09-10-2012, 04:10 PM
Multi-Resolution Analysis
Multi-Resolution.ppt (Size: 1.14 MB / Downloads: 105)
FFT Vs Wavelet
FFT, basis functions: sinusoids
Wavelet transforms: small waves, called wavelet
FFT can only offer frequency information
Wavelet: frequency + temporal information
Fourier analysis doesn’t work well on discontinuous, “bursty” data
music, video, power, earthquakes,…
Fourier versus Wavelets
Fourier
Loses time (location) coordinate completely
Analyses the whole signal
Short pieces lose “frequency” meaning
Wavelets
Localized time-frequency analysis
Short signal pieces also have significance
Scale = Frequency band
More on scaling
It lets you either narrow down the frequency band of interest, or determine the frequency content in a narrower time interval
Scaling = frequency band
Good for non-stationary data
Low scalea Compressed wavelet Rapidly changing detailsHigh frequency .
High scale a Stretched wavelet Slowly changing, coarse features Low frequency
Discrete Wavelet Transform
“Subset” of scale and position based on power of two
rather than every “possible” set of scale and position in continuous wavelet transform
Behaves like a filter bank: signal in, coefficients out
Down-sampling necessary (twice as much data as original signal)
Image Pyramids
Original image, the base of the pyramid, in the level J =log2N, Normally truncated to P+1 levels.
Approximation pyramids, predication residual pyramids
Steps: .1. Compute a reduced-resolution approximation (from j to j-1 level) by downsampling; 2. Upsample the output of step1, get predication image; 3. Difference between the predication of step 2 and the input of step1.
Subband Coding
Filters h1(n) and h2(n) are half-band digital filters, their transfer characteristics H0-low pass filter, output is an approximation of x(n) and H1-high pass filter, output is the high frequency or detail part of x(n)
Criteria: h0(n), h1(n), g0(n), g1(n) are selected to reconstruct the input perfectly.
Fundamental Requirements of MRA
The scaling function is orthogonal to its integer translate
The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales
The only function that is common to all Vj is f(x) = 0
Any function can be represented with arbitrary precision
Applications of wavelets
Pattern recognition
Biotech: to distinguish the normal from the pathological membranes
Biometrics: facial/corneal/fingerprint recognition
Feature extraction
Metallurgy: characterization of rough surfaces
Trend detection:
Finance: exploring variation of stock prices
Perfect reconstruction
Communications: wireless channel signals
Video compression – JPEG 2000