Resolution Enhancement of Geometric Phase Self-Interference Incoherent Digital Holography Using Synthetic Aperture

Abstract

Incoherent digital holography is a promising solution for acquiring three-dimensional information in the form of interference without a laser. Self-interference is the principal phenomenon of incoherent holography which splits the incident light wave into two waves and interferes with each one. The characteristics of incoherent holograms are defined by the optical configuration of wavefront dividing units. In this paper, we propose the synthetic aperture method to achieve super-resolution of geometric phase lens-based self-interference incoherent digital holography. We analyze the resolution of incoherent holograms acquired from geometric phase lenses and experimentally demonstrate the resolution enhancement via the synthetic-aperture method. Moreover, the sparse synthetic-aperture method is proposed to ensure effective data capturing and the numerical optimization method is also proposed to fill the gap between the empty space of sparsely acquired holograms.

1. Introduction

Holographic display technology, which leverages the wave nature of light to represent three-dimensional (3D) information, has emerged as a promising next-generation display solution due to its capability to provide genuine 3D visualization with complete depth cues. Recent advances in holographic displays have demonstrated remarkable progress in various applications from augmented reality (AR) and virtual reality (VR) systems [1,2,3,4,5]. The representation of complete light waves is implemented by amplitude or phase modulation with spatial light modulators (SLMs). The generation of hologram data for holographic displays relies on computer-generated holography (CGH) techniques [6,7,8,9]. Notable achievements include real-time photorealistic hologram generation through deep learning approaches and non-convex optimization-based algorithms [10,11,12,13,14]. However, the fundamental challenge in realizing high-fidelity holographic displays lies in their enormous spatial resolution requirements. The space-bandwidth product demand of holographic displays scales exponentially compared with conventional flat panel displays, and the resolution of generated holograms is also exponentially exceeded.

Incoherent holography is one of the promising holography techniques that can acquire the hologram under the condition of incoherent light source such as natural light or light-emitting diodes (LEDs) to address these challenges. Self-interference incoherent digital holography (SIDH) and Fresnel incoherent correlation holography (FINCH) are incoherent holography approaches that achieve the self-interference hologram by dividing the object light into two waves and modulating them to interfere with each other [15,16,17]. The key components of SIDH are a wavefront division device and a phase modulation device. The beam splitter or polarization-based optical components such as a phase-only SLM and a birefringence lens can be utilized as a wavefront division device, and the correlated phase modulation device must be determined according to the characteristics of the wavefront division method [18,19]. For instance, FINCH utilizes a liquid crystal-based phase-only SLM as a wavefront division device and a phase shift device at the same time with multiple exposures for the recording of stepwise phase-shifted interferograms [20,21,22]. Recent advanced FINCH and coded aperture correlation holography introduce high-level holographic microscopic solutions that enable 3D scanning with enhanced resolution and image quality [23,24,25].

Geometric phase (GP) lens-based self-interference incoherent digital holography (GP-SIDH) is introduced, offering several distinct advantages over conventional approaches [26,27]. Unlike traditional FINCH systems that rely on active SLMs, GP lenses are passive optical elements that can manipulate wavefronts and phase at the same time through geometric phase control, resulting in a more efficient system architecture. The passive nature of GP lenses also contributes to enhanced system stability, making them particularly suitable for portable applications [28,29]. Furthermore, GP lens-based holographic cameras are compatible with modern deep learning models to realize fully incoherent holographic streaming systems. The captured holographic data can be seamlessly integrated into neural network frameworks for real-time processing, enabling end-to-end pipelines from capture to display [30].

In this paper, we propose a resolution enhancement method using a sparsely acquired synthetic-aperture technique. Synthetic aperture is a straightforward super-resolution strategy that is generally used in conventional optics and imaging techniques. In incoherent holography, Katz et al. introduced the super-resolution technique of FINCH with synthetic aperture, and various studies to apply the idea of synthetic aperture on SIDH have been proposed [31,32,33,34]. Inspired by these studies, we analyze the resolution criteria of GP-SIDH and propose a synthetic-aperture method with sparse acquisition. Moreover, to solve the drawbacks of sparse acquisition, the gradient descent-based optimization algorithm is proposed for finding the empty signal of synthetic-aperture holograms.

