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Application:Signal Compression
Jyun-Ming Chen
Spring 2001
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
Wavelet Compression
• A function can be represented by linear combinations of any basis functions
• different bases yields different representation/approximation
Wavelet Compression (cont)
• Compression is defined by finding a smaller set of numbers to approximate the same function within the allowed error
Wavelet Compression
• : permutation of 1, …, m, then• L2 norm of approximation error
Assuming orthonormal bas
is
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
Results of Coarse Approximations (using Haar wavelets)
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
Application:Denoising Signals
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
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
Results (Haar)• Depend on how the wavelet transform compact
the signal
Haar vs. Coif30
Choosing a Threshold Value
Transform preserves the Gaussian nature of the noise
Removing Pop andBackground Static
• See description on pp. 63-4
Types of Thresholding
Soft vs. Hard Threshold on Image Denoising
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
signal edcontaminat:fsignal original:s
noise:nnsf
Error Measures (cont)
image original:f
imagenoisy :g
size image :,NM