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A hybrid imaging system is a simultaneous physical arrangement of a refractive lens and a multilevel phase mask (MPM) as a diffractive optical element (DOE). The favorable properties of the hybrid setup are improved extended-depth-of-field (EDoF) imaging and low chromatic aberrations. We built a fully differentiable image formation model in order to use neural network techniques to optimize imaging. At the first stage, the design framework relies on the model-based approach with numerical simulation and end-to-end joint optimization of both MPM and imaging algorithms. In the second stage, MPM is fixed as found at the first stage, and the image processing is optimized experimentally using the CNN learning-based approach with MPM implemented by a spatial light modulator. The paper is concentrated on a comparative analysis of imaging accuracy and quality for design with various basic optical parameters: aperture size, lens focal length, and distance between MPM and sensor. We point out that the varying aperture size, lens focal length, and distance between MPM and sensor are for the first time considered for end-to-end optimization of EDoF. We numerically and experimentally compare the designs for visible wavelength interval [400-700]nm and the following EDoF ranges: [0.5-100]m for simulations and [0.5-1.9]m for experimental tests. This study concerns an application of hybrid optics for compact cameras with aperture [5-9] mm and distance between MPM and sensor [3-10]mm., Comment: 16 pages, 11 figures, 1 table

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 artifacts 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, the broadband imaging system with phase-only spatial light modulator (SLM) as an encoded DOE is proposed and developed in this paper. A co-design of the SLM phase pattern and image reconstruction algorithm is produced following the end-to-end strategy, using for optimization a convolutional neural network equipped with quantitative and qualitative loss functions. The optics of the imaging system is hybrid consisting of 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 modeling and physical reality of image formation as light propagation is not numerically modeled but is physically done. In our experiments, the hybrid optics is implemented by the optical projection of the SLM phase-pattern on a lens plane for a depth range 0.4-1.9m. Comparison with compound multi-lens optics such as Sony A7 III and iPhone Xs Max cameras show that the proposed system is advanced in all-in-focus sharp imaging., Comment: 9 pages, 7 figures

Phase retrieval in optical imaging refers to the recovery of a complex signal from phaseless data acquired in the form of its diffraction patterns. These patterns are acquired through a system with a coherent light source that employs a diffractive optical element (DOE) to modulate the scene resulting in coded diffraction patterns at the sensor. Recently, the hybrid approach of model-driven network or deep unfolding has emerged as an effective alternative to conventional model-based and learning-based phase retrieval techniques because it allows for bounding the complexity of algorithms while also retaining their efficacy. Additionally, such hybrid approaches have shown promise in improving the design of DOEs that follow theoretical uniqueness conditions. There are opportunities to exploit novel experimental setups and resolve even more complex DOE phase retrieval applications. This paper presents an overview of algorithms and applications of deep unfolding for bootstrapped - regardless of near, middle, and far zones - phase retrieval., Comment: 13 pages, 11 figures, 1 table

We propose a novel approach and algorithm based on two preliminary tests of the optical system elements to enhance the super-resolved complex-valued imaging. The approach is developed for inverse phase imaging in a single-shot lensless optical setup. Imaging is based on wavefront modulation by a single binary phase mask. The preliminary tests compensate errors in the optical system and correct a carrying wavefront, reducing the gap between real-life experiments and computational modeling, which improve imaging significantly both qualitatively and quantitatively. These two tests are performed for observation of the laser beam and phase mask along, and might be considered as a preliminary system calibration. The corrected carrying wavefront is embedded into the proposed iterative Single-shot Super-Resolution Phase Retrieval (SSR-PR) algorithm. Improved initial diffraction pattern upsampling, and a combination of sparse and deep learning based filters achieves the super-resolved reconstructions. Simulations and physical experiments demonstrate the high-quality super-resolution phase imaging. In the simulations, we showed that the SSR-PR algorithm corrects the errors of the proposed optical system and reconstructs phase details 4x smaller than the sensor pixel size. In physical experiment 2um thick lines of USAF phase-target were resolved, which is almost 2x smaller than the sensor pixel size and corresponds to the smallest resolvable group of used test target. For phase bio-imaging, we provide Buccal Epithelial Cells reconstructed in computational super-resolution and the quality was of the same level as a digital holographic system with 40x magnification objective. Furthermore, the single-shot advantage provides the possibility to record dynamic scenes, where the framerate is limited only by the used camera. We provide amplitude-phase video clip of a moving alive single-celled eukaryote.

