CREDITS

Computer Vision : CISC 4/689

CREDITS

Rasmussen, UBC (Jim Little), Seitz (U. of Wash.), Camps (Penn. State), UC, UMD (Jacobs), UNC, CUNY


Multi-View GeometryRelates

• 3D World Points

• Camera Centers

• Camera Orientations


Multi-View GeometryRelates

• 3D World Points

• Camera Centers

• Camera Orientations

• Camera Intrinsic Parameters

• Image Points


Stereo

scene pointscene point

optical centeroptical center

image planeimage plane


Stereo

• Basic Principle: Triangulation– Gives reconstruction as intersection of two rays

– Requires

• calibration

• point correspondence


Stereo Constraints

p p’?

Given p in left image, where can the corresponding point p’in right image be?


Stereo Constraints

X1

Y1

Z1

O1

Image plane

Focal plane

M

p p’

Y2

X2

Z2O2

Epipolar Line

Epipole


Stereo

• The geometric information that relates two different viewpoints of the same scene is entirely contained in a mathematical construct known as fundamental matrix.

• The geometry of two different images of the same scene is called the epipolar geometry.


Stereo/Two-View Geometry

• The relationship of two views of a scene taken from different camera positions to one another

• Interpretations– “Stereo vision” generally means

two synchronized cameras or eyes capturing images

– Could also be two sequential views from the same camera in motion

• Assuming a static scene

http://www-sop.inria.fr/robotvis/personnel/sbougnou/Meta3DViewer/EpipolarGeo

http://www-sop.inria.fr/robotvis/personnel/sbougnou/Meta3DViewer/EpipolarGeo


3D from two-views

There are two ways of extracting 3D from a pair of images. • Classical method, called Calibrated route, we need to calibrate both

cameras (or viewpoints) w.r.t some world coordinate system. i.e, calculate the so-called epipolar geometry by extracting the essential matrix of the system.

• Second method, called uncalibrated route, a quantity known as fundamental matrix is calculated from image correspondences, and this is then used to determine the 3D.

Either way, principle of binocular vision is triangulation. Given a single image, the 3D location of any visible object point must lie on the straight line that passes through COP and image point (see fig.). Intersection of two such lines from two views is triangulation.


Mapping Points between Images

• What is the relationship between the images x, x’ of the

scene point X in two views?• Intuitively, it depends on:

– The rigid transformation between cameras (derivable from the

camera matrices P, P’)

– The scene structure (i.e., the depth of X)• Parallax: Closer points appear to move more


Example: Two-View Geometry

courtesy of F. Dellaert

x1x’1

x2x’2

x3 x’3

Is there a transformation relating the points xi to x’i ?


Epipolar Geometry

• Baseline: Line joining camera centers C, C’• Epipolar plane ¦: Defined by baseline and scene point X

from Hartley& Zisserman

baseline


Epipolar Lines

• Epipolar lines l, l’: Intersection of epipolar plane ¦ with image planes

• Epipoles e, e’: Where baseline intersects image planes– Equivalently, the image in one view of the other camera center.

C C’



Epipolar Pencil

• As position of X varies, epipolar planes “rotate” about the baseline (like a book with pages)

– This set of planes is called the epipolar pencil• Epipolar lines “radiate” from epipole—this is the pencil of epipolar lines



Epipolar Constraint

• Camera center C and image point define ray in 3-D space that projects to epipolar line l’ in other view (since it’s on the epipolar plane)

• 3-D point X is on this ray, so image of X in other view x’ must be on l’• In other words, the epipolar geometry defines a mapping x ! l’, of points in one image to

lines in the other


C C’

x’


Example: Epipolar Lines for Converging Cameras

from Hartley & ZissermanLeft view Right view

Intersection of epipolar lines = Epipole ! Indicates direction of other camera


Special Case: Translation Parallel to Image Plane

Note that epipolar lines are parallel and corresponding points lie on correspond-ing epipolar lines (the latter is true for all kinds of camera motions)


From Geometry to Algebra

O O’

P

pp’


Linear Constraint:Should be able to express as matrix multiplication.

