Approximation and compression with sparse orthonormal transforms.

We propose a new transform design method that targets the generation of compression-optimized transforms for next-generation multimedia applications. The fundamental idea behind transform compression is to exploit regularity within signals such that redundancy is minimized subject to a fidelity cost. Multimedia signals, in particular images and video, are well known to contain a diverse set of localized structures, leading to many different types of regularity and to nonstationary signal statistics.

Spatial sparsity-induced prediction (SIP) for images and video: a simple way to reject structured interference.

We propose a prediction technique that is geared toward forming successful estimates of a signal based on a correlated anchor signal that is contaminated with complex interference. The corruption in the anchor signal involves intensity modulations, linear distortions, structured interference, clutter, and noise just to name a few. The proposed setup reflects nontrivial prediction scenarios involving images and video frames where statistically related data is rendered ineffective for traditional methods due to cross-fades, blends, clutter, brightness variations, focus changes, and other complex transitions.

Weighted averaging for denoising with overcomplete dictionaries.

We consider the scenario where additive, independent, and identically distributed (i.i.d) noise in an image is removed using an overcomplete set of linear transforms and thresholding.

Linear, worst-case estimators for denoising quantization noise in transform coded images.

Transform-coded images exhibit distortions that fall outside of the assumptions of traditional denoising techniques. In this paper, we use tools from robust signal processing to construct linear, worst-case estimators for the denoising of transform compressed images. We show that while standard denoising is fundamentally determined by statistical models for images alone, the distortions induced by transform coding are heavily dependent on the structure of the transform used.

Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising--Part II: Adaptive algorithms.

We combine the main ideas introduced in Part I with adaptive techniques to arrive at a powerful algorithm that estimates missing data in nonstationary signals. The proposed approach operates automatically based on a chosen linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero. Unlike prevalent algorithms, our method does not necessitate any complex preconditioning, segmentation, or edge detection steps, and it can be written as a progression of denoising operations.

Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising--Part I: Theory.

We study the robust estimation of missing regions in images and video using adaptive, sparse reconstructions. Our primary application is on missing regions of pixels containing textures, edges, and other image features that are not readily handled by prevalent estimation and recovery algorithms. We assume that we are given a linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero.