Book contents
- Frontmatter
- Contents
- Acronyms
- Notation
- Preface
- 1 Introduction to the World of Sparsity
- 2 The Wavelet Transform
- 3 Redundant Wavelet Transform
- 4 Nonlinear Multiscale Transforms
- 5 The Ridgelet and Curvelet Transforms
- 6 Sparsity and Noise Removal
- 7 Linear Inverse Problems
- 8 Morphological Diversity
- 9 Sparse Blind Source Separation
- 10 Multiscale Geometric Analysis on the Sphere
- 11 Compressed Sensing
- References
- List of Algorithms
- Index
- Plate section
4 - Nonlinear Multiscale Transforms
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Acronyms
- Notation
- Preface
- 1 Introduction to the World of Sparsity
- 2 The Wavelet Transform
- 3 Redundant Wavelet Transform
- 4 Nonlinear Multiscale Transforms
- 5 The Ridgelet and Curvelet Transforms
- 6 Sparsity and Noise Removal
- 7 Linear Inverse Problems
- 8 Morphological Diversity
- 9 Sparse Blind Source Separation
- 10 Multiscale Geometric Analysis on the Sphere
- 11 Compressed Sensing
- References
- List of Algorithms
- Index
- Plate section
Summary
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:
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.
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.
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.
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- Information
- Sparse Image and Signal ProcessingWavelets, Curvelets, Morphological Diversity, pp. 75 - 88Publisher: Cambridge University PressPrint publication year: 2010