**2. Digital holographic microscopy (DHM)**

DHM is an interferometric modality where the recorded image data *H x*ð Þ , *y* represents interference between the object wave *O x*ð Þ , *y* representing the light which has interacted with the cell sample and the reference wave *R x*ð Þ , *y* that has not interacted with the sample is described as:

$$H = |\mathcal{R}|^2 + |\mathcal{O}|^2 + \mathcal{R}^\* \mathcal{O} + \mathcal{R} \mathcal{O}^\*. \tag{2}$$

Here ∗ represents the complex conjugation of the corresponding wave-function. In the image plane holography case as in the present study, *O x*ð Þ , *y* represents the

#### *Augmented Reality and Its Application*

resultant image field corresponding to the cell sample slide when observed through a 40*x* infinity corrected imaging system [23]. The interference is possible due to the use of a laser source which ensures that the object and reference waves remain temporally coherent at the detector plane and produce interference fringes with good contrast. Since our DHM system is also fitted with a white light LED illumination which allows recording of the cell sample in the usual bright-field mode for ease of interpretation by a clinician.

Reconstruction of single-shot holograms is traditionally performed using the Fourier transform method. However, due to the low-pass filtering nature of this method image plane phase recoveries with full pixel resolution cannot be obtained using this approach. This poses a problem as the bright-field images available will then seem to have higher resolution even though both have been recorded using the same microscope objective. In order to have both bright-field and phase images with same diffraction-limited resolution, we reconstruct of the complex object wave *O x*ð Þ , *y* using a sparse optimization method that has been developed by our group in recent years. In particular, recovery of the complex image field *O x*ð Þ , *y* is posed as an optimization problem where we minimize a cost function of the form:

$$\begin{aligned} \mathcal{C}(O, O^\*) &= \mathcal{C}\_1 + \mathcal{C}\_2 \\ &= \left\| H - \left( |\mathcal{R}|^2 + |O|^2 + \mathcal{R}^\* O + \mathcal{R} O^\* \right) \right\|^2 + \mathcal{y}(O, O^\*). \end{aligned} \tag{3}$$

Here k k … <sup>2</sup> denotes the squared L2-norm of the quantity inside. The reference beam *R x*ð Þ , *y* is estimated by a separate calibration step involving recording of a straight line interference fringe pattern without any sample followed by accurate estimation of carrier frequency to fractional fringe accuracy [28]. The first term of the cost function represents the least square data fit and the second term *<sup>ψ</sup> <sup>O</sup>*, *<sup>O</sup>*<sup>∗</sup> ð Þ is a suitable image domain constraint. We use the modified Huber penalty function as a constraint and use an adaptive alternating minimization scheme explained in detail elsewhere [23, 26] for recovering the complex object function *O x*ð Þ , *y* in the image plane. The modified Huber penalty is defined as:

$$\psi(O, O^\*) = \sum\_{k=all \ p pixels} \left[ \sqrt{1 + \frac{\left| \nabla O\_k \right|^2}{\delta^2}} - 1 \right]. \tag{4}$$

The tuning parameter *δ* is made proportional to the median of the gradient magnitudes of the image solution in a given iteration. The Huber penalty acts like the edge preserving Total Variation penalty at pixels where the gradient magnitude ∣∇*O*∣ is much larger than *δ* and acts like the smoothing quadratic penalty for pixels where the gradient magnitude is small compared to *δ*. Further the adaptive optimization strategy makes sure that the change in the solution due to error minimizing step is balanced by that due to Huber minimization step in every iteration. We point out that the optimization problem above involves real valued data (hologram *H*) whose solution is complex valued. The steepest descent directions evaluated in the algorithm need to be evaluated using Wirtinger derivatives with respect to *O*<sup>∗</sup> . In particular we note that the Wirtinger derivatives for the two terms of the cost function in Eq. (3) is given by:

$$\nabla\_{O^{\mathsf{T}}} \mathsf{C}\_{1} = -2 \left[ H - \left| R + O \right|^{2} \right] \cdot (R + O), \tag{5}$$

and

*Cytopathology Using High Resolution Digital Holographic Microscopy DOI: http://dx.doi.org/10.5772/intechopen.96459*

$$\nabla\_{O} \cdot \mathbf{C}\_{2} = -\nabla \cdot \left[\frac{\nabla O}{\sqrt{\mathbf{1} + \frac{|\nabla O|^{2}}{\delta^{4}}}}\right].\tag{6}$$

It is important to note that the optimization procedure operates fully in the image domain making it possible to employ it over a region of interest and thus allowing full resolution reconstruction in near real time. In our study, a 256 � 256 pixel ROI phase reconstruction requires few seconds (<25 iterations) in a MATLAB implementation on a desktop with 3*:*1 GHz processor. The data consistency error for the reconstructed solution is within 5% relative error. A user can therefore select a region of interest near a cell nucleus for object wave reconstruction. The resolution and noise advantage of this optimization procedure over traditional Fourier filtering approach has been shown in a series of publications [24–27, 29], as a result, we will not discuss this point here once again. However for completeness we summarize the advantages of the optimization method in comparison to the traditional methods for image plane hologram processing in **Table 1**.

The phase map *ϕ*ð Þ *x*, *y* as in Eq. (1) is the argument of the recovered complex object field *O x*ð Þ , *y* and is given by arctangent of the ratio of imaginary and real parts:

$$\phi(\mathbf{x}, \boldsymbol{y}) = \arctan\left(\frac{\operatorname{Im}[O(\mathbf{x}, \boldsymbol{y})]}{\operatorname{Re}\left[O(\mathbf{x}, \boldsymbol{y})\right]}\right). \tag{7}$$

Since the arctangent function is defined only over the range ½ � �*π*, *π* the phase map defined in Eq. (4) is wrapped. A 2D unwrapping procedure based on the transport of intensity equation (TIE) [30] has been employed in our work in order to associate physical meaning to the phase map in accordance to Eq. (1). The steps involved in imaging are summarized in supplementary (**Figure 1)**. A Pap-smear sample is first imaged in both bright-field and holographic modalities using a dual mode digital holographic microscope (fabricated by Holmarc Opto-Mechatronics Pvt. Ltd., Kochi, India). The holographic (or interferometric) image is used further


**Table 1.**

*Summary of resolution performance of image plane digital holographic methods.*

#### **Figure 1.**

*Steps in imaging chain (a) Pap-smear slide, (b) dual-mode DHM system, (c) illustrative example of a brightfield image and a hologram recorded using the DHM system, (d) computer used for reading image from camera, phase reconstruction and computing morphological parameters from the bright-field and phase images, (e) illustrative example of phase map of a cell nucleus rendered as a surface plot.*


*a M denotes a binary (0, 1) mask for individual cell nucleus, q denotes phase map. Both are defined over ROI of 256* � *256 pixels centered on cell nucleus.*

#### **Table 2.**

*Morphological parameters evaluated for each cell nucleus imaged in this study. More details about these parameters are provided in Table 1 of ref. [23].*

for phase reconstruction as explained above. **Table 2** provides details about a number of morphological parameters derived from the cell images in the brightfield and quantitative phase modes. The morphological parameters were decided in consultation with practicing cyto-pathologists who participated in this study. We summarize them in **Table 2** for convenience of the reader. The N/C ratio which is the ratio of nucleus to cytoplasm areas has been included as list of three labels (low = 1, medium = 2, high = 3). This is because we found that a number of cells in the patient samples appeared in clusters and it was difficult to find boundaries of cytoplasm in simple automated manner in such cases.
