Sparsity-Guided Phase Retrieval to Handle Concave- and Convex-Shaped Specimens in Inline Holography, Taking the Complexity Parameter into Account

Abstract

In this work, we explore an optimization idea for the complexity guidance of a phase retrieval solution for a single acquired hologram. This method associates free-space backpropagation with the fast iterative shrinkage-thresholding algorithm (FISTA), which incorporates an improvement in the total variation (TV) to guide the complexity of the phase retrieval solution from the complex diffracted field measurement. The developed procedure can provide excellent phase reconstruction using only a single acquired hologram.

1. Introduction

In various fields of physics, phase retrieval is one of the most important problems [1,2,3]. Phase measurement is a problem that is observed in many areas, such as the microscopy of transparent biological cells [4,5], electron microscopy [6], astronomical imaging [7], coherent X-ray imaging [8,9,10], etc. The phase retrieval technique is one way to measure the wavefront [11]. This phase estimation technique requires two or more intensity measurements. Phase recovery based on several intensity measurements in short distances has provided excellent results [12]. The phase estimation process requires a large number of measurements, which makes this approach difficult and inconvenient [3,13]. In recent years, many attractive approaches to building compact phase microscopes have been developed based on non-interferometric phase diversity techniques [3]. The compact phase microscopy devices reported so far have effectively required two or more holograms [14,15]. However, single-shot phase imaging in a compact free-space configuration is considered a difficult problem to address. The commonly used Gerchberg–Saxton (GS)-type phase retrieval methods show poor-quality phase recovery [16,17]. The incorporation of a sparsity-enhancing step in conventional phase retrieval algorithms eliminates the twin-image stagnation problem, as shown in [3,18]. As explained in [19], a variational reconstruction approach based on the Alternating Direction Method of Multipliers (ADMM) can be used for phase unwrapping. High-quality phase recovery can be achieved using the modified Huber function as a sparsity measure [3]. The present work explored the possibility of associating the phase retrieval approach with the fast iterative shrinkage-thresholding algorithm (FISTA) to measure a complex diffracted field [20] in the Fresnel zone. We proposed a technique that effectively combines free-space backpropagation with the FISTA model, which incorporates a modified total variation (TV) to guide the complexity of the phase retrieval solution. The developed procedure handles complex-valued objects from a single hologram acquired in short distances. Our approach numerically and experimentally explored a convex-shaped polystyrene microbead with a 10.2 µm diameter [20] and a concave-shaped red blood cell (RBC) sample from a diffracted field measurement. In a special case, we studied the complexity guidance of the solution for the proposed method against the one provided with free-space backpropagation [20,21], multi-height phase recovery [20,22] associated with the ADMM and multi-height phase recovery combined with the FISTA. The approach introduced here employs an inline holography configuration to record the complex diffracted field measurement. Additionally, it is based on an optimization idea to guide the complexity of the phase retrieval solution, unlike the work reported in [23].

This paper is organized as follows. In Section 2, we present the analysis and proposed sparsity-guided phase retrieval algorithm. The introduced technique is based on the angular spectrum analysis method. The angular spectrum method is employed to depict the free-space propagation of the light field back and forth between the object plane and the detector. We incorporate complexity guidance to control the convergence of the phase retrieval solution. In Section 3, we explain our main contribution in this work. In Section 4, we show the numerical and experimental results obtained. The numerical and experimental results obtained using free-space backpropagation as well as the proposed method are thus discussed. Finally, in Section 5, we provide conclusions and future frameworks.

2. Analysis and Proposed Sparsity-Guided Phase Retrieval

2.1. Total Variation (TV)

One of the approaches to sparsity-based image processing concerns nonlinear operation tools [24]. These operators carry out the operation on the data. The total variation (TV) is considered as one of the most common sparsifying transform tools. The TV evaluates the modulus of the gradient of the signal. This tool is highly useful in image processing tasks, specifically in holography. It is an efficient tool with which to address various artifacts. The TV-based constraint aims at the consideration of the sharp boundaries of the object. Also, the TV is considered a standard framework for improving the resolution in phase retrieval. In holography, many holograms are intrinsically sparse in some transform areas. In this framework, the basic motivation is that these holograms can be reconstructed with high accuracy. This is possible even with a low amount of measured data [24]. Here, the improved TV functional is incorporated into the sparsity-guided phase retrieval in order to address artefacts in the obtained phase reconstruction.

