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Complex Wavelet Wedgelets Curvelets

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 Wavelet Wedgelets Curvelets

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  1. Complex WaveletWedgeletsCurvelets Richard Baraniuk Rice University dsp.rice.edu

  2. (Dual Tree)Complex Wavelets Richard Baraniuk Rice University dsp.rice.edu

  3. Image Processing • Analyzing, modeling, processingimages • special class of 2-d functions • ex: digital photos • Problems: • approximation, compression, estimation, restoration, deconvolution, detection, classification, segmentation, …

  4. Image Structures • Prevalent image structures: • smooth regions - grayscale regularity • smooth edge contours - geometric regularity • Must exploit both for maximum performance

  5. Multiscale Image Analysis • Analyze an image at multiple scales …

  6. 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

  7. Wavelet-based Image Processing • Standard 2-D wavelet transform

  8. Wavelet Coefficient Sparsity • Most

  9. Wavelet Quadtree Structure • Wavelet analysis is self-similar • Image parent square divides into 4 children squares at next finer scale

  10. Wavelet Quadtree Correlations • Smooth regionsmallwc’s tumble down tree • Edge/ridge regionlarge wc’s tumble down tree

  11. 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

  12. 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

  13. Geometrical info not explicit • Modulations around singularities (geometry) • Inefficient-large number of significant WCs cluster around edge contours, no matter how smooth

  14. Wavelet Modulations • Wavelets are poor edge detectors • Severe modulation effects

  15. 2-D Wavelets: Poor Geometry Extraction

  16. Wavelets: Troubles In Paradise • Modulation effects around singularities • Shift varying • Poor geometry extraction in 2-D • Wavelet coefficients substantially aliased

  17. Wavelet Modulation Effects • Wavelet transform of edge • Seek amplitude/envelope • To extract amplitude need coherent representation

  18. 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)

  19. 1-D Dual-Tree CWT • Design g0[n] to be a ½ sample shift of h0[n]

  20. 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)

  21. Shift Invariance DWT CWT shifted step signal wavelet coefficients fine scale course scale

  22. 2-D Separable Real Wavelets 3 real wavelets Fourier domain

  23. 2-D Wavelets 3 real wavelets 6 complex wavelets Fourier domain Fourier domain

  24. 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

  25. Wavelet Image Processing

  26. Coherent Wavelet Processing real part +i imaginarypart

  27. Coherent Wavelet Processing |magnitude| x exp(iphase)

  28. Coherent Image Processing [Lina] magnitude FFT

  29. Coherent Image Processing [Lina] magnitude phase FFT

  30. Coherent Image Processing [Lina] magnitude phase FFT CWT

  31. Coherent Wavelet Processing feature magnitude phase 1 edge Lcoherent “speckle” Lincoherent > 1 edge smooth S undefined

  32. Coherent Segmentation feature magnitude phase 1 edge Lcoherent “speckle” Lincoherent > 1 edge smooth S undefined

  33. Local CWT Hilbert Transform

  34. Edge Geometry Extraction in 2-D  r • CWT magnitude encodes angleq • CWT phase encodes offsetr q

  35. Edge Geometry Extraction in 2-D estimated original

  36. Near Rotation Invariance DWT CWT

  37. Near Rotation Invariance DWT CWT

  38. Barbara

  39. Barbara Rotated CW 5.7o

  40. Barb’s Books

  41. Denoising Barb’s Books [Selesnick] noisy books DWT thresholding CWT thresholding

  42. Denoising Barb’s Books Wavelets and Subband Coding Vetterli and Kovacevic My Life as a DogParis Hilton Numerical Analysis of Wavelet Methods Albert Cohen

  43. 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

  44. X-letsWedgeletsCurvelets Richard Baraniuk Rice University dsp.rice.edu

  45. X-letsWedgeletsCurvelets Richard Baraniuk Rice University dsp.rice.edu

  46. Wavelet-based Image Processing

  47. Wavelet Challenges • Geometrical info not explicit • Inefficient-large number of large wc’s cluster around edge contours, no matter how smooth

  48. Wavelets and Cartoons 13 26 52 • Even for a smooth C2 contour, which straightens at fine scales…

  49. 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

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