WAVELETS ON MANIFOLDS

Mingzhen TanNational University of Singapore

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

E.g. Haar Wavelets in 1-dimension

WAVELETS IN EUCLIDEAN SPACES

E.g. Stereographic dilations for spheres

Admissible Function:

Mother Wavelet:

Wavelet coefficients:

BASES ON CLOSED MANIFOLDS

Inner Product on 2-manifolds:

We consider surfaces to be discretized meshes:

Decomposition of functions defined on the surface:

WAVELET BASES ON CLOSED MANIFOLDS

Definition of wavelet bases on closed manifolds:

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:

WEIGHT FUNCTIONS

Weight function for wavelet bases:

Weight function for scaling bases:

WEIGHT FUNCTIONS

LOCALIZATION IN BOTH FREQUENCY AND SPATIAL DOMAINS

Lemma (spatial localization):

SCALING EFFECTS

Simulation with varying

Simulation with varying

IMPLEMENTATION

EIGENFUNCTIONS OF LAPLACE-BELTRAMI OPERATOR

EIGENFUNCTIONS OF LAPLACE-BELTRAMI OPERATOR

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

PLOT OF THE SPECTRUM OF THE LAPLACE-BELTRAMI

(a)(b)(c)

EXAMPLES OF WAVELET BASES

Wavelet coefficients, Example 1

WAVELET TRANSFORM

Wavelet and scaling coefficients, Example 2

WAVELET TRANSFORM

WAVELET FRAMESQuestion: 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:

WAVELET FRAMES

Wavelet Transform of 𝑺𝟏 Inverse Wavelet Transform of wavelet coefficients of

𝑺𝟏

INVERSE WAVELET TRANSFORM

Reconstruction formula:

INVERSE WAVELET TRANSFORM

Reconstructing

Reconstruction formula:

RECONSTRUCTION ACCURACY OF WAVELET TRANSFORM

Fast Approximation Scheme using Chebyshev polynomials

Chebyshev polynomials:

Approximation of weighting functions:

WAVELET TRANSFORM (FAST APPROXIMATION)

Chebyshev expansion:

Fast Approximation Scheme using Chebyshev polynomials

Approximation of weighting functions:

Approximation of wavelet coefficients:

WAVELET TRANSFORM (FAST APPROXIMATION)

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.

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

CLASSIFICATION OF SUBJECTS AS AD OR CONTROL

PROBLEMS• Over-complete – Larger number of wavelet coefficients are used

• No multi-resolution analysis （MRA）