Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Levinson-Galerkin algorithm for regularized trigonometric approximation

Thomas Strohmer

TL;DR

The paper presents a robust method for regularized trigonometric approximation from nonuniform, noisy samples by adaptively selecting the polynomial degree via a multi-level least-squares framework. It introduces a Levinson-Galerkin algorithm that solves a nested sequence of Toeplitz normal equations with a stopping rule, achieving overall complexity ${\cal O}(r N_0 + N_0^2)$. The approach is accompanied by a principled choice of weights as a simple preconditioner, a clear extension to multivariate settings, and practical demonstrations in boundary recovery for echocardiography. The resulting method provides an efficient, stable way to recover smooth curves and surfaces from perturbed measurements, with direct applicability to medical imaging and related inverse problems.

Abstract

Trigonometric polynomials are widely used for the approximation of a smooth function $f$ from a set of nonuniformly spaced samples $\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multi-level algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most $\cal{O}(NM + M^2)$ operations ($M$ being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle.

A Levinson-Galerkin algorithm for regularized trigonometric approximation

TL;DR

The paper presents a robust method for regularized trigonometric approximation from nonuniform, noisy samples by adaptively selecting the polynomial degree via a multi-level least-squares framework. It introduces a Levinson-Galerkin algorithm that solves a nested sequence of Toeplitz normal equations with a stopping rule, achieving overall complexity . The approach is accompanied by a principled choice of weights as a simple preconditioner, a clear extension to multivariate settings, and practical demonstrations in boundary recovery for echocardiography. The resulting method provides an efficient, stable way to recover smooth curves and surfaces from perturbed measurements, with direct applicability to medical imaging and related inverse problems.

Abstract

Trigonometric polynomials are widely used for the approximation of a smooth function from a set of nonuniformly spaced samples . If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multi-level algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most operations ( being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle.

Paper Structure

This paper contains 15 sections, 8 theorems, 77 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multi-level least squares approximation
  3. A multi-level algorithm and an efficient stopping criterion
  4. Noisefree samples
  5. Uniformly spaced samples
  6. Noisy nonuniform samples
  7. A Toeplitz system and trigonometric approximation
  8. Weights as simple preconditioner
  9. A Levinson-Galerkin algorithm for trigonometric approximation
  10. Multivariate trigonometric approximation
  11. Line-type nonuniform sampling in 2-D
  12. Curve and surface approximation by trigonometric polynomials
  13. Object boundary recovery in Echocardiography
  14. Boundary recovery from a sequence of images
  15. Miscellaneous remarks

Key Result

lemma 1

If $N \ge N_{*}$ then $c^{(N)}$ satisfies $\|V_{N} c^{(N)} - s^{\varepsilon} \| \le \varepsilon \|s^{\varepsilon}\|$, hence stopping criterion stop always becomes active at some level $N_{0} \le N_{*}$.

Figures (5)

  • Figure 1: Controlling the smoothness of the solution is essential for trigonometric approximation from perturbed data in order to avoid overfitting and underfitting of the data. The proposed Levinson-Galerkin algorithm automatically adapts to the least squares solution of optimal degree.
  • Figure 2: Comparison of weights obtained by different approaches.
  • Figure 3: Line-type nonuniform sampling set
  • Figure 4: The recovery of the boundary of the Left Ventricle from 2-D ultrasound images is a basic step in echocardiography to extract relevant parameters of cardiac function.
  • Figure 5: The approximation in the left image results from using a too small polynomial degree, the approximation in the right image from a too large degree for the trigonometric approximation.

Theorems & Definitions (14)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • proposition 1
  • proof
  • theorem 1
  • proposition 2: Gröchenig, Gro93a
  • ...and 4 more