] Nobukawa T, Katano Y, Goto M, Muroi T, Kinoshita N, Iguchi Y, et al. Incoherent digital holography simulation based on scalar diffraction theory. Journal of Optical Society of America A. 2021;**38**:924-932

[45] Bulbul A, Rosen J. Coded aperture correlation holography (COACH) with a superior lateral resolution of FINCH and axial resolution of conventional direct imaging systems. Optics Express. 2021; **29**:42106-42118

[46] Bulbul A, Rosen J. Super-resolution imaging by optical incoherent synthetic aperture with one channel at a time. Photonics Research. 2021;**9**:1172-1181

[47] Merkle F. Synthetic-aperture imaging with the European very large telescope. Journal of Optical Society of America A. 1988;**5**:904-913

[48] Michelson AA, Pease FG. Measurement of the diameter of α-Orionis by the interferometer. Astrophysics Journal. 1921;**53**:249-259

[49] Ilovitsh A, Zach S, Zalevsky Z. Optical synthetic aperture radar. Journal of Modern Optics. 2013;**60**:803-807

[50] Gerchberg RW, Saxton WO. A practical algorithm for the determination of phase from image and diffraction plane pictures. Optik. 1972;**35**:227-246

[51] Narayanswamy R, Johnson GE, Silveira PE, Wach HB. Extending the imaging volume for biometric iris recognition. Applied Optics. 2005;**44**: 701-712

[52] Pieper RJ, Korpel A. Image processing for extended DOF. Applied Optics. 1983;**22**:1449-1453

[53] Li S, Kwok JT, Wang Y. Multifocus image fusion using artificial neural networks. Pattern Recognition Letters. 2002;**23**:985-997

*Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent… DOI: http://dx.doi.org/10.5772/intechopen.105962*

[54] Dowski ER, Cathey WT. Extended depth of field through wave-front coding. Applied Optics. 1995;**34**: 1859-1866

[55] Tucker S, Cathey WT, Dowski E Jr. Extended DOF and aberration control for inexpensive digital microscope systems. Optics Express. 1999;**4**:467-474

[56] Le VN, Chen S, Fan Z. Optimized asymmetrical tangent phase mask to obtain defocus invariant modulation transfer function in incoherent imaging systems. Optics Letters. 2014;**39**: 2171-2174

[57] Liao M, Lu D, Pedrini G, Osten W, Situ G, He W, et al. Extending the depthof-field of imaging systems with a scattering diffuser. Scientific Reports. 2019;**9**:7165

[58] Mikula G, Kolodziejczyk A, Makowski M, Prokopowicz C, Sypek M. Diffractive elements for imaging with extended depth of focus. Optical Engineering. 2005;**44**:058001

[59] Zhai Z, Ding S, Lv Q, Wang X, Zhong Y. Extended depth of field through an axicon. Journal of Modern Optics. 2009;**56**:1304-1308

[60] Rai MR, Rosen J. Depth-of-field engineering in coded aperture imaging. Optics Express. 2021;**29**:1634-1648

[61] Rosen J, Salik B, Yariv A. Pseudonondiffracting beams generated by radial harmonic functions. Journal of Optical Society of America. A. 1995;**12**: 2446-2457

[62] Rosen J, Salik B, Yariv A. Pseudonondiffracting beams generated by radial harmonic functions: Erratum. Journal of Optical Society of America A. 1996;**13**:387

[63] Rai MR, Rosen J. Noise suppression by controlling the sparsity of the point spread function in interferenceless coded aperture correlation holography (I-COACH). Optics Express. 2019;**27**: 24311-24323

[64] Habaza M, Kirschbaum M, Guernth-Marschner C, Dardikman G, Barnea I, Korenstein R, et al. Rapid 3D refractiveindex imaging of live cells in suspension without labeling using dielectrophoretic cell rotation. Advanced Science. 2017;**4**: 1600205

[65] Rivenson Y, Liu T, Wei Z, Zhang Y, de Haan K, Ozcan A. Phase Stain: The digital staining of label-free quantitative phase microscopy images using deep learning. Vol. 8. Light: Science and Application; 2019. pp. 1-11

[66] Charrière F, Kühn J, Colomb T, Montfort F, Cuche E, Emery Y, et al. Characterization of microlenses by digital holographic microscopy. Applied Optics. 2006;**45**:829-835

[67] Niu M, Luo G, Shu X, Qu F, Zhou S, Ho YP, et al. Portable quantitative phase microscope for material metrology and biological imaging. Photonics Research. 2020;**8**:1253-1259

[68] Kumar M, Vijayakumar A, Rosen J, Matoba O. Interferenceless coded aperture correlation holography with synthetic point spread holograms. Applied Optics. 2020;**59**:7321-7329

[69] Rai MR, Vijayakumar A, Rosen J. Superresolution beyond the diffraction limit using phase spatial light modulator between incoherently illuminated objects and the entrance of an imaging system. Optics Letters. 2019;**44**: 1572-1575

[70] Rai MR, Rosen J. Resolutionenhanced imaging using interferenceless coded aperture correlation holography with sparse point response. Scientific Reports. 2020;**10**:5033

[71] Rosen J, Hai N, Rai MR. Recent progress in digital holography with dynamic diffractive phase apertures [Invited]. Applied Optics. 2022;**61**:B171- B180

[72] Anand V, Rosen J, Juodkazis S. Review of engineering techniques in chaotic coded aperture imagers. Light. Advanced Manufacturing. 2022;**3**:24

## **Chapter 8**

## A Mapping Relationship-Based near-Field Acoustic Holography

*Haijun Wu and Weikang Jiang*

## **Abstract**

A mapping relationship-based near-field acoustic holography (MRS-based NAH) is a kind of innovative NAH by exploring the mapping relationship between modes on surfaces of the boundary and hologram. Thus, reconstruction is converted to obtain the coefficients of participant modes on holograms. The MRS-based NAH supplies an analytical method to determine the number of adopted fundamental solution (FS) as well as a technique to approximate a specific degree of mode on patches by a set of locally orthogonal patterns explored for three widely used holograms, such as planar, cylindrical, and spherical holograms. The NAH framework provides a new insight to the reconstruction procedure based on the FS in spherical coordinates. Reconstruction accuracy based on two types of errors, the truncation errors due to the limited number of participant modes and the inevitable measurement errors caused by uncertainties in the experiment, are available in the NAH. An approach is developed to estimate the lower and upper bounds of the relative error. It supplies a tool to predict the error for a reconstruction under the condition that the truncation error ratio and the signal-tonoise ratio are given. The condition number of the inverse operator is investigated to measure the sensitivity of the reconstruction to the input errors.

**Keywords:** near-field acoustic holography, mapping relationship, integral identity, acoustic measurement, spherical fundamental solutions

## **1. Introduction**

To locate the position and target the strength of noise for a vibrating structure, near-field acoustic holography (NAH) had been widely adopted as an effective tool. It has a significant influence on the noise diagnostics, which gives a permission to get all desired acoustic quantities, such as pressure, particle velocity, and sound power, from a number of discrete field measurement.

It was originally developed by Willams, Manynard, etc., to reconstruct surface velocity of a rectangular plane with Fourier transform technique [1–3]. Initially, the Fourier-based NAH decomposes the field pressure into k-space (wave number space) for baffled problems. In other words, the field pressure is expanded into plane waves, and the reconstruction procedure is to obtain coefficients of the plane waves based on measured pressure. Although different from the k-space decomposition, concept of

Fourier transformation was inherent to the 3D cylindrical and spherical NAH problems as the in-depth discussions in Ref. [4].

Since it was proposed [1], varieties of approaches had been proposed and their superiorities had been proven in various applications, which resulted in several categories according to their underlying theories. Statistical optimal NAH [5–7] uses the elemental waves to approach the acoustic field, in which the surface-to-surface projection of the sound field is performed by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy [6]. Boundary element method (BEM)-based NAH [8–13] is appropriate for arbitrarily shaped model in which a general transformer matrix between the surfaces of structure and hologram is derived from the integral equation. Among the BEM-based NAH, two types of integration equation are adopted: the directive formulation (Helmholtz integral equation) and indirect formulation (single- or double-layer integral equation). The quantities reconstructed by the NAH derived from directive formulation have clear physical meaning [8–10], while the ones obtained by NAH derived from the indirect formulation are not the real physical quantities [11–13]. The equivalent source method (ESM) [14–19], also named as wave superposition algorithm (WSA) [16, 20, 21], was proposed by Koopman [22] for solving acoustic radiation problems of closed sources. ESM assumes that the field is generated by a series of simple sources such as monopoles and dipoles, and numerical integration is not needed in determining the source strength for a set of prescribed positions. Despite versatility of the ESM and various successful applications, "retreat distance" between the actual source surface and the virtual source cannot be well defined and deserves more attention in the application [23]. The Helmholtz equation least square method (HELS) [24–26] adopted the spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating structure. Coefficients of the spherical wave function, the fundamental solution (FS) for the Helmholtz equation, are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points. Since the spherical wave functions solve the Helmholtz equation directly, it is immune to the nonuniqueness difficulty inherent in BEM-based NAH [27]. However, HELS works better for spherical or chunky model than elongated model due to the specific basis function [25].

