The Practicality of Stochastic Optimization in Imaging Inverse Problems

In this work we investigate the practicality of stochastic gradient descent and recently introduced variants with variance-reduction techniques in imaging inverse problems. Such algorithms have been shown in the machine learning literature to have optimal complexities in theory, and provide great improvement empirically over the deterministic gradient methods. Surprisingly, in some tasks such as image deblurring, many of such methods fail to converge faster than the accelerated deterministic gradient methods, even in terms of epoch counts. We investigate this phenomenon and propose a theory-inspired mechanism for the practitioners to efficiently characterize whether it is beneficial for an inverse problem to be solved by stochastic optimization techniques or not. Using standard tools in numerical linear algebra, we derive conditions on the spectral structure of the inverse problem for being a suitable application of stochastic gradient methods. Particularly, we show that, for an imaging inverse problem, if and only if its Hessain matrix has a fast-decaying eigenspectrum, then the stochastic gradient methods can be more advantageous than deterministic methods for solving such a problem. Our results also provide guidance on choosing appropriately the partition minibatch schemes, showing that a good minibatch scheme typically has relatively low correlation within each of the minibatches. Finally, we propose an accelerated primal-dual SGD algorithm in order to tackle another key bottleneck of stochastic optimization which is the heavy computation of proximal operators. The proposed method has fast convergence rate in practice, and is able to efficiently handle non-smooth regularization terms which are coupled with linear operators.

Hyperspectral holography and spectroscopy: computational features of inverse discrete cosine transform

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.

Nonlocality-Reinforced Convolutional Neural Networks for Image Denoising

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.

Single Image Super-Resolution based on Wiener Filter in Similarity Domain

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.

Blind estimation of white Gaussian noise variance in highly textured images

In the paper, a new method of blind estimation of noise variance in a single highly textured image is proposed. An input image is divided into 8x8 blocks and discrete cosine transform (DCT) is performed for each block. A part of 64 DCT coefficients with lowest energy calculated through all blocks is selected for further analysis. For the DCT coefficients, a robust estimate of noise variance is calculated. Corresponding to the obtained estimate, a part of blocks having very large values of local variance calculated only for the selected DCT coefficients are excluded from the further analysis. These two steps (estimation of noise variance and exclusion of blocks) are iteratively repeated three times. For the verification of the proposed method, a new noise-free test image database TAMPERE17 consisting of many highly textured images is designed. It is shown for this database and different values of noise variance from the set {25, 49, 100, 225}, that the proposed method provides approximately two times lower estimation root mean square error than other methods.

Statistical evaluation of visual quality metrics for image denoising

This paper studies the problem of full reference visual quality assessment of denoised images with a special emphasis on images with low contrast and noise-like texture. Denoising of such images together with noise removal often results in image details loss or smoothing. A new test image database, FLT, containing 75 noise-free "reference" images and 300 filtered ("distorted") images is developed. Each reference image, corrupted by an additive white Gaussian noise, is denoised by the BM3D filter with four different values of threshold parameter (four levels of noise suppression). After carrying out a perceptual quality assessment of distorted images, the mean opinion scores (MOS) are obtained and compared with the values of known full reference quality metrics. As a result, the Spearman Rank Order Correlation Coefficient (SROCC) between PSNR values and MOS has a value close to zero, and SROCC between values of known full-reference image visual quality metrics and MOS does not exceed 0.82 (which is reached by a new visual quality metric proposed in this paper). The FLT dataset is more complex than earlier datasets used for assessment of visual quality for image denoising. Thus, it can be effectively used to design new image visual quality metrics for image denoising.

Complex-valued image denosing based on group-wise complex-domain sparsity

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.