Rotation arrow should be at the other end, to get p’ in p coordinates


Review: Matrix Form of Cross Product


Matrix Form


The Essential Matrix

If un-calibrated, p gets multiplied by Intrisic matrix, K


The Fundamental Matrix F

• Mapping of point in one image to epipolar line in other image x ! l’ is

expressed algebraically by the fundamental matrix F

• Write this as l’ = F x• Since x’ is on l’, by the point-on-line definition we know that

x’T l’ = 0

• Substitute l’ = F x, we can thus relate corresponding points in the

camera pair (P, P’) to each other with the following:

x’T F x = 0

line point


The Fundamental Matrix F• F is 3 x 3, rank 2 (not invertible, in contrast to homographies)

– 7 DOF (homogeneity and rank constraint take away 2 DOF)

• The fundamental matrix of (P’, P) is the transpose FT


x’

NOW, can get implicit equation for any x, which is epipolar line)


Computing Fundamental Matrix

Fundamental Matrix is singular with rank 2

0uFuT

In principal F has 7 parameters up to scale and can be estimatedfrom 7 point correspondences

Direct Simpler Method requires 8 correspondences

(u’ is same as x in the prev. slide, u’ is same as x)


Estimating Fundamental Matrix

0uFuT

Each point correspondence can be expressed as a linear equation

0

1

1

333231

232221

131211

v

u

FFF

FFF

FFF

vu

01

33

32

31

23

22

21

13

12

11

F

F

F

F

F

F

F

F

F

vuvvvvuuvuuu

The 8-point algorithm


The 8-point Algorithm

Lot of squares, so numbers have varied range, from say 1000 to 1. So pre-normalize.And RANSaC!


Computing F: The Eight-point Algorithm

• Input: n point correspondences ( n >= 8)– Construct homogeneous system Ax= 0 from

• x = (f11,f12, ,f13, f21,f22,f23 f31,f32, f33) : entries in F

• Each correspondence gives one equation

• A is a nx9 matrix (in homogenous format)

– Obtain estimate F^ by SVD of A

• x (up to a scale) is column of V corresponding to the least singular value

– Enforce singularity constraint: since Rank (F) = 2

• Compute SVD of F^

• Set the smallest singular value to 0: D -> D’

• Correct estimate of F :

• Output: the estimate of the fundamental matrix, F’

• Similarly we can compute E given intrinsic parameters

0lT

r pFp

TUDVA

TUDVF ˆ

TVUDF' '


Locating the Epipoles from F

• Input: Fundamental Matrix F– Find the SVD of F– The epipole el is the column of V corresponding to the null singular

value (as shown above)– The epipole er is the column of U corresponding to the null singular

value

• Output: Epipole el and er

TUDVF

el lies on all the epipolar lines of the left image

0lT

r pFp

0lT

r eFp

F is not identically zero

True For every pr

0leF

pl pr

P

Ol Orel er

Pl Pr

Epipolar Plane

Epipolar Lines

Epipoles


Special Case: Translation along Optical Axis

• Epipoles coincide at focus of expansion

• Not the same (in general) as vanishing point of scene lines

from Hartley & Zisserman


Finding Correspondences

• Epipolar geometry limits where feature in one image can be in the other image– Only have to search along a line


Simplest Case

• Image planes of cameras are parallel.

• Focal points are at same height.

• Focal lengths same.

• Then, epipolar lines are horizontal scan lines.


We can always achieve this geometry with image rectification

• Image Reprojection– reproject image planes onto common

plane parallel to line between optical centers• Notice, only focal point of camera really matters

(Seitz)


Stereo Rectification

• Rectification – Given a stereo pair, the intrinsic and extrinsic parameters, find the image

transformation to achieve a stereo system of horizontal epipolar lines

– A simple algorithm: Assuming calibrated stereo cameras

p’lp’r

P

Ol Or

X’r

Pl Pr

Z’l

Y’l Y’r

TX’l

Z’r

Stereo System with Parallel Optical AxesEpipoles are at infinity

Horizontal epipolar lines



• Algorithm– Rotate both left and right

camera so that they share the same X axis : Or-Ol = T

– Define a rotation matrix Rrect for the left camera

– Rotation Matrix for the right camera is RrectRT

– Rotation can be implemented by image transformation

pl

pr

P

Ol Or

Xl

Xr

Pl Pr

Zl

Yl

Zr

Yr

R, T

TX’l

Xl’ = T_axis, Yl’ = Xl’xZl, Z’l = Xl’xYl’