2.2. FISTA

The fast iterative shrinkage-thresholding algorithm (FISTA) is in iterative soft thresholding algorithm (ISTA) domain [25]. The ISTA is used to solve linear inverse problems. The ISTA generally presents the worst-case complexity result of the optimization model. The introduction of a new ISTA improved the complexity result of the optimization model. The general technique consists of minimizing a smooth convex function. The ISTA is reduced to the gradient method. This method provides a complexity result that is an optimal first-order method for smooth problems [26]. The method described in [27] does not require more than one gradient calculation during each recursive operation. However, it considers just an additional point that is smartly chosen and easy to evaluate. The FISTA is attractive due to its simplicity. Additionally, the convergence rate is considerably the best, both theoretically and practically.

2.3. Proposed Sparsity-Guided Phase Retrieval

In this work, an approach that considers the optimization idea for the complexity guidance of phase retrieval solutions is explored. We retrieve the complex-valued object information from a single-shot digital hologram through a constrained optimization procedure. The complexity of the phase recovery solution exhibits a change in the nature of the solution as the iterations progress. The complexity is a numerical parameter that evaluates the fluctuations in the solution. The complexity of the suitable solution can be evaluated a priori from the Fourier magnitude information. The complexity measure of complex-valued object $o(x,y)$ provides an evaluation of fluctuation in its pixel values. It may be expressed as follows [23]:

$$\mathit{\zeta }={\sum }_{\mathit{i}=\mathit{a}\mathit{l}\mathit{l} \mathit{p}\mathit{i}\mathit{x}\mathit{e}\mathit{l}\mathit{s}}({\left|{\mathbf{\nabla }}_{\mathit{x}}{\mathit{o}}_{\mathit{i}}\right|}^{\mathbf{2}}+{\left|{\mathbf{\nabla }}_{\mathit{y}}{\mathit{o}}_{\mathit{i}}\right|}^{\mathbf{2}})$$

(1)

In this equation, ${\mathbf{\nabla }}_{\mathit{x}}$ and ${\mathbf{\nabla }}_{\mathit{y}}$ are the x and y gradient operators, respectively.

Indeed, the object itself should be known in order to calculate its complexity parameter. However, we have access to the Fourier intensity of the object in issues like phase retrieval. We cannot access the specimen itself. The complexity of the specimen can still be estimated using the Fourier magnitude data. The experimental configuration is shown in Figure 1. The holograms are acquired in short distances, so that only the propagating frequencies in the angular spectrum transfer function in Equation (2) [28,29] need to be taken into account for the visible wavelength range, as in [30].

$$H\left(X,Y\right)={\left|{FT}^{-1}[FT\left\{t(x,y\right\}\mathrm{e}\mathrm{x}\mathrm{p}({\displaystyle \frac{2\pi iz}{\lambda }}\sqrt{1-{\left(\lambda f_x\right)}^2+{\left(\lambda f_y\right)}^2}\left)\right]\right|}^2$$

(2)

where $FT$ and ${FT}^{-1}$ represent the Fourier transform and the inverse Fourier transform, respectively, and $(f_x,f_y)$ denote the spatial frequencies. The conventional Gerchberg–Saxton (GS) method involves propagating the optical field back and forth. This propagation is carried out between the input and output planes using the angular spectrum approach [3]. However, the traditional Gerchberg–Saxton (GS) method requires a considerable number of iterations to provide a significant solution [23]. The complexity parameter is an interesting framework of understanding the stagnation problems in phase retrieval techniques. In the sparsity-guided phase retrieval field, we show that the complexity guidance idea can be used along with the optimization-based approach. The proposed procedure is seen to significantly mitigate the number of total iterations required. The number of phase retrieval iterations is meaningfully reduced to reach a reasonable solution. The introduced methodology is additionally observed to address the twin-stagnation issue as illustrated in Figure 2.