Essentially speaking, NAH is to achieve the desired acoustic quantities by the measured physical quantities such as sound pressure in the field. Most of the methods explicitly require the transfer operator T **y**, **x** between desired acoustic quantities *f* **y** and measured physical quantities *<sup>p</sup>*ð Þ **<sup>x</sup>** . They built a linear system of *<sup>f</sup>* **<sup>y</sup>** <sup>¼</sup> *inv* <sup>T</sup> **<sup>y</sup>**, **<sup>x</sup>** *<sup>p</sup>*ð Þ **<sup>x</sup>** in which *inv*ð Þ <sup>⋆</sup> represents an inverse operator, by either a general numerical method (BEM-based NAH) [8–13], or specific basis spaces such as a general Fourier basis (Fourier-based NAH) [1–4], simplified monopoles, dipoles (ESM and WSA) [14–16, 18, 20, 21], and fundamental solutions (HELS) [24–27]. The reconstruction procedure is therefore to solve the linear system to obtain the physical quantities on the boundary, such as pressure or normal velocity in BEM-based NAH, the source strength of equivalent source in ESM, coefficients of basis functions in Fourier-based NAH and HELS, and following by an extrapolation process to achieve desired acoustic quantities.

Unfortunately, all the proposed methods are very sensitive to errors which may cause reconstruction to fail. It is primarily due to abundant adoption of basis functions in the transfer operator which amplifies the errors in the inverse process. That is the reason why there have been numerous studies focusing on the development of

## *A Mapping Relationship-Based near-Field Acoustic Holography DOI: http://dx.doi.org/10.5772/intechopen.108318*

regularization methods to stabilize this inverse problem, such as truncated singular value decomposition [28] and the Tikhonov regularization [29]. Thus, construction of transfer operator is not a trivial process but is crucial to the feasibility and accuracy of the NAH. Concerning the theory development and practical measurement, it naturally arises a question whether there exists a guideline to determine the number and location of generalized basis function as well as measurement to obtain their coefficients for a given shape of source surface and prescribed tolerance.

The number of FS as well as number and position of the microphones array in the measurement are not well studied for the category of NAH based on the FS. Thus, one advantage of the mapping relationship-based NAH (MRS-based NAH) is the available guideline to the determination of the number of FS and measurement configuration in the FS-based NAH by exploring the mapping relationship between the modes in FS between surface and hologram, and investigating approximation of the modes with a set of locally orthogonal patterns.

As errors are inevitable in the practical measurement, it is curious to know how the errors go through the inverse operation and what influence imposed on the accuracy of the reconstruction results. To the best knowledge of authors, few works are devoted to the errors analysis of the NAH by comparing with that for the regularization methods. It is because that the NAH was usually viewed as a very ill-posed inverse problem for which regularized solution is the primary task. Thus, it is difficult to predict or estimate the reconstruction accuracy. Instead of a predictable way, numerical simulation and experimental validation are two frequently adopted methods to investigate the performance of NAH for different parameters [19, 30, 31]. For practical problems, it is hard to estimate the accuracy of the reconstructed results. Thus, one merit of our approach is the availability for predicting the reconstructed accuracy for a specific setup of the MRS-based NAH.

## **2. The mapping relationship-based NAH**

## **2.1 Theorem development**

As shown in **Figure 1**, assume that the fluid is homogenous, inviscid, and compressible and only undergoes small translation movement. The time harmonic sound pressure radiated from a vibrating structure into an infinite domain Ω is described by the well-known Helmholtz equation:

**Figure 1.** *Exterior acoustic problem of a vibrating structure in free space.*

$$
\nabla^2 p(\mathbf{x}) + k^2 p(\mathbf{x}) = 0 \text{ for } \mathbf{x} \in \Omega \tag{1}
$$

where *k* is the wave number, relating to the acoustic speed *c* and angular frequency *ω* by *<sup>k</sup>* <sup>¼</sup> *<sup>ω</sup>=c*, and **<sup>x</sup>** is a point in the domain. The time component is assumed to be *<sup>e</sup>*�*iω<sup>t</sup>* .

The fundamental solution of the governing formulation Eq. (1) in the spherical coordinates is

$$\mathbf{S}\_{n}^{m}(k,\mathbf{x}) = h\_{n}(k||\mathbf{x}||) \mathbf{Y}\_{n}^{m}(\theta,\phi), |m| \le n\mathbf{n} = \mathbf{0}, \mathbf{1}, \mathbf{2}, \dots, \mathbf{N} \tag{2}$$

where variables *θ* and *ϕ* are the polar angles of a point in the spherical coordinates and *N* is the truncated degree in a series expansion. *j <sup>n</sup>* and *hn* are the *n*th spherical Bessel function and spherical Hankel of the first kind, respectively. Y*<sup>m</sup> <sup>n</sup>* is the normalized spherical harmonic function:

$$\mathbf{Y}\_{n}^{m}(\theta,\phi) = \frac{1}{\sqrt{2\pi}} \mathbf{P}\_{n}^{\bar{m}}(\cos\theta) e^{im\phi} \tag{3}$$

where P*<sup>m</sup> <sup>n</sup>* is the normalized associated Legendre function [32]. Normal gradient in the direction *n*ð Þ **x** for the fundament solution Eq. (3) is as follows:

$$q\_n^m(\mathbf{x}) = \frac{\partial \mathbf{S}\_n^m(\mathbf{x})}{\partial n(\mathbf{x})} = \frac{\partial h\_n(k||\mathbf{x}||) \mathbf{Y}\_n^m(\theta, \phi)}{\partial n(\mathbf{x})} \tag{4}$$

which is related to the normal velocity *vn* by the Euler formulation:

$$q(\mathbf{x}) = ik\rho c v\_n(\mathbf{x})\tag{5}$$

It should be noted that Eqs. (2) and (5) are related as a solution pair for exterior acoustic problems, which means giving one as the boundary condition, the other will be the solution. They form a set of pressure/velocity modes on the boundary of a vibrating structure, which are generally independent on nonspherical surfaces and orthogonal on spherical surfaces. To facilitate derivations, we refer the velocity modes as the normal gradient *q* instead of the normal velocity *vn*. Assume that a structure is vibrating in one of its velocity modes Eq. (5), and the radiated pressure must be in the form of Eq. (2), which can be derived by making use of the equivalent source method (ESM) and boundary integral equation (BIE) [33].

Based on the model decomposition theorem and the mapping relationship, the boundary velocity *v y* � � and the radiated sound pressure *<sup>p</sup>*ð Þ *<sup>x</sup>* on the hologram can be expressed for a given set of participant coefficients *an*<sup>0</sup> as:

$$\begin{cases} v\left(\mathbf{y}\right) = \sum\_{n'=0}^{N'} a\_{n'} v\_{n'}\left(\mathbf{y}\right), \mathbf{y} \in \mathcal{S} \\\ p(\mathbf{x}) = \sum\_{n'=0}^{N'} a\_{n'} p\_{n'}(\mathbf{x}), \mathbf{x} \in \Omega \end{cases} \tag{6}$$

where *vn*<sup>0</sup> <sup>¼</sup>*n*2þ*n*þ*m*þ<sup>1</sup> <sup>¼</sup> *<sup>q</sup><sup>m</sup> <sup>n</sup>* and *pn*<sup>2</sup>þ*m*þ<sup>1</sup> <sup>¼</sup> *<sup>S</sup><sup>m</sup> n* .

Eq. (6) is the basement of the MRS-based NAH but must be properly truncated. The subscription convention in Eq. (6) is convenient for the discretized linear operation. Obviously, the truncation number *N*<sup>0</sup> in Eq. (6) and *N* in Eq. (2) are related by

*<sup>N</sup>*<sup>0</sup> <sup>¼</sup> ð Þ *<sup>N</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> . The truncation number is crucial to the pressure evaluation as well as the number and positions of microphone array required in the NAH.

