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# Slides9 - Course 18.327 and 1.130 Course Wavelets and...

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Unformatted text preview: Course 18.327 and 1.130 Course Wavelets and Filter Banks Multiresolution Analysis (MRA): Requirements for MRA; Requirements Nested Spaces and Complementary Spaces; Scaling Functions and Wavelets Scaling Scaling Functions and Wavelets φ(t) Box function Continuous time: 1 0 1 0 1 φ(2t) Scaling (2t) Scaling 1/2 t t φ(2t - 1) (2t Scaling + Scaling Shifting Shifting 1 0 1/2 1 t 2 For this example: φ(t) = φ(2t) + φ(2t – 1) (t) (2t) (2t More generally: Refinement equation φ(t) = 2∑ h0[k]φ(2t – k) (2t or k=0 Two-scale difference equation N φ(t) is called a scaling function The refinement equation couples the representations The of a continuous-time function at two time scales. The of time continuous-time function is determined by a discretecontinuous time filter, h0[n]! For the above (Haar) example: h0[0] = h0[1] = ½ [1] (a lowpass filter) filter) 3 Note: (i) Solution to refinement equation may not Note: always exist. If it does… always (ii) φ(t) has compact support i.e. (ii) φ(t) = 0 outside 0 ≤ t < N (t) (comes from the FIR filter, h0[n]) (comes (iii) φ(t) often has no closed form solution. (iii) (iv) φ(t) is unlikely to be smooth. (iv) Constraint on h0[n]: N ∫ φ(t)dt = 2 ∑ h0[k] ∫ φ(2t – k)dt dt [k] (2t k=0 N = 2 ∑ h0[k] • ½ ∫ φ(τ)dτ [k] k=0 So N ∑ h0[k] = 1 [k] k=0 Assumes ∫ φ(t)dt ≠ 0 dt 4 1 Square wave of finite length of Haar wavelet w(t) Now consider: 1 0 1 t 1/2 φ(2t) Scaled 1/2 0 1/2 t 0 w(t) = φ(2t) - φ(2t – 1) (2t) (2t -φ(2t – 1) (2t Scaled + shifted Scaled + sign flipped sign 1 t 5 More generally: N w(t) = 2∑ h1[k] φ(2t – k) [k] (2t Wavelet equation k=0 For the Haar wavelet example: For wavelet h1[0] = ½ [0] h1[1] = -½ [1] (a highpass filter) (a filter) 6 Some observations for Haar scaling function and wavelet Some scaling 1. Orthogonality of integer shifts (translates): Orthogonality of 1 0 φ(t) 1 φ(t - 1) (t 1 t 0 1 2 t 1 if k = 0 ∫ φ(t) φ(t – k)dt = 0 otherwise (t) (t dt = δ[k] Similarly ∫ w(t) w(t – k)dt = δ[k] w(t) dt Reason: no overlap 7 2. Scaling function is orthogonal to wavelet: 1 φ(t) 1 w(t) + + 1 0 1 t 0 t 1/2 - ∫φ(t) w(t)dt = 0 dt Reason: +ve and –ve areas cancel each other. ve ve areas 8 3. Wavelet is orthogonal across scales: 1 w(2t) w(t) w(2t - 1) w(2t + + 1/2 1 0 1/2 - + t 0 ∫ w(t) w(2t)dt = 0 , dt - t - t ∫ w(t) w(2t – 1)dt = 0 w(t) dt Reason: finer scale versions change sign while Reason: coarse scale version remains constant. coarse 9 Wavelet Bases Our goal is to use w(t), its scaled versions (dilations) Our and their shifts (translates) as building blocks for continuous-time functions, f(t). Specifically, we are continuous time interested in the class of functions for which we can define the inner product: define ∞ <f(t) , g(t)> = ∫ f(t) g*(t)dt < ∞ <f(t) dt -∞ Such functions f(t) must have finite energy: ∞ ||f(t)|| = ∫f(t)2 dt ∫ dt 2 -∞ <∞ and they are said to belong to the Hilbert space, L2(ℜ). 10 10 Consider all dilations and translates of the Haar wavelet: Consider wavelet: /2 wj,k(t) = 2jj/2 w(2jt – k) ; -∞ ≤ j ≤ ∞ k) -∞ ≤ k ≤ ∞ Normalization factor so that ||wj,k(t)|| = 1 Normalization || J/2 ∫ wj,k(t) wJ,K(t) dt = ∫ 2j/2 w(2jt – k) . 