Cheirality Invariant Young Ki Baik Computer Vision Lab.

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


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

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


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


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


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


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


3D case ( Sign of depth with transformation )

MPX det);( wTsigndepthsign

]1 0, 0, 0,[TE CEXE TTwsign TT XE

4det CM



CEXEPX TTwsigndepthsign );(

1);( PH

1 CEHXEPHHX TTwsigndepthsign

1det);( HHCEHXEPHHX P1 TTwsigndepthsign



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

Cheirality Invariant Young Ki Baik Computer Vision Lab.

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