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Computational methods have been established as cornerstones in optical imaging and holography in recent years. Every year, the dependence of optical imaging and holography on computational methods is increasing significantly to the extent that optical methods and components are being completely and efficiently replaced with computational methods at low cost. This roadmap reviews the current scenario in four major areas namely incoherent digital holography, quantitative phase imaging, imaging through scattering layers, and super-resolution imaging. In addition to registering the perspectives of the modern-day architects of the above research areas, the roadmap also reports some of the latest studies on the topic. Computational codes and pseudocodes are presented for computational methods in a plug-and-play fashion for readers to not only read and understand but also practice the latest algorithms with their data. We believe that this roadmap will be a valuable tool for analyzing the current trends in computational methods to predict and prepare the future of computational methods in optical imaging and holography., Supplementary Information: The online version contains supplementary material available at 10.1007/s00340-024-08280-3., Competing Interests: Conflict of interestThe authors declare no competing interests., (© The Author(s) 2024.)

The race for miniature color cameras using flat meta-optics has rapidly developed the end-to-end design framework using neural networks. Although a large body of work has shown the potential of this methodology, the reported performance is still limited due to fundamental limitations coming from meta-optics, mismatch between simulated and resultant experimental point spread functions, and calibration errors. Here, we use a HIL optics design methodology to solve these limitations and demonstrate a miniature color camera via flat hybrid meta-optics (refractive + meta-mask). The resulting camera achieves high-quality full-color imaging for a 5-mm aperture optics with a focal length of 5 mm. We observed a superior quality of the images captured by the hybrid meta-optical camera compared to a compound multi-lens optics of a mirrorless commercial camera.

End-to-end optimization of diffractive optical elements (DOEs) profile through a digital differentiable model combined with computational imaging have gained an increasing attention in emerging applications due to the compactness of resultant physical setups. Despite recent works have shown the potential of this methodology to design optics, its performance in physical setups is still limited and affected by manufacturing artefacts of DOE, mismatch between simulated and resultant experimental point spread functions, and calibration errors. Additionally, the computational burden of the digital differentiable model to effectively design the DOE is increasing, thus limiting the size of the DOE that can be designed. To overcome the above mentioned limitations, a co-design of hybrid optics and image reconstruction algorithm is produced following the end-to-end hardware-in-the-loop strategy, using for optimization a convolutional neural network equipped with quantitative and qualitative loss functions. The optics of the imaging system consists on the phase-only spatial light modulator (SLM) as DOE and refractive lens. SLM phase-pattern is optimized by applying the Hardware-in-the-loop technique, which helps to eliminate the mismatch between numerical modelling and physical reality of image formation as light propagation is not numerically modelled but is physically done. Comparison with compound multi-lens optics of a last generation smartphone and a mirrorless commercial cameras show that the proposed system is advanced in all-in-focus sharp imaging for a depth range 0.4-1.9 m.

A power-balanced hybrid optical imaging system has a diffractive computational camera, introduced in this paper, with image formation by a refractive lens and multilevel phase mask (MPM). This system provides a long focal depth with low chromatic aberrations thanks to MPM and a high energy light concentration due to the refractive lens. We introduce the concept of optical power balance between the lens and MPM, which controls the contribution of each element to modulate the incoming light. Additional features of our MPM design are the inclusion of the quantization of the MPM's shape on the number of levels and the Fresnel order (thickness) using a smoothing function. To optimize the optical power balance as well as the MPM, we built a fully differentiable image formation model for joint optimization of optical and imaging parameters for the proposed camera using neural network techniques. We also optimized a single Wiener-like optical transfer function (OTF) invariant to depth to reconstruct a sharp image. We numerically and experimentally compare the designed system with its counterparts, lensless and just-lens optical systems, for the visible wavelength interval (400-700) nm and the depth-of-field range (0.5- ∞ m for numerical and 0.5-2 m for experimental). We believe the attained results demonstrate that the proposed system equipped with the optimal OTF overcomes its counterparts--even when they are used with optimized OTF--in terms of the reconstruction quality for off-focus distances. The simulation results also reveal that optimizing the optical power balance, Fresnel order, and the number of levels parameters are essential for system performance attaining an improvement of up to 5 dB of PSNR using the optimized OTF compared to its counterpart lensless setup.

