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Estimation
Direct Linear Transformations

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Mixed Reality Laboratory | Hanyang University http://mr.hanyang.ac.kr

Direct Linear Transformation


Our Goal?
● ??? ???? H :

● ????
– ??? : Scale Factor!

● ????


Then How?
● ?? :
– (Why? Wikipedia Link)

● ????

{ }


Reorganize
● ?? =

● ?? ???? ?? Linear

● ? 3? ?? 2?? Dependent
(? ?? ???? ?? X)


What is a Solution?
● ??? ? : =0
– Not Interested!
● ? ?? ???
– A is 12 x 9 (8 x 9) Matrix[Rank : 8, Null-
Space : 1] → Has a Solution
● ? ? Scale? ??? ????
|| ||=1? ??? ???.


What is the Problem?
● 4? ??? ???? ???? ??
Rank ?? ??? ????? h? ??
Exact Solution? ??
– ??? ?? ? 4?? ???? Error? ????
???? ? Exact Solution? ??? ???
Solution? ?? ???.
(Overdetermined)


Constraints
● What to minimize?
● Avoid =0
● || ||=1
● If No Solution? Minimize || ||

● 3??, 4??? ???? : || ||/|| ||
● ?? ??? ??? Basic DLT Algorithm.


Basic DLT Algorithm


Cost Function


Algebraic Distance
● ????, ??? ?? X
● Normalize? ??? ?? ??? ?
● Linear(Unique Solution)
● ??? ??
●


Geometric Distance
● Error in one Image
– H? ?? Transfer? ??, ??? ???
Euclidean Distance ??? ??

● Error in two Images (Symmetric Transfer
Error)
– ? ??? ??? Error? ??? ???? →
Forward? Backward? ?? ??


Algebraic vs. Geometric
● Algebraic :
● Geometric :

● Geometric Distance? Algebraic Distance
? ???? ?? (?? : )
– Affine Transformation


Reprojection Error
● True Point? Measured Point ??? ??
(Correction)
– ??? True Point? ??? Estimate ???
● Transfer?? ?? True Point? Measured
Point ??? ??(Correction)
● Measured Point? Transfer? Perfect?


Reprojection vs. Symmetric


Geometric Interpretation
of Reprojection Error
● 2?? ??? ?? Homography? 4?? ???
?? ???? Fitting?? ???? ??? ? ?
?.
● ??? ?? 4?? ???? ??? ?? ???
?? ???. (Perfect Fit)
● ??? ?? ??? ? ??? Geometric
Distance? ???

– ?? Reprojection Error? ??


Geometric Error Problem
● ?? H? ?????.

● ??? H? Fitting?? Point
?????

? ?? ???, Non-Linear, But ??


Sampson Error
● ? First-order Approximation? ??
– ??? ?? ??

● ( )

● ? ??? ?? ?


Sampson Error
● ??? ???? ??(? ??)
– Error :

● ? ?? ? Dependent?
– ??? ??? ?? ?? ??? ? ??

● ? ??? ?? Hyperplane?? Fitting?
?? ??? ??


Statistical Cost Function
● ??? ??? Error? ?? ??? ??? ?
? ??


Error in one Image
● Exact Value? ??

● Log-likelihood? ??

● Maximum Likelihood Estimate? Log-
likelihood? Maximize = Minimize


Error in both Images
● Exact Value? ??

● ????? Log-likelihood? Maximize
= Minimize


Mahalanobis Distance
● One Image? ??, Log-likelihood?
Maximize = Mahalanobis Distance?
Minimize
● Both Images? ??, ? ???? ????
????
● ? Point?? ?? ????? ??


Transformation Invariance & Normalization


Invariance to Image Coordinate
Transformations
1.? ???? ?? T? ??? ??
2.?? ? ???? T’? ???
3.?? ??? H? ?? ? ??

● ??, ????? ?? ??? ??? ???
??


Non-invariance
● ??? ??? ??, H? H’? ?? ?? ??
???
● H? H’? 1:1 ??? ?? ? ??

● Transform? ?? ??? ?? H? ???
?? ? ??


Invariance of Geometric Error
● Euclidean Transformation T, T’? ??? ?
?

● Euclidean Distance? Euclidean
Transformation? ??? ?? ???? ?
?
● H? ?? Geometric Error? ???? ??
??? Invarient


Normalizing Transformations
● DLT Algorithm? ???? ?? ??
● SVD ??? ? ? ????? ?? ???
Round-Off Error? ?? ? ??
● Point Near Infinity? ?? ??

1.Centroid? ??? ??? Translate
2.?????? ?? ??? ? ??? Scale
3.? ???? ?? ??


Normalized DLT for 2D
Homographies


Iterative Minimization Methods


Iterative Minimization Method
● Minimize Cost Function? ??? ??
● Cons : ??, Initial Estimate ??, Local
Minimum ??, Stop ?? ??

● Iterative?? Implement? ??? ?? ?
???


Technique of Iterative
Minimization
● Cost Function
– Minimization? ?? ??
● Parametrization
– Finite # of Parameters
● Function Specification
– Parameters? ?? Function ??
● Initialization
– ??? ?? (?? Linear Algorithm ??)
● Iteration
● Cost Function? ?? ?? ??????


Initialization
● Normalized DLT Algorithm?? ?? ???
– Outlier? ?? ??

● Robust Estimation?? ???

● Dense Sampling of Parameter Space

● Fixed Point in Parameter Space


Golden Standard Algorithm +
Sampson Error


Robust Estimation


Robust Estimation
● ??? ??? ???? ??, ??? ??
Mismatch? ???? ?? (Outlier ???)

● ?? : Inlier? ???? ??? ??(??)?
?


RANSAC
● ?? : ?? Outlier? ??? ??? ???
?
1.? 2?? ???? ?? (Line? ????)
2.?? Distance ?? ?? Point? ?? ??
3.? ??? ??? ?, ?? ?? Point? ??
?? ?? ?? Robust Fit?? ?? & ??
? Point?? Inlier


RANSAC Robust Algorithm


RANSAC Problems
● ?? Distance? ?? ??
– Error Probability Distribution (DOF, Variance)
● Sample ??
– ???? ??( )
● Acceptable Consensus Set?
–


Robust Maximum
Likelihood Estimation
● RANSAC?? ??? Inlier?? ???? re-
estimate? ??.
– Inlier? ??? Optimal? ?? ???
– ML Cost Function? Minimize
● Robust Cost Function
– ?? Data? ?? ??? Cost Function


Automatic Computation of
a Homography


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