Determination of the number of most efficient modes is converted to seeking the truncation number of radiation efficiency *σ<sup>n</sup>* making the upper bound of the radiated sound power converge to a given tolerance. A relative error of the upper bounded radiated sound power caused by new added modes for degree *N* in Eq. (2) with respective to the one produced by existing modes for degree less than *N* is defined as:

$$\varepsilon\_{N} = \frac{(2N+1)\sigma\_{N}(k\ddot{r})}{\sum\_{n=0}^{N-1} (2n+1)\sigma\_{n}(k\ddot{r})} \tag{7}$$

where ~*r* is the equivalent radius of the vibrating structure. The equivalent radius is the description of the spherical source which has the same average radiated sound power per unit area as that of the vibrating structure or holograms. Since the radiated sound power from the equivalent source and the vibrating structure as well as the holograms should be same, the requirement of same average radiated sound power per unit area makes the equivalent radius satisfy

$$
\tilde{r} = \sqrt{\mathbb{S}/4\pi} \tag{8}
$$

where *S* represents area of structure in the determination of equivalent radius ~*r*.

Fortunately, for a specific dimensionless value *k*~*r*, the radiation efficiency *σ<sup>N</sup>* <sup>&</sup>gt; *Nc* is a strictly decreasing function with respective to the degree *N* after a certain degree *Nc* which can be obtained by its closed-form expression. Radiation efficiency *σ<sup>N</sup>* and the relative error *ε<sup>N</sup>* for the dimensionless size 0*:*1≤*k*~*r*≤10 and the varying degree from 0 to 7 are presented in **Figure 2a** and **b**, respectively. It shows that *σ<sup>N</sup>* is a monotonously decreasing function and clearly distinguishes from each degree for *k*~*r*<2*:*0. There is a plateau on which *σ<sup>N</sup>* starts to overlap for *k*~*r*>2*:*0. It means that those degrees of modes contribute to the radiated sound power almost equally and no one can be neglected, which is verified by the relative errors in **Figure 2b**. Therefore, more degrees of modes are needed to make the radiated sound power converge for larger dimensionless value *k*~*r*. The relative error *ε<sup>N</sup>* presented in **Figure 2b** can be used as a reference to determine the degree of the most efficient modes for 0*:*1≤*k*~*r*≤10, or in other words the

#### **Figure 2.**

*(a) Radiation efficiency of a sphere, and (b) relative errors of upper bounded radiated sound power, for varying degrees and kr*~*:*

truncation number *N* for the FS-based NAH. However, Eq. (7) only relates to the size of the structure and has nothing to do with the field point or size of hologram.

## **2.2 The NAH procedure**

Suppose a set of at least independent velocity modes on the boundary, denoted as *vi*ð Þ **x ∈** *S* ,ð Þ *i* ¼ 1, 2, … , can produce a set of independent pressure modes *pi* ð Þ **x**∈Γ ,ð Þ *i* ¼ 1, 2, … on the measurement surface Γ, correspondingly, and they form a pair of bijective mapping relationship.

Generally, the pressure patterns *pi* ð Þ **x**∈Γ are non-orthogonal on the hologram, and an orthogonalization process is required, which can be done by the Gram-Schmidt approach as:

$$u\_i(\mathbf{x}) = p\_i(\mathbf{x}) - \sum\_{j=1}^{i-1} \frac{\langle p\_i(\mathbf{x}), u\_j(\mathbf{x}) \rangle}{||u\_j(\mathbf{x})||} u\_j(\mathbf{x}) \tag{9}$$

where the inner product h i *<sup>p</sup>*, *<sup>u</sup>* <sup>¼</sup> <sup>Ð</sup> <sup>Γ</sup>*p*ð Þ **<sup>x</sup>** *<sup>u</sup>*<sup>∗</sup> ð Þ **<sup>x</sup>** *<sup>d</sup>*Γð Þ **<sup>x</sup>** and the normk k*<sup>u</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi h i *<sup>u</sup>*, *<sup>u</sup>* <sup>p</sup> . After some algebraic operations, the normalized orthogonal modes *ei*ð Þ¼ **x** *ui*ð Þ **x** *=*k k *u*ð Þ **x** can be expressed in the following form:

$$\mathbf{E} = \mathbf{p}\mathbf{R} \tag{10}$$

where **<sup>E</sup>** <sup>¼</sup> ½ � *<sup>e</sup>*1ð Þ **<sup>x</sup>** , *<sup>e</sup>*2ð Þ **<sup>x</sup>** , <sup>⋯</sup> ,**<sup>p</sup>** <sup>¼</sup> *<sup>p</sup>*1ð Þ **<sup>x</sup>** , *<sup>p</sup>*2ð Þ **<sup>x</sup>** , <sup>⋯</sup> � � and **<sup>R</sup>** is a real upper triangular square matrix. As indicated in Eq. (10), the normalized orthogonal modes are actually a linear combination of the independent pressure modes *pi* . It is remarkable that evaluation of the inner produce is performed on holograms which are generally in smooth shapes, such as the three typical holograms in Section 0. Thus, they can be computed on exact geometries which are a void of discretization errors. Furthermore, orthogonalization of the modes on simple shaped holograms will yield a translation matrix **R** with good numerical characteristics, such as the condition number.

Assume the radiated pressure *p*ð Þ **x** is decomposed into the normalized orthogonal modes *ei*ð Þ **x** on the hologram as:

$$p(\mathbf{x}) = \sum\_{i=1}^{\infty} a\_i e\_i(\mathbf{x}) \tag{11}$$

where

$$a\_i = \langle p(\mathbf{x}), e\_i(\mathbf{x}) \rangle\_\Gamma \tag{12}$$

Once those participant coefficients are obtained by the measured pressure, substituting Eq. (10) into Eq. (12) yields

$$p(\mathbf{x}) = \sum\_{i=1}^{\infty} \lambda\_i p\_i(\mathbf{x}) \tag{13}$$

where the coefficients are

$$
\lambda\_i = \sum\_{j=i}^{\infty} R\_{ij} a\_j \tag{14}
$$

## *A Mapping Relationship-Based near-Field Acoustic Holography DOI: http://dx.doi.org/10.5772/intechopen.108318*

Due to the unique mapping relationship between surfaces of vibrating structure and hologram, reconstruction for the boundary quantities can be performed by multiplying the corresponding modes with the same set of participant coefficients *λ<sup>i</sup>* on the surface of the vibrating structure.

Thus, acoustic holography is converted to seek explicit descriptions of the mapping relationship between the modes on the boundary and modes on the field and design a proper experimental setup for obtaining the participant coefficients of the modes on the measurement surface. The modes on the boundary are free of restrictions for their form of expression, which could be in any well-studied analytical functions or in generally numerical representations. However, it should be expected to have a capacity of fast convergent ratio in the decomposing of boundary quantities and generate a radiated pressure on the hologram which is easy to be determined by the experiment. In the current work, the FS in spherical coordinates for the Helmholtz equation Eq. (2) and its normal gradient Eq. (5) are chosen as the pressure and velocity modes. Merits of choosing those forms of modes are twofold. First, the radiated modes on the field are also the in the same form; and second, the most effective modes contributing to the field pressure are easy to be determined. Henceforth, the radiated pressure modes in Eq. (13) are chosen as *pi* <sup>¼</sup> *pn*<sup>2</sup>þ*m*þ*n*þ<sup>1</sup> <sup>¼</sup> *<sup>S</sup><sup>m</sup> <sup>n</sup>* in our analysis to facilitate derivations. So do the velocity modes *qi* .

### **2.3 Setup of the microphone array**

Since the modes are distributed on an enclosing surface, the holograms should form an enveloping surface enclosing the vibrating structure. Otherwise, partially measured pressure cannot represent the modes completely and consequently cannot be applied to reconstruct the boundary information based on the mapping relationship.

The distribution of a specific mode varies on different holograms. Generally, the measurements are subject to the experimental resource such as microphones and permissible space. How to accurately recognize the field pressure modes is one of the crucial factors to NAH. In practice, microphones are preferred to be placed on planar, cylindrical, or spherical surfaces which are easy to be set up but generally not conformal to the vibrating structure, as shown in **Figure 3**.

