Wavelets

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 Decomposition: Wavelet Transform: [ 6 2 1 -1 ]

5. 2.1. Wavelets in one dimension 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 outto be very small in magnitude • Truncating,or removing, these small coefficients from the representationintroduces only small errors in the reconstructed image, givinga form of “lossy” image compression.

7. 2.1. Wavelets in one dimension

8. 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 entireinterval [0, 1). We’ll let V0 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 V1

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

11. 2.2. Haar basis functions (cont’d) • The basis functions for the spaces Vj are called scaling functions, and are usuallydenoted by the symbol ϕ. • A simple basis for Vj is given by theset of scaled and translated “box” functions:

12. 2.2. Haar basis functions (cont’d) The box basis for V2

13. 2.2. Haar basis functions (cont’d) • The next step is to choose an inner product defined on the vectorspaces Vj. The “standard” inner product,

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

15. 2.2. Haar basis functions (cont’d) • Acollection of linearly independent functionsspanningWjarecalled wavelets. These basis functions have two important properties • ThebasisfunctionsofWj, together with the basis functions of Vjform a basisforVj+1 • Everybasisfunctionof Wj is orthogonal to every basis functionofVj under the chosen inner product

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

17. 2.2. Haar basis functions (cont’d) The Haar wavelets for W1

18. 2.2. Haar basis functions (cont’d) We begin by expressing our original image I(x) as a linear combination of the box basis functions inV2:

19. 2.2. Haar basis functions (cont’d) We can rewrite the expression for I(x) in terms of basis functions in V1 and W1, 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 V0, W0, and W1:

21. 2.3. Properties of wavelets • Orthogonality: • An orthogonalbasis is one in which all of the basis functions. Note that orthogonalityis stronger than the minimum requirement for wavelets that be orthogonaltoallscalingfunctions of thesameresolutionlevel. areorthogonaltoeachother

22. 2.3. Properties of wavelets • Normalization: • Another property that is sometimes desirable is normalization. Abasis 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 valueswithin an image. Each is a generalization to two dimensions of the one-dimensional wavelet transform

25. 3.2. Standard Decomposition

26. 3.2. Standard Decomposition

27. 3.3. Non-Standard Decomposition

28. 3.3. Non-Standard Decomposition

29. 3.3. Non-Standard Decomposition Basis functions: Two dimensional scaling function: Vertical Horizontal Diagonal Three Wavelet functions: Haar basis functions:

30. 3.4. Comparison • The standard decomposition of an image requires performing one-dimensional transforms onall rows and then on all columns. For an m x m image it requires, 4(m2 – m) assignments whereas the non-standard decomposition requires only 8(m2 – 1) / 3 assignments.

31. 3.5. Application II The original image (a) can be represented using (b) 19% of its wavelet coefficients, with 5% relativeerror; (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..