1. W AVELETS ON MANIFOLDS Mingzhen Tan National University of Singapore

2. O VERVIEW 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. E.g. Haar Wavelets in 1-dimension W AVELETS IN E UCLIDEAN S PACES E.g. Stereographic dilations for spheres Admissible Function: Mother Wavelet: Wavelet coefficients:

4. B ASES ON C LOSED M ANIFOLDS Inner Product on 2-manifolds: We consider surfaces to be discretized meshes: Decomposition of functions defined on the surface:

5. W AVELET B ASES ON C LOSED M ANIFOLDS Definition of wavelet bases on closed manifolds:

6. W AVELET B ASES ON C LOSED M ANIFOLDS ‘Fourier’ transform: We consider dilation in the Fourier domain: Inverse ‘Fourier’ transform: Wavelet coefficients in the spatial domain: Wavelet Basis:

7. W EIGHT FUNCTIONS Weight function for wavelet bases: Weight function for scaling bases:

8. W EIGHT F UNCTIONS

9. L OCALIZATION IN BOTH FREQUENCY AND SPATIAL DOMAINS Lemma (spatial localization):

10. S CALING E FFECTS

11. IMPLEMENTATION

12. E IGENFUNCTIONS OF L APLACE -B ELTRAMI O PERATOR

13. References: Meyer et. al. Discrete differential-geometry operators for triangulated 2-manifolds

14. P LOT OF THE SPECTRUM OF THE L APLACE -B ELTRAMI (a) (b) (c)

15. E XAMPLES OF W AVELET B ASES

16. Wavelet coefficients, Example 1 W AVELET T RANSFORM

17. Wavelet and scaling coefficients, Example 2 W AVELET T RANSFORM

18. W AVELET F RAMES 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. Frames Bounds: Definition:

19. W AVELET F RAMES

20. Wavelet Transform of Inverse Wavelet Transform of wavelet coefficients of I NVERSE W AVELET T RANSFORM Reconstruction formula:

21. I NVERSE W AVELET T RANSFORM Reconstructing Reconstruction formula:

22. R ECONSTRUCTION ACCURACY OF WAVELET TRANSFORM

23. Fast Approximation Scheme using Chebyshev polynomials Chebyshev polynomials: Approximation of weighting functions: W AVELET T RANSFORM (F AST A PPROXIMATION ) Chebyshev expansion:

24. Fast Approximation Scheme using Chebyshev polynomials Approximation of weighting functions: Approximation of wavelet coefficients: W AVELET T RANSFORM (F AST A PPROXIMATION )

25. C LASSIFICATION 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. C LASSIFICATION 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. C LASSIFICATION OF SUBJECTS AS AD OR CONTROL

28. P ROBLEMS Over-complete – Larger number of wavelet coefficients are used No multi-resolution analysis （ MRA ）