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Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of Computer Science University of North Carolina, Chapel Hill

Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

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Page 1: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 1

ROBOT VISION Lesson 6: Shape from Stereo

Matthias Rüther

Slides partial courtesy of Marc Pollefeys Department of Computer Science

University of North Carolina, Chapel Hill

Page 2: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 2

Content

Two View Geometry– Epipolar Geometry

– 3D reconstruction

Computing F– Point Correspondences

– Interest Points

– Matching

Page 3: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 3

C, C’, x, x’ and X are coplanar

Epipolar Geometry

Page 4: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 4

(i) Correspondence geometry: Given an image point x in the first view, how does this constrain the position of the

corresponding point x’ in the second image?

(ii) Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views?

(iii) Scene geometry (structure): Given corresponding image points xi ↔x’i and cameras P, P’, what is the position of (their pre-image) X in space?

Three Questions

Page 5: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 5

What if only C, C’, x are known?

Epipolar Geometry

Page 6: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 6

All points on project on l and l’

Epipolar Geometry

Page 7: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 7

Family of planes and lines l and l’ Intersection in e and e’

Epipolar Geometry

Page 8: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 8

epipoles e,e’= intersection of baseline with image plane = projection of projection center in other image= vanishing point of camera motion direction

an epipolar plane = plane containing baseline (1-D family)

an epipolar line = intersection of epipolar plane with image(always come in corresponding pairs)

Epipolar Geometry

Page 9: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 9

Example: Converging Cameras

Page 10: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 10

Example: motion parallel with image plane

Page 11: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 11

e

e’

Example: forward motion

Page 12: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 12

algebraic representation of epipolar geometry

l'x

we will see that mapping is (singular) correlation (i.e. projective mapping from points to lines) represented by the fundamental matrix F

The Fundamental Matrix (F)

Page 13: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 13

geometric derivation

xHx' π

x'e'l' FxxHe' π

mapping from 2-D to 1-D family (rank 2)

The Fundamental Matrix (F)

0

0

0

12

13

23

aa

aa

aa

a

Page 14: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 14

algebraic derivation

λCxPλX IPP

PP'e'F

xPP'CP'l

(note: doesn’t work for C=C’ F=0)

xP

λX

The Fundamental Matrix (F)

Page 15: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 15

correspondence condition

0Fxx'T

The fundamental matrix satisfies the condition that for any pair of corresponding points x↔x’ in the two images

0l'x'T

The Fundamental Matrix (F)

Page 16: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 16

F is the unique 3x3 rank 2 matrix that satisfies x’TFx=0 for all x↔x’

(i) Transpose: if F is fundamental matrix for (P,P’), then FT is fundamental matrix for (P’,P)

(ii) Epipolar lines: l’=Fx & l=FTx’

(iii) Epipoles: on all epipolar lines, thus e’TFx=0, x e’TF=0, similarly Fe=0

(iv) F has 7 d.o.f. , i.e. 3x3-1(homogeneous)-1(rank2)

(v) F is a correlation, projective mapping from a point x to a line l’=Fx (not a proper correlation, i.e. not invertible)

The Fundamental Matrix (F)

Page 17: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 17

l,l’ epipolar lines, k line not through e l’=F[k]xl and symmetrically l=FT[k’]xl’

lke

kl lFk

e'

(pick k=e, since eTe≠0)

leFl' l'e'Fl T

Epipolar Line Geometry

Page 18: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 18

-1-T FHH'F̂ x'H''x̂ Hx,x̂

Derivation based purely on projective concepts

X̂P̂XHPHPXx -1

F invariant to transformations of projective 3-space

X̂'P̂XHHP'XP'x' -1

FP'P,

P'P,F

unique

not unique

canonical form

m]|[MP'0]|[IP

MmF

Invariance under projective transformation

Page 19: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 19

F matrix corresponds to P,P’ iff P’TFP is skew-symmetric

X0,FPXP'X TT

Possible choice:

]e'|F][[e'P' 0]|[IP

Canonical representation:

]λe'|ve'F][[e'P' 0]|[IP T

Canonical cameras given F

Page 20: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 20

~fundamental matrix for calibrated cameras (remove K)

RtE

0x̂E'x̂ T

FKK'E T

x'K'x̂ x;Kx̂ -1-1

5 d.o.f. (3 for R; 2 for t up to scale)

E is essential matrix if and only iftwo singular values are equal (and third=0)

