an image registration technique for recovering rotation, scale and translation parameters march 25,...
TRANSCRIPT
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
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Problem DomainSatellite, 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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New Technique
• Solves subproblem (practical case)
• O(ns(NlogN)/4k+Nk) compared to O(NlogN), O(N3)
• Correlations typically > .75 compared to .03
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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
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Differences Between Images
• Alignment
• Occlusion
• Noise
• Change
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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
N pixels
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RST Transformation
1100
cossin
sincos
1p
p
r
r
y
x
yss
xss
y
x
))cossin(,)sincos((, syxysyxxpyxr
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Fourier Rotation, Scale, and Translation Theorem†
ssFseF yxyxpsyxj
yxryx /cossin,/sincos, 2/)(
prr FPFRrF ,Let .DTFT Where
Pixel Domain Fourier Domain
p = rotate(r, ) P = rotate(R, )
p = dilate(r, s) Fp = s2 . dilate(Fr, 1/s)
p = translate(r, x, y) Fp = translate(Fr, x, y)
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†For Infinite Images
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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)
k
kXX )2(** sin0
k
kNtxx )(*0
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?
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Fourier Transform and Rotations
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Theorem
Infinite case: Fourier transform commutes with rotation
Folklore: It is true for the finite case
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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
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Folklore is wrongImage
Image
Tile
Tile Rotate
Rotate
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The Mathematical Proof
F f x y , ,
Transform, then rotate
The Finite Fourier transform
f x y,( , ) F
j x y N2 ( ) / D f x yyx
, ,
Windowing, sampling,
infinite tiling
D f x y eyx
j x y N , , ( )/2
continuous
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The Mathematical ProofRotate, then transform
F f x y , ,
D f x y eyx
, , j x y N2 /
D f x y eyx
, , j x y N2 /
x y
D f x y e j x y N , ( , ) ( )/2
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Finite-Transform Pairs
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The Artifacts
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Fourier Transforms
ts.coefficien nonzero with finite
are they so,~
,~ of periods single are and :
sampled.-sub is ~
. period ~
,~
][~][~
;][~
][~:
][;][:
;:
00
1
0
/21
0
1/21
21
21
N
XxXxDFT
XXNXx
enxkXekXnxDFS
enxXdeXnxDTFT
dtetxXdeXtxCTFT
N
n
NnkjN
kN
Nknj
N
n
njnj
jtjt
Oppenheim & Willsky Signals & Systems; Oppenheim and Schafer, Discrete-Time Signal Processing
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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.
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Correlation Computation
( , )x y
x yN
x y
xN
x yN
y
i i i i
i i i i
1
1 122
22
xy
x y
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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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An Empirical ObservationEven 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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Sources of Degradation
• Frequency– Aliasing (from Tiling)
– “+” Artifact
– Sampling Error
• Pixel– Image Window Occlusion
– Image Noise
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Algorithm Overview
Norm. Circ. Corr.
r p
G G
FMT
f,d f,d
f,logd
f,logd
J
Maximum Value Detector
Peak Detector
Norm. Corr.
List of scale factors (s)
exp
J
FMT
W W
HH
Coarse (x, y)
FFTFFT
Dilate
Rotate
FFT
Dilate
Rotate
FFT
(Pixel) Correlation
W W W W
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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Problem: “+” Artifact
None Rotation Dilation TranslationTransformation
DFT
Image
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Solution: “Edge-Blurring” Filter, G
Image
None
DFT
Disk BlurFilter
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Problem:Need Orthogonal Invariants
dxdyeyxrG
RR
yxj
yx
)(,
,sin,cos
In the “log-polar” (log,) domain:
Added NoiseTranslate
AliasingNoise,,logby Translateby Dilate
AliasingNoise,,by Shift Cyclicby Rotate
ss
Domain FMLPDomain Pixel
Fourier-Mellin transform:
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Mapping (x,y) to (log,)
y
x
x=8
y=8
log
log
/4
log
/4
x=4
y=4
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Sample Image Pair
G(r) G(p)
= 17.0o
s = 0.80
x = 10.0
y = -15.0
N = 65536
k = 2
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Nonzero Fourier Coefficients
R P
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Solution I: Rotation Signature
2/
0
sin,cosn
r dRJ
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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Signatures of r and p
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Correlations
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Solution II: Scale Signature
0
1 sin,coslog dRHSr
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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Raw S Signature
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Filtered S Signature
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S Correlation
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New Registration Algorithm
Norm. Circ. Corr.
r p
G G
FMT
f,d f,d
f,logd
f,logd
J
Maximum Value Detector
Peak Detector
Norm. Corr.
List of scale factors (s)
exp
J
FMT
W W
HH
Coarse (x, y)
FFTFFT
Dilate
Rotate
FFT
Dilate
Rotate
FFT
(Pixel) Correlation
W W W W
r m p h
Compute full-resolution Correlation for small neighborhood of Coarse (x, y) to refine.
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Recovered Parameters s (x, y)
Fish Actual 17.00 0.80 (10.00, -15.00)(natural) Recovered 17.25 0.75 (10.17, -14.98)
Peak Corr. 0.82 0.53 0.90
1d RMS = 3.42 2d RMS = 4.84
Image
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Disparity Map
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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 trials @ O(N) each
• Total algorithm is O((NlogN)/4k + Nk)
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Results & ConfidenceTrial s (x, y)
1 suburban Actual 31.00 1.00 (0.00, 0.00)(aerial) Recovered 31.00 1.00 (0.00, 0.00)
Peak Corr. 0.90 0.48 0.972 suburban Actual 0.00 1.40 (0.00, 0.00)
Recovered 0.25 1.37 (0.00, 0.00)Peak Corr. 0.86 0.31 0.84
3 suburban Actual 0.00 1.00 (10.00, -3.00)Recovered 0.00 1.00 (10.00, -3.00)Peak Corr. 0.91 0.83 1.00
4 fish Actual 24.00 0.70 (7.00, 12.00)(natural) Recovered 25.00 0.72 (7.06, 11.99)
Peak Corr. 0.78 0.45 0.955 mamogram Actual -10.00 1.20 (10.00, 9.00)
(x-ray) Recovered -10.25 1.15 (9.76, 8.75)Peak Corr. 0.94 0.73 0.58
6 essai Actual -15.00 1.50 (-10.00, -10.00)(satellite) Recovered -15.25 1.56 (-11.33, -11.40)
Peak Corr. 0.94 0.40 0.78
Image
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Analysis of Results
s (x, y) TotalMean Correlation 0.89 0.53 0.85 0.76
Mean RMS Error (1d) 1.92
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Future Directions
• Better scale signature
• Use occlusion masks for FM techniques?
• Combining FM technique with feature based techniques