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

Presentation on theme: "An image registration technique for recovering rotation, scale and translation parameters March 25, 1998 Morgan McGuire."— Presentation transcript:

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 3/25/98 Morgan McGuire

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

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

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

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

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

RST Transformation 3/25/98 Morgan McGuire

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

†For Infinite Images 3/25/98 Morgan McGuire

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

Fourier Transform and Rotations
3/25/98 Morgan McGuire

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

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

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

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

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

Finite-Transform Pairs
3/25/98 Morgan McGuire

The Artifacts 3/25/98 Morgan McGuire

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

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

Correlation Computation
3/25/98 Morgan McGuire

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

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

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

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

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

Solution: “Edge-Blurring” Filter, G
Image None DFT Disk Blur Filter 3/25/98 Morgan McGuire

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

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

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

Nonzero Fourier Coefficients
P 3/25/98 Morgan McGuire

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

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

F Correlations 3/25/98 Morgan McGuire

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

Raw S Signature 3/25/98 Morgan McGuire

Filtered S Signature 3/25/98 Morgan McGuire

S Correlation 3/25/98 Morgan McGuire

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

Recovered Parameters 3/25/98 Morgan McGuire

Disparity Map 3/25/98 Morgan McGuire

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

Results & Confidence 3/25/98 Morgan McGuire

Analysis of Results 3/25/98 Morgan McGuire

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

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

Similar presentations