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Complex Wavelet Wedgelets Curvelets. Richard Baraniuk Rice University dsp.rice.edu. (Dual Tree) Complex Wavelets. Richard Baraniuk Rice University dsp.rice.edu. Image Processing. Analyzing, modeling, processing images special class of 2-d functions ex: digital photos Problems:
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Complex WaveletWedgeletsCurvelets Richard Baraniuk Rice University dsp.rice.edu
(Dual Tree)Complex Wavelets Richard Baraniuk Rice University dsp.rice.edu
Image Processing • Analyzing, modeling, processingimages • special class of 2-d functions • ex: digital photos • Problems: • approximation, compression, estimation, restoration, deconvolution, detection, classification, segmentation, …
Image Structures • Prevalent image structures: • smooth regions - grayscale regularity • smooth edge contours - geometric regularity • Must exploit both for maximum performance
Multiscale Image Analysis • Analyze an image at multiple scales …
Multiscale Image Analysis • Analyze an image at multiple scales • How? Zoom out and record information lost in wavelet coefficients … info 1 info 2 info 3
Wavelet-based Image Processing • Standard 2-D wavelet transform
Wavelet Coefficient Sparsity • Most
Wavelet Quadtree Structure • Wavelet analysis is self-similar • Image parent square divides into 4 children squares at next finer scale
Wavelet Quadtree Correlations • Smooth regionsmallwc’s tumble down tree • Edge/ridge regionlarge wc’s tumble down tree
1-D Piecewise Smooth Signals • smooth except for singularities at a finite number of 0-D points Fourier sinusoids: suboptimal greedy approximation and extraction wavelets: near-optimal greedy approximation extract singularity structure
2-D Piecewise Smooth Signals • smooth except for singularities along a finite number of smooth 1-D curves • Challenge: analyze/approximategeometric structure geometry texture texture
Geometrical info not explicit • Modulations around singularities (geometry) • Inefficient-large number of significant WCs cluster around edge contours, no matter how smooth
Wavelet Modulations • Wavelets are poor edge detectors • Severe modulation effects
Wavelets: Troubles In Paradise • Modulation effects around singularities • Shift varying • Poor geometry extraction in 2-D • Wavelet coefficients substantially aliased
Wavelet Modulation Effects • Wavelet transform of edge • Seek amplitude/envelope • To extract amplitude need coherent representation
1-D Complex Wavelets [Grossman, Morlet, Lina, Abry, Flandrin, Mallat, Bernard, Kingsbury, Selesnick, Fernandes, van Spaendonck, Orchard, …] • real waveleteven symmetryimaginary waveletodd symmetry • Hilbert transform pair(complex Gabor atom) • Alias-free; shift invariant • Coherent wavelet representation (magnitude/phase)
1-D Dual-Tree CWT • Design g0[n] to be a ½ sample shift of h0[n]
1-D Complex Wavelet • Design g0[n] to be a ½ sample shift of h0[n] • In the limit, wavelets are 90 degrees out of phase(Hilbert transform pair)
Shift Invariance DWT CWT shifted step signal wavelet coefficients fine scale course scale
2-D Separable Real Wavelets 3 real wavelets Fourier domain
2-D Wavelets 3 real wavelets 6 complex wavelets Fourier domain Fourier domain
2-D Complex Wavelets[Lina, Kingsbury, Selesnick] • 4x redundant tight frame • 6 directional subbandsaligned along 6 1-D manifold directions • Magnitude/phase • Even/odd real/imag symmetry • Almost Hilbert transform pair (complex Gabor atom) • Almost shift invariant • Compute using 1-D CWT -75 +75 +45 +15 -15 -45 real imag
Coherent Wavelet Processing real part +i imaginarypart
Coherent Wavelet Processing |magnitude| x exp(iphase)
Coherent Image Processing [Lina] magnitude FFT
Coherent Image Processing [Lina] magnitude phase FFT
Coherent Image Processing [Lina] magnitude phase FFT CWT
Coherent Wavelet Processing feature magnitude phase 1 edge Lcoherent “speckle” Lincoherent > 1 edge smooth S undefined
Coherent Segmentation feature magnitude phase 1 edge Lcoherent “speckle” Lincoherent > 1 edge smooth S undefined
Edge Geometry Extraction in 2-D r • CWT magnitude encodes angleq • CWT phase encodes offsetr q
Edge Geometry Extraction in 2-D estimated original
Near Rotation Invariance DWT CWT
Near Rotation Invariance DWT CWT
Denoising Barb’s Books [Selesnick] noisy books DWT thresholding CWT thresholding
Denoising Barb’s Books Wavelets and Subband Coding Vetterli and Kovacevic My Life as a DogParis Hilton Numerical Analysis of Wavelet Methods Albert Cohen
CWT – Summary • Complex wavelets behave more like a “local Fourier Transform” than usual real wavelets • magnitude and phase representation • very useful image geometry information • Many formulations; one attractive one is viadual-tree filterbank • 2x redundant in 1-d, 4x redundant in 2-d [I. Selesnick, N. G. Kingsbury, and R. G. Baraniuk, “The Dual-Tree Complex Wavelet Transform – A Coherent Framework for Multiscale Signal and Image Processing,” IEEE Signal Processing Magazine, November 2005] • see also recent work by Mike Orchard
X-letsWedgeletsCurvelets Richard Baraniuk Rice University dsp.rice.edu
X-letsWedgeletsCurvelets Richard Baraniuk Rice University dsp.rice.edu
Wavelet Challenges • Geometrical info not explicit • Inefficient-large number of large wc’s cluster around edge contours, no matter how smooth
Wavelets and Cartoons 13 26 52 • Even for a smooth C2 contour, which straightens at fine scales…
2-D Wavelets: Poor Approximation 13 26 52 • Even for a smooth C2 contour, which straightens at fine scales… • Too many wavelets required! -term wavelet approximation not