Our approach begins with a random phase function. This phase map contains pixel values uniformly distributed in the image domain. It associates a random phase function with measured amplitude to generate a complex-valued field. The measured intensity of the optical field at z distance from the object plane provides these different components. The traditional Gerchberg–Saxton (GS) technique propagates the light field between the object plane and detection plane. The resultant field amplitude is replaced by the measured amplitude, leaving the phase unchanged in the output plane. So, the propagated light field from the detection plane backwards to the object plane is updated. The total variation (TV) of this intermediate solution is reduced. This operation is recursively achieved. A number of small gradient descent steps are applied to the group of pixels in the updated solution. The total variation (TV) of the solution is used as a sparsity criterion. The TV functional gradient is expressed as follows:

$$TV(o,o^{\ast })={\sum }_{i=all pixels}\sqrt{{\left|{\nabla }_x{o}_i\right|}^2+{\left|{\nabla }_y{o}_i\right|}^2}$$

(3)

The functional gradient of TV to be employed in a gradient descent process for TV reduction is calculated. The calculation is performed with respect to the conjugate image $o^{\ast }$ since $o$ is complex valued. The $\left(k+1\right)th$ gradient descent operation is as follows:

$${\widehat{o}}_{n+1}^{k+1}={\widehat{o}}_{n+1}^k-t{‖{\widehat{o}}_{n+1}^k‖}_2{\left[\widehat{u}\right]}_{o={\widehat{o}}_{n+1}^k}$$

(4)

where $\widehat{u}$ defined a unit vector in the direction of the conjugate image $o^{\ast }$ of TV and $t$ is the optimal step length. The nature of the solution changes when the iterations progress. The change in the solution is small in each gradient descent operation. Therefore, the parameter $t$ should be carefully chosen [24]. We set t to be 0:01 in this work. The TV reduction iteration begun with ${\widehat{o}}_{n+1}^0={\widehat{o}}_{n+1}$. The complexity parameter ${\zeta }_{n+1}^{k+1}$ is evaluated for ${\widehat{o}}_{n+1}^{k+1}$ in the image space. We compute this parameter after each gradient descent operation in the TV-reducing direction. The gradient descent steps for TV reduction are operated. This operation is stopped when the complexity parameter ${\zeta }_{n+1}^{k+1}$ for the updated solution ${\widehat{o}}_{n+1}^{k+1}$ is within 1% [23] of the suitable value ζ. In this way, the complexity parameter guided the numbers of sparsity-enhancing steps. The total variation (TV) can be easily incorporated into FISTA for solving more general inverse problems. The developed methodology aims to perform an optimal balance between the error and sparsity terms. We turn our attention to the following formulation used in References [31,32], which was employed as the total variation denoising approach [22].

$${min}_{\varphi }\left\{{\displaystyle \frac{1}{2}}{‖{\varphi }^{\ast }-\varphi ‖}_2^2+a\Phi \left(\varphi \right)\right\}$$

(5)

${\varphi }^{\ast }$ represents the object function that is projected to the sparsifying domain, $\varphi$ denotes the variable of the denoising technique, $\Phi \left(\varphi \right)$ is the regular value expressing prior information about the specimen, ${‖.‖}^2$ corresponds to the $l_2-norm$, which is introduced as a fidelity term and $a$ is the regularization parameter to control the phase retrieval solution. It controls the amount of regularization, which can be adaptively improved [31] or chosen a priori. Also, this parameter consists of guidance of the tradeoff between data fidelity and denoising. The FISTA procedure is considered to carry out the optimization process of Equation (5). This model can be described as follows:

$${\varphi }^k={prox}_{a\Phi /L}(g^k-{\displaystyle \frac{1}{L}}\nabla \epsilon \left(g^k\right))$$

(6)

$${\mathit{q}}_{\mathit{k}+\mathbf{1}}={\displaystyle \frac{\mathbf{1}+\sqrt{\mathbf{1}+{\mathit{q}}_{\mathit{k}}^{\mathbf{2}}}}{\mathbf{2}}}$$

(7)

$${\mathit{g}}^{\mathit{k}+\mathbf{1}}={\mathit{\varphi }}^{\mathit{k}}+\left({\displaystyle \frac{{\mathit{q}}_{\mathit{k}}-\mathbf{1}}{{\mathit{q}}_{\mathit{k}+\mathbf{1}}}}\right)({\mathit{\varphi }}^{\mathit{k}}-{\mathit{\varphi }}^{\mathit{k}-\mathbf{1}})$$

(8)

