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Wavelets on manifolds

Wavelets on manifolds. Mingzhen Tan National University of Singapore. Overview. W avelets on Euclidean spaces Continuation of the work from [Hammond, 2011] Defining wavelet bases on closed manifolds Using eigenfunctions of Laplace-Beltrami

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Wavelets on manifolds

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  1. Wavelets on manifolds Mingzhen Tan National University of Singapore

  2. Overview • Wavelets on Euclidean spaces • Continuation of the work from [Hammond, 2011] • Defining wavelet bases on closed manifolds • Using eigenfunctions of Laplace-Beltrami • Construction of wavelet transforms of functions defined on closed manifolds • Inverse wavelet transforms • Application on Alzheimer’s disease data

  3. Wavelets in Euclidean Spaces Admissible Function: Mother Wavelet: Wavelet coefficients: E.g. Haar Wavelets in 1-dimension E.g. Stereographic dilations for spheres

  4. Bases on Closed Manifolds Inner Product on 2-manifolds: We consider surfaces to be discretized meshes: Decomposition of functions defined on the surface:

  5. Wavelet Bases on Closed Manifolds Definition of wavelet bases on closed manifolds:

  6. Wavelet Bases on Closed Manifolds ‘Fourier’ transform: We consider dilation in the Fourier domain: Inverse ‘Fourier’ transform: Wavelet coefficients in the spatial domain: Wavelet Basis:

  7. Weight functions Weight function for wavelet bases: Weight function for scaling bases:

  8. Weight Functions

  9. Localization in both frequency and spatial domains Lemma (spatial localization):

  10. Scaling Effects Simulation with varying Simulation with varying

  11. Implementation

  12. Eigenfunctions of Laplace-Beltrami Operator

  13. Eigenfunctions of Laplace-Beltrami Operator References: Meyer et. al. Discrete differential-geometry operators for triangulated 2-manifolds

  14. Plot of the spectrum of the Laplace-Beltrami (c) (b) (a)

  15. Examples of Wavelet Bases

  16. Wavelet Transform Wavelet coefficients, Example 1

  17. Wavelet Transform Wavelet and scaling coefficients, Example 2

  18. Wavelet Frames Question: Is this set of bases well behaved for representing functions on the surface? To examine this, we consider the wavelets at discretized scales as a frame, and check the frame bounds. Definition: Frames Bounds:

  19. Wavelet Frames

  20. Inverse Wavelet Transform Reconstruction formula: Inverse Wavelet Transform of wavelet coefficients of Wavelet Transform of

  21. Inverse Wavelet Transform Reconstructing Reconstruction formula:

  22. Reconstruction accuracy of wavelet transform

  23. Wavelet Transform (Fast Approximation) Fast Approximation Scheme using Chebyshev polynomials Chebyshevpolynomials: Chebyshev expansion: Approximation of weighting functions:

  24. Wavelet Transform (Fast Approximation) Fast Approximation Scheme using Chebyshev polynomials Approximation of weighting functions: Approximation of wavelet coefficients:

  25. Classification of subjects as AD or control • Subjects selection: • Source of controls: community and clinics • Patient groups with dementia were recruited from the stroke service and memory clinics in Singapore • Normal controls were defined as subjects without any cognitive complaints or functional loss & • MMSE scores of at least 23 if they had secondary/tertiary education • MMSE scores of at least 21 if they had primary/no education on initial screening and had no significant cognitive impairments on formal neuropsychological testing • AD was diagnosed in accordance with the NINCDS-ADRDA critieria • All normal and AD subjects were required to have no history of stroke, and no evidence of severe cerebrovascular disease on MRI (no infarcts) and/or presence of significant white matter lesions, defined as a grade of at least 2 in more than 4 regions • From August 2010 to November 2012, a total of 172 subjects were recruited, out of which 25 were normal controls and 20 are AD subjects.

  26. Classification of subjects as AD or control • Experiment Objectives • A classifier constructed from a mix of hippocampi shapes could serve as an important biomarker to differentiate between the diseased and normal subjects • In particular, we want to improve the classification performance by using information on the possible hippocampal variations across the different resolutions brought about by the wavelet transform • Inputs • Jacobian determinants of transformations from a common template to the individual hippocampal shapes • Wavelet transform of the Jacobian determinants into 30 scales • Data reduction (each done separately for the Jacobian det. and their wavelet transforms) using PCA from d=1184 to d=4

  27. Classification of subjects as AD or control

  28. Problems • Over-complete • Larger number of wavelet coefficients are used • No multi-resolution analysis （MRA）

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