fundamental performance limits in image registration by dirk robinson and peyman milanfar ieee...
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Fundamental Performance Limits in Image Registration
By Dirk Robinson and Peyman Milanfar
IEEE Transactions on Image Processing Vol. 13, No. 9, 9/2004
CS679: Pattern RecognitionJosh Gleason and Rod Pickens
Topics Example Application of Performance Limits Image registration and errors Parameter estimation and errors Performance limits (bounds) of estimators Cramer-Rao lower bound
Robotic Helicopter toInspect Fukushima Reactors Purpose
Fly through damaged buildings Navigation Approach: SLAM
Install stereo sensors on craft Stereo vision 3D model Fly through 3D model
Critical Algorithm Image registration
Issue: Probability of collision How much bias in position? How much variance in position?
Analyze Accuracy of SLAM Errors in image registration
Decision: Will or will not helicopter successfully perform inspection?
Wiki Commons: Digital Globe
Fukushima Facility Building
Helicopter: http://flickrhivemind.net/Tags/apache,lego
SLAM: Simultaneous Localization and Mapping
Image Registration Errors
Errors Assume only translational
errors Δx and Δy
Higher order errors Not modeled
Asymptotic performance Bias
E(Δx) ≠ 0 and E(Δy) ≠ 0
Variance σ2 = E{(Δx+Δy)2} – E(Δx)E(Δy) >
0
Errors: Δx and Δy > 0
WikiCommons: Jazzjohn, 2012
Estimation: Accuracy and Precision
PDF: WikiCommons: PekajeTargets: www.caroline.com/teacher-resources
Target Practice 1D Error Distribution
(Bias)
(Variance)
LikelihoodFunction
ParameterValue
Truth
Performance Limits How accurate? Bias
Error about true position How precise? Variance
Error about mean of estimator What is best? Optimal
What are performance limits?
Is this best performance?
Targets: www.caroline.com/teacher-resources
Registration Errors Impact Navigation
[𝑥𝑦𝑧 ]𝑝
=[𝑥𝑦𝑧 ]𝑡𝑟𝑢𝑒
+[𝑏𝑥
𝑏𝑦
𝑏𝑧]𝑏𝑖𝑎𝑠
+[𝑛𝑥
𝑛𝑦
𝑛𝑧]𝑛𝑜𝑖𝑠𝑒
Stereo Vision
Navigation
3D Model
Image 1
Image 2[𝑥𝑦𝑧 ]
𝑝
Registration Errors Impact Navigation(Image registration errors cause 3D world model errors)
Small Bias, Small Variance
Large Bias, Small Variance
Small Bias, Large VarianceLarge Bias, Large Variance
Room 2: Fukushima Reactor
Damaged Wall Δ𝑥Δ𝑥 Δ𝑥
Enters Room 2: 3D mapping algorithm is a minimum variance, unbiased estimator (MVUE).
Room 1: Fukushima Reactor
Damaged Wall
Collides with Wall: not MVUE algo
Enters Room 2: MVUE algo
Minimum Variance, Unbiased Estimator: Cramer-Rao Lower Bound (CRLB)
Var(θ)
CRLB is a minimum variance unbiased estimator
Best
CRLBxp
E
2
2 ));(ln(
1)var(
CRLB is Best MVUE
Modeling Registration Errors and CRLB
CRLBxp
E
2
2 ));(ln(
1)var(
J=Fisher Information Matrix (FIM)
Log likelihood function as in Maximum Likelihood (ML) Estimation
)
BiasVariance
MSE = Mean Square Error used as measure of registration error
Registration, ML Estimation, and Objective Function
),(),(),( 11 nmnmfnmz ),(),(),( 2212 nmvnvmfnmz
cvnvmfnmznmfnmzvzPnm
2
,212
212
]),(),([)],(),([2
1));(log(
Log-likelihood function
f(m,n) = truth v = shift ε(m,n) = Gaussian noiseImage courtesy Matlab
Imagery
Objective Function
nm
nmDC vnvmf
vnvmfnmzvQ
, 212
, 212
),(
),(),()(
),(1 nm ),(2 nm
Deriving J(Φ) = FIM (Fisher Information)
𝜕2 ln (𝑃 (𝑧 ;𝑣 ))𝜕𝑣 𝑖
2 =𝜕𝜕𝑣 𝑖
[ 1𝜎2 ∑𝑚 ,𝑛
(𝑧 2−~𝑓 ) 𝜕
~𝑓𝜕𝑣 𝑖
]−𝐸 {𝜕2 ln (𝑃 (𝑧 ;𝑣 ))
𝜕 𝑣12 }= 1
𝜎2 ( 𝜕~𝑓
𝜕𝑣1 )2
Second partials of log likelihood
Expected value of second partials
−𝐸 {𝜕2 ln (𝑃 (𝑧 ;𝑣 ))𝜕 𝑣2
2 }= 1𝜎2 ( 𝜕
~𝑓
𝜕𝑣2 )2
−𝐸 {𝜕2 ln (𝑃 (𝑧 ;𝑣 ))𝜕𝑣1
❑𝑣2❑ }= 1
𝜎2 ( 𝜕~𝑓
𝜕𝑣1 )( 𝜕~𝑓
𝜕𝑣2 )
= =
Equating to x and to y
The Fisher Information Matrix (FIM)
= =
Given
The FIM is
𝐽 (𝑣 )= 1
𝜎2 [𝑎1 𝑎2𝑎3 𝑎4 ]
𝑎1=∑𝑚 ,𝑛
𝑓 𝑥2 (𝑚−𝑣1 ,𝑛−𝑣2) 𝑎3=∑
𝑚 ,𝑛
𝑓 𝑦2 (𝑚−𝑣1 ,𝑛−𝑣2)
𝑎2=∑𝑚 ,𝑛
𝑓 𝑥(𝑚−𝑣1 ,𝑛−𝑣2) 𝑓 𝑦(𝑚−𝑣1 ,𝑛−𝑣2)
Where
𝑟𝑒𝑐𝑎𝑙𝑙 :𝑀𝑆𝐸 (𝑣 )≥ 𝐽 (𝑣 )−1=𝐶𝑅𝐿𝐵(𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑)
Results: Registration Error Analysis
CRLB: MSE)ASD: average square distanceDC: maximum direct correlatorPyr: multiscale gradient-basedGB: gradient-based methodProj-GB: project GBPyr-Proj: Project PyrPhase: relative phase
Flat: Estimator biasSloping: Estimator variance
Conclusion Variance & Bias of an estimator Fisher Information Cramer-Rao lower bound (CRLB)
Quantitative measure of estimator performance Application of CRLB to image registration
BACKUP: Registration Algorithms Approximate Minimum Average Square Difference Approximate Maximum Direct Correlator (DC) Linear Gradient-Based Method (GB)
Fit and solve least squares
Multiscale (Pyramid) Gradient-Based Method (Pyr) Perform GB on multiresolution pyramid Iterate from coarsest to finest resolution
Projection Gradiant-Based Method (Proj-GB)
Projection Multiscale Gradient-Based Method (Pyr-Proj) Multiresolution version of Projection Gradiant-Based Method (Proj-GB)
Relative Phase (Phase) Match phases (in Fourier Domain) Solve using weighted least squares
BACKUP: Estimator Variance Example of estimator variance