Results 1  10
of
21
Performance guarantees of the thresholding algorithm for the cosparse analysis model
"... The cosparse analysis model for signals assumes that the signal of interest can be multiplied by an analysis dictionary Ω, leading to a sparse outcome. This model stands as an interesting alternative to the more classical synthesis based sparse representation model. In this work we propose a theore ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
The cosparse analysis model for signals assumes that the signal of interest can be multiplied by an analysis dictionary Ω, leading to a sparse outcome. This model stands as an interesting alternative to the more classical synthesis based sparse representation model. In this work we propose a theoretical study of the performance guarantee of the thresholding algorithm for the pursuit problem in the presence of noise. Our analysis reveals two significant properties of Ω, which govern the pursuit performance: The first is the degree of linear dependencies between sets of rows in Ω, depicted by the cosparsity level. The second property, termed the Restricted Orthogonal Projection Property (ROPP), is the level of independence between such dependent sets and other rows in Ω. We show how these dictionary properties are meaningful and useful, both in the theoretical bounds derived, and in a series of experiments that are shown to align well with the theoretical prediction.
Insights into analysis operator learning: from patchbased models to higherorder MRFs
 IEEE Transactions on Image Processing
, 2014
"... MRFs ..."
(Show Context)
M.: A joint intensity and depth cosparse analysis model for depth map superresolution
 In: Proceedings of the IEEE International Conference on Computer Vision, ICCV
, 2013
"... Highresolution depth maps can be inferred from lowresolution depth measurements and an additional highresolution intensity image of the same scene. To that end, we introduce a bimodal cosparse analysis model, which is able to capture the interdependency of registered intensity and depth informat ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Highresolution depth maps can be inferred from lowresolution depth measurements and an additional highresolution intensity image of the same scene. To that end, we introduce a bimodal cosparse analysis model, which is able to capture the interdependency of registered intensity and depth information. This model is based on the assumption that the cosupports of corresponding bimodal image structures are aligned when computed by a suitable pair of analysis operators. No analytic form of such operators exist and we propose a method for learning them from a set of registered training signals. This learning process is done offline and returns a bimodal analysis operator that is universally applicable to natural scenes. We use this to exploit the bimodal cosparse analysis model as a prior for solving inverse problems, which leads to an efficient algorithm for depth map superresolution. 1
Asymptotic Analysis of Inpainting via Universal Shearlet Systems
 SIAM J. Imaging Sci
"... ar ..."
(Show Context)
On MAP and MMSE Estimators for the Cosparse Analysis ModelI
"... The sparse synthesis model for signals has become very popular in the last decade, leading to improved performance in many signal processing applications. This model assumes that a signal may be described as a linear combination of few columns (atoms) of a given synthesis matrix (dictionary). The ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
The sparse synthesis model for signals has become very popular in the last decade, leading to improved performance in many signal processing applications. This model assumes that a signal may be described as a linear combination of few columns (atoms) of a given synthesis matrix (dictionary). The CoSparse Analysis model is a recently introduced counterpart, whereby signals are assumed to be orthogonal to many rows of a given analysis dictionary. These rows are called the cosupport. The Analysis model has already led to a series of contributions that address the pursuit problem: identifying the cosupport of a corrupted signal in order to restore it. While all the existing work adopts a deterministic point of view towards the design of such pursuit algorithms, this paper introduces a Bayesian estimation point of view, starting with a random generative model for the cosparse analysis signals. This is followed by a derivation of Oracle, MinimumMeanSquaredError (MMSE), and MaximumA’posterioriProbability (MAP) based estimators. We present a comparison between the deterministic formulations and these estimators, drawing some connections between the two. We develop practical approximations to the MAP and MMSE estimators, and demonstrate the proposed reconstruction algorithms in several synthetic and real image experiments, showing their potential and applicability.
