26 research outputs found
On starting and stopping criteria for nested primal-dual iterations
The importance of an adequate inner loop starting point (as opposed to a
sufficient inner loop stopping rule) is discussed in the context of a numerical
optimization algorithm consisting of nested primal-dual proximal-gradient
iterations. While the number of inner iterations is fixed in advance,
convergence of the whole algorithm is still guaranteed by virtue of a
warm-start strategy for the inner loop, showing that inner loop "starting
rules" can be just as effective as "stopping rules" for guaranteeing
convergence. The algorithm itself is applicable to the numerical solution of
convex optimization problems defined by the sum of a differentiable term and
two possibly non-differentiable terms. One of the latter terms should take the
form of the composition of a linear map and a proximable function, while the
differentiable term needs an accessible gradient. The algorithm reduces to the
classical proximal gradient algorithm in certain special cases and it also
generalizes other existing algorithms. In addition, under some conditions of
strong convexity, we show a linear rate of convergence.Comment: 18 pages, no figure
A duality based approach to the minimizing total variation flow in the space
We consider a gradient flow of the total variation in a negative Sobolev
space under the periodic boundary condition. If
, the flow is nothing but the classical total variation flow. If ,
this is the fourth order total variation flow. We consider a convex variational
problem which gives an implicit-time discrete scheme for the flow. By a duality
based method, we give a simple numerical scheme to calculate this minimizing
problem numerically and discuss convergence of a forward-backward splitting
scheme. Several numerical experiments are given
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
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
An Iterative Shrinkage Approach to Total-Variation Image Restoration
The problem of restoration of digital images from their degraded measurements
plays a central role in a multitude of practically important applications. A
particularly challenging instance of this problem occurs in the case when the
degradation phenomenon is modeled by an ill-conditioned operator. In such a
case, the presence of noise makes it impossible to recover a valuable
approximation of the image of interest without using some a priori information
about its properties. Such a priori information is essential for image
restoration, rendering it stable and robust to noise. Particularly, if the
original image is known to be a piecewise smooth function, one of the standard
priors used in this case is defined by the Rudin-Osher-Fatemi model, which
results in total variation (TV) based image restoration. The current arsenal of
algorithms for TV-based image restoration is vast. In the present paper, a
different approach to the solution of the problem is proposed based on the
method of iterative shrinkage (aka iterated thresholding). In the proposed
method, the TV-based image restoration is performed through a recursive
application of two simple procedures, viz. linear filtering and soft
thresholding. Therefore, the method can be identified as belonging to the group
of first-order algorithms which are efficient in dealing with images of
relatively large sizes. Another valuable feature of the proposed method
consists in its working directly with the TV functional, rather then with its
smoothed versions. Moreover, the method provides a single solution for both
isotropic and anisotropic definitions of the TV functional, thereby
establishing a useful connection between the two formulae.Comment: The paper was submitted to the IEEE Transactions on Image Processing
on October 22nd, 200
A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution
To overcome the weakness of a total variation based model for image
restoration, various high order (typically second order) regularization models
have been proposed and studied recently. In this paper we analyze and test a
fractional-order derivative based total -order variation model, which
can outperform the currently popular high order regularization models. There
exist several previous works using total -order variations for image
restoration; however first no analysis is done yet and second all tested
formulations, differing from each other, utilize the zero Dirichlet boundary
conditions which are not realistic (while non-zero boundary conditions violate
definitions of fractional-order derivatives). This paper first reviews some
results of fractional-order derivatives and then analyzes the theoretical
properties of the proposed total -order variational model rigorously.
It then develops four algorithms for solving the variational problem, one based
on the variational Split-Bregman idea and three based on direct solution of the
discretise-optimization problem. Numerical experiments show that, in terms of
restoration quality and solution efficiency, the proposed model can produce
highly competitive results, for smooth images, to two established high order
models: the mean curvature and the total generalized variation.Comment: 26 page
A variational method for tomographic reconstruction with few views
In this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We present a variational model and numerical analysis. We use a modified Nesterov algorithm to compute the solution. Numerical results are presented
Continuous Multiclass Labeling Approaches and Algorithms
We study convex relaxations of the image labeling problem on a continuous
domain with regularizers based on metric interaction potentials. The generic
framework ensures existence of minimizers and covers a wide range of
relaxations of the originally combinatorial problem. We focus on two specific
relaxations that differ in flexibility and simplicity -- one can be used to
tightly relax any metric interaction potential, while the other one only covers
Euclidean metrics but requires less computational effort. For solving the
nonsmooth discretized problem, we propose a globally convergent
Douglas-Rachford scheme, and show that a sequence of dual iterates can be
recovered in order to provide a posteriori optimality bounds. In a quantitative
comparison to two other first-order methods, the approach shows competitive
performance on synthetical and real-world images. By combining the method with
an improved binarization technique for nonstandard potentials, we were able to
routinely recover discrete solutions within 1%--5% of the global optimum for
the combinatorial image labeling problem