This procedure, taking as an initial assumption ${\varphi }^0=0, { g}^1={\varphi }^0$ and a stepsize $1/L$, makes it possible update the approximate solution ${\varphi }^k$ to be obtained. $\nabla \epsilon$ denotes the gradient of the fidelity term between the measurement ${\varphi }^{\ast }$ and the estimated value of (i.e., variable of the denoising model) [20,33]; ${prox}_{a\Phi /L}$ takes into account the regularization term $\Phi$ corresponding to the proximal operator [33,34]; it is defined as follows:

$${prox}_{a\Phi /L}\left(g\right)\triangleq {argmin}_{\varphi }({\displaystyle \frac{1}{2}}{‖\varphi -g‖}_2^2+{\displaystyle \frac{a}{L}}\Phi \left(\varphi \right))$$

(9)

The proximal operator cannot be defined closely [24] as in the present case when employing the $\Phi \left(\varphi \right)=TV\left(\varphi \right)$ approach. The problem (6) is then solved as a sub-problem at each iteration. We considered the dual minimization approach developed by Beck and Teboulle [20,31,34,35,36] to effectively solve the subsequent.

3. Main Contribution

This work proposes an approach to handle complex-valued objects from a single measurement of the light field in inline holography. The traditional Gerchberg–Saxton (GS) algorithm suffers from stagnation issues for complex-valued or phase objects [3,23]. In a special case, Fienup improved the GS algorithm [37,38]. He incorporated a suitable constraint in the recursive solution. This operation contributes to the convergence of the modified algorithm. In general, the iterative phase recovery techniques carry out operations by imposing constraints. These methods evaluate Fourier intensity constraints in the data area and appropriate constraints in the detection area. However, the GS and Fienup algorithms and their variants converge to a desirable solution after a considerable amount of iterative operations for uniform random phase initialization. Here, a complexity parameter provides interesting information that can be used to explore the stagnation problems in phase retrieval algorithms. We demonstrate that a new approach of total variation (TV) can significantly improve the performance of the phase retrieval [39] approach. Specially, the convex shape condition [40] and concave shape criteria [41] are combined with total variation (TV). Our modified total variation (TV) can be easily incorporated into the FISTA procedure for solving more general inverse problems. We introduced an approach based on the optimization procedure to control the complexity of phase retrieval solutions. We studied the impact of the free-space backpropagation combined with modified TV in the FISTA model to guide phase retrieval solution for a single hologram acquired.

4. Numerical and Experimental Results

4.1. Numerical Results

Numerical data were used in order to describe the performance of the proposed method. In the present work, a sparsity-guided phase retrieval approach based on single-shot diffraction intensity data is demonstrated. The introduced approach can successfully address the stagnation problem associated with the retrieval of phase objects. Here, a numerical reconstruction of the complex-valued objects was performed. The object area was selected to be 128 × 128 pixels and the wavelength was 650 nm. Two types of samples were considered. The first sample was a polystyrene microsphere with 10.2 µm diameter (see Figure 3b). This specimen was immersed in water. The second specimen was RBC. The refractive index value of the surrounding liquid medium was 1.34 as in [42] (see Figure 3c). For an object-to-detector distance of 0.5 mm, the obtained parameter was $\sigma =1.8\times {10}^{-4}$. The obtained Fresnel number was $N_F=0.32$. We obtained relatively low Fresnel numbers, which should provide reliable reconstructions as in [43].

The obtained phase reconstruction using the proposed approach was compared against the one reconstructed by free-space backpropagation, multi-height phase recovery combined with ADMM and multi-height phase recovery combined with FISTA. In Figure 4, the complexity measured against the amount of recursive operations for constant and random phase initializations is presented. This plot denotes that these complexities cannot reach the complexity of the ground truth phase. The ideal solution is described by the red solid line. We see that the artifacts in the free-space backpropagation solutions (Figure 5b,g and Figure 6b,g) can be associated with the higher complexity value for these solutions. This technique does not address the artifacts. Hence, this generally assumes that the ideal complexity level is a significant challenge to reach.

