Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Image Restoration Using Joint Statistical Modeling in Space-Transform Domain

This paper presents a novel strategy for high-fidelity image restoration by characterizing both local smoothness and nonlocal self-similarity of natural images in a unified statistical manner. The main contributions are three-folds. First, from the perspective of image statistics, a joint statistical modeling (JSM) in an adaptive hybrid space-transform domain is established, which offers a powerful mechanism of combining local smoothness and nonlocal self-similarity simultaneously to ensure a more reliable and robust estimation. Second, a new form of minimization functional for solving image inverse problem is formulated using JSM under regularization-based framework. Finally, in order to make JSM tractable and robust, a new Split-Bregman based algorithm is developed to efficiently solve the above severely underdetermined inverse problem associated with theoretical proof of convergence. Extensive experiments on image inpainting, image deblurring and mixed Gaussian plus salt-and-pepper noise removal applications verify the effectiveness of the proposed algorithm.Comment: 14 pages, 18 figures, 7 Tables, to be published in IEEE Transactions on Circuits System and Video Technology (TCSVT). High resolution pdf version and Code can be found at: http://idm.pku.edu.cn/staff/zhangjian/IRJSM

Generalized Inpainting Method for Hyperspectral Image Acquisition

A recently designed hyperspectral imaging device enables multiplexed acquisition of an entire data volume in a single snapshot thanks to monolithically-integrated spectral filters. Such an agile imaging technique comes at the cost of a reduced spatial resolution and the need for a demosaicing procedure on its interleaved data. In this work, we address both issues and propose an approach inspired by recent developments in compressed sensing and analysis sparse models. We formulate our superresolution and demosaicing task as a 3-D generalized inpainting problem. Interestingly, the target spatial resolution can be adjusted for mitigating the compression level of our sensing. The reconstruction procedure uses a fast greedy method called Pseudo-inverse IHT. We also show on simulations that a random arrangement of the spectral filters on the sensor is preferable to regular mosaic layout as it improves the quality of the reconstruction. The efficiency of our technique is demonstrated through numerical experiments on both synthetic and real data as acquired by the snapshot imager.Comment: Keywords: Hyperspectral, inpainting, iterative hard thresholding, sparse models, CMOS, Fabry-P\'ero

Development and implementation of efficient noise suppression methods for emission computed tomography

In PET and SPECT imaging, iterative reconstruction is now widely used due to its capability of incorporating into the reconstruction process a physics model and Bayesian statistics involved in photon detection. Iterative reconstruction methods rely on regularization terms to suppress image noise and render radiotracer distribution with good image quality. The choice of regularization method substantially affects the appearances of reconstructed images, and is thus a critical aspect of the reconstruction process. Major contributions of this work include implementation and evaluation of various new regularization methods. Previously, our group developed a preconditioned alternating projection algorithm (PAPA) to optimize the emission computed tomography (ECT) objective function with the non-differentiable total variation (TV) regularizer. The algorithm was modified to optimize the proposed reconstruction objective functions. First, two novel TV-based regularizers—high-order total variation (HOTV) and infimal convolution total variation (ICTV)—were proposed as alternative choices to the customary TV regularizer in SPECT reconstruction, to reduce “staircase” artifacts produced by TV. We have evaluated both proposed reconstruction methods (HOTV-PAPA and ICTV-PAPA), and compared them with the TV regularized reconstruction (TV-PAPA) and the clinical standard, Gaussian post-filtered, expectation-maximization reconstruction method (GPF-EM) using both Monte Carlo-simulated data and anonymized clinical data. Model-observer studies using Monte Carlo-simulated data indicate that ICTV-PAPA is able to reconstruct images with similar or better lesion detectability, compared with clinical standard GPF-EM methods, but at lower detected count levels. This implies that switching from GPF-EM to ICTV-PAPA can reduce patient dose while maintaining image quality for diagnostic use. Second, the 1 norm of discrete cosine transform (DCT)-induced framelet regularization was studied. We decomposed the image into high and low spatial-frequency components, and then preferentially penalized the high spatial-frequency components. The DCT-induced framelet transform of the natural radiotracer distribution image is sparse. By using this property, we were able to effectively suppress image noise without overly compromising spatial resolution or image contrast. Finally, the fractional norm of the first-order spatial gradient was introduced as a regularizer. We implemented 2/3 and 1/2 norms to suppress image spatial variability. Due to the strong penalty of small differences between neighboring pixels, fractional-norm regularizers suffer from similar cartoon-like artifacts as with the TV regularizer. However, when penalty weights are properly selected, fractional-norm regularizers outperform TV in terms of noise suppression and contrast recovery

Distributed Representation of Geometrically Correlated Images with Compressed Linear Measurements

This paper addresses the problem of distributed coding of images whose correlation is driven by the motion of objects or positioning of the vision sensors. It concentrates on the problem where images are encoded with compressed linear measurements. We propose a geometry-based correlation model in order to describe the common information in pairs of images. We assume that the constitutive components of natural images can be captured by visual features that undergo local transformations (e.g., translation) in different images. We first identify prominent visual features by computing a sparse approximation of a reference image with a dictionary of geometric basis functions. We then pose a regularized optimization problem to estimate the corresponding features in correlated images given by quantized linear measurements. The estimated features have to comply with the compressed information and to represent consistent transformation between images. The correlation model is given by the relative geometric transformations between corresponding features. We then propose an efficient joint decoding algorithm that estimates the compressed images such that they stay consistent with both the quantized measurements and the correlation model. Experimental results show that the proposed algorithm effectively estimates the correlation between images in multi-view datasets. In addition, the proposed algorithm provides effective decoding performance that compares advantageously to independent coding solutions as well as state-of-the-art distributed coding schemes based on disparity learning