INTRODUCTION

Some problems related to the wavelet transform may impact their use in certain applications. This motivates the development of other multiscale representations. Such problems include the following:

1. Negative values: By definition, the wavelet mean is zero. Every time we have a positive structure at a scale, we have negative values surrounding it. These negative values often create artifacts during the restoration process or complicate the analysis.

2. Point artifacts: For example, cosmic ray hits in optical astronomy can “pollute” all the scales of the wavelet transform because their pixel values are huge compared to other pixel values related to the signal of interest. The wavelet transform is nonrobust relative to such real or detector faults.

3. Integer values: The discrete wavelet transform (DWT) produces floating values that are not easy to handle for lossless image compression.

Section 4.2 introduces the decimated nonlinear multiscale transform, in particular, using the lifting scheme approach, which generalizes the standard filter bank decomposition. Using the lifting scheme, nonlinearity can be introduced in a straightforward way, allowing us to perform an integer wavelet transform or a wavelet transform on an irregularly sampled grid. In Section 4.3, multiscale transforms based on mathematical morphology are explored. Section 4.4 presents the median-based multiscale representations that handle outliers well in the data (non-Gaussian noise, pixels with high intensity values, etc.).

DECIMATED NONLINEAR TRANSFORM

Integer Wavelet Transform

When the input data consist of integer values, the (bi-)orthogonal wavelet transform is not necessarily integer valued.