Here, all the methods were initiated with the same initial random phase map. Figure 5 and Figure 6 illustrate the reconstructed phase image with 300 and 500 iterations. In fact, the free-space backpropagation technique provided the solution with a random phase guess. The obtained result is seen to have poor reconstruction quality. It is also observed to have poor numerical accuracy when compared with the ground truth phase. However, a sparsity-guided phase retrieval algorithm is considered. One can clearly see that the proposed method provided excellent results for short distances. We observe that the twin-image artifact is consequently eliminated in the single hologram acquired. The procedure significantly removes the background artifacts. The operation is effectively carried out for an average of 100 iterations. It provides visually distinguishable results of the traditional reconstruction technique. The introduced approach obtained complexity-guided phase retrieval solutions. Visually, the GS algorithm generally does not achieve good retrieval for phase objects even if the number of recursive operations is large.

The complexity measure is an interesting framework to explore the field of stagnation problems. Indeed, a sparsity-enhancing tool added to the free-space backpropagation method helps to eliminate the twin-image problem. In the proposed method, the sparsifying transform step is added to free-space backpropagation in a controlled fashion. This operation is achieved such that the complexity of the solution nearly corresponds to the complexity norm calculated as per Equation (1). Here, the proposed approach carries out a significantly reduced number of recursive operations to obtain a suitable solution.

4.2. Experimental Results

The holographic device that was used to acquire the optical field was an inline holography configuration. It features a light source that employs 650 nm wavelength illumination, a 15 µm diameter pinhole, and a Complementary Metal-Oxide-Semiconductor (CMOS) camera (THORLABS DCC1545M) ((THORLABS Inc., 56 Sparta Avenue. Newton, NJ, USA) (Delta Photonics Inc., Ottawa, ON, Canada)) to record images. The CMOS sensor records holograms of size 1024 × 1280 pixels with a 5.2 µm pixel pitch. In this experiment, two solutions of specimens were prepared. One consisted of 10.2 µm organic polymer microspheres. This sample included a polystyrene microbead with a refractive index $n=1.586$ at wavelength $\lambda =650 \mathrm{n}\mathrm{m}$ [44,45] at the temperature $T=20 °\mathrm{C}.$ This type of organic polymer microbead is considered as a convex-shape specimen. A small drop of the diluted solution was placed and spread on a glass slide that was very clean and resistant to corrosion. The glass slide was based on a standard 76 mm × 26 mm × 1 mm microscope slide. The sample dried for 30 min. After, it was used for microscopic examination. A light source with a wavelength 650 nm was employed to illuminate the sample. A single hologram acquisition of the complex diffracted field through the sample was made with a CMOS camera. The sample holder was positioned in the $z$ direction away at a distance from the detector of 0.5 mm. The second sample consisted of red blood cells. This specimen was in liquid medium with a 1.34 refractive index value as in [46]. Thin blood smears were prepared on the same type of glass slide using the standard procedure. In conventional blood smear preparation, cells are labelled with May–Grünwald Giemsa (MGG). However, the prepared samples were made without MGG. Unstained blood smears are preferred for microscopic analysis. They preserve the intrinsic optical properties of red blood cell components for investigation. In this analysis, we considered healthy blood cells, which are concave-shaped cells [47] with a constant average refractive index of 1.42 [42]. We retrieved the phase information of the single-shot complex diffracted field measurement. The reconstruction for the polystyrene microbead as well as for single blood cell was carried out.

Figure 7 presents a complexity value compared to the number of recursive operations for constant and random phase initializations. This explains that these complexities cannot break close to the complexity of the ground truth. The red solid line represents the suitable solution. The observed artifacts in the free-space backpropagation solutions (Figure 8b,g and Figure 9b,g) may be associated with the higher complexity norm for these solutions as explained in the section above.

The same initial random phase map was used as the initial guess for all methods. The reconstructed phase images with 300 and 500 iterations are illustrated in Figure 8 and Figure 9. The phase reconstructions were performed using the free-space backpropagation, proposed sparsity-guided phase retrieval and multi-height phase recovery when fed to optimization technique. Three holograms were used for multi-height phase recovery-based reconstructions as demonstrated in [20], while only single-shot diffraction intensity data were utilized for the free-space backpropagation technique and the new approach. The retrieved image of free-space backpropagation was wrapped with noise artifacts. The detailed information could not be clearly distinguished from the noisy background. This artifact type can be attributed to the interference patterns. The interference is created from the reference wave, which cannot be separated from the diffracted patterns. However, the specimen was clearly recovered by the proposed method. The proposed approach solution, however, does not suffer from the stagnation problem. Our introduced procedure with complexity guidance required a significantly reduced amount of iterative operations to reach a suitable solution. The operation was performed effectively in 100 iterations. The phase reconstruction using the proposed technique was also compared against the one obtained with multi-height phase recovery when using the optimization model. This result demonstrates the excellent performance of the single-shot free-space imaging technique. This finding indicate that it is possible to use a single-shot recording configuration to build a compact phase imaging device.

