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Complex Wavelet Wedgelets Curvelets

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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 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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