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/approxi
mation
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
basis
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 and
Background 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
f : contaminat ed signal
s : original signal
n : noise
f  sn
Error Measures (cont)
f : original image
g : noisy image
M , N : image size