Wavelets Shai Avidan Tel Aviv University Slide Credits (partial - - PowerPoint PPT Presentation

Wavelets

Shai Avidan Tel Aviv University

Slide Credits (partial list)

• Rick Szeliski
• Steve Seitz
• Alyosha Efros
• Yacov Hel-Or
• Miki Elad
• Hagit Hel-Or
• Marc Levoy
• Bill Freeman
• Fredo Durand
• Sylvain Paris
• Andrew Adams

Laplacian Pyramid

• Make the coarse level by downsampling
• Make the fine level by upsampling the coarse layer,

and taking the difference with the original

• Reconstruct by upsampling the coarse layer and

adding the fine layer

Problem

• Lapalacian Pyarmid is a redundant
• representation. Coarse level is blurry

and redundant. What about the fine layer?

The basic 2x2 transform

Lets start with a simple transform Given signal [a,b] we obtain [c,d] Given [c,d] we can recover [a,b] Observe that c is sum of elements and d is their difference The scaling function: [1 1] The wavelet function: [-1 1]

Extending to even length vector

Given sequence v[M] Example: v=[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16] c = 1/sqrt(2) [ (1+2) (3+4)…(15+16)] = 1/sqrt(2) [3 7 11 15 19 23 27 31] d =1/sqrt(2) [(1-2) (3-4)…(15-16)] = 1/sqrt(2) [-1 -1 …

• 1]

To reconstruct

Reconstruct: c = 1/sqrt(2)[0 3 0 7 0 11 0 15 0 19 0 23 0 27 0 31] d = 1/sqrt(2)[0 -1 0 -1 … 0 -1] LPF(c) = 1/2[(0+3) (3+0) (0+7) (7+0)…] = 1/2[3 3 7 7 … 31 31] HPF(d) = 1/2[(0+1) (-1-0) …] = 1/2[1 -1 …] LPF(c)-HPF(d) = 1/2([3 3 7 7 … 31 31]-[1 -1 …]) = 1/2([(3-1) (3+1) (7-1) (7+1)…]) = [1 2 3 4 …] Recall: LPF is: 1/sqrt(2)[1 1] HPF is: 1/sqrt(2)[1 -1]

Block diagrams

In Matrix form

A matrix representation of the transform on vector v:

Multi-Resolution

Repeat the process on the coarse level

And in Matrix form

The right matrix is the one we saw before, the left matrix keeps the last two elements and transforms the first two

2D Extension

Wavelets

• Memory efficient (unlike pyramids)
• Time efficient (O(n) and not O(nlogn) like Fourier)
• Simple to compute
• Orthogonal
• Not redundant

Wavelet basis functions

The Scaling function

The inner product

The wavelet function

The wavelet function

Example I

I = [9 7 3 5]

Example II

Example III

Wavelet Lifting

Here’s another way to build a wavelet transform: Given vector X of size N=2^n Denote P_G(0) = X P_G(1) = Reduce(P_G(0)) – Select even elements Use P_G(1) to predict P^odd_G(0) Now let: P_w(0) = P^odd_G(0)-P_G(1) And we can repeat this in a multi-scale fashion.

Version I

Version I - Example

Lifting Scheme (Version I)

So far we’ve used simple Reduce. Can we use a more complicated reduce function? The answer is yes and it’s called Lifting Scheme. It was proposed by Sweldens in the mid 90’s. Given X, split to even X_e and odd X_o. Use X_e to predict X_o, so: X^new_o = X_o – Pred(X_e) The advantage is that we can use better prediction schemes with larger support

Lifting Scheme (ExampleI)

Observe we have many more zeros

Lifting Scheme

Lifting Scheme

Lifting Scheme

Lifting Scheme

Predict and Update don’t have to be linear…

Wavelet example

Wavelet example

Wavelet Shrinkage Denoising

Noisy image Clean image

Wavelet Shrinkage Denoising

Wavelet Shrinkage Denoising

Wavelet coef. value New Wavelet coef. value

Wavelet Shrinkage Pipeline

Wavelet Compression (I)

Wavelet Compression (II)

Wavelet Compression (III)

Wavelet Compression (Result)

JPEG 2000

Wavelet transform used in JPEG 2000

JPEG 2000

JPEG 2000