Review Questions Jyun-Ming Chen Spring 2001

Wavelet Transform What is wavelet? How is wavelet transform different from Fourier transform? Wavelets are building blocks that can quickly decorrelate data. – Wim Sweldens Wavelets are optimal bases for compressing, estimating, and recovering functions … - David Donoho Both try to represent a function in other basis (transform into other domain) and hope this transformation can reveal some insights. Yet, unlike Fourier transform, wavelet can choose many different basis.

On Details of Wavelet Transform Describe the concepts of filter banks –Analysis –Synthesis MRA (multi- resolution analysis) VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

Formal Definition of an MRA An MRA consists of the nested linear vector space such that There exists a function  (t) (called scaling function) such that is a basis for V 0 If and vice versa ; Remarks: –Does not require the set of  (t) and its integer translates to be orthogonal (in general) –No mention of wavelet

Details (cont) The roles of scaling functions and wavelets –Basis functions in V and W Refinement (two-scale) relations Graphing by cascading Computing wavelet coefficients (orthogonal)

Important Properties of Fourier Transform Linearity: Time shifting: Time scaling: Parseval’s theorem

On Wavelet Coefficients

Orthogonal MRA VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

Biorthogonal MRA VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

Semiorthogonal MRA Common property: Differences: –if orthogonal: scaling functions (and wavelets) of the same level are orthogonal to each other –If semiorthogonal, wavelets of different levels are orthogonal (from nested space) VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3 Dual and primal are the same

Comparison of Different Types of Wavelet Transforms What’s the advantage of orthgonality? Why choose to design biorthogonal wavelets (instead of orthogonal wavelets)?

On Lifting What kind of wavelet transform does lifting produce? What are the advantages of lifting? In-place computation Easy inversion Extensible to 2 nd generation wavelets More efficient computations

Details of Lifting Types of predictors –Interpolating –Average-interpolating –B-spline –… Types of Update –Number of vanishing moments of the wavelets Characteristics of the transform –MRA order –Dual MRA order –Polynomial reproducibility and vanishing moments Cascading algorithms “Lifting” theory: –why it ensures biorthogonality –Exact reconstruction guaranteed

Applications of Wavelet Transform Denoising Compression Progressive Transmission Geometric simplification MR Editing Feature recognition Graphics related …

On Variations of Wavelet Transform What is continuous wavelet transform? What is fast wavelet transform? What is wavelet packet? What types of information does each one reveal?

Derivatives of phi, psi?