For the enclosing planar holograms, as shown in **Figure 3a**, each pressure mode is divided and projected onto six patches. On each patch, the measured pressure should be able to accurately represent the projected pressure modes. However, once the pressure is discretely sampled, the spectrum or the number of participant modes on

**Figure 3.**

*Three typical holograms: (a) planar holograms, (b) cylindrical holograms, and (c) spherical holograms.*

that patch is truncated. Therefore, the primary task in the measurement is to set up the microphone arrays properly with an aim to approximate all the projected pressure modes on each patch actually. On each planar surface, the pressure modes can be expressed by two sets of locally orthogonal polynomials such as polynomial *f <sup>n</sup>*ð Þ *x* and *gm*ð Þ*y* of degree *n* and *m* in each direction, respectively. The most significant degrees in each direction can be numerically obtained directly by approximating the analytical pressure mode, Eq. (2), with the polynomial expansions *f <sup>n</sup>gm*, as

$$S\_n^m(k, \mathbf{x}) = \sum\_{n'=0}^{N'} \sum\_{m'=0}^{M'} a\_{n'}^{m'} f\_{n'}(\mathbf{x}(\mathbf{x})) \mathbf{g}\_{m'}(\mathbf{y}(\mathbf{x})) \tag{15}$$

where *a<sup>m</sup>*<sup>0</sup> *<sup>n</sup>*<sup>0</sup> is the coefficients. Once the polynomial degrees *N*<sup>0</sup> and M<sup>0</sup> on each planar patch are determined, the microphones are placed at the abscissas of Gaussian quadrature points on the patch, which results in *N*<sup>0</sup> ð Þ� þ 1 *M*<sup>0</sup> ð Þ þ 1 microphone positions [33].

A closed cylindrical measurement surface, as shown in **Figure 3b**, has three patches, one left circular planar patch Γ*l*ð Þ **x**j � *b=*2 ¼ k k**x** cos *θ* , one right Γ*r*ð Þ **x**j*b=*2 ¼ k k**x** cos *θ* circular planar patch, and one cylindrical surface Γ*c*ð Þ **x**j*a* ¼ k k**x** sin *θ* in the central portion. Similar to the planar hologram, it needs to select a set of locally orthogonal patterns to approximate the pressure modes on each patch. In this case, arguments *θ* and *ϕ* in cylindrical coordinates are the two independent variables to define patches. All three patches possess a complete description for the variable *ϕ* in the range 0, 2 ½ � *π* . In light of the expression for pressure modes, Eq. (2), normalized basis *gm*ð Þ¼ *<sup>ϕ</sup>*ð Þ **<sup>x</sup>** *<sup>e</sup>imϕ*ð Þ **<sup>x</sup>** *<sup>=</sup>* ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> is selected as one set of the orthogonal patterns in the *ϕ* direction. Thus, the determination of the truncation number for another set of local basis function *f <sup>n</sup>*<sup>0</sup> is simplified to approximate the following function <sup>F</sup>*n*ð Þ¼ *<sup>k</sup>*, **<sup>x</sup>** *hn*ð Þ *<sup>k</sup>*k k**<sup>x</sup>** <sup>P</sup>*<sup>m</sup> <sup>n</sup>* ð Þ cos *θ* .

A spherical measurement surface is shown in **Figure 3c**, which is a conformal patch to the spherical coordinates upon which the FS is obtained. Field modes on the spherical surface are orthogonal. Determination of the field modes on the spherical hologram is actually to identify the spherical harmonic functions based on the measured pressure. Due to conformality of the hologram to the coordinate system of the spherical FS, an analytical way is available to determine the number and position of the measurement. The quadrature technique on a sphere is well studied and widely used in the computational acoustics [34]. Therefore, the participant coefficients in Eq. (12) can be accurately evaluated by ð Þ *N* þ 1 point Gaussian quadrature and ð Þ 2*N* þ 1 point square quadrature for variables of *θ* and *ϕ* on the spherical surface for the pressure mode *S<sup>m</sup> <sup>N</sup>*ð Þ j j *m* ≤ *N* . Thus, it is free of numerical searching in the determination of the truncation number. In addition, the total number of measurements is the smallest than that requested by the other two holograms.

## **3. Error analysis**

#### **3.1 Error bounds on pressure energy**

The NAH is an inverse problem and thus poses significant challenges to the stable and accurate solution. However, a practical measurement is prone to errors and always incorporates uncertainties, such as random fluctuations, effect of rapid decay

of the evanescent waves. Generally, the great affection to the reconstruction by the inevitable measurement errors is largely due to over-selected number of the basis (either in numerical or analytical form) which results in an ill-posed inverse operator. Fortunately, the number of basis or modes can be well estimated by an analytical way as introduced in Section 2. Thus, a pre-regularization process is embedded in the MRS-based NAH.

On the holograms, the error included pressure is simply modeled as:

$$\mathbf{p}(\mathbf{x}) = \mathbf{p}\_0(\mathbf{x}) + \mathbf{n}(\mathbf{x}), \text{or } \mathbf{P} = \mathbf{P}\_0 + \mathbf{n} \tag{16}$$

where **P0** represents the source pressure and **n** is the noise terms. Denote the signal-to-noise ratio (SNR) on the hologram as:

$$\text{SNR} = 10 \log\_{10} \frac{W\_{P\_0}}{W\_{Noise}} = 10 \log\_{10} \frac{\int\_{S\_k} |\text{Po}(\mathbf{x})|^2 dS(\mathbf{x})}{\int\_{S\_k} |\mathbf{n}(\mathbf{x})|^2 dS(\mathbf{x})} \tag{17}$$

where *WP*<sup>0</sup> and *WNoise* represent energies of source pressure and noise pressure, respectively. Under the condition that the source pressure and noise pressure can be completely decomposed by a set of modes, the SNR can be reformatted by the Parseval law as:

$$\text{SNR} = \mathbf{10} \log\_{10} \frac{||\mathbf{a}\_0||\_2^2}{||\mathbf{a}\_{\text{Noise}}||\_2^2} \tag{18}$$

where **α**<sup>0</sup> and **α***Noise* are coefficients of the locally normalized orthogonal patterns on the holograms for the source pressure and noise term, and correspondingly the coefficients for pressure **P** is *α* ¼ **α**<sup>0</sup> þ **α***Noise*.

By taking advantages of the mapping relationships, Eq. (11) is used to evaluate the reconstructed pressure on the surface of vibrating structure after the coefficients **λ** of FS are obtained on the holograms. Decompose the FS in Eq. (11) on the surface of vibrating structure as same as that in Eq. (10) but with symbol *S* substituting for pressure mode *p* as:

$$\mathbf{S}\_{\Gamma} = \mathbf{E}\_{\Gamma} \mathbf{R}\_{\Gamma}^{-} \tag{19}$$

where **E**<sup>Γ</sup> is the column normalized modes on the surface of vibrating structure and **R**<sup>Γ</sup> is a translation operator which is an upper triangular square matrix. In light of Eq. (14) and Eq. (19), reconstructed sound pressure on surface of the vibrating structure, **p**<sup>Γ</sup> in the Eq. (13) can be expressed as:

$$\mathbf{p}\_{\Gamma} = \mathbf{E}\_{\Gamma} \mathbf{R}\_{\Gamma}^{-} \mathbf{R} a \tag{20}$$

Therefore, it could be observed that the reconstruction process is to translate the local coefficients **α** obtained on the holograms to that on the surface of vibrating structure by the translator **R**� <sup>Γ</sup> **R**, and then the reconstructed pressure is evaluated by the modal decomposition method. Stability of the translation is largely dependent on the product of the two translators **R**� <sup>Γ</sup> and **R** which are closely related to the geometric information of the structure and holograms, respectively.