A novel phase retrieval algorithm for broadband hyperspectral phase imaging from noisy intensity observations is proposed. It utilizes advantages of the Fourier transform spectroscopy in the self-referencing optical setup and provides additional, beyond spectral intensity distribution, reconstruction of the investigated object's phase. The noise amplification Fellgett's disadvantage is relaxed by the application of a sparse wavefront noise filtering embedded in the proposed algorithm. The algorithm reliability is proved by simulation tests and by results of physical experiments for transparent objects. These tests demonstrate precise phase imaging and object depth (profile) reconstruction.

Design and optimization of lensless phase-retrieval optical system with phase modulation of free-space propagation wavefront is proposed for subpixel imaging to achieve super-resolution reconstruction. Contrary to the traditional super-resolution phase-retrieval, the method in this paper requires a single observation only and uses the advanced Super-Resolution Sparse Phase Amplitude Retrieval (SR-SPAR) iterative technique which contains optimized sparsity based filters and multi-scale filters. The successful object imaging relies on modulation of the object wavefront with a random phase-mask, which generates coded diffracted intensity pattern, allowing us to extract subpixel information. The system's noise-robustness was investigated and verified. The super-resolution phase-imaging is demonstrated by simulations and physical experiments. The simulations included high quality reconstructions with super-resolution factor of 5, and acceptable at factor up to 9. By physical experiments 3 μm details were resolved, which are 2.3 times smaller than the resolution following from the Nyquist-Shannon sampling theorem.

We investigated the peculiarities of the terahertz pulse time-domain holography principle in the case of raster scanning with the balance detection system. The noise in this system represents a Skellam distribution model, which differentiates it from systems based on a photoconductive antenna. We analyzed this Skellam model and provided both numerical and experimental investigations. We found that the variance of the noise in the balance detection system does not depend on the true signal. Complex-domain images obtained in this model are filtered by block-matching algorithms adapted for spatio-temporal and spatiospectral volumetric data. We presented a new cube complex-domain filter algorithm that uses block matching in all 3D data sets simultaneously in spatial and frequency coordinates. A combination of temporal and complex-domain filters allows us to expand the dynamic range of terahertz frequencies for which we can obtain amplitude/phase information. Experimental data demonstrate an improvement in the quality of the resultant images both in the time domain and complex-spectral domain. The simulation and experimental results are in good agreement.

In this paper, we have applied a recently developed complex-domain hyperspectral denoiser for the object recognition task, which is performed by the correlation analysis of investigated objects' spectra with the fingerprint spectra from the same object. Extensive experiments carried out on noisy data from digital hyperspectral holography demonstrate a significant enhancement of the recognition accuracy of signals masked by noise, when the advanced noise suppression is applied.

We investigated data denoising in hyperspectral terahertz pulse time-domain holography. Using the block-matching algorithms adapted for spatio-temporal and spatio-spectral volumetric data we studied and optimized parameters of these algorithms to improve phase image reconstruction quality. We propose a sequential application of the two algorithms oriented on work in temporal and spectral domains. Experimental data demonstrate the improvement in the quality of the resultant time-domain images as well as phase images and object's relief. The simulation results are proved by comparison with the experimental ones.

This paper proposes a novel algorithm for image phase retrieval, i.e., for recovering complex-valued images from the amplitudes of noisy linear combinations (often the Fourier transform) of the sought complex images. The algorithm is developed using the alternating projection framework and is aimed to obtain high performance for heavily noisy (Poissonian or Gaussian) observations. The estimation of the target images is reformulated as a sparse regression, often termed sparse coding, in the complex domain. This is accomplished by learning a complex domain dictionary from the data it represents via matrix factorization with sparsity constraints on the code (i.e., the regression coefficients). Our algorithm, termed dictionary learning phase retrieval (DLPR), jointly learns the referred to dictionary and reconstructs the unknown target image. The effectiveness of DLPR is illustrated through experiments conducted on complex images, simulated and real, where it shows noticeable advantages over the state-of-the-art competitors.

