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Application: Signal Compression Jyun-Ming Chen Spring 2001

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Signal Compression Lossless compression –Huffman, LZW, arithmetic, run-length –Rarely more than 2:1 Lossy Compression –Willing to accept slight inaccuracies Quantization/Encoding is not discussed here

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Wavelet Compression A function can be represented by linear combinations of any basis functions different bases yields different representation/approxi mation

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Wavelet Compression (cont) Compression is defined by finding a smaller set of numbers to approximate the same function within the allowed error

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Wavelet Compression : permutation of 1, …, m, then L2 norm of approximation error Assuming orthonormal basis

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Wavelet Compression If we sort the coefficients in decreasing order, we get the desired compression (next page) The above computation assumes orthogonality of the basis function, which is true for most image processing wavelets

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Results of Coarse Approximations (using Haar wavelets)

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Significance Map While transmitting, an additional amount of information must be sent to indicate the positions of these significant transform values Either 1 or 0 –Can be effectively compressed (e.g., run-length) Rule of thumb: –Must capture at least 99.99% of the energy to produce acceptable approximation

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Application: Denoising Signals

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Types of Noise Random noise –Highly oscillatory –Assume the mean to be zero Pop noise –Occur at isolated locations Localized random noise –Due to short-lived disturbance in the environment

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Thresholding For removing random noise Assume the following conditions hold: –Energy of original signal is effectively captured by values greater than Ts –Noise signal are transform values below noise threshold Tn –Tn < Ts Set all transformed value less than Tn to zero

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Results (Haar) Depend on how the wavelet transform compact the signal

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Haar vs. Coif30

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Choosing a Threshold Value Transform preserves the Gaussian nature of the noise

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Removing Pop and Background Static See description on pp. 63-4

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Types of Thresholding

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Soft vs. Hard Threshold on Image Denoising

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Quantitative Measure of Error Measure amount of error between noisy data and the original Aim to provide quantitative evidence for the effectiveness of noise removal Wavelet-based measure

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Error Measures (cont)

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