2. Materials and Methods

2.1. Resolution Analysis of GP-SIDH

GP-SIDH can be analyzed as a Fresnel hologram formed by the interference between two-point light sources. The polarization-selective characteristic of the GP lens enables wavefront division with positive and negative focal lengths ($f_{+GP}$ and $f_{-GP}$), allowing us to apply the Rayleigh resolution criterion:

$$Resolution=0.61{\displaystyle \frac{\lambda }{NA·M_T}}$$

(1)

where $\lambda$ is the wavelength, $NA$ is the numerical aperture determined by the point spread hologram (PSH) radius $r_H$ and reconstruction distance $z_r$ as $r_H/z_r$, and $M_T$ is the transverse magnification defined by the distance relation between GP lens and image sensor $z_h$ and GP lens and object $z_s$, denoted as $z_h/z_s$. The maximum spatial frequency of the reconstructed hologram is primarily determined by the PSH radius, which is constrained by three main factors:

$$r_H=min(r_{OPD},r_{Diff},r_{Aper})$$

(2)

where $r_{OPD}$ is limited by the optical path difference, $r_{Diff}$ is the diffraction limit of the image sensor, and $r_{Aper}$ is the aperture limitation of GP-SIDH. To achieve self-interference conditions, the optical path difference between two beams separated by the GP lens must be less than the coherence length of the light. This condition is determined by the geometric parameters of the system, including the focal length of the GP lens, and the distance between the object plane, the GP lens, and the image sensor plane. On the other hand, the diffraction limit $r_{Diff}$ follows the Nyquist–Shannon sampling theorem:

$$r_{Diff}=0.5·{\displaystyle \frac{\lambda z_r}{\Delta }}$$

(3)

where $\Delta$ is the image sensor pixel pitch. The reconstruction distance $z_r$ can be expressed as:

$$z_r={\displaystyle \frac{((z_s+z_h)f_{GP}-z_s{z}_h)((z_s+z_h)f_{GP}+z_s{z}_h)}{2z_s^2{f}_{GP}}}$$

(4)

This relationship demonstrates inherent trade-offs between numerical aperture, object distance, and field of view (FOV). The optimization of the optical system, particularly concerning the FOV and depth of field (DOF), is crucial for implementing a holographic camera as a practical imaging device. In the GP-SIDH configuration, the FOV is primarily determined by $z_h$. Among various system parameters, $r_{Diff}$ imposes the most stringent constraint when considering both the fabrication limitations of the GP lens and the image sensor dimensions. Upon establishing the target FOV and image sensor specifications for camera implementation, one can determine the optimal focal length conditions for the GP lens that maximize the achievable resolution [21,30].

2.2. Sparse Synthetic Aperture with GP-SIDH

To enhance the resolution of GP-SIDH beyond the previously calculated limitations, we propose a synthetic-aperture technique, which is a straightforward and effective approach for super-resolution. Synthetic aperture has been widely used in computer vision applications to create high-resolution images by combining multiple captures, effectively simulating image acquisition through an optical system with a larger aperture.

Previously reported SIDH-based synthetic-aperture techniques are conducted by mechanical shifts combined with the corresponding modulation of SLM phase patterns. Since GP-SIDH utilizes passive optical elements, the synthetic aperture can be realized through image sensor displacement only as illustrated in Figure 1. Moreover, leveraging the nature of the Fresnel hologram enables a substantial reduction in the number of exposures required for synthetic aperture implementation. The Fresnel hologram exhibits a characteristic where the central region contains the primary object information, while the peripheral regions mainly contain high-frequency components. This fundamental property enables the implementation of sparse synthetic aperture, where dense sampling is applied only to the central region of interest (ROI), while selective sparse sampling of high-frequency components in peripheral regions is sufficient to maintain image quality, as shown in Figure 1b.

