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An image registration technique for recovering rotation, scale and translation parameters

An image registration technique for recovering rotation, scale and translation parameters . March 25, 1998 Morgan McGuire. Acknowledgements. Dr. Harold Stone, NEC Research Institute Bo Tao, Princeton University NEC Research Institute. Problem Domain.

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An image registration technique for recovering rotation, scale and translation parameters

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  1. An image registration technique for recovering rotation, scale and translation parameters March 25, 1998 Morgan McGuire

  2. Acknowledgements • Dr. Harold Stone, NEC Research Institute • Bo Tao, Princeton University • NEC Research Institute Morgan McGuire

  3. Problem Domain Satellite, Aerial, and Medical sensors produce series images which need to be aligned for analysis. These images may differ by any transformation (possible noninvertible). Images courtesy of Positive Systems Morgan McGuire

  4. New Technique • Solves subproblem (practical case) • O(ns(NlogN)/4k+Nk) compared to O(NlogN), O(N3) • Correlations typically > .75 compared to .03 Morgan McGuire

  5. Structure of the Talk • Differences Between Images • Fourier RST Theorem • Degradation in the Finite Case • New Registration Algorithm • Edge Blurring Filter • Rotation & Scale Signatures • Experimental Results • Conclusions Morgan McGuire

  6. Differences Between Images • Alignment • Occlusion • Noise • Change Morgan McGuire

  7. ¥ n N pixels n Sub-problem Domain • Alignment = RSTL • Occlusion < 50% • Noise + Change = Small • Square, finite, discrete images • Image cropped from arbitrary infinite texture Morgan McGuire

  8. RST Transformation Morgan McGuire

  9. Fourier Rotation, Scale, and Translation Theorem† Pixel Domain Fourier Domain p = rotate(r, f) P = rotate(R, f) p = dilate(r, s) Fp = s2 . dilate(Fr, 1/s) p = translate(r, Dx, Dy) ÐFp = translate(ÐFr, Dx, Dy) Morgan McGuire

  10. †For Infinite Images Morgan McGuire

  11. In practice, we use the DFT Let X0 = DFT(x0) X0 and x0 are discrete, with N non-zero coefficients. Let X = DTFT(x) X0 and x0 are sub-sampled tiles (one period spans) of X and x. The Fourier RST theorem holds for X and x... does it also hold for X0 and x0? Morgan McGuire

  12. Fourier Transform and Rotations Morgan McGuire

  13. Theorem Infinite case: Fourier transform commutes with rotation Folklore: It is true for the finite case Morgan McGuire

  14. Using Fourier-Mellin Theory • Magnitude of Fourier Transform exhibits rotation, but not translation • Registration algorithm: • Correlate Fourier Transform magnitudes for rotation • Remove rotation, find translation • Generalizes to find scale factors, rotations, and translation as distinct operations Morgan McGuire

  15. Folklore is wrong Image Tile Rotate Tile Image Rotate Morgan McGuire

  16. The Mathematical Proof The Finite Fourier transform continuous Windowing, sampling, infinite tiling Transform, then rotate Morgan McGuire

  17. The Mathematical Proof Rotate, then transform Morgan McGuire

  18. Finite-Transform Pairs Morgan McGuire

  19. The Artifacts Morgan McGuire

  20. Fourier Transforms Oppenheim & Willsky Signals & Systems; Oppenheim and Schafer, Discrete-Time Signal Processing Morgan McGuire

  21. Tiling does not Commute with Rotation Tiled Image Rotated Tiled Image Tiled Rotated Image …so the Fourier RST Theorem does not hold for DFT transforms. Morgan McGuire

  22. Correlation Computation Morgan McGuire

  23. Prior Art • Alliney & Morandi (1986) • use projections to register translation-only in O(n), show aliasing in Fourier T theorem • Reddy & Chatterji (1996) • use Fourier RST theorem to register in O(NlogN) • Stone, Tao & McGuire (1997) • show aliasing in Fourier RST theorem Morgan McGuire