The paper is devoted to a computational super-resolution microscopy. A complex-valued wavefront of a transparent biological cellular specimen is restored from multiple intensity diffraction patterns registered with noise. For this problem, the recently developed lensless super-resolution phase retrieval algorithm [Optica, 4(7), 786 (2017)] is modified and tuned. This algorithm is based on a random phase coding of the wavefront and on a sparse complex-domain approximation of the specimen. It is demonstrated in experiments, that the reliable phase and amplitude imaging of the specimen is achieved for the low signal-to-noise ratio provided a low dynamic range of observations. The filterings in the observation domain and specimen variables are specific features of the applied algorithm. If these filterings are omitted the algorithm becomes a super-resolution version of the standard iterative phase retrieval algorithms. In comparison with this simplified algorithm with no filterings, our algorithm shows a valuable improvement in imaging with much smaller number of observations and shorter exposure time. In this way, presented algorithm demonstrates ability to work in a low radiation photon-limited mode., Competing Interests: The authors declare that there are no conflicts of interest related to this article.

Single image super-resolution (SISR) is an ill-posed problem aiming at estimating a plausible high-resolution (HR) image from a single low-resolution image. Current state-of-the-art SISR methods are patch-based. They use either external data or internal self-similarity to learn a prior for an HR image. External data-based methods utilize a large number of patches from the training data, while self-similarity-based approaches leverage one or more similar patches from the input image. In this paper, we propose a self-similarity-based approach that is able to use large groups of similar patches extracted from the input image to solve the SISR problem. We introduce a novel prior leading to the collaborative filtering of patch groups in a 1D similarity domain and couple it with an iterative back-projection framework. The performance of the proposed algorithm is evaluated on a number of SISR benchmark data sets. Without using any external data, the proposed approach outperforms the current non-convolutional neural network-based methods on the tested data sets for various scaling factors. On certain data sets, the gain is over 1 dB, when compared with the recent method A+. For high sampling rate (x4), the proposed method performs similarly to very recent state-of-the-art deep convolutional network-based approaches.

A variational algorithm to object wavefront reconstruction from noisy intensity observations is developed for the off-axis holography scenario with imaging in the acquisition plane. The algorithm is based on the local least square technique proposed in paper [J. Opt. Soc. Am. A21, 367 (2004)]. First, multiple reconstructions of the wavefront are produced for various size and various directional windows applied for localization of estimation. At the second stage, a special statistical rule is applied in order to select the best window size estimate for each pixel of the image and for each of the directional windows. At the third final stage the estimates of the different directions obtained for each pixel are aggregated in the final one. Simulation experiments and real data processing prove that the developed algorithm demonstrate the performance of the extraordinary quality and accuracy for both the phase and amplitude of the object wavefront.

Plenoptic cameras enable the capture of a light field with a single device. However, with traditional light field rendering procedures, they can provide only low-resolution two-dimensional images. Super-resolution is considered to overcome this drawback. In this study, we present a super-resolution method for the defocused plenoptic camera (Plenoptic 1.0), where the imaging system is modeled using wave optics principles and utilizing low-resolution depth information of the scene. We are particularly interested in super-resolution of in-focus and near in-focus scene regions, which constitute the most challenging cases. The simulation results show that the employed wave-optics model makes super-resolution possible for such regions as long as sufficiently accurate depth information is available.

Ptychography is a lensless coherent diffractive imaging that uses intensity measurements of multiple diffraction patterns collected with a localized illumination probe from overlapping regions of an object. An iterative algorithm is proposed that is targeted on optimal processing noisy measurements. The noise suppression is enabled by two instruments: first, the maximum-likelihood technique formulated for Poissonian (photon-counting) measurements, and, second, sparse approximation of the phase and magnitude of the object and probe. It is shown that the maximum-likelihood estimate of the wavefield at the sensor plane for noisy measurements is essentially different from the famous Gerchberg-Saxton-Fienup solution, where the magnitude of the estimate is replaced by the square root of the intensity measurement. The simulation experiments demonstrate the state-of-the-art performance of the proposed algorithm both numerically and visually.

