Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy

Holography is a method for storing and reconstructing both amplitude and phase information of a wave front. In digital holography the reconstruction process is accomplished by means of a computer (Yaroslavsky & Merzyalov, 1980) obtaining directly the phase distribution of the object wave front. Particularly with the improvement of the spatial resolution of CCD cameras and the increasing computational performance of personal computers digital holography has been widely applied in many fields such as deformation analysis (Schedin et al., 2001), object contouring (Wagner et al., 2000), microscopy (Takaki & Ohzu, 1999) and particle measurement (Murata & Yasuda, 2000). The technique of digital holography has been implemented in a configuration of an optical microscope (Schilling et al., 1997); the objective lens produces a magnified image of the object and the interference between this image and the reference beam is achieved by the integration of the microscope into one of the arms of a Mach-Zender interferometer. This configuration is called Digital Holographic Microscopy (DHM). DHM is a powerful technique for real-time quantitative phase contrast imaging, since a single intensity image, called a hologram, allows the reconstruction of the phase shift induced by a specimen. This property of holograms offers phase-contrast techniques, which can then be used for quantitative 3D imaging (Palacios et al., 2005). Quantitative phase imaging is important because it allows the determination of the optical thickness profile of a transparent object with sub-wavelength accuracy (Yu et al., 2009). Through numerical processing of the hologram one can filter out parasitic interferences and the components of the image reconstruction: zero-order and twin image terms (Cuche et al., 2000) or to compensate for curvature introduced by the microscope objective (MO) (Pedrini et al., 2001),


Introduction
Holography is a method for storing and reconstructing both amplitude and phase information of a wave front.In digital holography the reconstruction process is accomplished by means of a computer (Yaroslavsky & Merzyalov, 1980) obtaining directly the phase distribution of the object wave front.Particularly with the improvement of the spatial resolution of CCD cameras and the increasing computational performance of personal computers digital holography has been widely applied in many fields such as deformation analysis (Schedin et al., 2001), object contouring (Wagner et al., 2000), microscopy (Takaki & Ohzu, 1999) and particle measurement (Murata & Yasuda, 2000).The technique of digital holography has been implemented in a configuration of an optical microscope (Schilling et al., 1997); the objective lens produces a magnified image of the object and the interference between this image and the reference beam is achieved by the integration of the microscope into one of the arms of a Mach-Zender interferometer.This configuration is called Digital Holographic Microscopy (DHM).DHM is a powerful technique for real-time quantitative phase contrast imaging, since a single intensity image, called a hologram, allows the reconstruction of the phase shift induced by a specimen.This property of holograms offers phase-contrast techniques, which can then be used for quantitative 3D imaging (Palacios et al., 2005).Quantitative phase imaging is important because it allows the determination of the optical thickness profile of a transparent object with sub-wavelength accuracy (Yu et al., 2009).Through numerical processing of the hologram one can filter out parasitic interferences and the components of the image reconstruction: zero-order and twin image terms (Cuche et al., 2000) or to compensate for curvature introduced by the microscope objective (MO) (Pedrini et al., 2001),

Experimental set-up
Figure 1 shows the experimental set-up used in this work.It is a Digital Holographic Microscope designed for transmission imaging with transparent sample.The basic architecture is that of a Mach-Zehnder interferometer.A linearly polarized He-Ne laser (15 mW) is used as light source.The expanded beam from the laser is divided by the beam splitter BS1 into reference and object beams.The microscope produces a magnified image of the object and the hologram plane is located between the microscope objective MO and the image plane (x'-y') which is at a distance d' from the recording hologram plane (-).In digital holographic microscopy we can consider the object wave emerging from the magnified image and not from the object itself (VanLigten & Osterberg, 1966).With the combinations of the dual polarizer P 1 and P 2 the intensities are adjusted in the reference arm and the object arm of the interferometer and the same polarization state is also guaranteed for both arms improving their interference.The specimen S is illuminated by a plane wave and a microscope objective, that produces a wave front called object wave O, collects the transmitted light.A condenser, not shown, is used to concentrate the light or focus the light in order that the entire beam passes into the MO.At the exit of the interferometer the two beams are combined by beam splitter BS2 being formed at the CCD plane the interference pattern between the object wave O and the reference wave R, which is recorded as the hologram of intensity where R * and O * are the complex conjugates of the reference and object waves, respectively.The two first terms form the zero-order, the third and fourth terms are respectively the virtual (or conjugate image) and real image, which correspond to the interference terms.The off-axis geometry is considered; for this reason the mirror M2, which reflects the reference wave, is oriented so that the reference wave reaches the CCD camera with a small incidence angle with respect to the propagation direction of the object wave where j, l are integers defining the positions of the hologram pixels and  =  = 4.65 m defines the sampling intervals in the hologram plane.