According to the Parseval law, the reconstructed pressure energy on the surface of vibrating structure is

$$\mathcal{W}\_{P\_\Gamma} = \left\| \mathbf{R}\_\Gamma^- \mathbf{R} \mathbf{a} \right\|\_2^2 = \mathbf{a}^\* \, \mathbf{T} \mathbf{a} \mathbf{a} \tag{21}$$

where **TR** <sup>¼</sup> **<sup>R</sup><sup>∗</sup> <sup>R</sup>**� <sup>∗</sup> <sup>Γ</sup> **R**� <sup>Γ</sup> **R** is a Hermitian matrix. Therefore, there is an eigen dec omposition of **TR** <sup>¼</sup> **<sup>Q</sup> <sup>∗</sup>Λ<sup>Q</sup>** in which **<sup>Q</sup>** is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of **TR**, and **Λ** is a real diagonal matrix whose main diagonal entries are the corresponding eigenvalues. Assume the eigenvalues are sorted in a descending order, such as Λ*<sup>i</sup>* ≥Λ*<sup>j</sup>* for *j*>*i*. The lower and upper bounds of the reconstructed pressure energy are easy to be obtained

$$
\Lambda\_d \|\|\mathbf{a}\|\|\_2^2 \le \mathcal{W}\_{P\_\Gamma} \le \Lambda\_1 \|\|\mathbf{a}\|\|\_2^2 \tag{22}
$$

where *d* is the dimension of the matrix. In practice, the relative error of reconstructed pressure energy on the surface of the vibrating structure is more concerned. Obviously, the bounds of the exact pressure energies *WP*<sup>0</sup><sup>Γ</sup> and noise generated pressure energies *WP*noise<sup>Γ</sup> can be obtained with coefficients *α* replaced with *α*<sup>0</sup> and *αNoise* in Eq. (22), respectively. Thus, bounds for the relative errors *ε<sup>W</sup>*<sup>Γ</sup> ¼ *WP*noise<sup>Γ</sup> *=WP*<sup>0</sup><sup>Γ</sup> are

$$\varepsilon cond(\mathbf{T\_R})^{-}\mathbf{10}^{-\text{SNR}/10} \le \varepsilon\_{W\_\Gamma} \le cond(\mathbf{T\_R})\mathbf{10}^{-\text{SNR}/10} \tag{23}$$

where 10�*SNR=*<sup>10</sup> <sup>¼</sup> k k *<sup>α</sup>Noise* 2 <sup>2</sup>*=*k k *α*<sup>0</sup> 2 2, and *cond*ð Þ¼ **TR** *Λ*1*=***Λ***<sup>d</sup>* is the condition number of the translator matrix **TR**. Eq. (23) can be reexpressed as:

$$\text{SNR} - \log\_{10} \text{cond}(\mathbf{T}\_{\mathbf{R}}) \le \text{SNR}\_{\Gamma} \le \text{SNR} + \log\_{10} \text{cond}(\mathbf{T}\_{\mathbf{R}}) \tag{24}$$

in which *SNR*<sup>Γ</sup> ¼ � log <sup>10</sup>*ε<sup>W</sup>*<sup>Γ</sup> is the signal-to-noise ratio of the reconstructed pressure on the model's surface.

### **3.2 The modified error bounds**

Above analysis is based on an assumption that the pressure can be completely decomposed by a set of modes. Otherwise, the Parseval law cannot be applied equivalently in evaluating the pressure energy. However, the complete set of modes is hardly to be satisfied in decomposing the radiated pressure of a realistic radiator, but an incomplete set is applied to approximately decompose the radiated pressure within a given tolerance. Therefore, a compromise on accuracy and robustness is made by truncating the series expansion N<sup>0</sup> <sup>¼</sup> ð Þ *<sup>N</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> , Eq. (6), with a proper number N such as that given by Eq. (7). Due to the truncation, the measured pressure energy evaluated by the modal decomposition method on holograms is not equal to the true quantities, and so is the noise energy. It is also reasonable to define SNR on the hologram by Eq. (18), because once the radiated pressure is spatially measured by limited number of microphones, such as the way introduced in Section 2.2, the higher frequency components are filtered out which will not go through the inverse system.

Suppose the exact pressure on the surface of the vibrating structure is *p*0,<sup>Γ</sup> and the corresponding reconstructed pressure is *p*Γ. Therefore, the relative error of the reconstructed pressure energy in discretized form is

*A Mapping Relationship-Based near-Field Acoustic Holography DOI: http://dx.doi.org/10.5772/intechopen.108318*

$$\varepsilon\_{W\_{\Gamma}} = \frac{\left\| \mathbf{p}\_{\mathbf{0},\Gamma} - \mathbf{p}\_{\Gamma} \right\|\_{2}^{2}}{\left\| \mathbf{p}\_{\mathbf{0},\Gamma} \right\|\_{2}^{2}} \tag{25}$$

According to the derivation in the appendix in the ref. [35], the relative error*εW*<sup>Γ</sup> can be expressed as:

$$\varepsilon\_{W\_{\Gamma}}(c\_1, c\_2, c\_3, c\_4) = \frac{c\_4 + c\_3 \mathbf{1} \mathbf{0}^{-\text{SNR}/10} + c\_1 \sqrt{c\_4} \sqrt{c\_3} \mathbf{1} \mathbf{0}^{-\text{SNR}/20}}{\mathbf{1} + c\_4 + c\_2 \sqrt{c\_4}} \tag{26}$$

It reaches the lower bound at *c*<sup>1</sup> ¼ �2 and *c*<sup>2</sup> ¼ 2, and the upper bound at *c*<sup>1</sup> ¼ 2 and *c*<sup>2</sup> ¼ �2. However, Eq. (26) is a nonlinear function for variables *c*3, and constrained nonlinear optimization algorithm is sought to find the lower and upper bounds, as the minimum of a problem specified by:

$$\min\_{(c\_3)} f(c\_3) \\ \text{such that } cond(\mathbf{T\_R})^- \le c\_3 \le cond(\mathbf{T\_R}) \tag{27}$$

where the objective function is

$$f(c\_3) = \begin{cases} \varepsilon\_{W\_\Gamma}(-2, \ 2, \ c\_3, \ c\_4), for lower bound \\ -\varepsilon\_{W\_\Gamma}(2, \ -2, \ c\_3, \ c\_4), for upper bound \end{cases} \tag{28}$$

In the above analysis, the variables SNR and *c*<sup>4</sup> are supposed to be given. The SNR of the environment can be estimated by measurement. For the ideal cases in which there is no noise included, equivalent to SNR ¼ ∞, the lower bound of the relative error is easy to be obtained as:

$$
\varepsilon\_{W\_\Gamma} \ge \frac{c\_4}{1 + c\_4 + 2\sqrt{c\_4}} \tag{29}
$$

which is only related to the *c*<sup>4</sup> and in turn related to the number of adopted participant modes. The actual reconstructed error of a realistic problem or the case with small SNR are not expected to have a lower bound less than that estimated for no noise included case. Thus, the lower bound of the reconstructed pressure energy can be estimated by Eq. (29).

The variables *c*<sup>4</sup> is a crucial parameter for the error bounds estimation, which can either be evaluated by numerical simulation for specific problems or estimated by an analysis method. However, numerical simulation is hard to be realized for practical problems, since the source is not clear and it is the reason why the NAH is needed. Therefore, it is demand for developing a general analysis approach to properly estimate the variable *c*4. Whereas, it is out of our mathematical ability as well as the range of the current work. In the following numerical examples, variable *c*<sup>4</sup> is estimated by the numerical method, a combination of finite element method (FEM) and BEM.

#### **3.3 Characteristics of the translator**

To investigate how much the output value of a function, such as the reconstructed quantities, can change for a small variation, such as the errors introduced in the

experiment, in the input arguments, condition number of the function is one of the frequently used measure. Therefore, investigation of the condition number of translators in the NAH can somehow describe the stability of the reconstruction. Generally, numerical approach is applied to compute the condition number. However, if both shapes of the structure and holograms are conformal to sphere, a simple asymptotic expression of the condition number is available. The radii of spherical structure and holograms are denoted as *r*<sup>Γ</sup> and *r*. Translators **R**<sup>Γ</sup> and **R** are all diagonal matrices, taking the **R** as an example:

$$\mathbf{R} = \text{diag}\left\{ \bigcup\_{n=0}^{N} \left[ \overbrace{|rh\_n(kr)|^- \cdots \cdot |rh\_n(kr)|^-} \right] \right\} \tag{30}$$

where diagf g**v** means a square diagonal matrix with the elements of vector **v** on the main diagonal, and ⋃*<sup>N</sup> <sup>n</sup>*¼<sup>0</sup>**v***<sup>n</sup>* returns a vector combing all the subvector **<sup>v</sup>***n*. Thus,

$$\mathbf{T}\_{\mathbf{R}} = \text{diag}\left\{ \bigcup\_{n=0}^{N} \begin{bmatrix} \frac{2n+1}{\left| r\_{\Gamma} h\_{n}(kr\_{\Gamma}) \right|^{2}} & \cdots & \frac{\left| r\_{\Gamma} h\_{n}(kr\_{\Gamma}) \right|^{2}}{\left| r h\_{n}(kr) \right|^{2}} \end{bmatrix} \right\} \tag{31}$$

According to the analysis in Ref. [36], the asymptotic expression of j j *xhn*ð Þ *x* for *n* ≫ *x* is

$$|\varkappa h\_n(\varkappa)| \sim \sqrt{\frac{2}{e}} \left(\frac{2l+1}{e}\right)^l \varkappa^{-n} e^{\varkappa^2/4n} \tag{32}$$

which is actually the absolute value of the imagine part of the spherical Hankel function, since the real part goes rapidly to zero for *n* ≫ *x*; therefore,

$$cond(\mathbf{T}\_{\mathbf{R}}) = \frac{|r\_{\Gamma}h\_{N}(kr\_{\Gamma})|^{2}}{\left|rh\_{N}(kr)\right|^{2}} \sim \left(\frac{r}{r\mathbf{r}}\right)^{2N} e^{k^{2}\left(r\_{\Gamma}^{2}-r^{2}\right)/2N} \sim \left(\frac{r}{r\mathbf{r}}\right)^{2N} \tag{33}$$

due to that *e k*<sup>2</sup> *r*<sup>2</sup> <sup>Γ</sup>�*r*<sup>2</sup> ð Þ*<sup>=</sup>*2*<sup>N</sup>* approximate rapidly to one for *N* ≫ *kr*<sup>Γ</sup> and *N* ≫ *kr*.

