Abstract:A popular way to restore images comprising edges is to minimize a cost-function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularization term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with non-convex regularization. By promoting the idea of… Show more
“…In this section, we review a class of such functionals described in [26], [29] and defined in our framework as…”
Section: B Nonquadratic Regularizationmentioning
confidence: 99%
“…Consequently, the basic steepest-descent method applied to the given cost functional is equivalent to the corresponding gradient flow. Similarly, the iteratively reweighted leastsquares (IRLS) technique [25] that is used for nonquadratic regularization is associated with linearized versions of the gradient of the original functional [21], [26]. This provides an interpretation that relates IRLS to lagged-diffusivity fixedpoint iterations.…”
Abstract-Practical image-acquisition systems are often modeled as a continuous-domain prefilter followed by an ideal sampler, where generalized samples are obtained after convolution with the impulse response of the device. In this paper, our goal is to interpolate images from a given subset of such samples. We express our solution in the continuous domain, considering consistent resampling as a data-fidelity constraint. To make the problem well posed and ensure edge-preserving solutions, we develop an efficient anisotropic regularization approach that is based on an improved version of the edgeenhancing anisotropic diffusion equation. Following variational principles, our reconstruction algorithm minimizes successive quadratic cost functionals. To ensure fast convergence, we solve the corresponding sequence of linear problems by using multigrid iterations that are specifically tailored to their sparse structure. We conduct illustrative experiments and discuss the potential of our approach both in terms of algorithmic design and reconstruction quality. In particular, we present results that use as little as 2% of the image samples.
“…In this section, we review a class of such functionals described in [26], [29] and defined in our framework as…”
Section: B Nonquadratic Regularizationmentioning
confidence: 99%
“…Consequently, the basic steepest-descent method applied to the given cost functional is equivalent to the corresponding gradient flow. Similarly, the iteratively reweighted leastsquares (IRLS) technique [25] that is used for nonquadratic regularization is associated with linearized versions of the gradient of the original functional [21], [26]. This provides an interpretation that relates IRLS to lagged-diffusivity fixedpoint iterations.…”
Abstract-Practical image-acquisition systems are often modeled as a continuous-domain prefilter followed by an ideal sampler, where generalized samples are obtained after convolution with the impulse response of the device. In this paper, our goal is to interpolate images from a given subset of such samples. We express our solution in the continuous domain, considering consistent resampling as a data-fidelity constraint. To make the problem well posed and ensure edge-preserving solutions, we develop an efficient anisotropic regularization approach that is based on an improved version of the edgeenhancing anisotropic diffusion equation. Following variational principles, our reconstruction algorithm minimizes successive quadratic cost functionals. To ensure fast convergence, we solve the corresponding sequence of linear problems by using multigrid iterations that are specifically tailored to their sparse structure. We conduct illustrative experiments and discuss the potential of our approach both in terms of algorithmic design and reconstruction quality. In particular, we present results that use as little as 2% of the image samples.
“…This method has on the one hand the advantage of being very easy to implement, and on the other hand the disadvantage of being quite slow. To improve the convergence speed, quasi-Newton methods have been proposed [1,23,29,36,55,56,67]. Iterative methods have proved successful [16,18,35].…”
In this paper, we are interested in texture modeling with functional analysis spaces. We focus on the case of color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f , such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling.
“…Yet, the practical efficiency of the overall algorithm hinges on the choices of the scaling matrices H (k,k) θ and H (k+1,k) z and the subroutines for solving (13) and (15). In this work, our choices for H (k,k) θ and H (k+1,k) z (see formulas (18) and (21) below) will be structured Hessian approximations motivated from the iteratively reweighted least-squares (IRLS) method [13,14].…”
mentioning
confidence: 99%
“…In this work, our choices for H (k,k) θ and H (k+1,k) z (see formulas (18) and (21) below) will be structured Hessian approximations motivated from the iteratively reweighted least-squares (IRLS) method [13,14]. The solution of the resulting subproblem can be interpreted as a regularized Newton step; see [15,16,17].…”
Abstract. We tackle the nonlinear problem of photometric stereo under close-range pointwise sources, when the intensities of the sources are unknown (so-called semi-calibrated setup). A variational approach aiming at robust joint recovery of depth, albedo and intensities is proposed. The resulting nonconvex model is numerically resolved by a provably convergent alternating minimization scheme, where the construction of each subproblem utilizes an iteratively reweighted least-squares approach. In particular, manifold optimization technique is used in solving the corresponding subproblems over the rank-1 matrix manifold. Experiments on real-world datasets demonstrate that the new approach provides not only theoretical guarantees on convergence, but also more accurate geometry.
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