915 research outputs found
Faster gradient descent and the efficient recovery of images
Much recent attention has been devoted to gradient descent algorithms where
the steepest descent step size is replaced by a similar one from a previous
iteration or gets updated only once every second step, thus forming a {\em
faster gradient descent method}. For unconstrained convex quadratic
optimization these methods can converge much faster than steepest descent. But
the context of interest here is application to certain ill-posed inverse
problems, where the steepest descent method is known to have a smoothing,
regularizing effect, and where a strict optimization solution is not necessary.
Specifically, in this paper we examine the effect of replacing steepest
descent by a faster gradient descent algorithm in the practical context of
image deblurring and denoising tasks. We also propose several highly efficient
schemes for carrying out these tasks independently of the step size selection,
as well as a scheme for the case where both blur and significant noise are
present.
In the above context there are situations where many steepest descent steps
are required, thus building slowness into the solution procedure. Our general
conclusion regarding gradient descent methods is that in such cases the faster
gradient descent methods offer substantial advantages. In other situations
where no such slowness buildup arises the steepest descent method can still be
very effective
On Convergent Finite Difference Schemes for Variational - PDE Based Image Processing
We study an adaptive anisotropic Huber functional based image restoration
scheme. By using a combination of L2-L1 regularization functions, an adaptive
Huber functional based energy minimization model provides denoising with edge
preservation in noisy digital images. We study a convergent finite difference
scheme based on continuous piecewise linear functions and use a variable
splitting scheme, namely the Split Bregman, to obtain the discrete minimizer.
Experimental results are given in image denoising and comparison with additive
operator splitting, dual fixed point, and projected gradient schemes illustrate
that the best convergence rates are obtained for our algorithm.Comment: 23 pages, 12 figures, 2 table
Image Restoration using Total Variation with Overlapping Group Sparsity
Image restoration is one of the most fundamental issues in imaging science.
Total variation (TV) regularization is widely used in image restoration
problems for its capability to preserve edges. In the literature, however, it
is also well known for producing staircase-like artifacts. Usually, the
high-order total variation (HTV) regularizer is an good option except its
over-smoothing property. In this work, we study a minimization problem where
the objective includes an usual data-fidelity term and an overlapping
group sparsity total variation regularizer which can avoid staircase effect and
allow edges preserving in the restored image. We also proposed a fast algorithm
for solving the corresponding minimization problem and compare our method with
the state-of-the-art TV based methods and HTV based method. The numerical
experiments illustrate the efficiency and effectiveness of the proposed method
in terms of PSNR, relative error and computing time.Comment: 11 pages, 37 figure
Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization
Multiplicative noise (also known as speckle noise) models are central to the
study of coherent imaging systems, such as synthetic aperture radar and sonar,
and ultrasound and laser imaging. These models introduce two additional layers
of difficulties with respect to the standard Gaussian additive noise scenario:
(1) the noise is multiplied by (rather than added to) the original image; (2)
the noise is not Gaussian, with Rayleigh and Gamma being commonly used
densities. These two features of multiplicative noise models preclude the
direct application of most state-of-the-art algorithms, which are designed for
solving unconstrained optimization problems where the objective has two terms:
a quadratic data term (log-likelihood), reflecting the additive and Gaussian
nature of the noise, plus a convex (possibly nonsmooth) regularizer (e.g., a
total variation or wavelet-based regularizer/prior). In this paper, we address
these difficulties by: (1) converting the multiplicative model into an additive
one by taking logarithms, as proposed by some other authors; (2) using variable
splitting to obtain an equivalent constrained problem; and (3) dealing with
this optimization problem using the augmented Lagrangian framework. A set of
experiments shows that the proposed method, which we name MIDAL (multiplicative
image denoising by augmented Lagrangian), yields state-of-the-art results both
in terms of speed and denoising performance.Comment: 11 pages, 7 figures, 2 tables. To appear in the IEEE Transactions on
Image Processing
Inexact Bregman iteration with an application to Poisson data reconstruction
This work deals with the solution of image restoration problems by an
iterative regularization method based on the Bregman iteration. Any iteration of this
scheme requires to exactly compute the minimizer of a function. However, in some
image reconstruction applications, it is either impossible or extremely expensive to
obtain exact solutions of these subproblems. In this paper, we propose an inexact
version of the iterative procedure, where the inexactness in the inner subproblem
solution is controlled by a criterion that preserves the convergence of the Bregman
iteration and its features in image restoration problems. In particular, the method
allows to obtain accurate reconstructions also when only an overestimation of the
regularization parameter is known. The introduction of the inexactness in the iterative
scheme allows to address image reconstruction problems from data corrupted by
Poisson noise, exploiting the recent advances about specialized algorithms for the
numerical minimization of the generalized KullbackāLeibler divergence combined with
a regularization term. The results of several numerical experiments enable to evaluat
This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms - Theory and Practice
The observations in many applications consist of counts of discrete events,
such as photons hitting a detector, which cannot be effectively modeled using
an additive bounded or Gaussian noise model, and instead require a Poisson
noise model. As a result, accurate reconstruction of a spatially or temporally
