53,303 research outputs found
Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising
AbstractInspired by papers of Vese–Osher [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and Osher–Solé–Vese [Image decomposition and restoration using total variation minimization and the H−1 norm, Technical Report 02-57, 2002] we present a wavelet-based treatment of variational problems arising in the field of image processing. In particular, we follow their approach and discuss a special class of variational functionals that induce a decomposition of images into oscillating and cartoon components and possibly an appropriate ‘noise’ component. In the setting of [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the H−1 norm, Technical Report 02-57, 2002], the cartoon component of an image is modeled by a BV function; the corresponding incorporation of BV penalty terms in the variational functional leads to PDE schemes that are numerically intensive. By replacing the BV penalty term by a B11(L1) term (which amounts to a slightly stronger constraint on the minimizer), and writing the problem in a wavelet framework, we obtain elegant and numerically efficient schemes with results very similar to those obtained in [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimization and the H−1 norm, Technical Report 02-57, 2002]. This approach allows us, moreover, to incorporate general bounded linear blur operators into the problem so that the minimization leads to a simultaneous decomposition, deblurring and denoising
Shape Calculus for Shape Energies in Image Processing
Many image processing problems are naturally expressed as energy minimization
or shape optimization problems, in which the free variable is a shape, such as
a curve in 2d or a surface in 3d. Examples are image segmentation, multiview
stereo reconstruction, geometric interpolation from data point clouds. To
obtain the solution of such a problem, one usually resorts to an iterative
approach, a gradient descent algorithm, which updates a candidate shape
gradually deforming it into the optimal shape. Computing the gradient descent
updates requires the knowledge of the first variation of the shape energy, or
rather the first shape derivative. In addition to the first shape derivative,
one can also utilize the second shape derivative and develop a Newton-type
method with faster convergence. Unfortunately, the knowledge of shape
derivatives for shape energies in image processing is patchy. The second shape
derivatives are known for only two of the energies in the image processing
literature and many results for the first shape derivative are limiting, in the
sense that they are either for curves on planes, or developed for a specific
representation of the shape or for a very specific functional form in the shape
energy. In this work, these limitations are overcome and the first and second
shape derivatives are computed for large classes of shape energies that are
representative of the energies found in image processing. Many of the formulas
we obtain are new and some generalize previous existing results. These results
are valid for general surfaces in any number of dimensions. This work is
intended to serve as a cookbook for researchers who deal with shape energies
for various applications in image processing and need to develop algorithms to
compute the shapes minimizing these energies
A variable metric forward--backward method with extrapolation
Forward-backward methods are a very useful tool for the minimization of a
functional given by the sum of a differentiable term and a nondifferentiable
one and their investigation has experienced several efforts from many
researchers in the last decade. In this paper we focus on the convex case and,
inspired by recent approaches for accelerating first-order iterative schemes,
we develop a scaled inertial forward-backward algorithm which is based on a
metric changing at each iteration and on a suitable extrapolation step. Unlike
standard forward-backward methods with extrapolation, our scheme is able to
handle functions whose domain is not the entire space. Both {an convergence rate estimate on the objective function values and the
convergence of the sequence of the iterates} are proved. Numerical experiments
on several {test problems arising from image processing, compressed sensing and
statistical inference} show the {effectiveness} of the proposed method in
comparison to well performing {state-of-the-art} algorithms
Jump-sparse and sparse recovery using Potts functionals
We recover jump-sparse and sparse signals from blurred incomplete data
corrupted by (possibly non-Gaussian) noise using inverse Potts energy
functionals. We obtain analytical results (existence of minimizers, complexity)
on inverse Potts functionals and provide relations to sparsity problems. We
then propose a new optimization method for these functionals which is based on
dynamic programming and the alternating direction method of multipliers (ADMM).
A series of experiments shows that the proposed method yields very satisfactory
jump-sparse and sparse reconstructions, respectively. We highlight the
capability of the method by comparing it with classical and recent approaches
such as TV minimization (jump-sparse signals), orthogonal matching pursuit,
iterative hard thresholding, and iteratively reweighted minimization
(sparse signals)
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
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