Book contents
- Frontmatter
- Contents
- List of Acronyms
- Notation
- Foreword
- 1 Introduction to the World of Sparsity
- 2 The Wavelet Transform
- 3 Redundant Wavelet Transform
- 4 Nonlinear Multiscale Transforms
- 5 Multiscale Geometric Transforms
- 6 Sparsity andNoiseRemoval
- 7 Linear Inverse Problems
- 8 Morphological Diversity
- 9 Sparse Blind Source Separation
- 10 Dictionary Learning
- 11 Three-Dimensional Sparse Representations
- 12 Multiscale Geometric Analysis on the Sphere
- 13 Compressed Sensing
- 14 This Book's Take-Home Message
- Notes
- References
- Index
- Plate section
12 - Multiscale Geometric Analysis on the Sphere
Published online by Cambridge University Press: 05 October 2015
- Frontmatter
- Contents
- List of Acronyms
- Notation
- Foreword
- 1 Introduction to the World of Sparsity
- 2 The Wavelet Transform
- 3 Redundant Wavelet Transform
- 4 Nonlinear Multiscale Transforms
- 5 Multiscale Geometric Transforms
- 6 Sparsity andNoiseRemoval
- 7 Linear Inverse Problems
- 8 Morphological Diversity
- 9 Sparse Blind Source Separation
- 10 Dictionary Learning
- 11 Three-Dimensional Sparse Representations
- 12 Multiscale Geometric Analysis on the Sphere
- 13 Compressed Sensing
- 14 This Book's Take-Home Message
- Notes
- References
- Index
- Plate section
Summary
INTRODUCTION
Many wavelet transforms on the sphere have been proposed in past years. Using the lifting scheme, Schröder and Sweldens (1995) developed an orthogonal Haar wavelet transform on any surface, which can be directly applied on the sphere. Its interest is, however, relatively limited because of the poor properties of the Haar function and the problems inherent to orthogonal transforms.
More interestingly, many papers have presented new continuous wavelet transforms (Antoine 1999; Tenorio et al. 1999; Cayón et al. 2001; Holschneider 1996). These works have been extended to directional wavelet transforms (Antoine et al. 2002; McEwen et al. 2007). All these continuous wavelet decompositions are useful for data analysis, but cannot be used for restoration purposes because of the lack of an inverse transform. Freeden and Windheuser (1997) and Freeden and Schneider (1998) proposed the first redundant wavelet transform, based on the spherical harmonics transform, which presents an inverse transform. Starck et al. (2006) proposed an invertible isotropic undecimated wavelet transform (IUWT) on the sphere, also based on spherical harmonics, which has the same property as the starlet transform, that is, the sum of the wavelet scales reproduces the original image. A similar wavelet construction (Marinucci et al. 2008; Faÿ and Guilloux 2011; Fay et al. 2008) used the so-called needlet filters. Wiaux et al. (2008) also proposed an algorithm which permits the reconstruction of an image from its steerable wavelet transform. Since reconstruction algorithms are available, these new tools can be used for many applications such as denoising, deconvolution, component separation (Moudden et al. 2005; Bobin et al. 2008; Delabrouille et al. 2009), and inpainting (Abrial et al. 2007; Abrial et al. 2008).
Extensions to the sphere of 2-D geometric multiscale decompositions, such as the ridgelet transform and the curvelet transform,were presented in Starck et al. (2006).
The goal of this chapter is to overview these multiscale transforms on the sphere. Section 12.2 overviews the hierarchical equal area isolatitude pixelization (HEALPix) of a sphere pixelization scheme and the spherical harmonics transform. Section 12.3 shows how a fast orthogonal Haar wavelet transform on the sphere can be built using HEALPix. In Section 12.5, we present an isotropic wavelet transform on the sphere which has similar properties as the starlet transform and therefore should be very useful for data denoising and deconvolution.
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- Sparse Image and Signal ProcessingWavelets and Related Geometric Multiscale Analysis, pp. 321 - 372Publisher: Cambridge University PressPrint publication year: 2015
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