Basic principles of the alternative reconstruction method
In In our approach the reconstruction of the complex wave distribution o(x',y') (x',y';z=d') consists basically of two stages that involve two wavefield propagations.In the first stage, figure 3, we reconstruct the wave distribution õ(u,v) on the (u-v) plane at reconstruction distance z = D (first propagation).Applying the (ASA) algorithm the distance D is calculated, as shown in section 3.1.1.After the distance D has been calculated, the first propagation is carried out by means of the Fresnel approximation method, specifically the Single Fourier Transform Formulation (SFTF), where z is the reconstruction distance,  is an operator that denotes the Fourier transform,  (Cuche et al., 2000), As has been proven (Palacios et al., 2008), the complex field

 
,; f SFTF uvz D   is equivalent to the complex field distribution õ(u,v) on the back focal plane of the objective lens.From Abbe's theory of image formation (Lipson S. & Lipson H., 1981), the field on the back focal plane can be represented by the expression, where S o is the distance from the object to the lens, f the focal distance of the lens,  the wavelength of the incident plane wave.
From the wave theory of image formation, after the objective lens producing the diffraction pattern of the object in its back focal plane, a second Fourier transformation performed on the diffraction pattern it is associated with the image of the object (Goodman, 1968).Consequently, all the information about image wavefield at the hologram plane is contained in the complex wavefront o(u,v) on the back focal plane, therefore the reconstruction of the optical wavefield o(x',y';d') can be carried out from the (u-v) plane, instead of traditionally the hologram plane (ξ-η), figure 4. In the second stage of the method, the complex wavefield (x',y';d') at an arbitrary distance d' can be obtained by propagation of the wavefield õ(u,v) through a distance d' and the result is inverse Fourier transformed, where  -1 symbolizes the inverse Fourier Transform, k=2/ , k u and k v are corresponding spatial frequencies of u and v respectively.The numerical implementation of Eq. ( 8), that we call the Double-Propagation algorithm (DPA), is given by, where j, l, m, n are integers is the discrete formulation of Eq. ( 4).From Eq. ( 9) we can obtain the intensity image I DPA (x',y';d') by calculating | DPA (m,n;d')| 2 and the phase image  DPA (x',y';d') (m,n;d')].In short, the following steps describe the general procedure of the alternative reconstruction method: 1. Determining the distance D.

 
,; f SFTF uvz D   by filtering the spatial components that correspond to the complex amplitudes of the Fourier transform of the objects.
5. Determining the intensity image or the phase image by calculating the square modulus or the argument of Eq. ( 9).
As we will demonstrate in next section, the formulation based on Eq. ( 9) guarantees that the reconstructed image maintains its size independently of depth d' and the phase curvature compensation can be done easily by techniques of image background subtraction.At z = D the reconstruction of real image is different of the genuine appearance of object image because at this distance the Fraunhofer diffraction pattern of the objects is reconstructed, figure 6, rather than the image of the objects.Larger or lower spatial frequencies of object decomposition will be represented by intensity in the focal plane that is farther or closer from optical axis or equivalently farther or closer from the center of the pattern where is contained the undiffracted object wavefield.

Determination of the distance D
The distance D is determined only one time, and remains the same until there is some variation in the experimental set-up.Applying the (ASA) algorithm, the following steps were performed to calculate the distance D: 1.The angular spectrum A(k , k  ;z=0) of the hologram I H (,) at z = 0 is obtained by taking its Fourier transform.k  ,and k  are the corresponding spatial frequencies in the hologram plane -.
2. Filtering of the angular spectrum to suppress both the zero-order and the twin image.In this step a region of interest corresponding only to the object spectrum is selected and the modified angular spectrum

 
,; 0 The reconstructed complex wave field at any plane (x'-y') perpendicular to the propagating z axis is found by, 5. The reconstruction of the amplitude image I ASA (x',y';z)=| ASA (x',y';z)| 2 , is carried out varying z from 0 to 300 mm with 20 mm as incremental step.The punctual maximum value P(z) = [I ASA (x',y';z)] max is calculated and plotted for each z, figure 7. 6. Between the two points around the relative maximum the incremental step is reduced to 1 mm.The distance D is equal to the z value at which the absolute maximum of I ASA (x',y';z) max is reached; the value of z = D = 173 mm is shows in figure 7. From figure 8 it is corroborated that as the reconstruction plane approaches the focal plane, the phase jumps between the reference and object waves gradually disappear.These phase jumps totally disappear for z = D = 173 mm, where the focal plane is reconstructed.This behaviour allows us to conclude that the curvature of the wavefront has a minimum on the focal plane and it is increased as the wavefield propagates away from this plane.