To investigate how much the reconstructed coefficients and in turn the pressure can change for a small variation in the local coefficients *α*, condition number of the translator **R**� <sup>Γ</sup> **R** is the quantity needs to be studied. In light of the previous analysis, the condition number of the translation operator **R**� <sup>Γ</sup> **R** for spherical structure and holograms satisfies

$$\text{cond}\left(\mathbf{R}\_{\Gamma}^{-}\mathbf{R}\right) = \frac{|r\_{\Gamma}h\_{N}(kr\_{\Gamma})|}{|rh\_{N}(kr)|} \sim \left(\frac{r}{r\_{\Gamma}}\right)^{N} \tag{34}$$

Actually, the condition number of translator **TR** is a square power of that for the translator **R**� <sup>Γ</sup> **R**, i.e. *cond*ð Þ¼ **TR** cond **R**� <sup>Γ</sup> **<sup>R</sup>** � �<sup>2</sup> . Eq. (34) indicates that the condition number has a geometric growth with *N* under the condition *N* ≫ *kr*<sup>Γ</sup> and *N* ≫ *kr*. Correspondingly, the larger condition number of the translator will obviously increase the sensibility to the inevitable errors introduced in the experiment. In addition, the

## *A Mapping Relationship-Based near-Field Acoustic Holography DOI: http://dx.doi.org/10.5772/intechopen.108318*

large ratio of *r=r*<sup>Γ</sup> could also increase the condition number for a fixed degree *N*. It implies that the same distance of the measurement to a smaller size surface will result in a reconstruction which is more prone to be contaminated. The condition numbers of the translator **R**� <sup>Γ</sup> **R** for a spherical model with radius being 0.1 m and spherical holograms at frequency 601 Hz are given in **Figure 4**, which clearly validates the asymptotic form Eq. (34) to the exact one, for *N* ≫ *kr*<sup>Γ</sup> and *N* ≫ *kr*.

The asymptotic expression of the condition number of the translator **R**� <sup>Γ</sup> **R** for general models and holograms are hardly to be obtained. Numerical method is resorted to get the condition number once the frequency, geometrical information of the model as well as holograms are supplied. To explore the influence of shapes of the model and holograms to the condition number, case studies of a cubic model with planar holograms and spherical holograms, which are representatives as conformal measurement and non-conformal measurement, are performed. The cubic model is same as that used in the numerical examples in Section 4. The planar holograms and spherical hologram also have the same configuration as that in the numerical examples. Two frequencies 601 Hz and 1333 Hz are analyzed. The condition numbers obtained by numerical method and asymptotic are presented in **Figure 5**. It is noticeable that the asymptotic condition number of the cubic model with planar holograms are evaluated with their equivalent radii. It can be observed that the asymptotic expression for planar holograms, which is a conformal hologram to the model, approximates to that obtained numerically very well for the frequency 601 Hz. However, the approximation does not show up for the frequency 1333 Hz up to the degree *N* ¼ 10. It is because that the larger dimensionless value *k*~*r* needs larger degree to satisfy the condition of the asymptotic expression. The asymptotic estimation of the condition number does not work well for the spherical holograms due to the disparity in shapes with the cubic model. Even the ratio *r=r*~<sup>Γ</sup> for the spherical hologram is smaller than that of the planar hologram, the condition number for the spherical hologram is inclined to be larger than that for the planar holograms as illustrated in **Figure 5**. It implicates that a cubic model with spherical hologram, a representative case of non-conformal measurement, may be more sensitive to measurement errors than that with planar hologram.

#### **Figure 4.**

*The exact and asymptotic condition numbers of the translator R*� *<sup>Γ</sup> R for different ratio r=r<sup>Γ</sup> for spherical model and holograms.*

#### **Figure 5.**

*The asymptotic and numerical condition numbers of the translator R*� *<sup>Γ</sup> R for a cubic model with planar holograms and spherical holograms at two frequencies.*

## **4. Experiment study**

#### **4.1 Numerical simulation**

The necessary number of participant modes is hard to be obtained exactly for a realistic problem, and truncation error is introduced. In this case, the radiated source pressure on holograms is generated from a vibrating cubic model driven by a harmonic excitation. As shown in **Figure 6**, the cubic model is of size 0*:*2m � 0*:*2m � 0*:*2m and is excited by a harmonic force along z-axial at a specified position ð Þ 0*:*4*a*, 0*:*4*a*, �0*:*5*a* , and the four corners at the bottom are constrained. Thicknesses of the six walls are set as 0.004 m, and the steel material is assigned to the model. The harmonic response is obtained by a commercial finite element software at frequencies 601 Hz, which is chosen closely to the one modal frequency with an aim to obtain a uniformly distributed velocity on the surface. Once the boundary velocities are obtained, the radiated sound pressure at the measurement positions are computed by the boundary element method [37] as the inputs for the reconstruction. It is to simulate the realistic radiator whose exact number of efficient modes is hardly to be obtained but estimated by a reasonable guideline. Both of the cases include a specific amount of noise with prescribed SNR on the hologram. The noise is generated with

$$\mathbf{n} = \chi \mathbf{S} \mathbf{\hat{A}}\_{\text{Noise}} \tag{35}$$

where **<sup>λ</sup>***Noise* is a Gaussian random vector, and *<sup>γ</sup>* <sup>¼</sup> <sup>10</sup>�*SNR=*<sup>20</sup> **<sup>R</sup>**�<sup>1</sup> **λ***Noise* � � � �� 2 ffiffiffiffiffiffiffiffiffi *WP*<sup>0</sup> p is an energy-related variable to make sure the generated noise **n** can form a specified SNR.

In this section, investigations are devoted to the reconstructions of the NAH with spherical holograms. To use the MRS-based NAH, the primary task is to determine the number of necessary modes. According to the method introduced in Section 2.1, the equivalent radius of the cubic model is <sup>~</sup>*<sup>r</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi 3*=*2*π* p *a* which results in dimensionless variables *k*~*r* equal to 1.53. The energy criteria Eq. (7) are adopted to determine the necessary number of FS due to its reasonable compromise on the accuracy and efficiency in the reconstruction. The tolerance is set as *ε*<sup>T</sup> ¼1E-1, and it yields the necessary number of FS being 3 by referring to the diagram in **Figure 2**. The number and position of microphones are consequently determined by the approach in Section 2.3. To investigate the influence of the necessary number to the final results, reconstructions are performed for the necessary number ranging from 3 to 6 for each case.

There are errors due to the truncation of the participant modes. Therefore, reconstructions are firstly performed for the pressure *pnum* obtained numerically by the FEM and BEM which are also treated as the exact source pressures. The reconstructions are considered as the references of no-noise included measurement. Contour plots of the reconstructed pressure for reference cases are given in **Figure 7**. In spite of a slight disparity in the quantity, it can be observed that the reference reconstructions are very satisfied with the simulated pressure, because the significant pressure distributions are well reconstructed. Later on, different SNRs ranging from 4 dB to 28 dB with the increment being 4 dB are added to the simulated pressure to validate the robustness of the NAH to the noise which is unavoidable in the realistic experiment. The noise is obtained by Eq. (35).

