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Stereo vision and

two-view geometry

• The goal of a stereo system is to get 3D information• A stereo system consists at least of two ‘converging’ cameras rigidly attached

Chapter 7 of the textbook

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One real example:

• stereo geometry: epipolar geometrygeometric relation between two images

• correspondencepixels (pts) in different images from the same pt

• reconstruction (triangulation)3d coordinate of the pt

Three topics:

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Intuitive epipolar geomegtry

Entirely characterized by the so-called epipolar geometry

Geometric concepts: • epipole: the image of the other camera center• epipolar plane: plane defined by the two camera centers and the space pt• epipolar lines: intersection of the epipolar plane and image plane• pencil of epipolar lines and planes• baseline: distance between two camera centers

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u

O

u’

Oepipole

epipolar line

epipolar plane

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One real example of epipolar lines:

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Algebraic characterisation of the epipolar geometry: the

fundamental matrixGiven a correspondence pair u and u’,

where F is 3 by 30 Fu u

T

Proof sketch: follow the geometric construction. • compute the epipole in the second image• compute the pt at infinity (or ray direction), reproject it onto the second• define the epipolar line by these two pts• use anti-symmetric matrix for cross-product

Assume the camera projection matrices are P=(I 0) and P’=(A a), it can be shown that F = [a] A.The procedure is the same even if P is of general form.

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• rank of F• ker(F) • how many d.o.f.• l’=F u • l = … •

0T

Fu u

Properties of the fundamental matrix F:

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Stereo Vision by ‘traditional’ calibrated approach

• calibration of each camera w.r.t. the same object: P and P’• (optional) rectification• disparity map using F• 3D reconstruction

Traditional (calibrated) stereo approach:

Correspondence using F (computed from P and P’)

two (or more) cameras rigidly attached=stereo rig=stereo system

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Obtain F from the given P and P’:

b B P a A P3 3 3 3 ,

o P ea A

o

,1

1

u BA x P uu A

x1

1

,0

Fu u BA e u e l

1

] [

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• Correspondence (discussed later) • 3D reconstruction : trianglulation

),(),,(),( iii iiiiiii vuzyxvu uXu

34333231

24232221

34333231

14131211

czcycxc

czcycxcv

czcycxc

czcycxcu

iii

iiii

iii

iiii

34333231

24232221

34333231

14131211

czcycxc

czcycxcv

czcycxc

czcycxcu

iii

iiii

iii

iiii

Same equation as the calibration, but unknowns are nowxi, yi, zi instead of cij

4

4 41

0

i

i

i

i

t

z

y

x

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u

O

u’

O’

Triangulation:

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‘Modern’ uncalibrated approach:Epipolar geometry by point

correspondences – two-view geometry

0 Fu uT

Because of

NB: it is more powerful, ‘calibration’ needs 3d info, point-correspondence does not, but not 3d reconstruction

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• 8 pts algorithm• 7 pts algorithm (minimal data)• (optimal and robust sol.)

ii uu Given compute F

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8-point algorithm (unstable)

0 Fu uT

0 33 32 31 23 22 21 13 12 11 f vf uf f v vf v uf v f u vf u uf u

0 )1, , , , , , , , (9 f v u v v v u v u v u u u

9 9

9

1, , , , , , , ,0 f

n

i i i i i i i i i i i iv u v v v uv u v u u u

9 9 90 f A n

T Tv u v u)1, , ( )1, , ( u u

expand

rewrite

for N points

rewrite

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• linear sol by svd with ||f||=1: f=v9• F’, rank enforcement afterwards by svd!

33 32 31

23 22 21

13 12 11

3 3 33 32 31 23 22 21 13 12 11 9) , , , , , , , , (

f f f

f f f

f f f

f f f f f f f f fF f

F V U

V U

F

T recompose

T decompose SVD

f f f

f f f

f f f

0 0 0

0 0

0 0

0 0

0 0

0 0

3

2

1

3

2

1

33 32 31

23 22 21

13 12 11

3 3

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7-point algorithm• one parameter solution by svd• from the vanishing determinant, get a cubic equation

9 8 9v v fy x

0 0 ) det(3 2 2 3 dy cxy y bx ax F

* * *

* * *

* * 11 11y c x c

F

02 3

d cz bz az

y

xz

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Normalisation 8 pt algorithmTo make the average point as close as possible to (1,1,1)!