A novel formulation of the hyperspectral broadband phase retrieval is developed for the scenario where both object and modulation phase masks are spectrally varying. The proposed algorithm is based on a complex domain version of the alternating direction method of multipliers (ADMM) and Spectral Proximity Operators (SPO) derived for Gaussian and Poissonian observations. Computations for these operators are reduced to the solution of sets of cubic (for Gaussian) and quadratic (for Poissonian) algebraic equations. These proximity operators resolve two problems. Firstly, the complex domain spectral components of signals are extracted from the total intensity observations calculated as sums of the signal spectral intensities. In this way, the spectral analysis of the total intensities is achieved. Secondly, the noisy observations are filtered, compromising noisy intensity observations and their predicted counterparts. The ability to resolve the hyperspectral broadband phase retrieval problem and to find the spectrum varying object are essentially defined by the spectral properties of object and image formation operators. The simulation tests demonstrate that the phase retrieval in this formulation can be successfully resolved.

The power-balanced hybrid optical imaging system is a special design of 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 unique features of our MPM design are the inclusion of quantization of the MPM's shape on the number of levels and the Fresnel order (thickness) using a smoothing function. To optimize optical power-balance as well as the MPM, we build a fully-differentiable image formation model for joint optimization of optical and imaging parameters for the proposed camera using Neural Network techniques. Additionally, we optimize 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-$\infty$m for numerical and 0.5-2m for experimental). 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 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 5dB of PSNR using the optimized OTF compared with its counterpart lensless setup., Comment: 18 pages, 14 figures

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, additionally beyond spectral intensity distribution, reconstruction of the investigated object's phase. The noise amplification Fellgett's disadvantage is relaxed by the application of sparse wavefront noise filtering embedded in the proposed algorithm. The algorithm reliability is proved by simulation tests and results of physical experiments on transparent objects which demonstrate precise phase imaging and object depth (profile) reconstructions., Comment: 12 pages, 8 figures

Broadband hyperspectral digital holography and Fourier transform spectroscopy are important instruments in various science and application fields. In the digital hyperspectral holography and spectroscopy the variable of interest are obtained as inverse discrete cosine transforms of observed diffractive intensity patterns. In these notes, we provide a variety of algorithms for the inverse cosine transform with the proofs of perfect spectrum reconstruction, as well as we discuss and illustrate some nontrivial features of these algorithms., Comment: 20 pages, 9 figures

A new denoising algorithm for hyperspectral complex domain data has been developed and studied. This algorithm is based on the complex domain block-matching 3D filter including the 3D Wiener filtering stage. The developed algorithm is applied and tuned to work in the singular value decomposition (SVD) eigenspace of reduced dimension. The accuracy and quantitative advantage of the new algorithm are demonstrated in simulation tests and in the processing of the experimental data. It is shown that the algorithm is effective and provides reliable results even for highly noisy data.

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.

We introduce a paradigm for nonlocal sparsity reinforced deep convolutional neural network denoising. It is a combination of a local multiscale denoising by a convolutional neural network (CNN) based denoiser and a nonlocal denoising based on a nonlocal filter (NLF) exploiting the mutual similarities between groups of patches. CNN models are leveraged with noise levels that progressively decrease at every iteration of our framework, while their output is regularized by a nonlocal prior implicit within the NLF. Unlike complicated neural networks that embed the nonlocality prior within the layers of the network, our framework is modular, it uses standard pre-trained CNNs together with standard nonlocal filters. An instance of the proposed framework, called NN3D, is evaluated over large grayscale image datasets showing state-of-the-art performance., Comment: Accepted for publication in IEEE SPL

The phase retrieval from multi-frequency intensity (power) observations is considered. The object to be reconstructed is complex-valued. A novel algorithm is presented that accomplishes both the object phase (absolute phase) retrieval and denoising for Poissonian and Gaussian measurements. The algorithm is derived from the maximum likelihood formulation with Block Matching 3D (BM3D) sparsity priors. These priors result in two filtering: one is in the complex domain for complex-valued multi-frequency object images and another one in the real domain for the object phase. The algorithm is iterative with alternating projections between the object and measurement variables. The simulation experiments are produced for Fourier transform image formation and random phase modulations of the object, then the observations are random object diffraction patterns. The results demonstrate the success of the algorithm for reconstruction of the complex phase objects with the high-accuracy performance even for very noisy data., Comment: 5 pages, 3 figures

Phase imaging and wavefront reconstruction from noisy observations of complex exponent is a topic of this paper. It is a highly non-linear problem because the exponent is a 2{\pi}-periodic function of phase. The reconstruction of phase and amplitude is difficult. Even with an additive Gaussian noise in observations distributions of noisy components in phase and amplitude are signal dependent and non-Gaussian. Additional difficulties follow from a prior unknown correlation of phase and amplitude in real life scenarios. In this paper, we propose a new class of non-iterative and iterative complex domain filters based on group-wise sparsity in complex domain. This sparsity is based on the techniques implemented in Block-Matching 3D filtering (BM3D) and 3D/4D High-Order Singular Decomposition (HOSVD) exploited for spectrum design, analysis and filtering. The introduced algorithms are a generalization of the ideas used in the CD-BM3D algorithms presented in our previous publications. The algorithms are implemented as a MATLAB Toolbox. The efficiency of the algorithms is demonstrated by simulation tests., Comment: Submitted to Signal Processing