The sparse synthetic aperture approach inevitably creates empty regions in the final hologram. Since holographic fringe patterns inherently preserve some energy distribution even in regions without direct measurement due to interference effects, these gaps can lead to degraded resolution in the ROI and limited viewing angles in 3D display applications. Therefore, we propose a hologram completion algorithm to fill these empty regions while maintaining the optical reconstruction quality inspired by non-convex optimization-based hologram generation and compensation [11,14,35,36,37]. We utilize a stochastic gradient descent (SGD)-based optimization process, as illustrated in Figure 2. The objective function can be expressed as:

$$H^{\prime }=arg\underset{H_{com}}{min}\mathcal{L}(R(H+H_{com}),R\left(H\right)+w·\mathcal{S}\left(R\left(H\right)\right))$$

(5)

where $R(·)$ represents the angular spectrum numerical reconstruction operator, $\mathcal{S}(·)$ is the sharpness enhancement filter, and $\mathcal{w}$ is a weighting factor. The fundamental objective of this optimization approach is to extrapolate the missing information through the enhancement of the central primary signal. For sharpness enhancement, various edge detection and enhancement algorithms can be implemented, e.g., the Laplacian operator, Sobel filter, and unsharp masking. The process iteratively refines the empty regions until convergence, producing a complete hologram suitable for optical reconstruction.

3. Results

To demonstrate our proposal, we first conducted an experiment on the formation of PSH and the change in resolution according to the number of high-frequency components. In GP-SIDH, the Fresnel hologram captured on an image sensor has a high-frequency fringe depending on the positional relation of the GP lens and the object. Therefore, the location of objects affects the high-frequency fringe limits. To acquire the various location patterns, we utilized the motorized stage. We used a customized 2-inch GP lens with a focal length of 1000 mm at the central wavelength of 532 nm and wavelength selectivity by utilizing a full-color polarized image sensor (PHX050S-QNL, LUCID Vision Labs, Richmond, BC, Canada), whose central wavelengths of the color filter are 620, 550, and 450 nm. The object is 200 mm from the GP lens, and the gap between the GP lens and polarized image sensor is 8 mm.

Figure 3 shows a comprehensive comparison of hologram acquisition and reconstruction for three different spatial positions of a resolution target. The resolution of the smallest experimental target is 0.561 lp/mm. The first and second columns show numerical reconstruction results. The third column is the phase part of the hologram and the fourth column is the Fourier spectrum of the hologram. Compared with horizontally shifted targets, as shown in Figure 3a–h, the vertical frequency component is increased and phase distribution shows asymmetric interference patterns indicating the capture of additional high-frequency information in the vertical direction. The diagonal shift case in Figure 3i–l demonstrates the most comprehensive high-frequency capture, exhibiting rich interference patterns. The Fourier spectra provide a clear comparative visualization of the acquired high-frequency components corresponding to different hologram locations.

Figure 4 shows a conventional synthetic-aperture result for GP-SIDH. The experimental synthetic-aperture acquisition was performed utilizing a polarization image sensor with a resultant resolution of 1024 × 1024. The sensor was mechanically translated behind a 2-inch diameter GP lens, enabling the acquisition of up to 25 positions in a 5 × 5 synthetic-aperture position. Figure 4a,b show the phase distribution of the point source and resolution target hologram. The dotted yellow lines indicate the sampling grid used for synthetic aperture. Figure 4c–e show the numerical reconstruction results for the central region of the synthetic-aperture hologram. The numerical reconstruction demonstrates that the resolution increases with the number of synthetic-aperture positions.

To validate the 3D capturing of synthetic aperture, we conducted experiments with real-world object scenarios. Figure 5 shows the numerical reconstruction results of the toy vehicle, whose reconstruction was performed at two different focal planes. Compared with the single-capture reconstruction, 3 × 3 and 5 × 5 synthetic-aperture holograms demonstrate improvement in resolution with preserved depth information. However, a comparison between 3 × 3 and 5 × 5 hologram configurations reveals minimal improvement in resolution. This observation, in conjunction with the increasing complexity of hologram alignment at higher multiplexing apertures, establishes a trade-off criterion. Such findings can be instrumental in determining the optimal number of synthetic aperture positions during system design.