Relative errors of reconstructed pressure energy on the model's surface is defined as:

$$\varepsilon\_{W\_{\Gamma}} = \frac{||\mathbf{p}\_{\text{recon}} - \mathbf{p}\_{\text{num}}||\_{2}^{2}}{||\mathbf{p}\_{\text{num}}||\_{2}^{2}} \tag{36}$$

#### **Figure 7.**

*The reconstructed pressure on the boundary based on the no-noised included measurement on (a) the spherical holograms with the necessary number of modes N* ¼ 3 *for (b) the simulated results at frequency 601 Hz.*

where *precon* is the reconstructed pressure based on different SNRs. To estimate the error, variable *c*<sup>4</sup> is required to be supplied by:

$$\mathcal{L}\_4 = \frac{||\mathbf{p}\_{recon}||\_2^2}{||\mathbf{p}\_{recon} - \mathbf{p}\_{mm}||\_2^2} \tag{37}$$

based on the simulated results. According to bounds Eq. (26), normalized relative errors of pressure energy are defined as:

$$\tilde{\varepsilon}\_{W\_{\Gamma}} = \frac{\left(\mathbf{1}\mathbf{0}^{\mathrm{SNR}/10}\varepsilon\_{W\_{\Gamma}} - cond(\mathbf{T}\_{\mathbf{R}})^{-}\right)}{cond(\mathbf{T}\_{\mathbf{R}}) - cond(\mathbf{T}\_{\mathbf{R}})^{-}}\tag{38}$$

for different SNRs and holograms, which should satisfy ~*ε<sup>W</sup><sup>Γ</sup>* ∈ ½ � 0, 1 .

The errors are plotted in **Figure 8**. **Figure 8a** depict that the most necessary number of modes for the boundary pressure reconstruction is *N* ¼ 3. However, the reconstruction with *N* ¼ 4 is superior to that of *N* ¼ 5 in the sense of stability. Because the condition numbers of translators for *N* ¼ 5 are larger which results in an inverse operation sensitive to errors. Actually, condition number for *N* ¼ 5 is more than six times of that for *N* ¼ 4 which reduces the reconstruction accuracy by the amplified errors. It is also the reason why the results obtained with *N* ¼ 5 is worse than *N* ¼ 4 for SNRs smaller than 20 dB even that it has a smaller radiation efficiency error in **Figure 2**. The decreased results for the reconstructed pressure with increased number of necessary modes illustrate that over-selected number of modes may result in accuracy loss, especially for models with irregular shapes as well as small SNRs. It is because that over-selected number of modes will yield translators with larger condition numbers which are likely to amplify the errors in the reconstruction. It is also observed from **Figure 8** that the two types of holograms can deliver the same level of accuracy with proper selected number of modes. However, the spherical hologram is more preferable since it requires the smallest number of microphones as compared with others.

The normalized errors are presented in of **Figure 8b**. The normalized errors are all less than 1 and increase along with the SNRs. The small errors for small SNRs and participant number of modes are due to the fact their lower bounds are

#### **Figure 8.**

*Errors of the reconstructed pressure energy with spherical holograms at 601 Hz for (a) the relative errorεWT and (b) the normalized error* ~*εWT .*

underestimated, while the upper bounds are overestimated by the approach in Section 3.3. The extreme small normalized errors for case *N* ¼ 3 are due to the dominated truncation errors for small number of adopted participant modes. As indicated for the reference cases, also the no-noise included cases, the reconstructed errors are closer to 1, or in other words more approximated to the estimated upper bounds. More important than the cases with small SNRs for which the estimated lower bounds are almost 0, the no-noise included cases (with infinity large SNR) can supply more reasonable lower bounds by the approach in Section 3.3. The numerical examples clearly demonstrated the validity of the proposed bounds estimation.

## **4.2 A practical experiment**

An experiment is set up to explore the performance of the MRS-based NAH in this section. The source is in the same size and possesses the same material property as the one in the Section 4.1. Reconstruction is only preformed on the spherical hologram, since it requires the minimum number of microphones by comparing with the other two types of holograms.

An equipment is designed to facilitate the measurement. As shown in **Figure 9a**, the measurement on a spherical hologram is realized by rotating a half circular album arm, on which the microphones are mounted, around an axial which is the z-axial. The cubic model is placed at the center of the spherical hologram by hanging in a portal frame with a rigid hollow rod. A single-point drive is applied to the model by a small exciter on the top surface, as shown in **Figure 9b**. To make sure a uniform velocity distribution is generated on the surface, the analyzing frequency is selected closely to one of modal frequencies, which is 634 Hz. The model has an equivalent radius ~*r* ¼ 0*:*138m, and the measurements are performed on a spherical hologram apart from the equivalent sphere by Δ ¼ 0*:*1 m. In light of Eq. (13), the necessary degree of the FS is adopted as *N* ¼ 3. Correspondingly, the minimum required microphones along the *θ* and *ϕ* direction are 4 and 7, respectively. Here, we made an oversampling by placing five microphones along the θ direction and taking nine sequential measurements along ϕ direction.

To validate the reconstructed results, the same 5 by 9 measurements are performed on a spherical validation surface Ω with radius being 0.18 m. In light of Eq. (13), the

**Figure 9.** *The experimental setup: (a) overview of the configuration and (b) details of the model.*

**Figure 10.**

*The radiated pressure distribution on a spherical validation surface for (a) reconstructed from the hologram and (b) obtained from the direct measurement.*

relative difference of reconstructed pressure with respective to the measured one on the validation surface is denoted by:

$$\varepsilon\_{\mathsf{W}\_{\Omega}} = \frac{\left\| \sum\_{n=0}^{\left(N+1\right)^{2}} (\lambda\_{n} - \lambda\_{n,\Omega}) p\_{n}(k,\ \mathbf{x}) \right\|\_{2}}{\left\| \sum\_{n=0}^{\left(N+1\right)^{2}} \lambda\_{n,\Omega} p\_{n}(k,\ \mathbf{x}) \right\|\_{2}} \tag{39}$$

where *λ<sup>n</sup>* and *λ<sup>n</sup>*,<sup>Ω</sup> are coefficients obtained on the hologram and validation surfaces, respectively. **Figure 10** depicts a comparison of the radiated pressure distribution on a spherical surface between the one reconstructed from the hologram and the one measured directly on the validation surface. It is clearly observed that the reconstructed pressure has a very satisfactory distribution agreement with the measured one for which the *ε<sup>W</sup>*<sup>Ω</sup> is 4.5% and the relative error of the maximum pressure is 21.8%.

The experiment is done in a semi-anechoic chamber which can reduce the influences of the environmental noise. Even that, positional errors and some other uncertainties are inevitable to be included in the measured signals which in turn affects the reconstruction results. The error analysis in Section 3 ascribes them to the SNR. How those relate to the SNR is crucial to the error estimation, which needs more investigation.

## **5. Conclusions**

A NAH based on the mapping relationship between modes on surfaces of structure and hologram is introduced. The modes adopted in the NAH are FS of the Helmholtz equation in spherical coordinates which are generally independent and not orthogonal except on the spherical surface. The NAH framework provides a new insight to the reconstruction procedure based on the FS in spherical coordinates. The modes on the surface of structure and hologram form a bijective mapping. Number of modes prescribed in the MRS-based NAH is crucial to total number of measurement as well as the final reconstruction accuracy. An approach is proposed to estimate the necessary degree of effective modes. It is built on the energy criteria by exploring the radiation efficiency of the modes on the equivalent spherical source. An upper bounded error is derived for the radiated sound power of a vibrating structure with degree of modes up to a specific value. A relatively small value of degree is given by this approach.

Once the necessary degree of modes is determined, the number and position of microphones, which are also very crucial to the NAH, are investigated. Techniques to approximate the modes on three types of holograms by a set of locally orthogonal patterns are developed. A numerical algorithm is needed to determine the tight bounds for two locally orthogonal patterns on the planar patch. Due to the completeness of polar angles on cylindrical holograms, the algorithm is reduced by one dimension and the number of degree and positions are analytically determined for the local patterns along the polar direction. The number and position of measurement on the spherical hologram are determined by a purely analytical method because of its conformity to the coordinates of modes.

