**4.4 Calculation of nucleated cell dimensions**

198 Advanced Holography – Metrology and Imaging

(a) (b)

(a) (b)

drawn the reciprocal lattice (b). (*Inset: Representation of the real space*)

\* sin *K*

*L* is the constant of diffraction, where

*a* \*

quantities are related by the following expressions,

\*

*a*

where *K =* 

Fig. 18. Fraunhofer diffraction pattern (a), scale bar 2 mm-1. Over diffraction pattern it is

As shown in figure 18b, according to the theory of crystal diffraction, the reciprocal lattice edges of dimensions *a\** and *b\** are respectively perpendicular to the cell edges *b* and *a*, and

\* sin

radiation and *L* is the camera length, i.e., the distance from the specimen to the diffraction

 and \* <sup>180</sup>*<sup>o</sup>* 

(16)

the wavelength of monochromatic

*<sup>K</sup> <sup>b</sup> b* 

Fig. 17. Digital hologram (a) and intensity image (b) of the sample. Scale bar *5*

*m* We demonstrate in this section the potentialities of Digital Holographic Microscopy in the determination of morphological parameters of nucleated cells. The spectral analysis of the radial behaviour of the Fraunhofer diffraction pattern allows the correlation between the peaks observed in the spectra and lineal dimensions of the cell.

As an example of application a sample of oral mucosa epithelial cell was selected. These cells have a nucleus inside a regular cytoplasm, figure 19.

Fig. 19. Oral mucosa epithelial cell, optical microscopy image. Scale bar *20 m*

Two cases were analyzed, (i) the nucleus of the cell is located approximately at the centre of the cell and (ii) the nucleus is outside the centre.

Figure 20 shows the hologram (left) and the phase image reconstruction (right) of cell with the nucleus at the centre (A) and outside of centre (B). Both holograms were captured with the apparatus of figure 1 and the phase image was obtained by the calculation of the argument of Eq. (9) with *d' = 0.*

For both holograms the Fraunhofer pattern was calculated and is shown in figure 21 (left). Applying the spectral analysis of the Fraunhofer pattern radial intensity, the corresponding frequency spectra are obtained, figure 21 (right). In each spectrum a sequence of peaks are seen. In the case of cell with nucleus at the centre (A) four peaks appear in the spectrum, which is different for the case of the cell with nucleus outside the centre (B) where six peaks are observed in the spectrum.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 201

Fig. 21. Fraunhofer pattern (left) and the corresponding spectrum of their radial intensity

(a) (b)

These results agree with theoretical predictions and experimental test presented in (Türke et al., 1978), although with the proposed method simpler the recording process and data

Fig. 22. Cellular dimensions reflected in the diffraction patterns: diagram (a) and (b) correspond to lineal dimensions of spectrums represented in figure 21 (right).

(right) of cell with the nucleus at the centre (A) and outside of centre (B).

A

B

processing are simpler.

Fig. 20. Hologram (left) and phase image reconstruction (right) of cell with the nucleus at the centre (A) and outside of centre (B). Scale bar *20 m*

The correlation between the peaks observed in the spectra and the diameter of cytoplasm and nucleus as well as other dimensions of the cell is definite, i.e., the peak position in the spectra is related with a lineal dimension in the cell.

In the case of the cell with the nucleus at the centre the peaks mean: peak (1) is characteristic of the nucleus diameter Dn, peak (2) characterizes half of the difference of both diameters, (Dc-Dn)/2, peak (3) characterizes half of the sum of both diameters, (Dc+Dn)/2 and peak (4) is characteristic of the cytoplasm diameter Dc.

If we consider the case of a cell with eccentric nucleus, which is shifted a distance *e* from the centre of cytoplasm two peaks correspond to the nuclear and cytoplasm diameters, (1) and (6) respectively, and the other four to half of the sums and differences of these two plus/minus the eccentricity 'e': peak (2) → (Dc-Dn)/2 – e, peak (3) → (Dc+Dn)/2 – e, peak (4) → (Dc-Dn)/2 + e, peak (5) → (Dc+Dn)/2 + e. The equivalent intervals in the cell are shown in figure 22.