distributed phenomenon (f*) from Poisson data (y) cannot be effectively
accomplished by minimizing a conventional penalized least-squares objective
function. The problem addressed in this paper is the estimation of f* from y in
an inverse problem setting, where (a) the number of unknowns may potentially be
larger than the number of observations and (b) f* admits a sparse
approximation. The optimization formulation considered in this paper uses a
penalized negative Poisson log-likelihood objective function with nonnegativity
constraints (since Poisson intensities are naturally nonnegative). In
particular, the proposed approach incorporates key ideas of using separable
quadratic approximations to the objective function at each iteration and
penalization terms related to l1 norms of coefficient vectors, total variation
seminorms, and partition-based multiscale estimation methods.Comment: 11 pages, 7 figures, IEEE Transactions on Image Processing (2011), in
pres
Weighted Mean Curvature
In image processing tasks, spatial priors are essential for robust
computations, regularization, algorithmic design and Bayesian inference. In
this paper, we introduce weighted mean curvature (WMC) as a novel image prior
and present an efficient computation scheme for its discretization in practical
image processing applications. We first demonstrate the favorable properties of
WMC, such as sampling invariance, scale invariance, and contrast invariance
with Gaussian noise model; and we show the relation of WMC to area
regularization. We further propose an efficient computation scheme for
discretized WMC, which is demonstrated herein to process over 33.2
giga-pixels/second on GPU. This scheme yields itself to a convolutional neural
network representation. Finally, WMC is evaluated on synthetic and real images,
showing its superiority quantitatively to total-variation and mean curvature.Comment: 12 page
Compressed Sensing Parallel MRI with Adaptive Shrinkage TV Regularization
Compressed sensing (CS) methods in magnetic resonance imaging (MRI) offer
rapid acquisition and improved image quality but require iterative
reconstruction schemes with regularization to enforce sparsity. Regardless of
the difficulty in obtaining a fast numerical solution, the total variation (TV)
regularization is a preferred choice due to its edge-preserving and structure
recovery capabilities. While many approaches have been proposed to overcome the
non-differentiability of the TV cost term, an iterative shrinkage based
formulation allows recovering an image through recursive application of linear
filtering and soft thresholding. However, providing an optimal setting for the
regularization parameter is critical due to its direct impact on the rate of
convergence as well as steady state error. In this paper, a regularizer
adaptively varying in the derivative space is proposed, that follows the
generalized discrepancy principle (GDP). The implementation proceeds by
adaptively reducing the discrepancy level expressed as the absolute difference
between TV norms of the consistency error and the sparse approximation error. A
criterion based on the absolute difference between TV norms of consistency and
sparse approximation errors is used to update the threshold. Application of the
adaptive shrinkage TV regularizer to CS recovery of parallel MRI (pMRI) and
temporal gradient adaptation in dynamic MRI are shown to result in improved
image quality with accelerated convergence. In addition, the adaptive TV-based
iterative shrinkage (ATVIS) provides a significant speed advantage over the
fast iterative shrinkage-thresholding algorithm (FISTA).Comment: 27 pages,9 figure
Local Linear Convergence of the ADMM/Douglas--Rachford Algorithms without Strong Convexity and Application to Statistical Imaging
We consider the problem of minimizing the sum of a convex function and a
convex function composed with an injective linear mapping. For such problems,
subject to a coercivity condition at fixed points of the corresponding Picard
iteration, iterates of the alternating directions method of multipliers
converge locally linearly to points from which the solution to the original
problem can be computed. Our proof strategy uses duality and strong metric
subregularity of the Douglas--Rachford fixed point mapping. Our analysis does
not require strong convexity and yields error bounds to the set of model
solutions. We show in particular that convex piecewise linear-quadratic
functions naturally satisfy the requirements of the theory, guaranteeing
eventual linear convergence of both the Douglas--Rachford algorithm and the
alternating directions method of multipliers for this class of objectives under
mild assumptions on the set of fixed points. We demonstrate this result on
quantitative image deconvolution and denoising with multiresolution statistical
constraints.Comment: Revised manuscript: 30 pages including 9 figures, one appendix and 57
references. Difference from version 2: title and abstract changed, one new
figure added, and a posteriori error estimates in numerical experiments
reporte
A multilevel based reweighting algorithm with joint regularizers for sparse recovery
Sparsity is one of the key concepts that allows the recovery of signals that
are subsampled at a rate significantly lower than required by the
Nyquist-Shannon sampling theorem. Our proposed framework uses arbitrary
multiscale transforms, such as those build upon wavelets or shearlets, as a
sparsity promoting prior which allow to decompose the image into different
scales such that image features can be optimally extracted. In order to further
exploit the sparsity of the recovered signal we combine the method of
reweighted , introduced by Cand\`es et al., with iteratively updated
weights accounting for the multilevel structure of the signal. This is done by
directly incorporating this approach into a split Bregman based algorithmic
framework. Furthermore, we add total generalized variation (TGV) as a second
regularizer into the split Bregman algorithm. The resulting algorithm is then
applied to a classical and widely considered task in signal- and image
processing which is the reconstruction of images from their Fourier
measurements. Our numerical experiments show a highly improved performance at
relatively low computational costs compared to many other well established
methods and strongly suggest that sparsity is better exploited by our method
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