Errors are inevitable to be encountered in the NAH experiment. It is found that the reconstruction accuracy is subjected to two kinds of errors, one is the SNR and another one is the truncation error due to the limited number of participant modes adopted in the MRS-based NAH. An error model is built, and the relative error of the reconstructed pressure energy on the surface of the vibrating structure is derived. The lower and upper bounds of the relative error can be achieved numerically by a constrained nonlinear optimization algorithm. However, the approach generally yields underestimation of the lower bound and the overestimation of the upper bound, especially for MRS-NAH with large condition numbers. Alternatively, a reasonable lower bound is obtained by considering the case without noise or equivalently with positive infinite SNR. It eliminates the influence of the condition number of the inverse translator and is only related to the truncation errors. Thus, it is feasible to predicate the lower error of a reconstruction with the MRS-based NAH once the truncation error is given, which is validated by numerical examples. Proper estimation of the truncation errors is highly related to the reasonable estimation of the lower bound, which deserves more investigation.

Numerical examples are set up to validate the error analysis of the MRS-based NAH. It clearly demonstrates that the reconstructed results agree well with the simulated results. Physical experiment is designed to further demonstrate the feasibility and performance of the MRS-based NAH. The reconstructed results demonstrate a very satisfactory agreement with the direct measured one with respective to the quantities as well as the distribution on the validated surface. However, to estimate the performance with respective to the actual quantities by the proposed approach, it is desirable to investigate the influences of positional errors and other uncertainties on the SNR.

## **Acknowledgements**

The work is supported by the National Natural Science Foundation of China (Grant No. 11404208).

## **Notes/thanks/other declarations**

Place any other declarations, such as "Notes," "Thanks," etc. in before the References section. Assign the appropriate heading. Do NOT put your short biography in this section. It will be removed.

## **Author details**

Haijun Wu1,2\* and Weikang Jiang1,2

1 State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, China

2 Institute of Vibration, Shock and Noise, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai Jiao Tong University, Shanghai, China

\*Address all correspondence to: haijun.wu@sjtu.edu.cn

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Williams EG, Maynard JD, Skudrzyk E. Sound source reconstructions using a microphone array. Journal of the Acoustical Society of America. 1980;**68**(1):340-344

[2] Maynard JD, Williams EG, Lee Y. Nearfield acoustic holography: I. theory of generalized holography and the development of NAH. Journal of the Acoustical Society of America. 1985; **78**(4):1395-1413

[3] Veronesi WA, Maynard JD. Nearfield acoustic holography (NAH) II. Holographic reconstruction algorithms and computer implementation. Journal of the Acoustical Society of America. 1987;**81**(5):1307-1322

[4] Williams EG. Fourier Acoustics Sound Radiation and Nearfield Acoustical Holography. San Diego, Calif: Academic Press; 1998

[5] Steiner R, Hald J. Near-field acoustical holography without the errors and limitations caused by the use of spatial DFT. International Journal of Acoustics and Vibration. 2001;**6**(2): 83-8989

[6] Cho YT, Bolton JS, Hald J. Source visualization by using statistically optimized near-field acoustical holography in cylindrical coordinates. Journal of the Acoustical Society of America. 2005;**118**(4):2355-2364

[7] Hald J. Basic theory and properties of statistically optimized near-field acoustical holography. Journal of the Acoustical Society of America. 2009; **125**(4):2105-2120

[8] Kim GT, Lee BH. 3-D sound source reconstruction and field reprediction using the Helmholtz integral equation. Journal of Sound and Vibration. 1990; **136**(2):245-261

[9] Bai MR. Application of BEM (boundary element method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries. Journal of the Acoustical Society of America. 1992;**92**(1):533-549

[10] Veronesi WA, Maynard JD. Digital holographic reconstruction of sources with arbitrarily shaped surfaces. Journal of the Acoustical Society of America. 1989;**85**(2):588-598

[11] Zhang Z et al. A computational acoustic field reconstruction process based on an indirect boundary element formulation. Journal of the Acoustical Society of America. 2000;**108**(5 I): 2167-2178

[12] Zhang Z et al. Source reconstruction process based on an indirect variational boundary element formulation. Engineering Analysis with Boundary Elements. 2001;**25**(2):93-114

[13] Schuhmacher A et al. Sound source reconstruction using inverse boundary element calculations. Journal of the Acoustical Society of America. 2003; **113**(1):114-127

[14] Johnson ME et al. An equivalent source technique for calculating the sound field inside an enclosure containing scattering objects. Journal of the Acoustical Society of America. 1998; **104**(3 I):1221-1231

[15] Jeon IY, Ih JG. On the holographic reconstruction of vibroacoustic fields using equivalent sources and inverse boundary element method. Journal of the Acoustical Society of America. 2005; **118**(6):3473-3482

[16] Sarkissian A. Method of superposition applied to patch near-field acoustic holography. Journal of the Acoustical Society of America. 2005; **118**(2):671-678

[17] Bi CX et al. Nearfield acoustic holography based on the equivalent source method. Science in China, Series E: Technological Sciences. 2005;**48**(3): 338-353

[18] Bi CX, Chen XZ, Chen J. Sound field separation technique based on equivalent source method and its application in nearfield acoustic holography. Journal of the Acoustical Society of America. 2008;**123**(3): 1472-1478

[19] Bi CX, Bolton JS. An equivalent source technique for recovering the free sound field in a noisy environment. Journal of the Acoustical Society of America. 2012;**131**(2):1260-1270

[20] Song L, Koopmann GH, Fahnline JB. Numerical errors associated with the method of superposition for computing acoustic fields. Journal of the Acoustical Society of America. 1991;**89**(6): 2625-2633

[21] Fahnline JB, Koopmann GH. A numerical solution for the general radiation problem based on the combined methods of superposition and singular-value decomposition. Journal of the Acoustical Society of America. 1991; **90**(5):2808-2819

[22] Koopmann GH, Song L, Fahnline JB. A method for computing acoustic fields based on the principle of wave superposition. Journal of the Acoustical Society of America. 1989;**86**(6): 2433-2438

[23] Bai MR, Chen CC, Lin JH. On optimal retreat distance for the

equivalent source method-based nearfield acoustical holography. Journal of the Acoustical Society of America. 2011;**129**(3):1407-1416

[24] Wu SF. On reconstruction of acoustic pressure fields using the Helmholtz equation least squares method. Journal of the Acoustical Society of America. 2000;**107**(5 I): 2511-2522

[25] Wang Z, Wu SF. Helmholtz equation-least-squares method for reconstructing the acoustic pressure field. Journal of the Acoustical Society of America. 1997;**102**(4):2020-2032

[26] Wu SF, Yu JY. Reconstructing interior acoustic pressure fields via Helmholtz equation least-squares method. Journal of the Acoustical Society of America. 1998;**104**(4):2054-2060

[27] Wu SF. Methods for reconstructing acoustic quantities based on acoustic pressure measurements. Journal of the Acoustical Society of America. 2008; **124**(5):2680-2697

[28] Thite AN, Thompson DJ. The quantification of structure-borne transmission paths by inverse methods. Part 1: Improved singular value rejection methods. Journal of Sound and Vibration. 2003;**264**(2):411-431

[29] Williams EG. Regularization methods for near-field acoustical holography. Journal of the Acoustical Society of America. 2001;**110**(4): 1976-1988

[30] Lu HC, Wu SF. Reconstruction of vibroacoustic responses of a highly nonspherical structure using Helmholtz equation least-squares method. Journal of the Acoustical Society of America. 2009;**125**(3):1538-1548

*A Mapping Relationship-Based near-Field Acoustic Holography DOI: http://dx.doi.org/10.5772/intechopen.108318*

[31] Bi CX et al. Reconstruction of the free-field radiation from a vibrating structure based on measurements in a noisy environment. Journal of the Acoustical Society of America. 2013; **134**(4):2823-2832

[32] Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: U.S. Govt. Print. Off; 1964

[33] Wu HJ, Jiang WK, Zhang HB. A mapping relationship based near-field acoustic holography with spherical fundamental solutions for Helmholtz equation. Journal of Sound and Vibration. 2016;**373**(7):66-88

[34] Wu HJ, Liu YL, Jiang WK. A fast multipole boundary element method for 3D multi-domain acoustic scattering problems based on the Burton-miller formulation. Engineering Analysis with Boundary Elements. 2012;**36**(5):779-788

[35] Wu HJ, Jiang WK. Experimental study of the mapping relationship based near-field acoustic holography with spherical fundamental solutions. Journal of Sound and Vibration. 2017;**394**: 185-202

[36] Rahola J. Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems. BIT Numerical Mathematics. 1996;**36**(2):333-358

[37] Wu HJ, Liu YJ, Jiang WK. A lowfrequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems. Engineering Analysis with Boundary Elements. 2013; **37**(2):309-318

Section 2
