Single-frame reconstruction by fractional Fourier-transform domain filtering in off-axis digital holographic microscopy

Wanying Cui; Yiwei Liu; Junjie Zhang; Yang Han; Zhuqing Jiang

doi:10.1364/OPTCON.501159

1. Introduction

Digital holography can achieve a quantitative phase imaging by recording and reconstructing the light field distribution of a three-dimensional sample with a charge-coupled device (CCD) or CMOS camera, based on optical interference and diffraction [1–3]. Unlike in-line digital holograms in which the three spectral terms overlap each other, the centers of the zero-order term, signal term and conjugate term of an off-axis hologram are separated from each other in its Fourier transform domain [4–7]. For the complex amplitude reconstruction of an object wave in off-axis digital holography, the spatial-frequency filtering is carried out typically in the Fourier transform domain, by intercepting the positive or negative first-order spectrum of the hologram. However, the separation of the three spectrum terms may reduce the utilization of the spatial bandwidth of CCD/CMOS, which further limits the effective bandwidth of the reconstructed complex amplitude distribution after spectrum filtering. Consequently, the imaging resolution of an off-axis digital holographic reconstruction is lower than that of in-line digital holography under the same conditions.

For reconstruction resolution improvement, the phase shifting method is used to remove the zero-order term and the conjugate term in the spatial-frequency spectrum of hologram [8,9]. The zero-order term can be eliminated by performing phase shifts in recording and numerically processing the holograms for different recording parameters [10]. The nonlinear spectrum filtering method realizes the suppression of the zero-order term and conjugate term for the off-axis hologram by effectively intercepting the signal spectrum [5]. Numerical iteration methods also work in selectively suppress the zero-order term to improve the resolution of reconstructed images [11–13]. Furthermore, convolving the single-frame hologram with a Laplacian kernel function can enhance the image quality of off-axis digital holography [14].

Fractional Fourier transform is the promotion of Fourier transform that is commonly used in the field of communication [15]. In the recent decade, the fractional Fourier transform has received more attention in the field of the image processing. With a fractional Fourier transform and iterative phase recovery algorithms, the high-quality quantitative phase imaging can be achieved in a coherent diffraction imaging system [16]. Talbot effect is described under the theoretical system of the fractional Fourier transform [17]. The fractional Fourier transform is widely applied in many aspects such as digital image encryption and other information security due to its additional parameter, namely the order of fractional Fourier transform [18–20]. Additionally, fractional Fourier transform can also address the issue about resolution improvement in off-axis digital holography [21]. The conventional spectrum domain filtering in off-axis digital holographic reconstruction may affect the imaging resolution due to the limitation of the filtering range. A fractional Fourier transform domain is a mixed signal domain, consisting of both space domain information and frequency domain information. Fractional domain filtering can extract more high-frequency components from the hologram that conventional Fourier spectrum domain filtering cannot get, although they have been contained in the hologram. The more high-frequency components extracted from the hologram by fractional Fourier transform domain filtering make a contribution to improve the imaging resolution of off-axis digital holographic reconstruction. However, the reconstruction with fractional Fourier transform domain filtering is more severely impacted by the zero-order term, which limits its single-frame reconstruction in imaging applications.

In this paper, we propose a single-frame off-axis digital holographic reconstruction method based on Laplacian-hologram and fractional Fourier transform domain filtering, which can realize the zero-order-eliminated reconstruction just using single frame hologram. The method involves in two major procedures, where an off-axis hologram is operated with Laplacian to generate the Laplacian hologram, and then the fractional Fourier domain filtering is applied on it.

2. Method

The fractional Fourier transform of an off-axis digital hologram $I$($x$, $y$) can be expressed as:

(1)$$\begin{aligned} \mathcal{F}^{p}\{I(x,y)\}=&\mathcal{F}^{p}\{|O|^2+|R|^2\}\\ &+\mathcal{F}^p\{OR{\exp}[-{\rm j}2\mathrm{\pi} (f_xx+f_yy)]\}\\ &+\mathcal{F}^p\{O^{{\ast}}R{\exp}[{\rm j}2\mathrm{\pi} (f_xx+f_yy)]\} \end{aligned}$$

where $O$ and $R{\exp}[{\rm j}2\mathrm{\pi} (f_xx+f_yy)]$ represent the complex amplitude distributions of the object wave and reference wave, respectively. $f_x$ and $f_y$ are the spatial frequency shifts of the reference wave relative to the object wave along the $x$ and $y$ directions on the recording plane. $|O|^2$ and $|R|^2$ are together referred as the zero-order term in the hologram, which consist of a self-correlation and a DC terms. $\mathcal {F}^{p}\{\ \}$ denotes the $p$-order fractional Fourier transform, where $p$ typically is a fraction between 0 and 1. The fractional Fourier transform distribution of a hologram is equal to the sum of the fractional Fourier transforms of all the terms in the hologram. The zero-order Fourier transform of a hologram is still its original distribution in the spatial domain, while the first-order Fourier transform of a hologram corresponds to its spatial frequency spectrum distribution. As the fractional order $p$ increases, the zero-order term and $\pm$1-order terms of the hologram, which overlap in the spatial domain, gradually separate from each other. Figure 1 shows the intensity distributions of several fractional Fourier transform domains of a hologram.

