Graph-based regularization for spherical signal interpolation

This paper addresses the problem of the interpolation of 2-d spherical signals from non-uniformly sampled and noisy data. We propose a graph-based regularization algorithm to improve the signal reconstructed by local interpolation methods such as nearest neighbour or kernel-based interpolation algorithms. We represent the signal as a function on a graph where weights are adapted to the particular geometry of the sphere. We then solve a total variation (TV) minimization problem with a modified version of Chambolle's algorithm. Experimental results with noisy and uncomplete datasets show that the regularization algorithm is able to improve the result of local interpolation schemes in terms of reconstruction quality.

Nonideal sampling and regularized interpolation of noisy data

Sathish Ramani

Conventional sampling (Shannon's sampling formulation and its approximation-theoretic counterparts) and interpolation theories provide effective solutions to the problem of reconstructing a signal from its samples, but they are primarily restricted to the noise-free scenario. The purpose of this thesis is to extend the standard techniques so as to be able to handle noisy data. First, we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling, and the samples are corrupted by additive noise. In order to counterbalance the effect of noise, the reconstruction problem is formulated in a variational framework where the solution is obtained by minimizing a continuous-domain Tikhonov-like L2-regularization subject to a ℓp-based data fidelity constraint. We present theoretical justification for the minimization of this cost functional and show that the global-minimum solution belongs to a shift-invariant space generated by a function that is generally not bandlimited. The optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a B-spline-like basis. We also consider stochastic formulations – min-max and minimum mean-squared error (MMSE/Wiener) formulations – of the nonideal sampling problem and show that they yield the same type of estimators and point towards the existence of optimal shift-invariant spaces for certain classes of stochastic processes. In the stochastic context, we also derive an exact formula for the error of approximating a stationary stochastic signal in the presence of discrete additive noise and justify the noise-reducing effect of regularization through illustrations. Next, we focus on the use of a much wider class of non-quadratic regularization functionals for the problem of interpolation in the presence of noise. Starting from the afine-invariance of the solution, we show that the Lp-norm (p ≠ 2) is the most suitable type of non-quadratic regularization for our purpose. We give monotonically convergent numerical algorithms to carry out the minimization of the non-quadratic cost criterion. We also demonstrate experimentally that the proposed regularized interpolation scheme provides superior interpolation performance compared to standard methods in the presence of noise. Finally, we address the problem of selecting an appropriate value for the regularization parameter which is most crucial for the working of variational methods in general including those discussed in this thesis. We propose a practical scheme that is based on the concept of risk estimation to achieve minimum MSE performance. In this context, we first review a well known result due to Stein (Stein's unbiased risk estimate — SURE) that is applicable for data corrupted by additive Gaussian noise and also derive a new risk estimate for a Poisson-Gaussian mixture model that is appropriate for certain biomedical imaging applications. Next, we introduce a novel and efficient Monte-Carlo technique to compute SURE for arbitrary nonlinear algorithms. We demonstrate experimentally that the proposed Monte-Carlo SURE yields regularization parameter values that are close to the oracle-optimum (minimum MSE) for all methods considered in this work. We also present results that illustrate the applicability of our technique to a wide variety of algorithms in denoising and deconvolution.

EPFL2009

TV-Regularized Generation of Planar Images from Omnicams

Pascal Frossard, Laurent Jacques, Didier Raboud, Pierre Vandergheynst

This paper addresses the problem of mapping images between different vision sensors. Such a mapping could be modeled as a sampling problem that has to encompass the change of geometry between the two sensors and the specific discretization of the real scene observed by the two different imaging systems. We formulate the problem in a general framework that can be cast as a minimization regularized problem with a linear operator, that applies to any image geometry. We then focus on the particular problem of the generation of planar images from omnidirectional images, in any viewing direction and for any size and resolution. In this regularized approach, the ﬁdelity term is expressed in the original omnicam geometry and the regularization is based on Total Variation (TV) solved here with proximal methods. Experimental results demonstrate the superiority of this approach with respect to alternative schemes based on linear interpolation or TV inpainting.

2009

Tree-Based Pursuit: Algorithm and Properties

Pascal Frossard, Philippe Jost, Pierre Vandergheynst

This paper proposes a tree-based pursuit algorithm that efficiently trades off complexity and approximation performance for overcomplete signal expansions. Finding the sparsest representation of a signal using a redundant dictionary is, in general, a NP-Hard problem. Even sub-optimal algorithms such as Matching Pursuit remain highly complex. We propose a structuring strategy that can be applied to any redundant set of functions, and which basically groups similar atoms together. A measure of similarity based on coherence allows for representing a highly redundant sub-dictionary of atoms by a unique element, called molecule. When the clustering is applied recursively on atoms and then on molecules, it naturally leads to the creation of a tree structure. We then present a new pursuit algorithm that uses the structure created by clustering as a decision tree. This tree-based algorithm offers important complexity reduction with respect to Matching Pursuit, as it prunes important parts of the dictionary when traversing the tree. Recent results on incoherent dictionaries are extended to molecules, while the true highly redundant nature of the dictionary stays hidden by the tree structure. We then derive recovery conditions on the structured dictionary, under which tree-based pursuit is guaranteed to converge. Experimental results finally show that the gain in complexity offered by tree-based pursuit does in general not have a high penalty on the approximation performance. They show that the dimensionality of the problem is reduced thanks to the tree construction, without significant loss of information at hand.

2005

Nonideal sampling and regularized interpolation of noisy data

Sathish Ramani

EPFL2009