We introduce a method to infer a variational approximation to the posterior distribution of solutions in computational imaging inverse problems. Machine learning methods applied to computational imaging have proven very successful, but have so far largely focused on retrieving a single optimal solution for a given task. Such retrieval is arguably an incomplete description of the solution space, as in ill-posed inverse problems there may be many similarly likely reconstructions. We minimise an upper bound on the divergence between our approximate distribution and the true intractable posterior, thereby obtaining a probabilistic description of the solution space in imaging inverse problems with empirical prior. We demonstrate the advantage of our technique in quantitative simulations with the CelebA dataset and common image reconstruction tasks. We then apply our method to two of the currently most challenging problems in experimental optics: imaging through highly scattering media and imaging through multi-modal optical fibres. In both settings we report state of the art reconstructions, while providing new capabilities, such as estimation of error-bars and visualisation of multiple likely reconstructions.
However, despite its theoretical benefits and wide adoption, sparsity regularisation is often too generic to accurately model the desired assumptions in a given imaging setting @cite_37 .
@cite_37
2018990310 2803204576 2061033783 2045079989
Sparsity has been widely exploited for exact reconstruction of a signal from a small number of random measurements. Recent advances have suggested that structured or group sparsity often leads to more powerful signal reconstruction techniques in various compressed sensing (CS) studies. In this paper, we propose a nonlocal low-rank regularization (NLR) approach toward exploiting structured sparsity and explore its application into CS of both photographic and MRI images. We also propose the use of a nonconvex log det ( X) as a smooth surrogate function for the rank instead of the convex nuclear norm and justify the benefit of such a strategy using extensive experiments. To further improve the computational efficiency of the proposed algorithm, we have developed a fast implementation using the alternative direction multiplier method technique. Experimental results have shown that the proposed NLR-CS algorithm can significantly outperform existing state-of-the-art CS techniques for image recovery. Two important elements have driven recent innovation in the field of regression: sparsity-inducing regularization, to cope with high-dimensional problems; multi-task learning through joint parameter estimation, to augment the number of training samples. Both approaches complement each other in the sense that a joint estimation results in more samples, which are needed to estimate sparse models accurately, whereas sparsity promotes models that act on subsets of related variables. This idea has driven the proposal of block regularizers such as L1 Lq norms, which however effective, require that active regressors strictly overlap. In this paper, we propose a more flexible convex regularizer based on unbalanced optimal transport (OT) theory. That regularizer promotes parameters that are close, according to the OT geometry, which takes into account a prior geometric knowledge on the regressor variables. We derive an efficient algorithm based on a regularized formulation of optimal transport, which iterates through applications of Sinkhorn's algorithm along with coordinate descent iterations. The performance of our model is demonstrated on regular grids and complex triangulated geometries of the cortex with an application in neuroimaging. Iterative image reconstruction with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is nontrivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified. Where does the sparsity in image signals come from? Local and nonlocal image models have supplied complementary views toward the regularity in natural images — the former attempts to construct or learn a dictionary of basis functions that promotes the sparsity; while the latter connects the sparsity with the self-similarity of the image source by clustering. In this paper, we present a variational framework for unifying the above two views and propose a new denoising algorithm built upon clustering-based sparse representation (CSR). Inspired by the success of l 1 -optimization, we have formulated a double-header l 1 -optimization problem where the regularization involves both dictionary learning and structural structuring. A surrogate-function based iterative shrinkage solution has been developed to solve the double-header l 1 -optimization problem and a probabilistic interpretation of CSR model is also included. Our experimental results have shown convincing improvements over state-of-the-art denoising technique BM3D on the class of regular texture images. The PSNR performance of CSR denoising is at least comparable and often superior to other competing schemes including BM3D on a collection of 12 generic natural images.