Fig. 1. (a)-(f) Intensity distributions of fractional Fourier transform domains of a hologram, when $p$ = 0.4, 0.5, 0.6, 0.8, 0.9, and 1.

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Fractional Fourier transform, as a generalized form of Fourier transform, shares similar phase shift property as Fourier transform. The fractional Fourier transform of signal term can be expressed as follows:

(2)$$\begin{aligned} \hspace{-0.45cm}\mathcal{F}^{p}\{OR{\exp}[-{\rm j}2\mathrm{\pi} (f_xx+f_yy)]\}={\exp}[-{\rm j}\mathrm{\pi} (f_x^2+f_y^2){\rm sin} \ \alpha {\rm cos} \ \alpha]\\ {\exp}[-{\rm j}2 \mathrm{\pi} (f_xu+f_yv) {\rm cos} \ \alpha]\\ G_p (u+f_x {\rm sin} \ \alpha,\ v+f_y {\rm sin} \ \alpha) \end{aligned}$$

where, $G_p$($u$, $v$) represents the $p$-order fractional Fourier transform distribution of $OR$, $u$ and $v$ are two mutually perpendicular coordinates in the $p$-order fractional Fourier transform domain. If $g$($x$, $y$) denotes the distribution of $OR$, then $G_p$($u$, $v$) can be expressed as:

(3)$$G_p (u,v)=\mathcal{F}^{p}\{g(x, y)\}=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(x,y) K_p (x,y;u,v)\, dx \, dy$$

where $K_p$($x$, $y$; $u$, $v$) is the transform kernel function, and can be written as:

(4)$$K_p (x,y;u,v)=\frac{1- {\rm j}{\rm cot}\ \alpha}{2 \mathrm{\pi}} {\exp}[\frac{{\rm j}(x^2+u^2)}{2{\rm tan}\ \alpha}-\frac{{\rm j}x u}{{\rm sin}\ \alpha}] {\exp}[\frac{{\rm j}(y^2+v^2)}{2{\rm tan}\ \alpha}-\frac{{\rm j}y v}{{\rm sin}\ \alpha}]$$

Fractional Fourier transform can be considered as an intermediate transform domain between spatial domain and frequency domain. In the coordination using the space domain as a horizontal axis and the frequency domain as a vertical axis, Fourier transform is typically described as a rotating from the spatial axis to the frequency axis, i.e. rotating the angle $\mathrm{\pi} /2$, and the $p$-order fractional Fourier transform is viewed as changing into another transform domain by rotating at an arbitrary angle $\alpha =p\mathrm{\pi} /2$. When a signal undergoes a phase shift, the fractional Fourier transform of the signal will result in the position shift $(-f_x{\rm sin}\ \alpha,\ -f_y{\rm sin}\ \alpha )$ of its distribution in the fractional Fourier domain. As the order $p$ changes from 0 to 1, the distance between the centers of the signal’s fractional-transform distribution and the fractional transform domain increases from 0 to $(f_x^2+f_y^2)^{1/2}$. This also means that when $p$=1, the center of the signal component is farthest from the center of the zero-order term in the traditional Fourier transform domain. In other words, although the fractional domain filtering can improve the resolution of reconstructed images, it will also be more affected by the zero-order term. On the other hand, because the zero-order term and $\pm$1-order terms of a hologram consist of the low to high frequencies, higher frequency components can get greater shift than lower frequency components, and the amount of the shift decreases with increases of the order number. Thus, when $p$ is properly selected, the fractional Fourier transform domain filtering can extract more high-frequency information to yield the reconstruction resolution improvement.

