We investigate the problem of reconstructing signals from a subsampled convolution of their modulated versions and a known filter. The problem is studied as applies to a specific imaging architecture that relies on spatial phase modulation by randomly coded “masks.” The diversity induced by the random masks is deemed to improve the conditioning of the deconvolution problem while maintaining sampling efficiency. We analyze a linear model of the imaging system, where the joint effect of the spatial modulation, blurring, and spatial subsampling is represented concisely by a measurement matrix. We provide a bound on the conditioning of this measurement matrix in terms of the number of masks @math , the dimension (i.e., the pixel count) of the scene image @math , and certain characteristics of the blurring kernel and subsampling operator. The derived bound shows that the stable deconvolution is possible with high probability even if the number of masks (i.e., @math ) is as small as @math , meaning that the total number of (scalar) measurements is within a logarithmic factor of the image size. Furthermore, beyond a critical number of masks determined by the extent of blurring and subsampling, use of every additional mask improves the conditioning of the measurement matrix. We also consider a more interesting scenario where the target image is known to be sparse. We show that under mild conditions on the blurring kernel, with high probability the measurement matrix is a restricted isometry when the number of masks is within a logarithmic factor of the sparsity of the scene image. Therefore, the scene image can be reconstructed using any of the well-known sparse recovery algorithms such as the basis pursuit. The bound on the required number of masks grows linearly in sparsity of the scene image but logarithmically in its ambient dimension. The bound provides a quantitative view of the effect of the blurring and subsampling on the required number of masks, which is critical for designing efficient imaging systems.
Classical deconvolution techniques can be broadly categorized in two frameworks based on their approaches to regularization of the inverse problem. Methods of the first category, including Wiener filtering and a variety of Bayesian methods, assume some stochastic model for the image or the blurring kernel that is often application specific. Methods of the second category, that are essentially some variants of the least squares, only use the deterministic spatial or spectral structures of the image such as smoothness for regularization. For a comprehensive survey of classic deconvolution methods for image restoration and reconstruction we refer the interested readers to @cite_1 and @cite_4 .
@cite_1 @cite_4
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Non-blind image deconvolution is an ill-posed problem. The presence of noise and band-limited blur kernels makes the solution of this problem non-unique. Existing deconvolution techniques produce a residual between the sharp image and the estimation that is highly correlated with the sharp image, the kernel, and the noise. In most cases, different restoration models must be constructed for different blur kernels and different levels of noise, resulting in low computational efficiency or highly redundant model parameters. Here we aim to develop a single model that handles different types of kernels and different levels of noise: general non-blind deconvolution. Specifically, we propose a very deep convolutional neural network that predicts the residual between a pre-deconvolved image and the sharp image rather than the sharp image. The residual learning strategy makes it easier to train a single model for different kernels and different levels of noise, encouraging high effectiveness and efficiency. Quantitative evaluations demonstrate the practical applicability of the proposed model for different blur kernels. The model also shows the state-of-the-art performance on synthesized blurry images. Image deconvolution is the ill-posed problem of recovering a sharp image, given a blurry one generated by a convolution. In this work, we deal with space-invariant non-blind deconvolution. Currently, the most successful methods involve a regularized inversion of the blur in Fourier domain as a first step. This step amplifies and colors the noise, and corrupts the image information. In a second (and arguably more difficult) step, one then needs to remove the colored noise, typically using a cleverly engineered algorithm. However, the methods based on this two-step approach do not properly address the fact that the image information has been corrupted. In this work, we also rely on a two-step procedure, but learn the second step on a large dataset of natural images, using a neural network. We will show that this approach outperforms the current state-of-the-art on a large dataset of artificially blurred images. We demonstrate the practical applicability of our method in a real-world example with photographic out-of-focus blur. Many fundamental image-related problems involve deconvolution operators. Real blur degradation seldom complies with an ideal linear convolution model due to camera noise, saturation, image compression, to name a few. Instead of perfectly modeling outliers, which is rather challenging from a generative model perspective, we develop a deep convolutional neural network to capture the characteristics of degradation. We note directly applying existing deep neural networks does not produce reasonable results. Our solution is to establish the connection between traditional optimization-based schemes and a neural network architecture where a novel, separable structure is introduced as a reliable support for robust deconvolution against artifacts. Our network contains two submodules, both trained in a supervised manner with proper initialization. They yield decent performance on non-blind image deconvolution compared to previous generative-model based methods. Statistical image reconstruction using penalized weighted least-squares (PWLS) criteria can improve image-quality in X-ray computed tomography (CT). However, the huge dynamic range of the statistical weights leads to a highly shift-variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly. We propose to alleviate the problem by using a variable-splitting scheme that separates the shift-variant and ("nearly") invariant components of the statistical data model and also decouples the regularization term. This leads to an equivalent constrained problem that we tackle using the classical method-of-multipliers framework with alternating minimization. The specific form of our splitting yields an alternating direction method of multipliers (ADMM) algorithm with an inner-step involving a "nearly" shift-invariant linear system that is suitable for FFT-based preconditioning using cone-type filters. The proposed method can efficiently handle a variety of convex regularization criteria including smooth edge-preserving regularizers and non- smooth sparsity-promoting ones based on the l1-norm and total variation. Numerical experiments with synthetic and real in vivo human data illustrate that cone-filter preconditioners accelerate the proposed ADMM resulting in fast convergence of ADMM compared to conventional (nonlinear conjugate gradient, ordered subsets) and state-of-the-art (MFISTA, split-Bregman) algorithms that are applicable for CT.