Download presentation

Presentation is loading. Please wait.

Published byAvery Raynolds Modified over 4 years ago

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 3/25/98 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 3/25/98 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 3/25/98 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 3/25/98 Morgan McGuire

6
**Differences Between Images**

Alignment Occlusion Noise Change 3/25/98 Morgan McGuire

7
**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

8
RST Transformation 3/25/98 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) 3/25/98 Morgan McGuire

10
†For Infinite Images 3/25/98 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? 3/25/98 Morgan McGuire

12
**Fourier Transform and Rotations**

3/25/98 Morgan McGuire

13
**Theorem Infinite case: Fourier transform commutes with rotation**

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

15
**Folklore is wrong Image Tile Rotate Tile Image Rotate 3/25/98**

Morgan McGuire

16
**The Mathematical Proof**

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

17
**The Mathematical Proof**

Rotate, then transform 3/25/98 Morgan McGuire

18
**Finite-Transform Pairs**

3/25/98 Morgan McGuire

19
The Artifacts 3/25/98 Morgan McGuire

20
**Fourier Transforms 3/25/98 Morgan McGuire**

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

22
**Correlation Computation**

3/25/98 Morgan McGuire

23
**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

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 3/25/98 Morgan McGuire

25
**Sources of Degradation**

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

26
**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

27
**Problem: “+” Artifact Transformation**

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

28
**Solution: “Edge-Blurring” Filter, G**

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

29
**Problem:Need Orthogonal Invariants**

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

30
**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

31
**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

32
**Nonzero Fourier Coefficients**

P 3/25/98 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. 3/25/98 Morgan McGuire

34
F Signatures of r and p 3/25/98 Morgan McGuire

35
F Correlations 3/25/98 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. 3/25/98 Morgan McGuire

37
Raw S Signature 3/25/98 Morgan McGuire

38
Filtered S Signature 3/25/98 Morgan McGuire

39
S Correlation 3/25/98 Morgan McGuire

40
**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

41
Recovered Parameters 3/25/98 Morgan McGuire

42
Disparity Map 3/25/98 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 O(N) each Total algorithm is O((NlogN)/4k + Nk) 3/25/98 Morgan McGuire

44
Results & Confidence 3/25/98 Morgan McGuire

45
Analysis of Results 3/25/98 Morgan McGuire

46
**Future Directions Better scale signature**

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

Similar presentations

OK

Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.

Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google