For comparison, the filters used in different order fractional Fourier transform domain filtering are set at the same size, such as $h_x\times h_y$ pixels. The hologram and its distribution in fractional Fourier transform domain are discretized into $2M\times 2N$ pixels. The fractional Fourier transform distribution is denoted as $G_p(m, n)$, where $m$ and $n$ are integers and $m\in [-M, M], n\in [-N, N]$. According to the above-mentioned phase shift property of fractional Fourier transform, the center positions of the different terms change with the fractional order $p$. Therefore, the filter function for the $p$-order transform domain filtering can be expressed as:

(5)$$\hspace{-0.2cm} H(m,n)=\left\{\begin{array}{ll} \hspace{-0.1cm} 1, & |m-\dfrac{f_x{\rm sin} \ \alpha}{M\Delta x}|\leq \dfrac{h_x}{2},\ |n-\dfrac{f_y{\rm sin} \ \alpha}{N\Delta y}|\leq \dfrac{h_y}{2} \\ \hspace{-0.1cm} 0, & else\\ \end{array}\right.$$

where $\Delta x$ and $\Delta y$ represent the pixel sizes along the $x$ and $y$ directions, and the central coordinates of the $\pm$1-order spectrum of the hologram are $(\pm f_x,\ \pm f_y)$.

It should be indicated that the holographic reconstruction procedure in fractional Fourier transform domain filtering is some different from traditional filtering reconstruction. Because the coordinates in fractional Fourier transform domain does not represent the spatial-frequencies, the position shift of the signal term in the fractional Fourier transform domain has completely different physical meaning from that in the spatial frequency domain. The signal term directly being shifted to the domain center after performing a filtering interception in the fractional Fourier transform domain cannot achieve holographic reconstruction. In the presented reconstruction procedure, the result after the $p$-order transform filtering have to be subjected to a (1-$p$) order transform, to convert it into a spatial frequency distribution. Then, the next processing follows the conventional holographic reconstruction process, including shifting the positive or negative first-order spectrum to the frequency domain center and performing an inverse Fourier transform on it to obtain the complex amplitude distribution of the sample.

Fractional Fourier transform domain filtering can intercept more high-frequency components than conventional Fourier transform domain filtering if the same filter size is used, which is beneficial for improving the reconstruction resolution. However, the fractional domain filtering is more affected by the zero-order term, which is not conducive to single-frame digital holographic reconstruction. Herein, we use a Laplacian convolution kernel as Ref. [14] for suppression of the zero-order term in fractional Fourier transform.

The hologram using Laplacian in the spatial domain can be expressed as:

(6)$$\begin{aligned} I_L(x,y)&=W_O|O|^2+W_R|R|^2\\ &+W_{{+}1}OR{\exp}[-{\rm j}2\mathrm{\pi} (f_xx+f_yy)]\\ &+W_{{-}1}O^{{\ast}}R{\exp}[{\rm j}2\mathrm{\pi} (f_xx+f_yy)] \end{aligned}$$

where $W_O$ and $W_R$ each denote the weights of two parts of the zero-order term, while $W_{+1}$ and $W_{-1}$ represent the weights of the positive and negative first-order terms, respectively. These weight factors are written as:

(7)$$\left\{{\begin{array}{c} W_O=\frac{\sum\limits_{k=1,-1}{[|O(x+k \Delta x, \ y)|^2+|O(x, \ y+k \Delta y)|^2]}}{|O(x, \ y)|^2}-4 \\ W_R=\frac{\sum\limits_{k=1,-1}{[|R(x+k \Delta x, \ y)|^2+|R(x, \ y+k \Delta y)|^2]}} {|R(x, \ y)|^2}-4 \\ W_{{\pm} 1}=-4[{\rm sin}^2(\mathrm{\pi} \Delta x f_x)+{\rm sin}^2(\mathrm{\pi} \Delta y f_y)] \end{array}}\right.$$

Equation (6) implies that Laplace operator acting on a hologram will have different effects on the various frequency components in the original hologram, and in particular, it can conduct stronger suppression on lower frequency components. For an off-axis hologram, both $O(x, y)$ and $R(x, y)$ can be viewed as low-frequency signals as compared with the high-frequency carrier of the first-order spectra, so according to the approximation in Ref. [14], there have $W_O$=0 and $W_R$=0. Accordingly, the zero-order term is suppressed by the Laplace operation of hologram. The frequency band occupied by the zero-order spectrum is a single-connected domain centered at the zero carried-frequency, while the first-order spectra are centered at higher carried-frequencies. As a result, the zero-order spectrum is subjected to stronger suppression.

