Wavelet Shrinkage on Paths for Denoising of Scattered Data

Dennis Heinen,Gerlind Plonka

doi:10.1007/s00025-012-0285-3

Abstract

We propose a new algorithm for denoising of multivariate function values given at scattered points in $${\mathbb{R}^{d}}$$ . The method is based on the one-dimensional wavelet transform that is applied along suitably chosen path vectors at each transform level. The idea can be seen as a generalization of the relaxed easy path wavelet transform by Plonka (Multiscale Model Simul 7:1474–1496, 2009) to the case of multivariate scattered data. The choice of the path vectors is crucial for the success of the algorithm. We propose two adaptive path constructions that take the distribution of the scattered points as well as the corresponding function values into account. Further, we present some theoretical results on the wavelet transform along path vectors in order to indicate that the wavelet shrinkage along path vectors can really remove noise. The numerical results show the efficiency of the proposed denoising method.

Highlights

Within the last years, wavelet threshold methods have been shown to be a suitable tool for denoising of functions and images
Supposing that the noise corresponds to wavelet coefficients with a small amplitude, the application of a thresholding procedure to the wavelet expansion of f removes it from the signal
The basic idea of our scheme is very similar to the easy path wavelet transform (EPWT), namely to employ the usual one-dimensional wavelet transform along suitable path vectors at each level

Summary

Introduction

Wavelet threshold methods have been shown to be a suitable tool for denoising of functions and images. [2,7,11,19] These wavelet constructions adaptively depend on the scattered points and the corresponding function values and lose much of the simplicity and efficiency of the traditional wavelet transform. The EPWT employs the one-dimensional wavelet transform along path vectors through the image values. We want to propose a new adaptive wavelet threshold scheme for scattered data denoising. The basic idea of our scheme is very similar to the EPWT, namely to employ the usual one-dimensional wavelet transform along suitable path vectors at each level. Vol 62 (2012) Wavelet Shrinkage on Paths for Denoising of Scattered Data 339 scheme several times along different path vectors and compute the average of the results. For the special case of images, we present some comparisons with other image denoising methods

Description of the Algorithm

Construction of Path Vectors

Adaptive Random Path Construction

Properties of Wavelet Transform on Paths

Numerical Results