Reconstruction of Digital Hologram by use of the Wavelet Transform

In combination with charge coupled device (CCD) sensor technology and image processing systems, real time dynamic analysis becomes one of the significant advantages of digital holography. In order to achieve the dynamic analysis, lots of holograms should be recorded and every hologram should be numerical reconstructed. The most popular numerical reconstruction methods for digital holography include the well-known Fresnel diffraction integral method, the angular spectrum method and the convolution-based method. In order to filter out the zero-order term, the twin image term and the parasitic interferences, the process of the spatial filtering must be carried out in these methods. When some noises and parasitic interferences are introduced into the hologram, the spectrum of the virtual image would be disturbed by some other spectrum. It brings difficulties to define the spatial filter because of the blurry boundary and non-regular distribution of the spectrum. Manual spatial filters are often employed. However, defining different manual spatial filters would consume plenty of time for dynamic analysis. Therefore, the process of the spatial filtering limits the application of the digital holography for the dynamic and automatic analysis. The wavelet transform, a tool excelling for its multi-resolution and localization in the spacefrequency domains, is applied to the reconstruction of the object wave from the digital hologram in this chapter. Here the one-dimensional (1D) and two-dimensional (2D) Gabor wavelet transform (GWT) are employed for the analysis. By calculating the wavelet coefficients of the hologram at the ridge and the peak automatically, the object wave at the hologram plane can be reconstructed. At the same time the effect of the zero-order and the twin-image terms are eliminated without the process of the spatial filtering. The reconstruction approach for digital holography by use of the GWT is described in theory. And the experimental results are shown.


Introduction
In combination with charge coupled device (CCD) sensor technology and image processing systems, real time dynamic analysis becomes one of the significant advantages of digital holography.In order to achieve the dynamic analysis, lots of holograms should be recorded and every hologram should be numerical reconstructed.The most popular numerical reconstruction methods for digital holography include the well-known Fresnel diffraction integral method, the angular spectrum method and the convolution-based method.In order to filter out the zero-order term, the twin image term and the parasitic interferences, the process of the spatial filtering must be carried out in these methods.When some noises and parasitic interferences are introduced into the hologram, the spectrum of the virtual image would be disturbed by some other spectrum.It brings difficulties to define the spatial filter because of the blurry boundary and non-regular distribution of the spectrum.Manual spatial filters are often employed.However, defining different manual spatial filters would consume plenty of time for dynamic analysis.Therefore, the process of the spatial filtering limits the application of the digital holography for the dynamic and automatic analysis.The wavelet transform, a tool excelling for its multi-resolution and localization in the spacefrequency domains, is applied to the reconstruction of the object wave from the digital hologram in this chapter.Here the one-dimensional (1D) and two-dimensional (2D) Gabor wavelet transform (GWT) are employed for the analysis.By calculating the wavelet coefficients of the hologram at the ridge and the peak automatically, the object wave at the hologram plane can be reconstructed.At the same time the effect of the zero-order and the twin-image terms are eliminated without the process of the spatial filtering.The reconstruction approach for digital holography by use of the GWT is described in theory.And the experimental results are shown.

1D-GWT
The continuous one-dimensional wavelet transform is defined as follows: where 0 s > is the scale parameter related to the frequency; b is the shift parameter related to the position; () f x is the signal to be analyzed; , () sb x ψ is the analyzing wavelet obtained by shifting and scaling the mother wavelet () x ψ ; * indicates the complex conjugate.The Gabor wavelet is employed as the mother wavelet.The Gabor wavelet function and its Fourier transform are given as: The reasons for the choice of the Gabor wavelet are as follows: (1) The Gabor wavelet consists of a sine wave and a cosine wave that have been modulated by a Gaussian function.
(2) It can represent a complex sinusoidal function that correlates well with the sinusoidal characteristics of the interferogram.
(3) The Gaussian function has the least spread in both domains of space and frequency.
Computing the 1D-GWT, the modulus of the wavelet coefficients can be obtained by: ⎦ indicate the imaginary part and the real part of the (, ) Wsb.At the position b , the ridge of the wavelet transform is defined as the location where the modulus (, ) Wsb reaches its local maximum along the scaling direction s .