We present an extension of the BM3D filter to volumetric data. The proposed algorithm, BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar d-dimensional patches are stacked together in a (d+1)-dimensional array and jointly filtered in transform domain. While in BM3D the basic data patches are blocks of pixels, in BM4D we utilize cubes of voxels, which are stacked into a 4-D "group." The 4-D transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes. Thus, the spectrum of the group is highly sparse, leading to very effective separation of signal and noise through coefficient shrinkage. After inverse transformation, we obtain estimates of each grouped cube, which are then adaptively aggregated at their original locations. We evaluate the algorithm on denoising of volumetric data corrupted by Gaussian and Rician noise, as well as on reconstruction of volumetric phantom data with non-zero phase from noisy and incomplete Fourier-domain (k-space) measurements. Experimental results demonstrate the state-of-the-art denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction.

Generally, wave field reconstructions obtained by phase-retrieval algorithms are noisy, blurred, and corrupted by various artifacts such as irregular waves, spots, etc. These distortions, arising due to many factors, such as nonidealities of the optical system (misalignment, focusing errors), dust on optical elements, reflections, and vibration, are hard to localize and specify. It is assumed that there is a cumulative disturbance called "background," which describes mentioned distortions in the coherent imaging system manifested at the sensor plane. Here we propose a novel iterative phase-retrieval algorithm compensating for these distortions in the optical system. An estimate of this background is obtained via special calibration experiments, and then it is used for the object reconstruction. The algorithm is based on the maximum likelihood approach targeting on the optimal object reconstruction from noisy data and imaging enhancement using a priori information on the object amplitude. In this work we demonstrate the compensation of the distortions of the optical trace for a complex-valued object with a binary amplitude. The developed algorithm results in state-of-the-art filtering, and sharp reconstruction imaging of the object amplitude can be achieved.

The computational ghost imaging with a phase spatial light modulator (SLM) for wave field coding is considered. A transmission-mask amplitude object is reconstructed from multiple intensity observations. Compressive techniques are used in order to gain a successful image reconstruction with a number of observations (measurement experiments), which is smaller than the image size. Maximum likelihood style algorithms are developed, respectively, for Poissonian and approximate Gaussian modeling of random observations. A sparse and overcomplete modeling of the object enables the advanced high accuracy and sharp imaging. Numerical experiments demonstrate that an approximative Gaussian distribution with an invariant variance results in the algorithm that is efficient for Poissonian observations.

A family of the block matching 3-D (BM3D) algorithms for various imaging problems has been recently proposed within the framework of nonlocal patchwise image modeling , . In this paper, we construct analysis and synthesis frames, formalizing BM3D image modeling, and use these frames to develop novel iterative deblurring algorithms. We consider two different formulations of the deblurring problem, i.e., one given by the minimization of the single-objective function and another based on the generalized Nash equilibrium (GNE) balance of two objective functions. The latter results in the algorithm where deblurring and denoising operations are decoupled. The convergence of the developed algorithms is proved. Simulation experiments show that the decoupled algorithm derived from the GNE formulation demonstrates the best numerical and visual results and shows superiority with respect to the state of the art in the field, confirming a valuable potential of BM3D-frames as an advanced image modeling tool.

We apply a nonlocal adaptive spectral transform for sparse modeling of phase and amplitude of a coherent wave field. The reconstruction of this wave field from complex-valued Gaussian noisy observations is considered. The problem is formulated as a multiobjective constrained optimization. The developed iterative algorithm decouples the inversion of the forward propagation operator and the filtering of phase and amplitude of the wave field. It is demonstrated by simulations that the performance of the algorithm, both visually and numerically, is the current state of the art.

