Course and Wavelets and Filter Banks

Transcription

1 Course and.30 Wavelets and Filter Banks Multiresolution Analysis (MRA): Requirements for MRA; Nested Spaces and Complementary Spaces; Scaling Functions and Wavelets

2 Scaling Functions and Wavelets Continuous time: φ(t) Box function 0 t φ(2t) Scaling φ(2t - ) Scaling + Shifting 0 /2 t 0 /2 t 2

3 For this example: More generally: φ(t) = φ(2t) + φ(2t ) φ(t) = 2 2 h 0 [k]φ(2t k) φ(t) is called a scaling function N k=0 The refinement equation couples the representations of a continuous-time time function at two time scales. The continuous-time time function is determined by a discrete- time filter, h 0 [n]! For the above (Haar( Haar) ) example: h 0 [0] = h 0 [] = ½ (a lowpass filter) Refinement equation or Two-scale difference equation 3

4 Note: (i) Solution to refinement equation may not always exist. If it does (ii) φ(t) has compact support i.e. φ(t) = 0 outside 0 t < N (comes from the FIR filter, h 0 [n]) (iii) φ(t) often has no closed form solution. (iv) φ(t) is unlikely to be smooth. Constraint on h 0 [n]: So N φ(t) (t)dt = 2 h 0 [k] φ(2t k)dt N k=0 N = 2 h 0 [k] ½ φ(τ)dτ k=0 h 0 [k] = k=0 Assumes φ(t) (t)dt 0 4

5 Now consider: 0 /2 w(t) Square wave of finite length - Haar wavelet t φ(2t) Scaled /2 -φ(2t ) Scaled + shifted + sign flipped 0 /2 t 0 t w(t) = φ(2t) - φ(2t ) 5

6 More generally: N w(t) = 2 2 h [k] φ(2t k) Wavelet equation k=0 For the Haar wavelet example: h [0] = ½ h [] = -½ (a highpass filter) 6

7 Some observations for Haar scaling function and wavelet. Orthogonality of integer shifts (translates): φ(t) φ(t - ) 0 t 0 2 t φ(t) φ(t k)dt = if k = 0 0 otherwise Similarly = δ[k] w(t) w(t k)dt = δ[k] Reason: no overlap 7

8 2. Scaling function is orthogonal to wavelet: φ(t) w(t) + 0 t + 0 /2 - t φ(t) w(t)dt = 0 Reason: +ve + and ve areas cancel each other. 8

9 3. Wavelet is orthogonal across scales: 0 + /2 - w(t) t 0 + /2 - w(2t) t + w(2t - ) - t w(t) w(2t)dt = 0, w(t) w(2t )dt = 0 Reason: finer scale versions change sign while coarse scale version remains constant. 9

10 Wavelet Bases Our goal is to use w(t), its scaled versions (dilations) and their shifts (translates) as building blocks for continuous-time time functions, f(t). Specifically, we are interested in the class of functions for which we can define the inner product: <f(t), g(t)> = f(t) g*(t)dt < Such functions f(t) must have finite energy: f(t) 2 = f(t) f(t) 2 dt < - - and they are said to belong to the Hilbert space, L 2 (R). 0

11 Consider all dilations and translates of the Haar wavelet: w j,k,k(t) = 2 j/2 j/2 w(2 (2 j t k) ; - j - k Normalization factor so that w j,k,k(t) = w j,k,k(t) w J,K,K(t) dt = 2 j/2 w(2 j t k). 2 J/2 if j = J and k = K = 0 otherwise = δ[ [ j J ] δ[ [ k K ] J/2 w(2 (2 J t K)dt

12 v 2 M L L t w -,k,k(t) L L w 0,k (t) t v 2 L L t w,k (t) 2

13 w jk (t) form an orthonormal basis for L 2 (R). f(t) = b jk b jk j,k jk w jk jk (t) ; jk = f(t) w jk (t) dt - w jk jk (t) = 2 j/2 w(2 j t k) 3

14 Multiresolution Analysis Key ingredients:. A sequence of embedded subspaces: {0} V - V 0 V V j V j+ L 2 (R) L 2 (R)) = all functions with finite energy = {ƒ(t):{ ƒ(t) 2 dt < } } Hilbert - space Requirements: Completeness as j. If ƒ(t) belongs to L 2 (R)) and ƒ j (t) is the portion of ƒ(t) that lies in lim V j, then ƒ j (t) = ƒ(t) j 4

15 Restated as a condition on the subspaces: V j = L 2 (R) j = - Emptiness as j - lim j - f j (t) = 0 Restated as a condition on the subspaces: V j = {0} j = - 5

16 2. A sequence of complementary subspaces, W j, such that V j + W j = V j+ and V j W j = {0} (no overlap) This is written as V j W j = V j+ (Direct sum) Note: An orthogonal multiresolution will have W j orthogonal to V j : W j? V j. So orthogonality will ensure that V j W j = {0} 6

17 We thus have V = V 0 W 0 V 2 = V W = V 0 W 0 W V 3 = V 2 W 2 = V 0 W 0 W W 2 M J- V J = V J- W J- = V 0 W j j = 0 M L 2 (R)) = V 0 W j j = 0 We can also write the recursion for j < 0 V 0 = V - W - = V -2 W -2 W - M - = V -k W j j = - k M - = W j L 2 (R)) = W j j = - j = - 7

18 3. A scaling (dilation) law: If ƒ(t) V j then ƒ(2t) V j+ 4. A shift (translation) law: If ƒ(t) V j then ƒ(t-k) V j k integer 5. V 0 has a shift-invariant invariant basis, {φ(t{ (t-k) : - k } W 0 has a shift-invariant invariant basis, {w(t-k) : - k } We expect that V = V 0 + W 0 will have twice as many basis functions as V 0 alone. First possibility: {φ(t{ (t-k), w(t-k) : - k } Second possibility: use the scaling law i.e. if φ(t- k) V 0, then φ(2t- k) V 8

19 So V has a shift-invariant invariant basis, {v2{ φ(2t-k): - k } Can we relate this basis for V to the basis for V 0? We know that V 0 V So any function in V 0 can be written as a combination of the basic functions for V. In particular, since φ(t) V 0, we can write φ(t) = 2 2 h 0 [k] φ(2t k) k This is the Refinement Equation (a.k.a. the Two- Scale Difference Equation or the Dilation Equation). 9

20 We also know that W 0 = V V 0 So W 0 V This means that any function in W 0 can also be written as a combination of the basic functions for V. Since w(t) W 0, we can write w(t) = 2 2 h [k] φ(2t k) k Wavelet Equation 20

21 Multiresolution Representations Functions: L 2 ( R) = V0 W0 W W2... Images: Finite energy functions V0 W0 Level 0 detail Coarse approximation Level 2 detail Level detail V W V

22 Multiresolution Representations Geometry: Mesh courtesy of Igor Guskov (Caltech) 22