cheirality invariant young ki baik computer vision lab
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Cheirality InvariantCheirality Invariant
Young Ki Baik
Computer Vision Lab.
ContentsContents
Introduce Cheirality Quasi-affine reconstruction
Cheirality invariant property 2D / 3D case
Cheiral inequalities Algorithm Conclusion and Future work
IntroduceIntroduce Convex, Convex hull
A subset B of R is called convex if the line segment joining any two points in B also lies entirely within B.
Convex hull of B is the smallest convex set containing B.
R
BA
C
Convex hull
Convex
IntroduceIntroduce
What is Cheirality? Phenomenon of breaking the Convex property by
transformation.
H
IntroduceIntroduce
Why cheirality is occurred? Plane at infinity segments convex by H.
Hπ
IntroduceIntroduce
Quasi-affine transformation
Euclidean
Similarity
Affine
Projective
Euclidean
Similarity
Affine
Projective
Quasi-affine
IntroduceIntroduce
Quasi-affine transformation
R t 0 1λR t 0
1
KR t
L
KR t 0 1
λ
K
L
Euclidean
Similarity
Affine
Quasi-affine
Projective
IntroduceIntroduce
Quasi-affine transformation
Quasi-affine
Projective
L
L
KR t
L
IntroduceIntroduce
Quasi-affine reconstruction (QUARC) We have to perform QUARC before Metric
reconstruction.
Quasi-affine
Metric
IntroduceIntroduce
Quasi-affine reconstruction (QUARC) Search for safe region and transform plane at
infinity.
Quasi-affine
LL’
L’
I 0 L’
Cheirality invariant propertyCheirality invariant property
2D case Suppose that {xi} and {yi} are corresponding
points in two view and h representing a planar projectivity such that h( xi ) = wi yi .
To ensure convex hull, all wi have the same sign.
h (L) = L∞
h (B) ∩ L∞ = 0
Cheirality invariant propertyCheirality invariant property
2D case To ensure convex hull, all wi have the same sign.
x∞ = [ a, b, c, 0 ]T
L
x = [ a, b, c, +w ]T x = [ a, b, c, -w ]T
Cheirality invariant propertyCheirality invariant property
3D case ( Depth of points )
3d point
2d point
Camera center
3
det );(
m
MPX
T
wsigndepth
C
X
3m
3mX
]|[ 4pMP
X ofcomponent Last :T
x
xPX w
Cheirality invariant propertyCheirality invariant property
3D case ( Sign of depth ) Positive
Negative
Zero or infinite
,0);( PXdepth
3d point
2d point
Camera center
CX
0);( PXdepth
X
X
L
C
CX
0);( PXdepth
3)0(
det );(
m
MPX
T
wsigndepth
Cheirality invariant propertyCheirality invariant property
3D case ( Sign of depth ) We only concern with the sign of depth.
3
det );(
m
MPX
T
wsignsigndepthsign
]|[ 4pMP
X ofcomponent Last :T
xPX w Mdet wTsign
Cheirality invariant propertyCheirality invariant property
3D case ( Sign of depth with transformation )
MPX det);( wTsigndepthsign
]1 0, 0, 0,[TE CEXE TTwsign TT XE
4det CM
Cheirality invariant propertyCheirality invariant property
3D case ( Sign of depth with transformation )
CEXEPX TTwsigndepthsign );(
1);( PH
1 CEHXEPHHX TTwsigndepthsign
1det);( HHCEHXEPHHX P1 TTwsigndepthsign
Cheirality invariant propertyCheirality invariant property
3D case ( Sign of depth with transformation )
1det);( HHCEHXEPHHX P1 TTwsigndepthsign
PCπXπ TTwsign
]1 0, 0, 0,[TETTπHE
Hdetsign
Cheiral inequalitiesCheiral inequalities
Solving the cheiral inequalities
0);( PCvXvPX TTwsigndepthsign
0);( PCvXvPX TTsigndepthsign
j
ijT
Ti
allfor 0
allfor 0
vC
vX
suppose that w > 0
Cheiral inequalitiesCheiral inequalities
Solving the cheiral inequalities We can solve inequalities using linear programming
(such as the simplex method).
We can perform QUARC using v.
j
ijT
Ti
allfor 0
allfor 0
vC
vX
Quasi-affine
I 0 vT
AlgorithmAlgorithm
Summary of algorithm Obtain set ( X, P ) For each pair, search w from PX = wx Replace sign of P or X to ensure that each w > 0 Form the cheiral inequalities : For each of value δ = ±1, choose a solution with
maximum d (Av > d >0) using linear programming (Simplex method)
Define H having last row equal to v and sign of det(H) = δ
Implement QUARC using H
0 , 0 vCvX jTTi
Conclusion and Future workConclusion and Future work
Conclusion QUARC using cheiral inequalities Untwisted 3D object reconstruction with QUARC
Future work Linear programming problem (Simplex method) Implementation of QUARC algorithm
The End
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