If the fractional-Fourier-transform of the Laplacian hologram is denoted as $\mathcal {F}^{p}\{I_L(x, y)\}$, we can have the expression as

(8)$$\begin{aligned} \mathcal{F}^{p}\{I_L(x, y)\}=&-4[{\rm sin}^2(\mathrm{\pi} \Delta x f_x)+{\rm sin}^2(\mathrm{\pi} \Delta y f_y)]\\ &\mathcal{F}^{p}\{OR{\exp}[-{\rm j}2\mathrm{\pi} (f_xx+f_yy)]\\ &+O^{{\ast}}R{\exp}[{\rm j}2\mathrm{\pi} (f_xx+f_yy)]\} \end{aligned}$$

3. Experiment and results

An optical setup for recording off-axis digital holograms of the amplitude-type and phase-type USAF-1951 resolution targets is shown in Fig. 2. A laser beam at 532 nm passes through a beam expander and collimation system (BEC), and then is split into an $s$-polarized object beam and a $p$-polarized reference beam via a polarizing beam splitter (PBS). Two half-wave plates HWP$_1$ and HWP$_2$ are set to adjust the intensity ratio of the two beams while keeping their polarization orientation in parallel. After setting an off-axis interference angle between the two beams via a beam splitter (BS), the object wave and the reference wave generate optical interference on a CCD sensor, to record an image-plane digital hologram. So, the propagation distance in holographic reconstruction is taken as zero. Both microscope objectives MO$_1$ and MO$_2$ are same with 10$\times$ magnification, and the CCD pixel size is 3.45$\times$3.45 $\mathrm{\mu}$m$^2$.

Fig. 2. Diagram of an optical setup of off-axis digital holographic microscopy with 10$\times$ magnification: BEC, beam expander and collimation; PBS, polarizing beam splitter; HWP, half-wave plate; BS, non-polarized beam splitter; M, mirror; MO, microscope objective.

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As for an amplitude-type target, Fig. 3 shows the Fourier spectrums of the recorded hologram and its Laplacian hologram both with the size $900\times 900$ pixels, and their 0.7-order fractional Fourier transform distributions. The Fourier spectrums of the original hologram and its Laplacian-hologram are shown in Figs. 3(b) and 3(e). The weights of the zero-order term and $\pm$1-order terms can be calculated by dividing the central intensities of the zero-order spectrum and $\pm$1-order spectrum in Fig. 3(e) by those in Fig. 3(b), respectively, resulting as $W_0=0.01$ and $W_{\pm 1}=11.12$. The smaller the weight value means the stronger the suppression effect. After such suppression processing, the zero-order weight is three orders of magnitude smaller than the $\pm$1-order ones. Meanwhile, the distributions in Figs. 3(c) and 3(f) also exhibit that the zero-order-term of the Laplacian-hologram in its fractional Fourier transform domain is suppressed in effect.

Fig. 3. (a) Original hologram of an amplitude-type USAF-1951, (b) its Fourier transform spectrum, and (c) its fractional transform distribution when $p$=0.7; (d) Laplacian hologram, (e) its Fourier transform spectrum, and (f) its fractional transform distribution when $p$=0.7.

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In addition, when the fractional order $p$ is small such as taking 0.5$\sim$0.6, there may be some overlapping between the signal term and the conjugate term. Herein, we separate the signal term away from conjugate term in fractional transform domain by zero-padding the hologram.

Figure 4 shows the fractional Fourier transform distributions of the Laplacian-hologram when $p$=0.4, 0.5 and 0.6, and their corresponding distributions after a zero-padding processing. After zero-padding an original hologram to the size $1800\times 1800$ pixels and then operating Laplacian on this zero-padded hologram, its fractional Fourier transform distributions exhibit the well separated between the signal and conjugate terms for $p$=0.5 and 0.6, as seen in Figs. 4(e)–4(f).

Fig. 4. Fractional Fourier transform distributions of the Laplacian-hologram: (a)-(c) with the size $900\times 900$ pixels when $p$=0.4, 0.5, and 0.6; (d)-(f) with the zero-padded size $1800\times 1800$ pixels when $p$=0.4, 0.5, and 0.6, respectively.

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Figure 5 shows the amplitude images reconstructed from the Laplacian-hologram in fractional Fourier transform domains and in Fourier spectrum domain, after zero-padding original hologram. In the imaging reconstruction, a filter window marked with a red frame in Fig. 4(e) is used for filtering, and the amplitude distribution is reconstructed with the fractional-domain reconstruction algorithm, referred as in Ref. [21]. The reconstructed amplitude images of the Laplacian zero-padding hologram with fractional Fourier transform domain filtering when $p$=0.5 and 0.6 are shown in Figs. 5(a) and 5(b), respectively. The amplitude image reconstructed from the same one Laplacian zero-padded hologram by filtering in conventional Fourier spectrum domain, i.e. when $p$=1, is shown in Fig. 5(c). The curves of the amplitude distributions at the solid lines marked in Figs. 5(a) to 5(c) are given in Fig. 5(d).

