283 research outputs found
A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems
Non-Local Total Variation (NLTV) has emerged as a useful tool in variational
methods for image recovery problems. In this paper, we extend the NLTV-based
regularization to multicomponent images by taking advantage of the Structure
Tensor (ST) resulting from the gradient of a multicomponent image. The proposed
approach allows us to penalize the non-local variations, jointly for the
different components, through various matrix norms with .
To facilitate the choice of the hyper-parameters, we adopt a constrained convex
optimization approach in which we minimize the data fidelity term subject to a
constraint involving the ST-NLTV regularization. The resulting convex
optimization problem is solved with a novel epigraphical projection method.
This formulation can be efficiently implemented thanks to the flexibility
offered by recent primal-dual proximal algorithms. Experiments are carried out
for multispectral and hyperspectral images. The results demonstrate the
interest of introducing a non-local structure tensor regularization and show
that the proposed approach leads to significant improvements in terms of
convergence speed over current state-of-the-art methods
Epigraphical Projection and Proximal Tools for Solving Constrained Convex Optimization Problems: Part I
We propose a proximal approach to deal with convex optimization problems involving nonlinear constraints. A large family of such constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For this class of constraints, the associated projection operator generally does not have a closed form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set. In this paper, we focus on a family of constraints involving linear transforms of l_1,p balls. Our main theoretical contribution is to provide closed form expressions of the epigraphical projections associated with the Euclidean norm and the sup norm. The proposed approach is validated in the context of image restoration with missing samples, by making use of TV-like constraints. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems
Structure tensor total variation
This is the final version of the article. Available from Society for Industrial and Applied Mathematics via the DOI in this record.We introduce a novel generic energy functional that we employ to solve inverse imaging problems
within a variational framework. The proposed regularization family, termed as structure tensor
total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both
grayscale and vector-valued images. It generalizes several existing variational penalties, including
the total variation seminorm and vectorial extensions of it. Meanwhile, thanks to the structure
tensor’s ability to capture first-order information around a local neighborhood, the STV functionals
can provide more robust measures of image variation. Further, we prove that the STV regularizers
are convex while they also satisfy several invariance properties w.r.t. image transformations. These
properties qualify them as ideal candidates for imaging applications. In addition, for the discrete
version of the STV functionals we derive an equivalent definition that is based on the patch-based
Jacobian operator, a novel linear operator which extends the Jacobian matrix. This alternative
definition allow us to derive a dual problem formulation. The duality of the problem paves the
way for employing robust tools from convex optimization and enables us to design an efficient
and parallelizable optimization algorithm. Finally, we present extensive experiments on various
inverse imaging problems, where we compare our regularizers with other competing regularization
approaches. Our results are shown to be systematically superior, both quantitatively and visually
Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients
We address the denoising of images contaminated with multiplicative noise,
e.g. speckle noise. Classical ways to solve such problems are filtering,
statistical (Bayesian) methods, variational methods, and methods that convert
the multiplicative noise into additive noise (using a logarithmic function),
shrinkage of the coefficients of the log-image data in a wavelet basis or in a
frame, and transform back the result using an exponential function. We propose
a method composed of several stages: we use the log-image data and apply a
reasonable under-optimal hard-thresholding on its curvelet transform; then we
apply a variational method where we minimize a specialized criterion composed
of an data-fitting to the thresholded coefficients and a Total
Variation regularization (TV) term in the image domain; the restored image is
an exponential of the obtained minimizer, weighted in a way that the mean of
the original image is preserved. Our restored images combine the advantages of
shrinkage and variational methods and avoid their main drawbacks. For the
minimization stage, we propose a properly adapted fast minimization scheme
based on Douglas-Rachford splitting. The existence of a minimizer of our
specialized criterion being proven, we demonstrate the convergence of the
minimization scheme. The obtained numerical results outperform the main
alternative methods
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