Fig. 20. Hologram (left) and phase image reconstruction (right) of cell with the nucleus at the

The correlation between the peaks observed in the spectra and the diameter of cytoplasm and nucleus as well as other dimensions of the cell is definite, i.e., the peak position in the

In the case of the cell with the nucleus at the centre the peaks mean: peak (1) is characteristic of the nucleus diameter Dn, peak (2) characterizes half of the difference of both diameters, (Dc-Dn)/2, peak (3) characterizes half of the sum of both diameters, (Dc+Dn)/2 and peak (4)

If we consider the case of a cell with eccentric nucleus, which is shifted a distance *e* from the centre of cytoplasm two peaks correspond to the nuclear and cytoplasm diameters, (1) and (6) respectively, and the other four to half of the sums and differences of these two plus/minus the eccentricity 'e': peak (2) → (Dc-Dn)/2 – e, peak (3) → (Dc+Dn)/2 – e, peak (4) → (Dc-Dn)/2 + e, peak (5) → (Dc+Dn)/2 + e. The equivalent intervals in the cell are

*m*

centre (A) and outside of centre (B). Scale bar *20* 

spectra is related with a lineal dimension in the cell.

is characteristic of the cytoplasm diameter Dc.

shown in figure 22.

**A** 

**B** 

Fig. 21. Fraunhofer pattern (left) and the corresponding spectrum of their radial intensity (right) of cell with the nucleus at the centre (A) and outside of centre (B).

Fig. 22. Cellular dimensions reflected in the diffraction patterns: diagram (a) and (b) correspond to lineal dimensions of spectrums represented in figure 21 (right).

These results agree with theoretical predictions and experimental test presented in (Türke et al., 1978), although with the proposed method simpler the recording process and data processing are simpler.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 203

The hologram of a microscopic character with zero shape, figure 23a, is used to calculate the Fraunhofer diffraction pattern shown in figure 25a. The radial intensity curve *IR(u)* and the

corresponding frequency spectrum are shown in figure 25b.

(a) (b)

and the smallest radii are *140 μm* and *90 μm* respectively.

lower.

figure 26 (right).

spectrum is associated with each form.

Fig. 25. Fraunhofer diffraction pattern of the microscopic character with zero shape (a) and the radial intensity curve (b)-upper and the corresponding frequency spectrum (b)-

According to figure 25a, the Fraunhofer diffraction pattern consists of elliptically shaped fringes with similar forms to that of the zero character. The fringes' width variation along their perimeter is detected by the radial intensity curve and with the subsequent spectral analysis the largest and smallest radius of the zero character can be obtained. The two peaks are signaled with arrows in the spectrum, figure 25b-down. They are positioned at the spatial frequencies *7.12 mm-1* and *11.10 mm-1*, that represent the biggest and smallest radius of the zero character respectively. According to the peaks' spatial frequencies, the largest

For object detection the holograms of a microscopic characters with '20' and '10' shapes, figure 24 (left), were used to calculate the Fraunhofer diffraction patterns shown in figure 26 (left). The radial intensity curve *IR(u)* and the corresponding frequency spectrum is shown in

In the frequency spectrum shown in figure 26 (right) appear, mixed with other peaks, two peaks at the same spatial frequency as those that are characteristic of the zero shape frequency spectrum. In this way, the object detection can be generalized for other objects with irregular forms, because the spectrum of the radial curve of the object's diffraction pattern presents a sequence of peaks that characterize the form of the object. A unique

As has been shown, this method of object detection is similar to the qualitative analysis in xray diffraction, i.e. the presence of an object is characterized by the presence of peaks in appropriate positions in the spectrum. This analogy is very important because all the

developed tools for the qualitative analysis in x-rays can be used for object detection.
