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Handout12 - Course 18.327 and 1.130 Wavelets and Filter...

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Unformatted text preview: Course 18.327 and 1.130 Wavelets and Filter Banks Mallat pyramid algorithm Pyramid Algorithm for Computing Wavelet Coefficients Goal: Given the series expansion for a function f j (t) in V j f j (t) = a j [k] j,k (t) k how do we find the series f j-1 (t) = a j-1 [k] j-1,k (t) k in V j-1 and the series g j-1 (t) = b j-1 [k]w j-1,k (t) k in W j-1 such that f j (t) = f j-1 (t) + g j-1 (t) ? 2 1 3 Example: suppose that (t) = box on [0,1]. Then functions in V 1 can be written either as a combination of (2t) 1 0 , 1 0 1 , (2t-1) or as a combination of (t) 1 0 1 , , 1 0 1 2 (t-1) 4 plus a combination of 1 0 1 , , w(t) w(t-1) 1 0 1 2 Easy to see because (2t) = [ (t) + w(t)] (2t 1) = [ (t) - w(t)] 2 5 k Suppose that f(t) is a function in L 2 (R). What are the coefficients, a j [k], of the projection of f(t) on to V j ? Call the projection f j (t), f j (t) = a j [k] j,k (t) a j [k] must minimize the distance between f(t) and f j (t) {f(t) f...
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This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.

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Handout12 - Course 18.327 and 1.130 Wavelets and Filter...

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