The 4f optical setup is considered with a wave field modulation by a spatial light modulator located in the focal plane of the first lens. Phase as well as amplitude of the wave field are reconstructed from noisy multiple-intensity observations. The reconstruction is optimal due to a constrained maximum likelihood formulation of the problem. The proposed algorithm is iterative with decoupling of the inverse of the forward propagation of the wave field and the filtering of phase and amplitude. The sparse modeling of phase and amplitude enables the advanced high-accuracy filtering and sharp imaging of the complex-valued wave field. Artifacts typical for the conventional algorithms (wiggles, ringing, waves, etc.) and attributed to optical diffraction can be suppressed by the proposed algorithm.

A complex-valued wave field is reconstructed from intensity-only measurements given at multiple observation planes parallel to the object plane. The phase-retrieval algorithm is obtained from the constrained maximum likelihood approach provided that the additive noise is gaussian. The forward propagation from the object plane to the measurement plane is treated as a constraint in the proposed variational setting of reconstruction. The developed iterative algorithm is based on an augmented lagrangian technique. An advanced performance of the algorithm is demonstrated by numerical simulations.

We consider reconstruction of a wave field distribution in an input/object plane from data in an output/diffraction (sensor) plane. We provide digital modeling both for the forward and backward wave field propagation. A novel algebraic matrix form of the discrete diffraction transform (DDT) originated in Katkovnik et al. [Appl. Opt. 47, 3481 (2008)] is proposed for the forward modeling that is aliasing free and precise for pixelwise invariant object and sensor plane distributions. This "matrix DDT" is a base for formalization of the object wave field reconstruction (backward propagation) as an inverse problem. The transfer matrices of the matrix DDT are used for calculations as well as for the analysis of conditions when the perfect reconstruction of the object wave field distribution is possible. We show by simulation that the developed inverse propagation algorithm demonstrates an improved accuracy as compared with the standard convolutional and discrete Fresnel transform algorithms.

The paper attacks absolute phase estimation with a two-step approach: the first step applies an adaptive local denoising scheme to the modulo-2 pi noisy phase; the second step applies a robust phase unwrapping algorithm to the denoised modulo-2 pi phase obtained in the first step. The adaptive local modulo-2 pi phase denoising is a new algorithm based on local polynomial approximations. The zero-order and the first-order approximations of the phase are calculated in sliding windows of varying size. The zero-order approximation is used for pointwise adaptive window size selection, whereas the first-order approximation is used to filter the phase in the obtained windows. For phase unwrapping, we apply the recently introduced robust (in the sense of discontinuity preserving) PUMA unwrapping algorithm [IEEE Trans. Image Process.16, 698 (2007)] to the denoised wrapped phase. Simulations give evidence that the proposed algorithm yields state-of-the-art performance, enabling strong noise attenuation while preserving image details., ((c) 2008 Optical Society of America)

We present a simple and usable noise model for the raw-data of digital imaging sensors. This signal-dependent noise model, which gives the pointwise standard-deviation of the noise as a function of the expectation of the pixel raw-data output, is composed of a Poissonian part, modeling the photon sensing, and Gaussian part, for the remaining stationary disturbances in the output data. We further explicitly take into account the clipping of the data (over- and under-exposure), faithfully reproducing the nonlinear response of the sensor. We propose an algorithm for the fully automatic estimation of the model parameters given a single noisy image. Experiments with synthetic images and with real raw-data from various sensors prove the practical applicability of the method and the accuracy of the proposed model.

A discrete diffraction transform (DDT) is a novel discrete wavefield propagation model that is aliasing free for a pixelwise invariant object distribution. For this class of distribution, the model is precise and has no typical discretization effects because it corresponds to accurate calculation of the diffraction integral. A spatial light modulator (SLM) is a good example of a system where a pixelwise invariant distribution appears. Frequency domain regularized inverse algorithms are developed for reconstruction of the object wavefield distribution from the distribution given in the sensor plane. The efficiency of developed frequency domain algorithms is demonstrated by simulation.

The local polynomial approximation (LPA) is a nonparametric regression technique with pointwise estimation in a sliding window. We apply the LPA of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data. Using the intersection of confidence interval (HCI) algorithm, the window size is selected as adaptive pointwise varying. This adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense. For calculations, we use a Gauss-Newton recursive procedure initiated by the phase estimates obtained for the neighboring points. It enables tracking properties of the algorithm and its ability to go beyond the principal interval [-pi, pi] and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values. The algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state-of-the-art algorithms developed for noisy phase unwrap. The theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the HCI adaptation algorithm.