T0)VUdiag(1,1,E

The Essential Matrix

t]|[RP''P̂0],|[IPˆ t]|[R'P'0],|[K 11 KKPKP

Page 21: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 22

C1

C2

l2

P

l1

e1

e2

0m m 1T2 F

Fundamental matrix (3x3 rank 2

matrix)

1. Computable from corresponding points

2. Simplifies matching3. Allows to detect wrong matches4. Related to calibration

Underlying structure in set of matches for rigid scenes

l2

C1m1

L1

m2

L2

M

C2

m1

m2

C1

C2

l2

P

l1

e1

e2

m1

L1

m2

L2

M

l2lT1

Canonical representation:

]λe'|ve'F][[e'P' 0]|[IP T

Epipolar Geometry

Page 22: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 23

given xi↔x‘i , compute P,P‘ and Xi

reconstruction problem:

ii PXx x PXi i for all i

without additional information possible up to projective ambiguity

3D reconstruction of cameras and structure

Page 23: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 24

(i) Compute F from correspondences(ii) Compute camera matrices from F(iii) Compute 3D point for each pair of

corresponding points

computation of Fuse x‘iFxi=0 equations, linear in coeff. F8 points (linear), 7 points (non-linear), 8+ (least-squares)

computation of camera matricesuse ]λe'|ve'F][[e'P' 0]|[IP T

triangulationcompute intersection of two backprojected rays

Outline of reconstruction

Page 24: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 25

iii XHPHPXx S-1S

λt]t'RR'|K[RR'λ0

t'R'-R' t]|K[RPH TTTT

1-S

Reconstruction ambiguity: similarity

Page 25: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 26

iii XHPHPXx P-1

P

Reconstruction ambiguity: projective

Page 26: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 27

xi↔x‘i

Original scene Xi

Projective, affine, similarity reconstruction = reconstruction that is identical to original up to projective, affine, similarity transformation

Literature: Metric and Euclidean reconstruction = similarity reconstruction

Terminology

Page 27: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 28

If a set of point correspondences in two views determine the fundamental matrix uniquely, then the scene and cameras may be reconstructed from these correspondences alone, and any two such reconstructions from these correspondences are projectively equivalent

i111 X,'P,P i222 X,'P,Pii xx -1

12 HPP -112 HPP

12 HXX 0FxFx :except ii

iiiii 22111-1

112 XPxXPHXHPHXP along same ray of P2, idem for P‘2

two possibilities: X2i=HX1i, or points along baseline

key result: allows reconstruction from pair of uncalibrated images

The projective reconstruction theorem

Page 28: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 29

Page 29: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 30

(i) Projective reconstruction

(ii) Affine reconstruction

(iii) Metric reconstruction

Stratified reconstruction

Page 30: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 31

ii HXXE

use control points XEi with known coordinatesto go from projective to metric

Eii XPHx -1(2 lin. eq. in H-1

per view, 3 for two views)

Direct metric reconstruction using ground truth

Page 31: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 32

C1

C2

l2

l1

e1

e20m m 1

T2 F

Fundamental matrix (3x3 rank 2

matrix)

1. Computable from corresponding points

2. Simplifies matching3. Allows to detect wrong matches4. Related to calibration

Underlying structure in set of matches for rigid scenes

l2

C1m1

L1

m2

L2

M

C2

m1

m2

C1

C2

l2

l1

e1

e2

m1

L1

m2

L2

M

l2lT1

Canonical representation:

]λe'|ve'F][[e'P' 0]|[IP T

1T

1 lPπ

1T

1T

22 lPPl

0lm 2T

2

111 mel

0m][ePPm 1x1T

1T

2T2

Epipolar Geometry: computation of F

Page 32: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 33

0Fxx'T

separate known from unknown

0'''''' 333231232221131211 fyfxffyyfyxfyfxyfxxfx

0,,,,,,,,1,,,',',',',',' T333231232221131211 fffffffffyxyyyxyxyxxx

(data) (unknowns)(linear)

0Af

0f1''''''

1'''''' 111111111111

nnnnnnnnnnnn yxyyyxyxyxxx

yxyyyxyxyxxx

Computation of F: basic equation

Page 33: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 34

0Fe'T 0Fe 0detF 2Frank

T333

T222

T111

T

3

2

1

VσUVσUVσUVσ

σσ

UF

SVD from linearly computed F matrix (rank 3)

T222

T111

T2

1

VσUVσUV0

σσ

UF'