2J/2 w(2Jt – K)dt (t) (t) 1 if j = J and k = K = 0 otherwise = δ[ j – J ] δ[ k – K ] 11 11 M 1 v2 L L w-1,k(t) 4 2 1 3 t 1 L 1 4 3 2 L w0,k(t) t v2 L 1 2 3 4 L w1,k(t) t 12 12 wjk(t) form an orthonormal basis for L2(ℜ). (t) basis f(t) = ∑ bjk wjk(t) ; f(t) (t) j,k ∞ /2 w(2 wjk(t) = 2jj/2 w(2jt – k) bjjk = -∞ f(t) wjk(t) dt ∫ f(t) (t) dt k 13 13 Multiresolution Analysis Multiresolution Analysis Key ingredients: 1. A sequence of embedded subspaces: {0} ⊂ … ⊂ V-1 ⊂ V0 ⊂ V1 ⊂ … ⊂ Vj ⊂ Vj+1 ⊂ … ⊂ L2(ℜ) {0} L2(ℜ) = all functions with finite energy ∞ = {ƒ(t): ∫ ƒ(t) 2 dt < ∞} Hilbert (t): (t) dt - space space Requirements: • Completeness as j → ∞ . If ƒ(t) belongs to Completeness If (t) L2(ℜ) and ƒj(t) is the portion of ƒ(t) that lies in and (t) (t) Vj, then lim∞ ƒj(t) = ƒ(t) then j→ (t) (t) 14 14 Restated as a condition on the subspaces: ∞ ∪ j=-∞ • Vj = L2 (ℜ) Emptiness as j → - ∞ Emptiness lim || j → - ∞ || fj(t) || = 0 Restated as a condition on the subspaces: ∞ ∩ Vj = {0} j=-∞ 15 15 2. A sequence of complementary subspaces, Wj, sequence such that Vj + Wj = Vj+1 and and Vj ∩ Wj = {0} {0} (no overlap) This is written as Vj ⊕ Wj = Vj+1 (Direct sum) Note: An orthogonal multiresolution will have Wj Note: orthogonal to Vj : Wj ? Vj . orthogonal So orthogonality will ensure that Vj ∩ Wj = {0} So 16 16 We thus have V1 = V 0 ⊕ W 0 V2 = V 1 ⊕ W 1 = V 0 ⊕ W 0 ⊕ W 1 V3 = V 2 ⊕ W 2 = V 0 ⊕ W 0 ⊕ W 1 ⊕ W 2 M J-1 VJ = VJ-1 ⊕ WJ-1 = V0 ⊕ ∑ Wj j=0 M ∞ 2(ℜ) = V ⊕ ∑ W L 0 j j=0 We can also write the recursion for j < 0 We V0 = V-1 ⊕ W-1 = V-2 ⊕ W-2 ⊕ W-1 M -1 = V-k ⊕ ∑ Wj j=-k M ∞ -1 = ∑ Wj ⇒ L2(ℜ) = ∑ Wj j = -∞ j = -∞ 17 17 3. A scaling (dilation) law: If ƒ(t) ∈ Vj then ƒ(2t) ∈ Vj+1 If (t) then (2t) 4. A shift (translation) law: If ƒ(t) ∈ Vj then ƒ(t-k) ∈ Vj k integer If (t) then k) integer 5. V0 has a shift-invariant basis, {φ(t-k) : - ∞ ≤ k ≤ ∞} k) W0 has a shift-invariant basis, {w(t-k) : - ∞ ≤ k ≤ ∞} k) We expect that V1 = V0 + W0 will have twice as will many basis functions as V0 alone. many First possibility: {φ(t-k) , w(t-k) : - ∞ ≤ k ≤ ∞} k) Second possibility: use the scaling law i.e. if φ(t- k) ∈ V0 , then φ(2t- k) ∈ V1 if k) k) 18 18 So V1 has a shift-invariant basis, {v 2 φ(2t-k): - ∞ ≤ k ≤ ∞} k): Can we relate this basis for V1 to the basis for V0? We know that V0 ⊂ V1 So any function in V0 can be written as a combination can of the basic functions for V1. of In particular, since φ(t) ∈ V0, we can write In (t) φ(t) = 2∑ h0[k] φ(2t – k) [k] (2t k This is the Refinement Equation (a.k.a. the TwoScale Difference Equation or the Dilation Equation). 19 19 We also know that W0 = V1 – V0 So W0 ⊂ V1 This means that any function in W0 can also be written can as a combination of the basic functions for V1. as Since w(t) ∈ W0, we can write w(t) = 2∑ h1[k] φ(2t – k) [k] (2t k Wavelet Equation 20 20 Multiresolution Representations Functions: L2 (ℜ) = V0 ⊕ W0 ⊕ W1 ⊕ W2 ⊕ ... Level 2 detail Level 1 detail Level 0 detail Finite energy Finite functions functions Coarse Coarse approximation approximation Images: V0 W0 + V1 W1 V2 + 21 21 Multiresolution Representations Geometry: Mesh courtesy of Igor Guskov (Caltech) 22 22 ...
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