Figure 6 shows the sparsely acquired hologram and its numerical reconstruction results. The sparse acquisition modeling was conducted through numerical reconstruction of the resolution target hologram with the resolution of 1024 × 1024 with selective spatial masking. The implementation of a custom GP lens necessitated wavefront correction algorithms, resulting in characteristic contour-like artifacts [29]. The masking strategy employed a diagonal mask of approximately 400 × 400 pixels, systematically attenuating peripheral signal components. Comparative evaluation of numerical reconstructions between masked and unmasked holograms demonstrates that, despite minor degradation in the bias component manifesting as slight blur, the overall holographic reconstruction quality remains largely preserved. These results substantiate that the essential holographic information can be maintained through sparse acquisition protocols, suggesting opportunities for more efficient synthetic-aperture implementations.

To evaluate different approaches for hologram reconstruction enhancement, we implemented various image filtering techniques using the scikit-image library [38] and denoising algorithms including BM3D [39] and Noise2Noise [40]. Figure 7 shows a comparative analysis of different filtering methods applied to the reconstructed hologram of a resolution target. The reference masked hologram reconstruction as shown in Figure 7a serves as our baseline for comparison. We then applied five different enhancement filters:

• Laplacian operator (Figure 7b):

  $$L(x,y)={\displaystyle \frac{{\partial }^{2}I}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}I}{\partial y^2}}$$

  (6)
• Unsharp masking (Figure 7c):

  $$I_{enhanced}=I+\lambda (I-G_{\sigma }\ast I)$$

  (7)
• Roberts operator

  $$\begin{array}{cc}\hfill & R=\sqrt{{(R_x\ast I)}^2+{(R_y\ast I)}^2},\hfill \\ \hfill \mathrm{where}& R_x=\left[\begin{array}{cc}+1& 0\\ 0& -1\end{array}\right],R_y=\left[\begin{array}{cc}0& -1\\ +1& 0\end{array}\right]\hfill \end{array}$$

  (8)
• Sobel operator (Figure 7e):

  $$\begin{array}{cc}\hfill & S=\sqrt{{(S_x\ast I)}^2+{(S_y\ast I)}^2},\hfill \\ \hfill \mathrm{where}& S_x=\left[\begin{array}{ccc}-1& 0& +1\\ -2& 0& +2\\ -1& 0& +1\end{array}\right]S_y=\left[\begin{array}{ccc}-1& -2& -1\\ 0& 0& 0\\ +1& +2& +1\end{array}\right]\hfill \end{array}$$

  (9)
• Prewitt operator (Figure 7f):

  $$\begin{array}{cc}\hfill & P=\sqrt{{\left(P_{x}I\right)}^2+{\left(P_{y}I\right)}^2},\hfill \\ \hfill \mathrm{where}& P_x=\left[\begin{array}{ccc}-1& 0& +1\\ -1& 0& +1\\ -1& 0& +1\end{array}\right],P_y=\left[\begin{array}{ccc}-1& -1& -1\\ 0& 0& 0\\ +1& +1& +1\end{array}\right]\hfill \end{array}$$

  (10)

  where $G_{\sigma }$ represents a Gaussian blur kernel and $\lambda$ is the enhancement strength. The Laplace operator takes absolute values to compare with other operations. The Sobel and Prewitt operators demonstrate the best overall performance, offering an optimal balance between edge enhancement and noise suppression. The unsharp mask and Roberts operator also perform well, while the Laplacian filters show more sensitivity to noise.