Kplane clustering algorithm for analysis dictionary learning
 in IEEE International Workshop on Machine Learning for Signal Processing
, 2013
"... Analysis dictionary learning (ADL) aims to adapt dictionaries from training data based on an analysis sparse representation model. In a recent work, we have shown that, to obtain the analysis dictionary, one could optimise an objective function defined directly on the noisy signal, instead of on ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Analysis dictionary learning (ADL) aims to adapt dictionaries from training data based on an analysis sparse representation model. In a recent work, we have shown that, to obtain the analysis dictionary, one could optimise an objective function defined directly on the noisy signal, instead of on the estimated version of the clean signal as adopted in analysis KSVD. Following this strategy, a new ADL algorithm using Kplane clustering is proposed in this paper, which is based on the observation that, the observed data are coplaner in the analysis sparse model. In other words, the columns of the observed data form multidimensional subspaces (hyperplanes), and the rows of the analysis dictionary are the normal vectors of the hyperplanes. The normal directions of the Kdimensional concentration hyperplanes can be estimated using the Kplane clustering algorithm, and then the rows of the analysis dictionary which are the normal vectors of the hyperplanes can be obtained. Experiments on natural image denoising demonstrate that the Kplane clustering algorithm provides comparable performance to the baseline algorithms, i.e. the analysis KSVD and the subset pursuit based ADL. Index Terms — Kplane clustering; Analysis dictionary learning; Cosparse; Image denoising
OMP with Highly Coherent Dictionaries
"... Abstract—Recovering signals that has a sparse representation from a given set of linear measurements has been a major topic of research in recent years. Most of the work dealing with this subject focus on the reconstruction of the signal’s representation as the means to recover the signal itself. Th ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract—Recovering signals that has a sparse representation from a given set of linear measurements has been a major topic of research in recent years. Most of the work dealing with this subject focus on the reconstruction of the signal’s representation as the means to recover the signal itself. This approach forces the dictionary to be of lowcoherence and with no linear dependencies between its columns. Recently, a series of contributions show that such dependencies can be allowed by aiming at recovering the signal itself. However, most of these recent works consider the analysis framework, and only few discuss the synthesis model. This paper studies the synthesis and introduces a new mutual coherence definition for signal recovery, showing that a modified version of OMP can recover sparsely represented signals of a dictionary with very high correlations between pairs of columns. We show how the derived results apply to the plain OMP. I.
Gradient Histogram Estimation and Preservation for Texture Enhanced Image Denoising
, 2013
"... Natural image statistics plays an important role in image denoising, and various natural image priors, including gradient based, sparse representation based and nonlocal selfsimilarity based ones, have been widely studied and exploited for noise removal. In spite of the great success of many denois ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Natural image statistics plays an important role in image denoising, and various natural image priors, including gradient based, sparse representation based and nonlocal selfsimilarity based ones, have been widely studied and exploited for noise removal. In spite of the great success of many denoising algorithms, they tend to smooth the fine scale image textures when removing noise, degrading the image visual quality. To address this problem, in this paper we propose a texture enhanced image denoising method by enforcing the gradient histogram of the denoised image to be close to a reference gradient histogram of the original image. Given the reference gradient histogram, a novel gradient histogram preservation (GHP) algorithm is developed to enhance the texture structures while removing noise. Two regionbased variants of GHP are proposed for the denoising of images consisting of regions with different textures. An algorithm is also developed to effectively estimate the reference gradient histogram from the noisy observation of the unknown image. Our experimental results demonstrate that the proposed GHP algorithm can well preserve the texture appearance in the denoised images, making them look more natural.
CAN WE ALLOW LINEAR DEPENDENCIES IN THE DICTIONARY IN THE SPARSE SYNTHESIS FRAMEWORK?
"... Signal recovery from a given set of linear measurements using a sparsity prior has been a major subject of research in recent years. In this model, the signal is assumed to have a sparse representation under a given dictionary. Most of the work dealing with this subject has focused on the reconstruc ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Signal recovery from a given set of linear measurements using a sparsity prior has been a major subject of research in recent years. In this model, the signal is assumed to have a sparse representation under a given dictionary. Most of the work dealing with this subject has focused on the reconstruction of the signal’s representation as the means for recovering the signal itself. This approach forced the dictionary to be of low coherence and with no linear dependencies between its columns. Recently, a series of contributions that focus on signal recovery using the analysis model find that linear dependencies in the analysis dictionary are in fact permitted and beneficial. In this paper we show theoretically that the same holds also for signal recovery in the synthesis case for the ℓ0synthesis minimization problem. In addition, we demonstrate empirically the relevance of our conclusions for recovering the signal using an ℓ1relaxation. Index Terms — Sparse representations, compressed sensing, analysis versus synthesis, inverse problems.
Analysis SimCO: A new algorithm for analysis dictionary learning
 in Proc. Int. Conf. Acoust., Speech, and Signal Process
"... We consider the dictionary learning problem for the analysis model based sparse representation. A novel algorithm is proposed by adapting the synthesis model based simultaneous codeword optimisation (SimCO) algorithm to the analysis model. This algorithm assumes that the analysis dictionary cont ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We consider the dictionary learning problem for the analysis model based sparse representation. A novel algorithm is proposed by adapting the synthesis model based simultaneous codeword optimisation (SimCO) algorithm to the analysis model. This algorithm assumes that the analysis dictionary contains unit 2norm atoms and trains the dictionary by the optimisation on manifolds. This framework allows one to update multiple dictionary atoms in each iteration, leading to a computationally efficient optimisation process. We demonstrate the competitive performance of the proposed algorithm using experiments on both synthetic and real data, as compared with three baseline algorithms, Analysis KSVD, analysis operator learning (AOL) and learning overcomplete sparsifying transforms (LOST), respectively. Index Terms — Analysis model, SimCO, analysis dictionary learning.