67 research outputs found
On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
Among the many ways to model signals, a recent approach that draws
considerable attention is sparse representation modeling. In this model, the
signal is assumed to be generated as a random linear combination of a few atoms
from a pre-specified dictionary. In this work we analyze two Bayesian denoising
algorithms -- the Maximum-Aposteriori Probability (MAP) and the
Minimum-Mean-Squared-Error (MMSE) estimators, under the assumption that the
dictionary is unitary. It is well known that both these estimators lead to a
scalar shrinkage on the transformed coefficients, albeit with a different
response curve. In this work we start by deriving closed-form expressions for
these shrinkage curves and then analyze their performance. Upper bounds on the
MAP and the MMSE estimation errors are derived. We tie these to the error
obtained by a so-called oracle estimator, where the support is given,
establishing a worst-case gain-factor between the MAP/MMSE estimation errors
and the oracle's performance. These denoising algorithms are demonstrated on
synthetic signals and on true data (images).Comment: 29 pages, 10 figure
Power-Constrained Sparse Gaussian Linear Dimensionality Reduction over Noisy Channels
In this paper, we investigate power-constrained sensing matrix design in a
sparse Gaussian linear dimensionality reduction framework. Our study is carried
out in a single--terminal setup as well as in a multi--terminal setup
consisting of orthogonal or coherent multiple access channels (MAC). We adopt
the mean square error (MSE) performance criterion for sparse source
reconstruction in a system where source-to-sensor channel(s) and
sensor-to-decoder communication channel(s) are noisy. Our proposed sensing
matrix design procedure relies upon minimizing a lower-bound on the MSE in
single-- and multiple--terminal setups. We propose a three-stage sensing matrix
optimization scheme that combines semi-definite relaxation (SDR) programming, a
low-rank approximation problem and power-rescaling. Under certain conditions,
we derive closed-form solutions to the proposed optimization procedure. Through
numerical experiments, by applying practical sparse reconstruction algorithms,
we show the superiority of the proposed scheme by comparing it with other
relevant methods. This performance improvement is achieved at the price of
higher computational complexity. Hence, in order to address the complexity
burden, we present an equivalent stochastic optimization method to the problem
of interest that can be solved approximately, while still providing a superior
performance over the popular methods.Comment: Accepted for publication in IEEE Transactions on Signal Processing
(16 pages
Image Deblurring and Super-resolution by Adaptive Sparse Domain Selection and Adaptive Regularization
As a powerful statistical image modeling technique, sparse representation has
been successfully used in various image restoration applications. The success
of sparse representation owes to the development of l1-norm optimization
techniques, and the fact that natural images are intrinsically sparse in some
domain. The image restoration quality largely depends on whether the employed
sparse domain can represent well the underlying image. Considering that the
contents can vary significantly across different images or different patches in
a single image, we propose to learn various sets of bases from a pre-collected
dataset of example image patches, and then for a given patch to be processed,
one set of bases are adaptively selected to characterize the local sparse
domain. We further introduce two adaptive regularization terms into the sparse
representation framework. First, a set of autoregressive (AR) models are
learned from the dataset of example image patches. The best fitted AR models to
a given patch are adaptively selected to regularize the image local structures.
Second, the image non-local self-similarity is introduced as another
regularization term. In addition, the sparsity regularization parameter is
adaptively estimated for better image restoration performance. Extensive
experiments on image deblurring and super-resolution validate that by using
adaptive sparse domain selection and adaptive regularization, the proposed
method achieves much better results than many state-of-the-art algorithms in
terms of both PSNR and visual perception.Comment: 35 pages. This paper is under review in IEEE TI
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
A Monte Carlo framework for noise removal and missing wedge restoration in cryo-electron tomography
In this paper, we describe a statistical method to address an important issue in cryo-electron tomography image analysis: reduction of a high amount of noise and artifacts due to the presence of a missing wedge (MW) in the spectral domain. The method takes as an input a 3D tomo-gram derived from limited-angle tomography, and gives as an output a 3D denoised and artifact compensated volume. The artifact compensation is achieved by filling up the MW with meaningful information. To address this inverse problem, we compute a Minimum Mean Square Error (MMSE) estimator of the uncorrupted image. The underlying high-dimensional integral is computed by applying a dedicated Markov Chain Monte-Carlo (MCMC) sampling procedure based on the Metropolis-Hasting (MH) algorithm. The proposed computational method can be used to enhance visualization or as a pre-processing step for image analysis, including segmentation and classification of macromolecules. Results are presented for both synthetic data and real 3D cryo-electron images
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