  24. An Empirical Observation Even though the Fourier RST Theorem does not hold for finite images, we observe the DFT does have a “signature” that transforms in a method predicted by the Theorem. Image DFT Magnitude Morgan McGuire

  25. Frequency Aliasing (from Tiling) “+” Artifact Sampling Error Pixel Image Window Occlusion Image Noise Sources of Degradation Morgan McGuire

  26. p r r m p h W W W W Dilate Dilate W W G G Rotate Rotate H H FMT FMT fq,logrdq fq,logrdq FFT FFT FFT FFT J J (Pixel) Correlation fq,rd fq,rd Norm. Corr. Coarse (Dx, Dy) Peak Detector Maximum Value Detector exp List of scale factors (s) q Algorithm Overview 1. Pre-Process 5. Recover Translation Parameters 2. FMLP Transform 4. Recover Rotation Parameter 3. Recover Scale Parameter Norm. Circ. Corr. Morgan McGuire

  27. None Rotation Dilation Translation Transformation Image DFT Problem: “+” Artifact Morgan McGuire

  28. Filter None Disk Blur Image DFT Solution: “Edge-Blurring” Filter, G Morgan McGuire

  29. Problem:Need Orthogonal Invariants Fourier-Mellin transform: In the “log-polar” (logr,q) domain: Morgan McGuire

  30. logr wy wx=8 wy=8 wx -q logr=3, q=p/4 logr=2, q=3p/4 Mapping (wx,wy) to (logr,q) wx=4 wy=4 Morgan McGuire

  31. Sample Image Pair f = 17.0o s = 0.80 Dx = 10.0 Dy = -15.0 N = 65536 k = 2 G(r) G(p) Morgan McGuire

  32. Nonzero Fourier Coefficients P R Morgan McGuire

  33. Solution I: Rotation Signature 1. Selectively weight “edge coefficients” (J filter) 2. Integrate along r axis F is Scale and Translation Invariant. Pixel rotation appears as a cyclic shift => use simple 1d O(nlogn) correlation to recover rotation parameter. Morgan McGuire

  34. F Signatures of r and p Morgan McGuire

  35. F Correlations Morgan McGuire

  36. Solution II: Scale Signature 1. Integrate along q axis (rings) 2. Normalize by r (area) 3. Enhance S/N ratio (H filter) S is Rotation and Translation Invariant. Pixel dilation appears as a translation => use simple 1d O(nlogn) correlation to recover scale parameter. Morgan McGuire

  37. Raw S Signature Morgan McGuire

  38. Filtered S Signature Morgan McGuire

  39. S Correlation Morgan McGuire

  40. p r r m p h W W W W Dilate Dilate W W G G Rotate Rotate H H FMT FMT fq,logrdq fq,logrdq FFT FFT FFT FFT J J (Pixel) Correlation fq,rd fq,rd Norm. Corr. Coarse (Dx, Dy) Peak Detector Maximum Value Detector exp List of scale factors (s) q New Registration Algorithm Norm. Circ. Corr. Compute full-resolution Correlation for small neighborhood of Coarse (Dx, Dy) to refine. Morgan McGuire

  41. Recovered Parameters Morgan McGuire

  42. Disparity Map Morgan McGuire

  43. Multiresolution for Speed • Algorithm is O(NlogN) because of FFT’s • With kth order wavelet, O((NlogN)/4k) • To refine, search 22k = 4k positions • Using binary search, k extra trials @ O(N) each • Total algorithm is O((NlogN)/4k + Nk) Morgan McGuire

  44. Results & Confidence Morgan McGuire

  45. Analysis of Results Morgan McGuire

  46. Future Directions • Better scale signature • Use occlusion masks for FM techniques? • Combining FM technique with feature based techniques Morgan McGuire

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