In Figure 8 and Figure 9, the top row presents the experimental reconstruction of the 10.2 µm polystyrene microbead in a $n=1.33$ [48] refractive index background, which was illuminated at a wavelength of 650 nm. This specimen has a refractive index contrast of $∆\mathrm{n}=0.256,$ leading to a phase shift of $∆\mathsf{\phi }=4.017\times 2\mathsf{\pi } (∆\mathsf{\phi }=25.25 \mathrm{r}\mathrm{a}\mathrm{d})$. The reconstruction was performed by free-space backpropagation when a random initial guess was used (Figure 8b and Figure 9b). The second row describes the experimental reconstruction of RBC in a $n=1.34$ [42] refractive index surrounding medium, which was also illuminated at a wavelength of 650 nm. It has a refractive index contrast of $∆\mathrm{n}=0.08,$ which leads to a phase shift of $∆\mathsf{\phi }=0.270\times 2\mathsf{\pi } (∆\mathsf{\phi }=1.702 \mathrm{r}\mathrm{a}\mathrm{d}).$ The reconstruction was equally carried out using free-space backpropagation when a random initial guess was used (Figure 8g and Figure 9g). All images are XY cross sections.

In many holograms, the specimens of interest are relatively isolated and intrinsically sparse. In this case, the observed twin-image may be easily addressed [46]. Therefore, a simple backpropagation procedure is sufficient for the reconstructions. However, the image quality is compromised by the artefacts for the dense and connected specimens [49]. Thus, the image quality is enhanced by using phase retrieval techniques. Figure 10 shows again the complexity norm against a number of recursive operations for constant and random object initializations. This denotes that these complexities cannot break close to the complexity of the suitable solution. The desired result is defined by the red solid line. We observe that the artifacts in the free-space backpropagation solutions (Figure 11b,g and Figure 12b,g) described the same case as that previously mentioned.

Indeed, the recorded hologram has only amplitude information and its phase information is initially missing [50]. More advanced algorithms consider the phase retrieval for sample scenes. To overcome the challenge of phase reconstructions, here we explored a different approach. This elaborated approach achieves excellent elimination of twin-image and self-interference artifacts as illustrated in Figure 11 and Figure 12. The proposed method even eliminated these artifacts using a single hologram recorded in short distances.

The performance of the proposed method is evaluated. The error between the experimental measurements and numerical predictions is shown in Figure 13. It is plotted as a function of the number of iterations for free-space backpropagation, the proposed technique, the multi-height phase recovery associated with ADMM and the multi-height phase recovery associated with FISTA. We observe that with sparsity-guided phase retrieval, the error value falls rapidly compared to the free-space backpropagation method. Indeed, in our proposed sparsity-guided phase retrieval, the specimen is reconstructed very well after an average of 100 iterations. However, free-space backpropagation iteration alone does not similar findings for the same case. The free-space backpropagation technique cannot provide a good reconstruction for phase specimens even for a significant number of recursive operations. The proposed sparsity-guided phase retrieval contributes to providing a suitable solution after a fixed amount of iterative operations. Additionally, the proposed method is also seen to efficiently mitigate the twin image. We equally show that the imaging performance of the proposed sparsity-guided phase retrieval using a single hologram is comparable to that of multi-height phase reconstruction techniques combined with ADMM and FISTA using three holograms as in [21].

5. Conclusions

In the present work, we propose a sparsity-guided phase retrieval technique. Our proposed approach takes the complexity parameter into account. Appropriate complexity guidance is integrated into the reconstruction of complex diffracted fields of 10.2 µm polystyrene microbeads with a convex shape and RBCs with a concave shape, respectively. The performance of the proposed approach is compared against the performance of the free-space backpropagation technique, multi-height phase recovery method associated with ADMM and multi-height phase recovery method combined with FISTA. The developed procedure can achieve excellent phase reconstruction using only a single acquired hologram.