Intensity and phase image reconstruction
Calculating the square modulus and the argument of Eq. ( 9) in figure 9   As can be appreciated from figure 11b, a good fitting of  B is achieved when a median filter with a large kernel size is applied over  DPA .
Taking into consideration that the reconstructed complex wavefield  DPA can be expressed For comparison figure 12b shows the phase image calculated from the same hologram and reconstructed with the ASA, but using a reference hologram (Colomb et al., 2006).

Refocusing at different reconstruction distances
With the analysis of the intensity image reconstruction we demonstrate the ability of DPA to refocus at different distances d'.
Figure 13  According to figure 13, as the reconstruction distance d' increases, the size of the reconstructed intensity image is constant for both algorithm until d' = 10 mm.For values of d' bigger than 20 mm, the size of the reconstructed intensity image is reduced when the ASA is used, whereas it remain constant when is used the DPA.The effect is most obvious for d' = 70 mm.Summarizing, two main advantages can be attributed to the DPA algorithm: capability to maintain the size of a reconstructed image, independent of the reconstruction distance and wavelength for objects larger than a CCD and phase reconstruction with curvature compensation without the necessity of either a reference hologram or parameter adjustment.For a hologram of 1024 x 768 pixels using a standard PC computer (Pentium IV, 3.2 GHz) the required time for the calculation of the phase image, the intensity image and the distance D is 2.3, 1.2 and 2.6 seconds respectively.The limitations are related to the two manual filtering stages that exist in the reconstruction process.After the first propagation, the manual selection of Fourier transform components at the back focal plane can introduce others element that disturb the reconstructed plane.

Microscopic object analysis using DHM
In this section we study microscopic objects with regular forms starting from their Fraunhofer diffraction patterns obtained with DHM.Two types of analysis are considered: (i) analysis of objects according to their spatial distribution and (ii) analysis of individual objects.

Fourier transformation at the back focal plane
From Eq. ( 5) it can be seen that the wave field at the back focal plane õ(u,v) is proportional to the Fourier transform of the objects except for the spherical wave front S  , which represents a quadratic phase curvature factor that causes a phase error if the optical Fourier transformation is computed (Poon, 2007).
Using DHM it is possible to find the exact Fourier Transform of objects at the back focal plane.The complex conjugate of the constant phase factor S  (u,v), can be expressed through the well-known parameters of the experimental design presented in figure 1, i.e. d', D and f, Multiplying Eq. ( 12) by Eq. ( 5), the constant phase factor is eliminated and the exact Fourier Calculating the intensity distribution from Eq. ( 13), the object's Fraunhofer diffraction pattern I FDP (u,v) is obtained, Eq. ( 14) offers a powerful tool in microscopic analysis because the Fourier Transform plane can be manipulated and different techniques of Fourier optics can be applied digitally, such as pattern recognition, image processing and others.

System magnification
The knowledge of system magnification is important when quantitative relations between lineal dimensions of the enlarged image and the microscopic object have to be known.In DHM the total system magnification depends on where the camera CCD is placed.

Analysis of objects according to their distribution
We consider objects with regular forms and two different forms of spatial distribution: randomly and periodically distributed.The objects' parameters can be determined by diffraction pattern manipulation in a simple and accurate way.This is an example of objects analysis that is important to biological and materials sciences.

Similar objects in random arrangement
We consider a sample of mouse blood cell as an example of random distribution of similar objects and use the proposed methodology to determine the diameter of cells.Figure 15a shows the hologram recorded with the experimental set-up.Applying Eq. ( 14) the Fraunhofer pattern is obtained, figure 15b.As predicted by theory, the Fraunhofer diffraction pattern has a 'spotty' interference pattern, with a central peak and intensity in the diffraction plane that shows random fluctuations on a general background.
The radial intensity distribution I R (r), figure 16-upper, is measured by scanning the Fraunhofer diffraction pattern along radial lines (Palacios et al., 2001).For each r value the intensity I R (r) is the result of averaging the intensity values I FDP (u,v) along the circumference from 0º to 360º, mathematically this operation can be represented by the expression, where, The spatial coordinates in the Fraunhofer diffraction pattern are defined on basis of the Fraunhofer diffraction pattern pixel resolution Δu, which is determined directly from the Fresnel diffraction formula at the reconstruction distance z = D.In this way, the radial distance u=ju, j=0,1,…,Np, where Np is the points number of the radial intensity curve and u= D/M.In the frequency spectrum, the spatial frequency is f u =j/Npu, j=0,1,..Np.
The spectral analysis of the radial intensity curve I R (r) is carried out by the calculation of the square of the modulus of its 1D Fourier transform.In the resulting spectrum, figure 16lower, the harmonic components are seen.As seen, only one fundamental harmonic that characterizes the diameter (r o = 6.3 m) of the mouse blood cell appears.