Fig. 5. (a)-(b) Reconstructed amplitude images in fractional Fourier transform domain filtering when $p$=0.5 and 0.6 with the single-frame reconstruction method, respectively; (c) Reconstructed amplitude image by filtering in conventional Fourier spectrum domain for the Laplacian hologram; (d) Curves of the amplitude distributions at the solid lines in (a) to (c).

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According to the nominal resolution of each line-pair element of USAF-1951 target, the reconstructed amplitude image by using fractional Fourier transform domain filtering with the Laplacian convolution processing when $p$=0.5 and 0.6 can be resolved up to the line-pairs element 8-6 of the resolution 456.1 lp/mm. The reconstruction only using the Laplacian of the hologram to eliminate zero-order-term can resolve line-pairs element 8-4, corresponding to the resolution of 362 lp/mm. As seen in Fig. 5(d), the amplitude distribution curve in the case of $p$=0.5 shows three distinct peaks, indicating that the three horizontal line pairs on this line-pairs element can be resolved. However, by just using Laplacian of the hologram and the spectrum domain filtering, the three line pairs are not distinguishable. The above results demonstrate that the proposed method improves the resolution of the reconstructed amplitude image just using a single frame hologram.

Further, the phase reconstruction imaging for a phase-type USAF-1951 resolution target is performed with the presented method. The recorded hologram has the size of $900\times 900$ pixels, as shown in Fig. 6(a). The phase maps reconstructed from its Laplacian zero-padded hologram with the size of $1800\times 1800$ pixels, by filtering in the 0.5-order fractional Fourier transform domain and in the Fourier spectrum domain, are shown in Fig. 6(b) and 6(c). The curves of their phase distributions along the solid lines in the phase maps are exhibited in Fig. 6(d).

Fig. 6. (a) Recorded hologram of a phase-type USAF-1951 target; (b) Reconstructed phase map by using the 0.5-order fractional Fourier transform domain filtering for the Laplacian zero-padded hologram; (c) Reconstructed phase map by filtering in Fourier spectrum domain for the Laplacian hologram; (d) Curves of the phase distributions at the solid lines in (b) and (c).

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For above phase reconstruction, the phase map by using the 0.5-order fractional Fourier transform domain filtering with the Laplacian convolution processing can be distinguished up to the line-pairs element 8-6, corresponding to the resolution of 456.1 lp/mm. The phase reconstruction from the Laplacian hologram by filtering in Fourier spectrum domain can resolve up to the line-pair element 8-3 of the resolution 322.5 lp/mm. By comparing two curves in Fig. 6(d), the phase distribution obtained by fractional Fourier transform domain filtering exhibit three peaks, which clearly distinguish the three bright bars in the line-pair element 8-6. But, the other curve just with conventional filtering cannot show the bright bar structure of the line pair 8-6. In addition, while fractional Fourier transform filtering improves the reconstruction resolution, it may also introduce some degree of edge loss around the image edges, as seen at the right side in Fig. 6(b). Such loss of the field of view is caused by the limitation of the filtering range. So, in the fractional Fourier transform domain filtering, the choice of the optimal fractional order should have a trade-off between two factors of the higher local resolution and the complete field of view.

4. Conclusion

A method of single frame reconstruction in off-axis digital holographic microscopy by fractional Fourier-transform domain filtering is presented, which can improve the resolution of the reconstructed amplitude and phase images. After obtaining the Laplacian hologram of the zero-order suppression, the optimum resolution of single-frame reconstruction can be achieved by performing a fractional Fourier-transform domain filtering on this Laplacian hologram. In the experiments, the reconstruction results of the amplitude-type and phase-type object targets demonstrate its capability of improving the reconstruction resolution from a single-frame hologram in effect. It should be noted that the resolution improvement is not due to the operation of fractional Fourier transform itself but rather the filtering performed in fractional Fourier transform domains. The fractional order can be selected empirically according to the separation between the signal and conjugate terms in corresponding fractional domains as a criterion. If the conjugate term in the fractional transform domain overlaps partially on the signal term, the reconstruction of the complex amplitude is unable to be achieved effectively, which will appear the noise or the edge loss brought from the overlapping portion of the conjugate term. Since the presented off-axis digital holographic reconstruction method can achieve high-resolution imaging only with single-frame acquisition, it will have a promising application in real-time imaging for biological cells and moving objects.

Funding

National Natural Science Foundation of China (61575009).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Single-frame reconstruction by fractional Fourier-transform domain filtering in off-axis digital holographic microscopy

Abstract

1. Introduction

2. Method

3. Experiment and results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (8)

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