Reconstruction principle by means of the 1D-GWT
The hologram is created by the interference, in off-axis geometry, between two coherent waves: on one side the wave of interest, called object wave (,) Oxy , coming from the object, and on the other side a reference wave (,) Rxy being plane: (,) (,)e x p (,) Oxy oxy j xy where (,) xy are the coordinates of the hologram plane; (,) oxy and 0 R are the amplitude of the object and the reference waves; (,) xy φ is the phase of the object wave; λ is the wavelength; θ is the angle between the propagation direction of the object and the reference waves.The intensity of the hologram (,) Ixy can be written as: The first and the second terms form the zero-order term, the third and the fourth terms are the virtual and the real image terms respectively.
When the higher order terms of () xb − are neglected, the phase () x ϕ can be expressed as: Perform the above calculation by employing the Gabor wavelet as follows: where, ( ) where, Therefore, it can be got as follows: ( ) Because of the feature of 0 s > , the modulus (, ) Wsb reaches its maximum at: Therefore, the wavelet coefficients at the ridge of the 1D-GWT, which is described as () ridge Wb , become: ( )  21) can be rewritten as: www.intechopen.com A conclusion can be obtained from Eqs. ( 23) that the reconstructed wave at the ridge of the 1D-GWT is equal to the object wave at the hologram plane multiplied by a constant coefficient.Therefore the numerical reconstruction of the object wave can be carried out by means of the 1D-GWT.At the same time the effect of the zero-order and the twin-image terms are eliminated without the process of the spatial filtering.
It should be noted that the conclusion here is obtained by assuming the higher order terms of () xb − in Eqs.(9) being neglected.In the following two cases this approximation can be satisfied.First, if () φ′ is of slow variation, it can be set that ≈≈ A , subsequently the higher order terms of () xb − can be neglected.The phase of the object wave () x φ corresponds to the optical path length (OPL) when a plane wave propagates through a phase object.Therefore, the slow variation of () x φ′ denotes that the first derivative of OPL is of slow variation, or the first derivative of refraction-index distribution of the phase object is of slow variation in physics.Second, because the function , () sb x ψ is concentrated around xb = with a full width of half maximum 2 xs Δ = , the higher order terms of () xb − can be neglected when () 1 xb − < in the full width.The reconstructed wave at the object plane perpendicular to the propagation axis can be calculated by the Fresnel diffraction integral operation or by the angular spectrum method.Suppose that the wave front at the hologram plane is described as (,; 0 ) Exy , and its angular spectrum (,; 0 ) {(,; 0 ) } AE x y ξη =ℑ can be obtained by taking the Fourier transform.Here {} ℑ denotes the Fourier transform, and (,) ξ η are the corresponding spatial frequencies of (,) xy .The angular spectrum of the reconstructed wave at the plane z perpendicular to the propagation axis, (,;) A z ξη , can be calculated from (,; 0 ) A ξη as:

2D-GWT
There is a deficiency of the 1D-GWT method that the two-dimensional hologram must be divided into a series of one-dimensional signals along x direction when computing the 1D-GWT on x direction with y fixed.Thereby, the spatial frequency on y direction would be lost.It means that it would fail to analyze the fringes along x direction.Especially when there are some interference fringes of the hologram are along x direction but others along y direction proposed, or there are some parasitic interference signals along x direction, the two-dimensional wavelet transform is necessary.
The continuous two-dimensional wavelet transform is defined as follows: is the analyzing wavelet obtained by shifting, scaling and rotating the mother wavelet (,) xy ψ , and it can be considered as a spatial window; * indicates the complex conjugate.Here the one-dimensional Gabor wavelet is extended to the twodimensional situation to create a two-dimensional Gabor wavelet as a mother wavelet for its optimal joint space and frequency resolution for a signal.The two-dimensional Gabor wavelet function is given as: where 2l n2 γπ = , and j is the imaginary unit.A series of analyzing wavelets, namely "daughter wavelets", are built up by shifting, scaling and rotating the two-dimensional Gabor wavelet, namely "mother wavelet", as follows: (, , , ) (, , , ) a r c t a n (, , , ) .The result of the 2D-GWT is a four-dimensional array related to a parameter a , a parameter b , a scale parameter s and a rotation angle parameter θ .Fig. 2 shows an illustration of a four-dimensional map of size 512×512 pixel, in schematic form, for a specific example of a 2D-GWT at five different scale values { } 2,3,4,5,6 and three different rotation angle values 000 {0 ,30 ,60 } .At a position (,) ab , the peak of the 2D-GWT is defined as the location where the modulus (, , , ) f Ws a b θ reaches its local maximum along both the scaling direction s and the rotating direction θ .The determination of the peak of the 2D-GWT can be carried out by the following process.First, an array is constructed by picking up the modulus of the 2D-GWT Fig. 2. Illustration of a four-dimensional array produced by the 2D-GWT.coefficients corresponding to the position (,) ab at each scale and rotation angle value.For the above specific example including five different scale values and three different rotation angle values, a array of size 5×3 is obtained as shown in Fig. 3. Second, the maximum value of this array is found which is pointed out by an arrow in Fig. 3. Finally, the corresponding scale value and rotation angle value are determined, and the peak of the 2D-GWT at the position (,) ab is determined.