Similar objects in periodical arrangement
In this section we consider a sample of periodically hexagonal structures inscribed on a plastic material with an ion beam as an example of regularly repeated identical objects.
In figure 17   9.69 m.This value coincides with that obtained by AFM.

Calculation of nucleated cell dimensions
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.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 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.The microscopic character with zero shape will be detected when mixed with other microscopic characters.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 spectrum is associated with each form.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.

Conclusion
In
Fig. 2. Schematic diagram of traditional image reconstruction methods in DHM.

Fig. 3 .
Fig. 3. Reconstruction of the wave distribution õ(u,v) on the (u-v) plane at reconstruction distance z = D (first propagation).
Fig. 5. (a) Hologram of a USAF resolution target, (b) Components of the reconstructed wavefield at z = D.The circle delimits the real image and the rectangle the conjugate image.

Fig. 6 .
Fig. 6.Intensity of field distribution on the focal plane.

Fig. 7 .
Fig. 7. Distance D determination using ASA.Because the reference wave is plane, the distance D is the physical distance from the hologram to back focal point of the objective lens.

Fig. 9 .
Fig. 9. Reconstruction of the intensity image (a) and the phase image (b) at z =D=173 mm Phase curvature compensation: It should be noted from figure 9b that mixed phase images  o and  B appear, which are related with the phase of the objects and the quadratic constant phase factor S  respectively.Both phase images are shown more clearly in figure 10.

Fig. 10 .
Fig. 10.Pseudo 3D rendering of the phase image  DPA presented in figure 9b.Due to the slow variation of  B , it can be considered as a background contribution to the phase image of the objects  o , thus the problem of phase curvature compensation can be treated as a problem of phase image background subtraction.This way, alternative image background subtraction methods can be used.A procedure(Ankit & Rabinkin, 2007), that consists of applying a median filter with a large kernel size to the phase image represents a quick and simple way to obtain  B , figure 11.

Fig. 13 .
Fig. 13.Comparison between DPA (row A) and ASA (row B) in the intensity image reconstruction for different distances d'.
Fig. 17.Digital hologram (a) and intensity image (b) of the sample.Scale bar 5m

Fig. 19 .B
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.Figure20shows 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 figure1and 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 figure21 (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.

Fig. 21 .Fig. 22 .
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. 23.Hologram of a microscopic character with zero shape (a), scale bar: 50 m.Phase image reconstruction (b)

Fig. 26 .
Fig. 26.Fraunhofer diffraction patterns of corresponding hologram of figure 24 (left) and the radial intensity curve with the corresponding frequency spectrum (right).
this chapter, an alternative methodology for image reconstruction and object analysis in digital holographic microscopy is discussed.The model we use for image reconstruction is specific for digital holographic microscopy; it includes a first propagation from the hologram plane to obtain the Fourier Transform plane of the object at the back focal plane of the microscope objective lens and then a second propagation is applied to find out the image wave field at the reconstruction plane.Using digital hologram reconstruction as a method for calculating the amplitude and intensities distributions of the optical field on the Fourier Transform plane microscopic objects with regular forms are studied.Random and periodical distributions of objects are considered.The lineal dimensions of the objects are determined by manipulating the diffraction pattern in a simple and accurate way.The advantages of this methodology are summarized as follows: (a) using a single hologram the phase image is calculated with simple operation for phase curvature correction, (b) there are no limitations in the minimum reconstruction distance, (c) capability to maintain the size of a reconstructed image, independent of the reconstruction distance and wavelength for objects larger than a CCD, and (d) the spectral analysis of the radial intensity curve of the Fraunhofer diffraction pattern allows the determination of the lineal dimensions of the objects.
. A digital hologram is recorded by the CCD camera HDCE-10 with 1024x768 square pixels of size 4.65 µm, and transmitted to the computer by means of the IEEE 1394 interface.The digital hologram I H (j,l) is an array of M x N = 1024 x 768 8-bit-encoded numbers that results from the twodimensional sampling of I H (,) by the CCD camera,