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Wavelets on manifolds

This work explores the definition and construction of wavelet bases on closed manifolds, building on previous research. Utilizing eigenfunctions of the Laplace-Beltrami operator, it introduces wavelet transforms for functions defined on closed manifolds, focusing on their application in analyzing Alzheimer’s disease data. The study examines wavelet coefficients and the reconstruction of functions from these wavelet transforms, with a particular emphasis on classification accuracy between Alzheimer’s patients and normal controls, leveraging shape variations in hippocampi.

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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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