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slides15 - 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 Signal and Image Processing: finite length signals; boundary filters and boundary wavelets; wavelet compression algorithms. Finite-Length Signals y-2 y-1 y0 y1 o yN-1 o = h0 h1 h2 h0 h1 y[n] h-1 h0 h-1 h1 h0 h-1 rr r x-2 x-1 x0 x1 o xN-1 o unknown LMN H(ω) x[n] length N (finite length) unknown 2 1) zero-padding x[n] m -2 -1 0 1 2 3 m n filtered by [1, 1] y[n] m n -2 -1 0 1 2 3 filtered by [1, -1] y[n] m -2 -1 0 1 2 3 n artificial edge resulting from zero-padding 3 2) Periodic Extension … x[N] = x[0] x[n] … … n - 1 0 1 2 … N-1 N wrap-around o y0 y1 o = yN-1 N-output h1 h0 h-1 h-2 m h2 h1 h1 h0 h-1 m h3 h2 h-1 h-2 m h2 h1 h0 xN-2 xN-1 x0 x1 x2 o xN-1 x0 o circulant matrix = H 4 What is the eigenvector for the circulant matrix? [ 1 eiω ei2ω m ei(N-1)ω ] T We need eiNω = 1 = ei0ω ∴ Nω = 2πk , ω= 2πk N discrete set of ω’s For the 0th row, N-1 H[k] = ∑ h[n] e-i 2πk Nn n=0 5 111m 1 w w2 [H] 1 w2 w4 oo o 1 wN-1 1 H[0] H[1] wN-1 r w2(N-1) = [F] o H[N-1] k=0k=1 k=N-1 LOOOOMOOOON 2π iN w=e F HF = FΛ Λ contains the Fourier coefficients 2πk Nn H[k] = ∑ n h[n]e-i ∑ ∑ h[n - x[e-i If x[ = ei n n 2πk0 N 2πk Nn = H[k]X[k] H[k]X[k] ⇒ X[k] = δ[k – k0] ⇒ H[k]X[k] = H[k0]X[k] 6 3) Symmetric Extension 1) Whole point symmetry – when filter is whole point symmetric. 2) Half point symmetry – when filter is half point symmetric. e.g. Whole point symmetry: filter and signal h1x2 + h0x1 + h1x0 h0 h1 h1x1 + h0x0 + h1x1 h1 h0 h1 = h1x0 + h0x1 + h1x2 h1 h0 h1 r rr x2 x1 x0 x1 x2 7 e.g. whole point symmetry – filter, half-point symmetry - signal h1x2 + h1x1 + h1x0 + h1x0 + h0x1 + h1x0 h0x0 + h1x0 h0x0 + h1x1 h0x1 + h1x2 Half point symmetry h1 h0 h1 h1 h0 h1 = h1 h0 h1 r rr x2 x1 x0 x0 x1 x2 Whole point symmetry 8 Downsampling a whole-point symmetric signal with even length N at the left boundary: x x -2 -1 0 1 ⇒ still whole-point symmetric after ↓2. 2 at the right boundary: x x x ⇒ half-point symmetric after ↓2. N-1 odd E.g. 9/7 filter: whole-point symmetric use the above extension for signal ⇒ N N/2 exactly N/2 9 Downsample a half-point symmetric signal x ⇒ nothing guaranteed x x -3 -2 -1 0 1 2 Linear-phase filters H(ω) = A(ω)e-iωα 1) half-point symmetric, α = fraction 2) whole-point symmetric, α = integer Symmetric extension of finite-length signal X(ω) = B(ω)e-iωβ 10 10 The output: Y(ω) = H(ω)X(ω) W W H H W H H W W H W H W = whole-point symmetry H = half-point symmetry The above extensions ensure the continuity of function values at boundaries, but not the continuity of derivatives at boundaries. 11 11 4) Polynomial Extrapolation (not useful in image processing) • Useful for PDE with boundary conditions. x0 x1 01 x2 x3 4 coefficients ⇒ fits up to 3rd order polynomials. 23 a + bn + cn2 + dn3 = x(n) 1000 1111 1 2 22 23 1 3 32 33 LOMON A a x0 b = x1 c x2 d x3 12 12 Then, x–1 = [1 -1 1 -1] PDE a b = [1 -1 1 -1] A–1 c d x0 x1 x2 x3 f(x) = ∑ ckφ(x – k) k Assume f(x) has polynomial behavior near boundaries p-1 ∑ αixi = f(x) = ∑ ckφ(x – k) k i=0 {φ (• - k)} orthonormal p-1 ⇒ ∑ αi ∫ φ(x – k)xidx = ck i=0 LOMON µik 13 13 µ0 µ1 m µ p-1 0 0 0 µ0 1 µ1 µ2 1 1 m o α0 c0 α1 c1 = o αp-1 o cp-1 Using the computed αi’s, we can extrapolate, α0 e.g. c–1 = [µ01 µ1 1 m µp-1] o – – –1 αp-1 DCT idea of symmetric extension cf. DFT X[k] = ∑ n complex-valued 2πk x[n]e-i N n Want real-valued results. Want 14 14 1m 0 N-1 N m 2N-1 2N DFT of this extended signal: N-1 2πk –i 2N n ∑ x[n]e 2N-1 +∑ LOOMOON n=N n=0 N-1 N-1 2πk π -i 2N (2N-1-m) ∑ x[m]e m=0 = ∑ x[n] n=0 2πk x[2N-1-n]e-i 2N n 2πk -i 2πk n {e 2N + 2πk e-i 2N (2N-1-n)} N-1 X(k) ck ∑ √2 x[n]cos πk (n+½) m DCT – II used in JPEG N N n=0 1 ck = √2 k=0 1 k = 1, 2, …, N - 1 15 15 ...
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