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

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3/25/98Morgan McGuire2 Acknowledgements Dr. Harold Stone, NEC Research Institute Bo Tao, Princeton University NEC Research Institute

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3/25/98Morgan McGuire3 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

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3/25/98Morgan McGuire4 New Technique Solves subproblem (practical case) O(n s (NlogN)/4 k +Nk) compared to O(NlogN), O(N 3 ) Correlations typically >.75 compared to.03

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3/25/98Morgan McGuire5 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

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3/25/98Morgan McGuire6 Differences Between Images Alignment Occlusion Noise Change

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3/25/98Morgan McGuire7 Sub-problem Domain Alignment = RSTL Occlusion < 50% Noise + Change = Small Square, finite, discrete images Image cropped from arbitrary infinite texture n n N pixels

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3/25/98Morgan McGuire8 RST Transformation

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3/25/98Morgan McGuire9 Fourier Rotation, Scale, and Translation Theorem Pixel DomainFourier Domain p = rotate(r, )P = rotate(R, ) p = dilate(r, s)F p = s 2. dilate(F r, 1/s) p = translate(r, x, y) F p = translate( F r, x, y)

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3/25/98Morgan McGuire10 For Infinite Images

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3/25/98Morgan McGuire11 In practice, we use the DFT Let X 0 = DFT(x 0 ) X 0 and x 0 are discrete, with N non-zero coefficients. Let X = DTFT(x) X 0 and x 0 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 X 0 and x 0 ?

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3/25/98Morgan McGuire12 Fourier Transform and Rotations

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3/25/98Morgan McGuire13 Theorem Infinite case: Fourier transform commutes with rotation Folklore: It is true for the finite case

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3/25/98Morgan McGuire14 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

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3/25/98Morgan McGuire15 Folklore is wrong Image Tile Rotate

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3/25/98Morgan McGuire16 The Mathematical Proof Transform, then rotate The Finite Fourier transform Windowing, sampling, infinite tiling continuous

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3/25/98Morgan McGuire17 The Mathematical Proof Rotate, then transform

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3/25/98Morgan McGuire18 Finite-Transform Pairs

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3/25/98Morgan McGuire19 The Artifacts

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3/25/98Morgan McGuire20 Fourier Transforms Oppenheim & Willsky Signals & Systems; Oppenheim and Schafer, Discrete-Time Signal Processing

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3/25/98Morgan McGuire21 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.

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3/25/98Morgan McGuire22 Correlation Computation

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3/25/98Morgan McGuire23 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

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3/25/98Morgan McGuire24 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

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3/25/98Morgan McGuire25 Sources of Degradation Frequency –Aliasing (from Tiling) –+ Artifact –Sampling Error Pixel –Image Window Occlusion –Image Noise

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3/25/98Morgan McGuire26 Algorithm Overview Norm. Circ. Corr. r p GG FMT f, d f,log d J Maximum Value Detector Peak Detector Norm. Corr. List of scale factors (s) exp J FMT WW HH Coarse ( x, y) FFT Dilate Rotate FFT Dilate Rotate FFT (Pixel) Correlation WW WW r m p h 1. Pre-Process 3. Recover Scale Parameter 4. Recover Rotation Parameter 5. Recover Translation Parameters 2. FMLP Transform

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3/25/98Morgan McGuire27 Problem: + Artifact None Rotation Dilation Translation Transformation DFT Image

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3/25/98Morgan McGuire28 Solution: Edge-Blurring Filter, G Image None DFT DiskBlurFilter

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3/25/98Morgan McGuire29 Problem:Need Orthogonal Invariants In the log-polar (log, ) domain: Fourier-Mellin transform:

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3/25/98Morgan McGuire30 Mapping ( x, y ) to (log, ) y x x =8 y =8 log /4 log /4 x =4 y =4

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3/25/98Morgan McGuire31 Sample Image Pair G(r)G(p) =17.0 o s =0.80 x =10.0 y = -15.0 N=65536 k=2

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3/25/98Morgan McGuire32 Nonzero Fourier Coefficients R P

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3/25/98Morgan McGuire33 Solution I: Rotation Signature 1. Selectively weight edge coefficients (J filter) 2. Integrate along axis is Scale and Translation Invariant. Pixel rotation appears as a cyclic shift => use simple 1d O(nlogn) correlation to recover rotation parameter.

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3/25/98Morgan McGuire34 Signatures of r and p

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3/25/98Morgan McGuire35 Correlations

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3/25/98Morgan McGuire36 Solution II: Scale Signature 1. Integrate along axis (rings) 2. Normalize by (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.

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3/25/98Morgan McGuire37 Raw S Signature

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3/25/98Morgan McGuire38 Filtered S Signature

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3/25/98Morgan McGuire39 S Correlation

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3/25/98Morgan McGuire40 New Registration Algorithm Norm. Circ. Corr. r p GG FMT f, d f,log d J Maximum Value Detector Peak Detector Norm. Corr. List of scale factors (s) exp J FMT WW HH Coarse ( x, y) FFT Dilate Rotate FFT Dilate Rotate FFT (Pixel) Correlation WW WW r m p h Compute full-resolution Correlation for small neighborhood of Coarse ( x, y) to refine.

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3/25/98Morgan McGuire41 Recovered Parameters

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3/25/98Morgan McGuire42 Disparity Map

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3/25/98Morgan McGuire43 Multiresolution for Speed Algorithm is O(NlogN) because of FFTs With k th order wavelet, O((NlogN)/4 k ) To refine, search 2 2k = 4 k positions Using binary search, k extra trials @ O(N) each Total algorithm is O((NlogN)/4 k + Nk)

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3/25/98Morgan McGuire44 Results & Confidence

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3/25/98Morgan McGuire45 Analysis of Results

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3/25/98Morgan McGuire46 Future Directions Better scale signature Use occlusion masks for FM techniques? Combining FM technique with feature based techniques

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