A variational approach to reconstruction of phase and amplitude of a complex-valued object from Poissonian intensity observations is developed. The observation model corresponds to the typical optical setups with a phase modulation of wavefronts. The transform domain sparsity is applied for the amplitude and phase modeling. It is demonstrated that this modeling results in the essential advantage of the developed algorithm for heavily noisy observations corresponding to a short exposure time in optical experiments. We consider also two simplified versions of this algorithm where the sparsity modeling of phase and amplitude is omitted. In the simulation study we compare the developed algorithms versus the Gerchberg-Saxton and truncation Wirtinger flow algorithms. The latter algorithm being the maximum likelihood based is the state-of-the-art for the phase retrieval from Poissonian observations. For noisy and very noisy observations the proposed algorithm demonstrates a valuable advantage., Comment: 16 pages, 19 figures

This paper introduces a two-and multi-level phase mask design for improved depth of focus. A novel technique is proposed incorporating cubic and generalized cubic wavefront coding (WFC). The obtained system is optical-electronic requiring computational deblurring post-processing, in order to obtain a sharp image from the observed blurred data. A midwave infrared (MWIR) system is simulated showing that this design will produce high quality images even for large amounts of defocus. It is furthermore shown that this technique can be used to design a flat, single optical element, systems where the phase mask performs both the function of focusing and phase modulation. It is demonstrated that in this lensless design the WFC coding components can be omitted and WFC effects are achieved as a result of the proposed algorithm for phase mask design which uses the quadratic phase of the thin refractive lens as the input signal., Comment: 15 pages, 12 Figures

Single image super resolution (SISR) is an ill-posed problem aiming at estimating a plausible high resolution (HR) image from a single low resolution (LR) 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 a HR image. External data based methods utilize 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 collaborative filtering of patch groups in 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 datasets. Without using any external data, the proposed approach outperforms the current non-CNN based methods on the tested datasets for various scaling factors. On certain datasets, the gain is over 1 dB, when compared to 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., Comment: Paper accepted for publication on IEEE Transactions on Image Processing

Generally, wave field reconstructions obtained by phase-retrieval algorithms are noisy, blurred and corrupted by various artifacts such as irregular waves, spots, etc. These disturbances, arising due to many factors such as non-idealities of optical system (misalignment, focusing errors), dust on optical elements, reflections, vibration, are hard to be localized and specified. It is assumed that there is a generalized pupil function at the object plane which describes aberrations in the coherent imaging system manifested at the sensor plane. Here we propose a novel two steps phase-retrieval algorithm to compensate these distortions. We first estimate the cumulative disturbance, called background, using special calibration experiments. Then, we use this background for reconstruction of the object amplitude and phase. The second part of the algorithm is based on the maximum likelihood approach and, in this way, targeted on the optimal amplitude and phase reconstruction from noisy data. Numerical experiments demonstrate that the developed algorithm enables the compensation of various typical distortions of the optical track so sharp object imaging for a binary test-chart can be achieved., Comment: to be published in Applied Optics OCIS: 030.4280, 050.1960, 070.2025, 070.6120, 100.3010, 100.3190, 100.5070

Sparse modeling is one of the efficient techniques for imaging that allows recovering lost information. In this paper, we present a novel iterative phase-retrieval algorithm using a sparse representation of the object amplitude and phase. The algorithm is derived in terms of a constrained maximum likelihood, where the wave field reconstruction is performed using a number of noisy intensity-only observations with a zero-mean additive Gaussian noise. The developed algorithm enables the optimal solution for the object wave field reconstruction. Our goal is an improvement of the reconstruction quality with respect to the conventional algorithms. Sparse regularization results in advanced reconstruction accuracy, and numerical simulations demonstrate significant enhancement of imaging., Comment: Submitted to the 10th IMEKO Symposium LMPMI (Laser Metrology for Precision Measurement and Inspection in Industry) on May 31, 2011

A family of the Block Matching 3-D (BM3D) algorithms for various imaging problems has been recently proposed within the framework of nonlocal patch-wise image modeling [1], [2]. In this paper we construct analysis and synthesis frames, formalizing the BM3D image modeling and use these frames to develop novel iterative deblurring algorithms. We consider two different formulations of the deblurring problem: one given by minimization of the single objective function and another based on the Nash equilibrium balance of two objective functions. The latter results in an algorithm where the denoising and deblurring operations are decoupled. The convergence of the developed algorithms is proved. Simulation experiments show that the decoupled algorithm derived from the Nash equilibrium 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., Comment: Submitted to IEEE Transactions on Image Processing on May 18, 2011. implementation of the proposed algorithm is available as part of the BM3D package at http://www.cs.tut.fi/~foi/GCF-BM3D