Non-Ideal Sampling and Regularized Interpolation of Noisy Data

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 work 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 pre-filtered prior to sampling, and the samples are corrupted by additive 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 an lp-based data fidelity constraint. We show that the global-minimum solution belongs to a shift-invariant space generated by a function that is generally not band-limited. We also consider stochastic formulations (min-max and minimum mean-squared error (MMSE/Wiener) formulations) of the non-ideal 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. Next, we focus on the use of a much wider class of non-quadratic regularization functions for the problem of interpolation in the presence of noise. Starting from the affine-invariance of the solution, we show that the Lp-norm is the most suitable type of non-quadratic regularization for our purpose. We give monotonically convergent numerical algorithms to carry out the minimization of non-quadratic convex cost criteria. 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 work. 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 bioimaging 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 de-noising and de-convolution.
Sathish Ramani received the M.Sc. degree in Physics from Sri Sathya Sai Institute of Higher Learning, Puttaparthy, India, in 2000, the M.Sc. degree in electrical communication engineering from Indian Institute of Science, Bangalore, India, in 2004, and the Ph.D. degree in electrical communication engineering from Ecole Polytechnique F éd érale de Lausanne (Swiss Federal Institute of Technology, Lausanne), Switzerland, in 2009. He is currently visiting EECS, University of Michigan as a post-doc. His research interests are in splines, interpolation and risk estimation with applications to biomedical imaging.