Reconstruction principle by means of the 2D-GWT The object wave (,)
Oxy and the reference wave (,) Rxy being plane can be written as: (,) (,)e x p (,) Oxy oxy j xy α and β are the angles between the propagation direction of the object wave and the reference wave on x and y direction respectively.The intensity of the hologram (,) Ixy can be written as: The first and the second terms form the zero-order term, namely the background intensity, which is written as The instantaneous frequency is defined as follows: And the spatial frequency is: where T represents the spatial period of the hologram pattern around the position (,) ab .Therefore, Eqs. ( 37) can be rewritten as: Because of the reference wave being plane, 0 R is constant.And the phase object can be considered to be nearly transparent, i.e. it does not modulate the intensity of the illuminated wave but the phase, therefore it can be set that (,) oxy o ≈ and (,) Axy A ≈ .For the nonintensity-only object, when the amplitude of the object wave (,)  oxy is of slow variation compared with the phase term exp ( , ) ⎦ would has one periodic variation.For the intensity-only object, it seems that this approximation cannot be satisfied.However, there is not strict intensity-only wave due to the diffraction effect.Therefore the mentioned approximation condition can be satisfied in general.Setting (,) oxy o ≈ , (,) Axy A ≈ , and computing the 2D-GWT of (,) Performing the above calculation by employing the two-dimensional Gabor wavelet as follows:       direction but also the spatial frequency on y direction, which cannot be obtained by the 1D-GWT, are determined according to Eqs. (39).Therefore, the wavelet coefficients at the peak of the 2D-GWT are obtained according to not only x f but also y f , which is described as (,)  to suffer from horizontal "ripples".This "ripples" is likely to be an additional interference signal produced by the reflected wave on the slides, which appear in the image of the onion specimen shown in Fig. 5(a).Because 1D-GWT is carried out on the horizontal direction, it fails to eliminate this parasitic interference signal.It might be eliminated by employing the 2D-GWT.After removal of the 2π ambiguity by a phase unwrapping process, the unwrapped phase is obtained, as shown in Fig. 6(f).In order to eliminate the "ripples" as show in Fig. 6(d), the 2D-GWT is employed for the analysis.Table 1 shows

Conclusion
In this chapter, the theory of the 1D-GWT and 2D-GWT techniques for the analysis of the digital holography to obtain the quantitative phase especially for the phase object are demonstrated in detail.The results of the experiment of an onion specimen show the conclusion that the numerical reconstruction of the phase object can be carried out by the analysis of the GWT.At the same time the effect of the zero-order term and the twin-image term are eliminated without the spatial filtering.One of the significant advantages of the technique here we presented is that it can be applied to the dynamic live biological specimen analysis in the digital holographic phase-contrast microscopy.
that the position parameter b relates to x , so that Eqs. ( 24)Subsequently the reconstructed wave can be obtained by taking the inverse Fourier transform θ are the scale and the rotation parameters related to the frequency; (,) ab is the shift parameter related to the position;(,)   f xy is a two-dimensional real signal to be analyzed
at sT = and θ α = in theory according to Eqs. (50).It means that not only the spatial frequency on x

Fig. 6 .
Fig. 5. Hologram.(a) the image of the onion specimen; (b), (c) are the respectively hologram with and without the onion specimen; (d) the spectrum of the hologram with the onion specimen on the logarithmic coordinates.First, the 1D-GWT is employed for the analysis.Fig. 6(a) and (b) show the modulus and the phase of the wavelet coefficients at the 256 th row of the hologram.By detecting the ridge of the 1D-GWT and multiplying the wavelet coefficients at the ridge of the 1D-GWT by the numerical reference wave, the object wave is reconstructed.Fig. 6(c) shows the spectrum of the reconstructed wave by employing the 1D-GWT.It shows that the high frequency information of the object wave is reserved, which might be filtered out by the process of the filtering.Fig. 6(d) and (e) show the amplitude and the phase modulo 2π , coded to 256 gray levels, of the reconstructed wave.The amplitude image shown in Fig. 6(d) seems the modulus of the wavelet coefficients with nine different scale values from 2 to 10 with interval of 1, and ten different rotation angle values from 0 e. the peak of the 2D-GWT of the hologram at this position is determined.Fig.7(a), (b) and (c) show the spectrum on the logarithmic coordinates, the amplitude and the unwrapped phase of the reconstructed wave by the analysis of the 2D-GWT.From the comparison of Fig.5(d), Fig.6(c) and Fig.7(a), i.e. the spectrum distribution of the hologram, the reconstructed wave by the analysis of the 1D-GWT and the 2D-GWT, it can be seen that there is some unwanted spectrum by the analysis of the 1D-GWT, which results in failing to eliminate the parasitic interference signals along x direction like "ripples" as show in Fig.6(d).While there is a more accurate spectrum by the analysis of the 2D-GWT as shown in Fig.7(a), and the result is improved by employing the 2D-GWT as show in Fig.7(b).

Fig. 7 .
Fig. 7. Holography of an onion specimen by the analysis of the 2D-GWT.(a) the spectrum, (d) the amplitude, (b) the wrapped phase and (c) the unwrapped phase of the reconstructed wave.
Take one row of the hologram to analyze.The one-dimensional intensity distribution of the hologram can be written as:

Table 1 .
The modulus of the wavelet coefficients at the position (256, 256) .