Lecture11_Wavelets

# too many d0n coefcients h0n 2 h1n x n a0n 2 a0n d0n m

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Unformatted text preview: ts h0[n] 2 h1[n] x [ n] a0n? 2 a0n d0n M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 25 26 Fast DWT with Filter Banks h0[n] x [ n] Decomposition complexity: N + N/2 + N/4 + N/8 +...+ = 2N =O(N) h1[n] h0[n] 2 h1[n] 2 a0n d0n h0[n] 2 h1[n] 2 h0[n] h1[n] d1n h0[n] 2 h1[n] x [ n] a1n 2 a0n d0n h0[n] 2 h1[n] 2 a1n d1n M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 27 28 Reconstruction h0[n] Haar DWT Example g0[n] h1[n] g0[n] 2 g1[n] x [ n] 2 a0n d0n x[n] g1[n] g0[n] 2 g1[n] 2 a1n Haar d1n Just ﬂip arrows, replace h with g a2n d2n d1n d0n M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 29 30 Approximation from 25/256 coefﬁcients Example: Denoising Noisy Signals Haar Haar DFT M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 31 32 Example: Denoising by Thresholding Compression - JPEG2000 vs JPEG noisy Jpeg2000 - Wavelet Jpeg - DCT denoised largest 25 coefficients @ 66 fold compression ratio M. Lustig, EECS UC Berkeley M. Lustig, EECS UC Berkeley 33 34 Compression - JPEG2000 vs JPEG Jpeg2000 - Wavelet Jpeg - DCT @ 66 fold compression ratio M. Lustig, EECS UC Berkeley 35 36 a2 d2y d2x d1x d0x Noisy Wavelet Denoised d2xy d1y d1xy d0y d0xy 37 38 Approximation/Compression M. Lustig, EECS UC Berkeley 39 40 Example in Research Robust 4D Flow Denoising using Divergence-free Wavelet Transform Frank Ong1...
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