• normalisation by transformations

• linear solution for

• rank enforcement

• denormalisation

Warning: unnormalised 8 pt algorithm is unstable!!!

u T u Tu u ˆ , ˆu uˆ ˆ

0 ˆ ˆ ˆ uF uT

F F ˆ ˆ

T F T F ˆ T

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Data normalisation: each image data is normlised independently!

1 0 0

0

0

) ( ) ( ,2

2 )) ( ( )) ( (1

1,

1

2 2

2 2

vs s

us s

v v u u dd

ns

v v s u u sn

vn

v un

u

i i ii

i i

i i

T

Tu uˆ

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Summary (or a unified view) of all methods of computation of the fundamental matrix

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Stereo correspondence

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• disparity: difference in image position of the same space pt• disparity map: dense pixel-to-pixel corrrespondences• stereo rectification: make the epipolar lines horizontal

an option to speed up the computation of disparity map

The epipolar geometry gives only a constraint, but not yet a unique solution to the question: where is the corresponding point in the second image of a given point in the first image?

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Rectification of a stereo pair of images: two images are transformed (by a projective transformation in image plane or by a camera rotation around the center).

u T u Tu u ˆ , ˆ

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New rectified image plane

• equivalent to a plane parallel to the base line• T and T’ can be computed from F, but many possibilities• only an option, simplify the computation!

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N

2n n p p n pd )) ( I ) I( ( min

Nn n p p n pd) ( I) I( max

or

Very often ZNCC (Zero Normalized Cross Correlation), on normalised images instead of I and I’,

Matching by correlation:

| || |) ( , ) () 1 2( ) 1 2( ) 1 2( ) 1 2(

n n

n nI I n I I n

n n n n

Convert all (2n+1)(2n+1) elements from a matrix into a vector of dim (2n+1)(2n+1)

n

) 1 2( ) 1 2( ) ( n n I I

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Two points u and u’ are in correspondence if • ZNCC(u,u’) is big enoug (close to 1)• dist(u’, Fu) is small enough (a few pixels)

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Correspondence by correlation:

• For each point u, compute all correlations in a neighborhood u+d with a window size s

• Take the pixel having the highest correlation score as the correspondence

• Cross-validate the correspondence in the opposite direction from the second to the first image

Correlation window

neighborhood

Cross-validate

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When applied to ‘interest points’, sparse correspondenceWhen applied to every pixel, dense disparity map

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Using more cameras to remove match ambiguity: a system of 3 cameras

1

2

3

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What can we do more with F?

• Without calibration, what can we get?

• When calibrated, essential matrix, its decomposition

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0 Fu uT

x K u Kx u ,

0 ) ( ) ( Kx F x KT

0 ) ( x FK K xT T

FK K E Ex xT T , 0

From uncalibrated F

Calibrated E

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Essential matrix: fundamental matrix for the calibrated points

The extra algebraic constraint: the equal singular values (more complicated) for E

Decomposition of the essential matrix into R and t

Relationship between E and F

E = [t] R From F = [a] A

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Decomposition of E

T(T)0

T0

VURR

UUSS

100

001

010

0R

000

001

010

0S

Two factorisation

3)ker( uSt Two translation

Twisted pair

SR[t]REUSVE ,T

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• Given internal calibration K and K’

(more advanced studies allow us to remove this step by self-calibration that we will not handle)

• Compute F from point correspondences• Compute E• Decompose E to obtain R and t• Obtain P and P’• Triangulation

Summary of modern two-view approach:

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O

When space points are planar, a homography relating u and u’

u

O

u’

x

It is therefore a ‘collineation’ for COPLANAR points!

Never forget the coplanar case!

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From P=(I 0) , P’=(A a) and a known plane p^Tx=0,to get H

Or, the homography can be computed from at least 4 correspondingPoints, do it!

The homography uniquely determines point correspondences Unlike the fundamental matrix! But only for coplanar points.

