1. Wavelets Chapter 7 Serkan ERGUN

2. 1.Introduction Wavelets are mathematical tools for hierarchically decomposing functions. Regardless of whether the function of interest is an image, a curve, or a surface, wavelets offer an elegant technique for representing the levels of detail present.

3. 2.1. Wavelets in one dimension The Haar basis is the simplest wavelet basis. We will first discuss how a one- dimensional function can be decomposed using Haar wavelets Finally, we show how using the Haar wavelet decomposition leads to a straightforward technique for compressing a one-dimensional function

4. 2.1. Wavelets in one dimension ResolutionAveragesDetail coefficients 4 [ 9 7 3 5 ] 2 [ 8 4 ][ 1 -1 ] 1 [ 6 ][ 2 ] Decomposition: Wavelet Transform: [ 6 2 1 -1 ]

5. 2.1. Wavelets in one dimension ResolutionAverages 1 [ 6 2 1 -1 ] 2 [ 6+2 6-2 1 -1] 4 [ 8+1 8-1 4+(-1) 4-(-1) ] Recomposition:

6. 2.1. Wavelets in one dimension Storing the image’s wavelet transform, rather than the image itself, has a number of advantages. Often a large number of the detail coefficients turn out to be very small in magnitude Truncating, or removing, these small coefficients from the representation introduces only small errors in the reconstructed image, giving a form of “lossy” image compression.

7. 2.1. Wavelets in one dimension

9. 2.2. Haar basis functions A one-pixel image is just a function that is constant over the entire interval [0, 1). We’ll let V 0 be the vector space of all these functions. A two-pixel image has two constant pieces over the intervals [1, ½) and [½, 1) We’ll call the space containing all these functions V 1

10. 2.2. Haar basis functions (cont’d) The space V j will include all piecewise-constant functions defined on the interval [0, 1) with constant pieces over each of 2 j equal subintervals

11. 2.2. Haar basis functions (cont’d) The basis functions for the spaces V j are called scaling functions, and are usually denoted by the symbol ϕ. A simple basis for V j is given by the set of scaled and translated “box” functions:

12. 2.2. Haar basis functions (cont’d) The box basis for V 2

13. The next step is to choose an inner product defined on the vector spaces V j. The “standard” inner product, 2.2. Haar basis functions (cont’d)

14. Vector space W j as the orthogonal complement of V j in V j+1. In other words, we will let W j be the space of all functions inV j+1 that are orthogonal to all functions in V j under the chosen inner product. W j represents the parts of a function inV j+1 that cannot be represented in V j. 2.2. Haar basis functions (cont’d)

15. A collection of linearly independent functions spanning W j are called wavelets. These basis functions have two important properties The basis functions of W j, together with the basis functions of V j form a basis for V j+1 Every basis function of W j is orthogonal to every basis function of Vj under the chosen inner product 2.2. Haar basis functions (cont’d)

16. The wavelets corresponding to the box basis are known as theHaar wavelets, given by: 2.2. Haar basis functions (cont’d)

17. The Haar wavelets for W 1 2.2. Haar basis functions (cont’d)

18. We begin by expressing our original image I(x) as a linear combination of the box basis functions inV 2 :

19. 2.2. Haar basis functions (cont’d) We can rewrite the expression for I(x) in terms of basis functions in V 1 and W 1, using pairwise averaging and differencing:

20. 2.2. Haar basis functions (cont’d) Finally, we’ll rewrite I(x) as a sum of basis functions in V 0, W 0, and W 1 :

21. 2.3. Properties of wavelets Orthogonality: An orthogonal basis is one in which all of the basis functions. Note that orthogonality is stronger than the minimum requirement for wavelets that be orthogonal to all scaling functions of the same resolution level. are orthogonal to each other

22. 2.3. Properties of wavelets Normalization: Another property that is sometimes desirable is normalization. A basis function u(x) is normalized if Normalized Haar basis:

23. 2.4. Application 1

24. 3.1. Wavelets in two dimensions There are two ways we can use wavelets to transform the pixel values within an image. Each is a generalization to two dimensions of the one-dimensional wavelet transform

25. 3.2. Standard Decomposition

27. 3.3. Non-Standard Decomposition

29. Basis functions: Two dimensional scaling function: Three Wavelet functions: Haar basis functions: Vertical Horizontal Diagonal

30. 3.4. Comparison The standard decomposition of an image requires performing one- dimensional transforms on all rows and then on all columns. For an m x m image it requires, 4(m 2 – m) assignments whereas the non- standard decomposition requires only 8(m 2 – 1) / 3 assignments.

31. 3.5. Application II The original image (a) can be represented using (b) 19% of its wavelet coefficients, with 5% relative error; (c) 3% of its coefficients, with 10% relative error; and (d) 1% of its coefficients, with 15% relative error

32. 4. Conclusion There are many wavelet functions other than Haar like, Daubechies, Coiflets, Symlets, Discreet Meyer, biorthogonal and reverse biorthogonal Wavelets are not only used in Image Processing but also in fast image queries, surface editing, multiresolution curves etc..