To demonstrate the quantitative comparison of our proposed complete optimization approach, we present unsharp masking results as the optimal enhancement filter. Figure 8a shows the phase distribution of the reconstructed hologram after the proposed optimization algorithm. The enhancement reveals clear fringe patterns while maintaining the phase consistency of the original hologram. The amplitude reconstruction in Figure 8b shows improvement in edge definition and contrast. The resolution target pattern is resolved, with group numbers and element patterns distinctly visible. The orange box highlights a region of particular interest where fine details are well preserved. A quantitative analysis of the intensity profile is presented in Figure 8c, comparing the original masked reconstruction (blue line) with our proposed enhancement method (orange line). However, the deteriorated central bias signal is still not conserved by iteration.

Table 1. Quantitative comparison of PSNR values for different filtering methods.

Method	PSNR (dB)	SSIM
Reference masked	18.17	0.11
Laplace	19.02	0.25
Unsharp mask	19.38	0.27
Roberts	20.76	0.37
Sobel	21.64	0.42
Prewitt	21.94	0.44
BM3D	22.29	0.46
Noise2Noise	20.81	0.32

indicates the performance of different sharpness enhancement methods using both peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM). To calculate the sharpness metric, the reconstruction of the full-resolution hologram is utilized as a reference. This approach allows for a greater focus on the accurate reconstruction of the hologram. Consequently, a higher PSNR score indicates that the numerical reconstruction is closer to the original hologram. The Laplace filter is affected by unfiltered noise and shows the lowest increase. The unsharp mask and BM3D provide a good balance while maintaining stable performance. The gradient-based operators (Prewitt and Sobel) show the highest metric improvement, suggesting better edge enhancement performance. As the degree of sharpening can be adjusted through the use of weights, it is anticipated that the implementation of these image sharpening algorithms will result in enhanced outcomes. This can be achieved by setting the weights according to the distinct characteristics of each algorithm. However, the deep learning-based method (Noise2Noise) has not been trained on holographic reconstructions, so the improvement in PSNR appears to be minimal. We expect to see better performance when using deep learning models that target hologram reconstruction-based noise reduction or image quality refinement.

4. Discussion

In this study, we demonstrated that the GP-SIDH system can achieve enhanced resolution through sparse synthetic-aperture and optimized image enhancement techniques. However, several considerations need to be addressed for real-time camera system implementation. The current mechanical translation stage for synthetic aperture, while effective, introduces inherent limitations in capture speed. A potential solution could involve the development of a multi-sensor array system, where multiple polarization image sensors are arranged in a sparse configuration. This would eliminate the need for mechanical movement while maintaining the benefits of synthetic aperture, though careful consideration would need to be given to the precise alignment and synchronization of multiple sensors. Furthermore, the computational overhead of hologram completion and enhancement could be significantly reduced through hardware acceleration. The current image enhancement pipeline, particularly the unsharp masking and phase optimization steps, could be implemented on field-programmable gate arrays or graphics processing units. This would enable real-time processing of the holographic data, making it feasible for video-rate capture and reconstruction. Additionally, the development of specialized hardware for sparse synthetic aperture processing could further improve system performance. Such optimizations would make the GP-SIDH system more practical for applications requiring real-time 3D imaging, such as medical imaging, industrial inspection, and augmented reality displays.

5. Conclusions

We have demonstrated a comprehensive resolution enhancement approach for GP-SIDH. Our method combines sparse synthetic-aperture techniques with optimized image enhancement algorithms to achieve improved resolution while maintaining practical implementation feasibility. The theoretical analysis of GP-SIDH resolution limits was validated through experimental results, showing that strategic sparse sampling can achieve comparable results to full synthetic aperture while significantly reducing the number of required captures. We quantitatively evaluated various image enhancement techniques, demonstrating that appropriate filtering methods can effectively improve reconstruction quality without introducing significant artifacts. For future work, deep learning-based approaches could potentially further optimize the synthetic aperture process. Neural networks could be trained to predict missing holographic information from sparse captures, potentially reducing the number of required measurements while maintaining or even improving reconstruction quality. Additionally, learning-based approaches could adaptively optimize sampling patterns based on specific scene characteristics, leading to more efficient holographic capture systems. These developments would further advance the practical implementation of high-resolution incoherent holographic imaging systems.