)1,(, npanAH T

0,),( unxpaAuuuxP TT

So that Huu

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Panoramic image or image mosaicing

• the 3D scene is planar• the camera is rotating around the center (similar to rectification)

Example at HK airport (virtual tour), QuicktimeVRRealviz, stitcher, step-by-stephttp://iris.usc.edu/home/iris/elkang/iris-04/reports/2/techreport2.html

This homography leads to one important application:

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The images are related by a homography if the 3d scene is planar:

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x

A pencil of lines cut by two lines

A star of lines cut by two planes

Rotating the camera around the center is equivalent to a homographical Transformation of the image plane:

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• Compute point correspondences

• Compute the projective transformation between the two views

• Warp the first image onto the second

• Color-blend the overlapping areas

Compositing algorithm or mosacing algorithm:

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Quick-time VR

Inward-looking small object

Outward-looking large-scale environnement

Example of the virtual tour of HK airport: http://www.hkairport.com/eng/index.jsp

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Automatic computation of the fundamental matrix

Chicken-egg problem: we need corresponding points to compute F, we need F to establish correspondences …

Simultaneous automatic computation of correspondences and F

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Illustrative example of fitting a line to a set of 2D points

• the least squares solution (orthogonal regression) is optimal when no outliers• but it is becoming very fragile to outliers

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Robust line fitting

Fit a line to 2D data containing ‘bad points’---outliers

Solving two pbs: 1. A line fit to the data; 2. A classification of the data into ‘inliers’ and ‘outliers’‘inliers’: valid or good data satisfying the ‘line’ model‘outliers’: bad data not satisfying the model

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• randomly draw 2 data points• compute a line Li from these 2 points• compute the distance to the line Li for each data point• determine inliers/outliers by a threshold t• compute the number of inliers Si

• select the Li having the largest Si• re-estimate the final line using all inliers

How to find the best line?

repeating

then

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• randomly draw a sample of s data• initiate the model Mi• compute the distance to the model for each data pt• determine inliers/outliers by the threshold t• compute the size of inliers Si

• select the Mi having the largest Si• re-estimate using all inliers

Fischler and Bolles 1981

repeating

then

RANSAC (random sample consensus)

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The complete algorithm of automatic computation of F:

• detect points of interest in each image• compute the correspondences using correlation based method• RANSAC using 7-pt algo.• (non-linear optimal estimation on the final inliers: this is unnecessary in many cases, so just an option)

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• randomly draw a sample of 7 corresponding points• compute Fi• compute the distance to Fi for each corresponding pt• determine inliers/outliers by the threshold t• compute the size of inliers Si

• select the Fi having the largest Si• re-estimate the final F using all inliers

repeating

then

RANSAC using 7-pt algo to compute F

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))1(1log(/)1log( spN

Outlier proportion

s Sample size

Probability of success (only inliers)p

Ex: 99.9% success rate, 50% outliers s=2, N=17 s=4, N=72 s=6, N=293 s=7, N=588 s=8, N=1177

How many times to repeat?

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Robust statistics

From least-squares method to robust statistics (Ransac,least median of squares (LMS))

Handle ‘big errors’---outliers!

This is useful not only for computing F, but also for automatically computing a mosaicing of images!

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What tells us the F, the epipolar geometry?

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• Given internal calibration K and K’

(more advanced studies allow us to remove this step by self-calibration that we will not handle)

• Compute F from point correspondences• Compute E• Decompose E to obtain R and t• Obtain P and P’• Triangulation

Summary of ‘semi-modern’ two-view approach:

What do we obtain?

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What is a projective reconstruction?

What is a Euclidean (similarity) reconstruction?

A 3D reconstruction up to a 3D projective transformation.

A 3D reconstruction up to a 3D Euclidean transformation.

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Projective reconstruction without calibration

),]([),,( eFeP0IPF

2. Or the algebraic approach by epipolar geometry (cf. Faugeras’92 ECCV,What can be seen from an uncalibrated stereo rig?)

1. This is not unique, as for any v and lambda, we have

),]([ T eveFeP

3. The bottome line of numerical schema is the true ‘optimal’ of F

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Self-calibration

from only point correspondences using geometric self-consistency constraints

The original idea of Maybank&Faugeras91 key components: absolute conic, fundamental matrix and Kruppa equation

Later on: absolute quadric …

Projective metric reconstruction (uncalibrated)

Generally at least 3 images for self-calibrationwith constraints such as K=K’=K’’