We propose a novel image denoising strategy based on an enhanced sparse representation in transform domain. The enhancement of the sparsity is achieved by grouping similar 2-D image fragments (e.g., blocks) into 3-D data arrays which we call "groups." Collaborative filtering is a special procedure developed to deal with these 3-D groups. We realize it using the three successive steps: 3-D transformation of a group, shrinkage of the transform spectrum, and inverse 3-D transformation. The result is a 3-D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and, at the same time, it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions. Because these blocks are overlapping, for each pixel, we obtain many different estimates which need to be combined. Aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy. A significant improvement is obtained by a specially developed collaborative Wiener filtering. An algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail; an extension to color-image denoising is also developed. The experimental results demonstrate that this computationally scalable algorithm achieves state-of-the-art denoising performance in terms of both peak signal-to-noise ratio and subjective visual quality.

The shape-adaptive discrete cosine transform ISA-DCT) transform can be computed on a support of arbitrary shape, but retains a computational complexity comparable to that of the usual separable block-DCT (B-DCT). Despite the near-optimal decorrelation and energy compaction properties, application of the SA-DCT has been rather limited, targeted nearly exclusively to video compression. In this paper, we present a novel approach to image filtering based on the SA-DCT. We use the SA-DCT in conjunction with the Anisotropic Local Polynomial Approximation-Intersection of Confidence Intervals technique, which defines the shape of the transform's support in a pointwise adaptive manner. The thresholded or attenuated SA-DCT coefficients are used to reconstruct a local estimate of the signal within the adaptive-shape support. Since supports corresponding to different points are in general overlapping, the local estimates are averaged together using adaptive weights that depend on the region's statistics. This approach can be used for various image-processing tasks. In this paper, we consider, in particular, image denoising and image deblocking and deringing from block-DCT compression. A special structural constraint in luminance-chrominance space is also proposed to enable an accurate filtering of color images. Simulation experiments show a state-of-the-art quality of the final estimate, both in terms of objective criteria and visual appearance. Thanks to the adaptive support, reconstructed edges are clean, and no unpleasant ringing artifacts are introduced by the fitted transform.

We propose a novel nonparametric regression metthod for deblurring noisy images. The method is based on the local polynomial approximation (LPA) of the image and the paradigm of intersecting confidence intervals (ICI) that is applied to define the adaptive varying scales (window sizes) of the LPA estimators. The LPA-ICI algorithm is nonlinear and spatially adaptive with respect to smoothness and irregularities of the image corrupted by additive noise. Multiresolution wavelet algorithms produce estimates which are combined from different scale projections. In contrast to them, the proposed ICI algorithm gives a varying scale adaptive estimate defining a single best scale for each pixel. In the new algorithm, the actual filtering is performed in signal domain while frequency domain Fourier transform operations are applied only for calculation of convolutions. The regularized inverse and Wiener inverse filters serve as deblurring operators used jointly with the LPA-design directional kernel filters. Experiments demonstrate the state-of-art performance of the new estimators which visually and quantitatively outperform some of the best existing methods.

Passive microwave remote sensing from satellites and ground stations has contributed uniquely, and substantially, to the study of atmospheric chemistry, meteorology, and environmental monitoring. As user requirements are raised, in terms of the accuracy and the spatial resolution, a mechanically scanning radiometer, with a real aperture, becomes impractical due to the requirement for a very large antenna size. However, an aperture synthesis interferometric radiometer presents a valuable alternative. The work presented in this paper was devoted to high spatial resolution imaging, using the 37 GHz band interferometric radiometer, developed by ourselves. The spatially adaptive Capon beamforming method was exploited for the imaging, which outperformed the conventional Fourier Transform method. We concluded that the high spatial resolution imaging of the brightness temperature of the atmosphere could be accomplished with an interferometric radiometer equipped with the developed Capon beamforming imaging algorithm.