Nonideal Sampling and Regularization Theory

Abstract

Abstract-Shannon's sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some "shift-invariant" space, which may or may not be bandlimited. In this paper, we present some further justification for this type of representation, while addressing the issue of the specification of the best reconstruction space. We consider a realistic setting where a multidimensional signal is prefiltered prior to sampling, and the samples are corrupted by additive noise. We adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a Tikhonov-like (continuous domain) 2 -regularization to obtain the continuous-space solution. We present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous-space solution belongs to a shift-invariant space generated by a function (generalized B-spline) that is generally not bandlimited. When the sampling is ideal, we recover some of the classical smoothing spline estimators. 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. To make the scheme practical, we specify the generating functions corresponding to the most popular families of regularization operators (derivatives, iterated Laplacian), as well as a new, generalized one that leads to a new brand of Matérn splines. We conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error (MMSE/Wiener) solutions of the generalized sampling problem.Index Terms-Matérn class, minimax and minimum mean square error (MMSE) reconstruction, nonideal sampling, regularization, shift-invariant spaces, smoothing splines, Tikhonov criterion, Wiener solution.

Sampling and Reconstruction of Spatial Fields Using Mobile Sensors

“…Hence the reconstruction in (20) becomes accurate and the expression in (21) reduces to that in (18). This means that we have for , and .…”

Regularized Interpolation for Noisy Images

“…The primary argument is statistical: It can be proved that the minimization of a proper version of (5) will yield the minimum-mean-squared-error reconstruction of the signal (under the assumption that the signal is a stationary Gaussian process), provided that the basis functions are matched to the regularization operator [27], [29]. More generally, we may adopt a Bayesian point of view and use (5) to specify the maximum a posteriori estimator of the unknown signal, which is continuously defined.…”

“…Various authors have formulated interpolation as a variational problem to accommodate regularization constraints [21]- [29]. The resulting scheme is often termed regularized interpolation where the objective is to obtain the solution by minimizing a cost criterion that jointly measures the data-fitting error and the regularity of the solution.…”

“…The former refers to the case where the solution is a discrete entity defined on a grid that is finer than that of the data-these methods specifically cater to the image-upsampling problem [21]- [23]. In the latter case, a continuously defined solution-a smoothing-spline-is obtained by minimizing the -norm of some scalar derivative of the solution-Tikhonov-like quadratic regularization-subject to certain data-fitting requirements [24]- [29].…”

“…In this paper, we concentrate on analog-domain regularized interpolation and propose to extend smoothing-spline-like approaches [24]- [29] by considering the use of nonquadratic regularization-the motivation is to overcome the shortcoming of Tikhonov-like quadratic regularization which tends to smear important signal features (e.g., edges in images). We also want the solution to be invariant to translation, rotation and scaling of the coordinates.…”