FF'-FminCompute closest rank-2 approximation

Imposing the singularity constraint

Page 34: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 35

Page 35: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 36

0

1´´´´´´

1´´´´´´

1´´´´´´

33

32

31

23

22

21

13

12

11

222222222222

111111111111

f

f

f

f

f

f

f

f

f

yxyyyyxxxyxx

yxyyyyxxxyxx

yxyyyyxxxyxx

nnnnnnnnnnnn

~10000 ~10000 ~10000 ~10000~100 ~100 1~100 ~100

!Orders of magnitude differenceBetween column of data matrix least-squares yields poor results

The NOT normalized 8-point algorithm

Page 36: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 37

Transform image to ~[-1,1]x[-1,1]

(0,0)

(700,500)

(700,0)

(0,500)

(1,-1)

(0,0)

(1,1)(-1,1)

(-1,-1)

1

1500

2

10700

2

Least squares yields good results (Hartley, PAMI´97)

The normalized 8-point algorithm

Page 37: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 38

Maximum Likelihood Estimation

i

iiii dd 22 'x̂,x'x̂,x

(= least-squares for Gaussian noise)

0x̂F'x̂ subject to T

iXt],|[MP'0],|[IP Parameterize:

Initialize: normalized 8-point, (P,P‘) from F, reconstruct Xi

iiii XP'x̂,PXx̂

Minimize cost using Levenberg-Marquardt(preferably sparse LM, see book)

(overparametrized)

The Gold Standard Algorithm

Page 38: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 39

Examples

Page 39: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 40

Examples

Page 40: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 41

Examples

Page 41: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 42

1. Do not use unnormalized algorithms

2. Quick and easy to implement: 8-point normalized

3. Better: enforce rank-2 constraint during minimization

4. Best: Maximum Likelihood Estimation (minimal parameterization, sparse implementation)

Recommendations

Page 42: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 43

Extract feature points to relate images

Required properties:– Well-defined

(i.e. neigboring points should all be different)

– Stable across views

(i.e. same 3D point should be extracted as feature for neighboring viewpoints)

The correspondence problem: feature points

Page 43: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 44

homogeneous

edge

corner

M should have large eigenvalues

(e.g.Harris&Stephens´88; Shi&Tomasi´94)

Find points that differ as much as possible from all neighboring points

Feature = local maxima (subpixel) of F(1, 2)

Feature points

Page 44: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 45

Select strongest features (e.g. 1000/image)

Feature points

Page 45: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 46

Evaluate NCC for all features with

similar coordinates

Keep mutual best matchesKeep mutual best matches

Still many wrong matches!Still many wrong matches!

10101010 ,,´´, e.g. hhww yyxxyx

?

Feature matching

Page 46: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 47

0.96 -0.40 -0.16 -0.39 0.19

-0.05 0.75 -0.47 0.51 0.72

-0.18 -0.39 0.73 0.15 -0.75

-0.27 0.49 0.16 0.79 0.21

0.08 0.50 -0.45 0.28 0.99

1 5

24

3

1 5

24

3

Gives satisfying results for small image motions

Similarity Example

Page 47: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 48

Step 1. Extract features

Step 2. Compute a set of potential matches

Step 3. doStep 3.1 select minimal sample (i.e. 8 matches)

Step 3.2 compute F

Step 3.3 determine inliers

until (#inliers,#samples)<95%

samples#7)1(1

matches#inliers#

#inliers 90% 80% 70% 60% 50%

#samples 5 13 35 106 382

Step 4. Compute F based on all inliersStep 5. Look for additional matchesStep 6. Refine F based on all correct matches

(generate hypothesis)

(verify hypothesis)

RANSAC

Page 48: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 49

restrict search range to neighborhood of epipolar line (1.5 pixels)

relax disparity restriction (along epipolar line)

Finding more matches

Page 49: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 50

Degenerate cases– Planar scene

– Pure rotation

No unique solution– Remaining DOF filled by noise

– Use simpler model (e.g. homography)

Model selection (Torr et al., ICCV´98, Kanatani, Akaike)

– Compare H and F according to expected residual error (compensate for model complexity)

Degenerate Cases

Page 50: Robot Vision SS 2008 Matthias Rüther 1 ROBOT VISION Lesson 6: Shape from Stereo Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of

Robot Vision SS 2008 Matthias Rüther 51

Absence of sufficient features (no texture)

Repeated structure ambiguity

(Schaffalitzky and Zisserman, BMVC‘98)

• Robust matcher also finds Robust matcher also finds support for wrong hypothesissupport for wrong hypothesis• solution: detect repetition solution: detect repetition

More problems