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

March 25, 1998 Morgan McGuire

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**Acknowledgements Dr. Harold Stone, NEC Research Institute**

Bo Tao, Princeton University NEC Research Institute 3/25/98 Morgan McGuire

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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 3/25/98 Morgan McGuire

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**New Technique Solves subproblem (practical case)**

O(ns(NlogN)/4k+Nk) compared to O(NlogN), O(N3) Correlations typically > .75 compared to .03 3/25/98 Morgan McGuire

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**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 3/25/98 Morgan McGuire

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**Differences Between Images**

Alignment Occlusion Noise Change 3/25/98 Morgan McGuire

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**Sub-problem Domain Alignment = RSTL Occlusion < 50%**

Noise + Change = Small Square, finite, discrete images Image cropped from arbitrary infinite texture n N pixels 3/25/98 Morgan McGuire

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RST Transformation 3/25/98 Morgan McGuire

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**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) 3/25/98 Morgan McGuire

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†For Infinite Images 3/25/98 Morgan McGuire

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**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? 3/25/98 Morgan McGuire

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**Fourier Transform and Rotations**

3/25/98 Morgan McGuire

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**Theorem Infinite case: Fourier transform commutes with rotation**

Folklore: It is true for the finite case 3/25/98 Morgan McGuire

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**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 3/25/98 Morgan McGuire

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**Folklore is wrong Image Tile Rotate Tile Image Rotate 3/25/98**

Morgan McGuire

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**The Mathematical Proof**

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

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**The Mathematical Proof**

Rotate, then transform 3/25/98 Morgan McGuire

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**Finite-Transform Pairs**

3/25/98 Morgan McGuire

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The Artifacts 3/25/98 Morgan McGuire

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**Fourier Transforms 3/25/98 Morgan McGuire**

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

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**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. 3/25/98 Morgan McGuire

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**Correlation Computation**

3/25/98 Morgan McGuire

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**Prior Art Alliney & Morandi (1986) Reddy & Chatterji (1996)**

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 3/25/98 Morgan McGuire

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**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 3/25/98 Morgan McGuire

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**Sources of Degradation**

Frequency Aliasing (from Tiling) “+” Artifact Sampling Error Pixel Image Window Occlusion Image Noise 3/25/98 Morgan McGuire

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**5. Recover Translation Parameters**

Algorithm Overview r p G FMT fq,rd fq,logrdq J q Maximum Value Detector Peak Detector Norm. Corr. List of scale factors (s) exp W H Coarse (Dx, Dy) FFT Dilate Rotate (Pixel) Correlation W r m p h 1. Pre-Process 5. Recover Translation Parameters 2. FMLP Transform 4. Recover Rotation Parameter 3. Recover Scale Parameter Norm. Circ. Corr. 3/25/98 Morgan McGuire

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**Problem: “+” Artifact Transformation**

None Rotation Dilation Translation Transformation DFT Image 3/25/98 Morgan McGuire

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**Solution: “Edge-Blurring” Filter, G**

Image None DFT Disk Blur Filter 3/25/98 Morgan McGuire

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**Problem:Need Orthogonal Invariants**

Fourier-Mellin transform: In the “log-polar” (logr,q) domain: 3/25/98 Morgan McGuire

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**Mapping (wx,wy) to (logr,q)**

q=p/4 logr=2, q=3p/4 wx=4 wy=4 wy wx wx=8 wy=8 3/25/98 Morgan McGuire

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**Sample Image Pair f = 17.0o s = 0.80 Dx = 10.0 Dy = -15.0 G(r) G(p)**

N = 65536 k = 2 G(r) G(p) 3/25/98 Morgan McGuire

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**Nonzero Fourier Coefficients**

P 3/25/98 Morgan McGuire

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**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. 3/25/98 Morgan McGuire

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F Signatures of r and p 3/25/98 Morgan McGuire

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F Correlations 3/25/98 Morgan McGuire

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**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. 3/25/98 Morgan McGuire

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Raw S Signature 3/25/98 Morgan McGuire

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Filtered S Signature 3/25/98 Morgan McGuire

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S Correlation 3/25/98 Morgan McGuire

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**New Registration Algorithm**

p G FMT fq,rd fq,logrdq J q Maximum Value Detector Peak Detector Norm. Corr. List of scale factors (s) exp W H Coarse (Dx, Dy) FFT Dilate Rotate (Pixel) Correlation W r m p h Norm. Circ. Corr. Compute full-resolution Correlation for small neighborhood of Coarse (Dx, Dy) to refine. 3/25/98 Morgan McGuire

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Recovered Parameters 3/25/98 Morgan McGuire

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Disparity Map 3/25/98 Morgan McGuire

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**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 O(N) each Total algorithm is O((NlogN)/4k + Nk) 3/25/98 Morgan McGuire

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Results & Confidence 3/25/98 Morgan McGuire

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Analysis of Results 3/25/98 Morgan McGuire

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**Future Directions Better scale signature**

Use occlusion masks for FM techniques? Combining FM technique with feature based techniques 3/25/98 Morgan McGuire

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