• Algorithm– Rotate both left and right

camera so that they share the same X axis : Or-Ol = T

– Define a rotation matrix Rrect for the left camera

– Rotation Matrix for the right camera is RrectRT

– Rotation can be implemented by image transformation

Zr

p’lp’r

P

Ol Or

X’r

Pl Pr

Z’l

Y’l Y’r

R, T

TX’l

T’ = (B, 0, 0),

Computer Vision : CISC 4/689Public Library, Stereoscopic Looking Room, Chicago, by Phillips, 1923

Computer Vision : CISC 4/689Teesta suspension bridge-Darjeeling, India

Computer Vision : CISC 4/689Mark Twain at Pool Table", no date, UCR Museum of Photography

Computer Vision : CISC 4/689Woman getting eye exam during immigration procedure at Ellis

Island, c. 1905 - 1920 , UCR Museum of Phography


Stereo matching

• attempt to match every pixel

• use additional constraints


A Simple Stereo System

Zw=0

LEFT CAMERA

Left image:reference

Right image:target

RIGHT CAMERA

Elevation Zw

disparity

Depth Z

baseline


Let’s discuss reconstruction with this geometry before correspondence, because it’s much easier.

OOll OOrr

PP

ppll pprr

TT

ZZ

xxll xxrr

ff

T T is the stereo baselineis the stereo baselined d measures the difference in retinal position between corresponding measures the difference in retinal position between corresponding pointspoints (Camps)

Then given Z, we can compute X and Y.

Disparity:

xl,yl=(f X/Z, f Y/Z)Xr,yr=(f (X-T)/Z, f Y/Z)d=xl-xr=f X/Z – f (X-T)/Z

( -ve, +ve, referprevious slide fig.)


Correspondence: What should we match?

• Objects?

• Edges?

• Pixels?

• Collections of pixels?


Extracting Structure

• The key aspect of epipolar geometry is its linear constraint on where a point in one image can be in the other

• By correlation-matching pixels (or features) along epipolar lines and measuring the disparity between them, we can construct a depth map (scene point depth is inversely proportional to disparity)

View 1 View 2 Computed depth mapcourtesy of P. Debevec


Correspondence: Photometric constraint

• Same world point has same intensity in both images.– Lambertian fronto-parallel

– Issues:

• Noise

• Specularity

• Foreshortening


Using these constraints we can use matching for stereo

For each epipolar lineFor each pixel in the left image

• compare with every pixel on same epipolar line in right image

• pick pixel with minimum match cost

• This will never work, so:

Improvement: match windows

(Seitz)


Aggregation• Use more than one pixel

• Assume neighbors have similar disparities*

– Use correlation window containing pixel

– Allows to use SSD, ZNCC, etc.


Comparing Windows: ==??

ff gg

MostMostpopularpopular

(Camps)

For each window, match to closest window on epipolar line in other image.


Compare intensities pixel-by-pixel

Comparing image regions

I(x,y) I´(x,y)

Sum of Square Differences

Dissimilarity measures


Compare intensities pixel-by-pixel

Comparing image regions

I(x,y) I´(x,y)

Zero-mean Normalized Cross Correlation

Similarity measures


Aggregation window sizes

Small windows

• disparities similar

• more ambiguities

• accurate when correct

Large windows

• larger disp. variation

• more discriminant

• often more robust

• use shiftable windows to deal with discontinuities

(Illustration from Pascal Fua)


Window size

W = 3 W = 20

Better results with adaptive window• T. Kanade and M. Okutomi,

A Stereo Matching Algorithm with an Adaptive Window: Theory and Experiment,, Proc. International Conference on Robotics and Automation, 1991.

• D. Scharstein and R. Szeliski. Stereo matching with nonlinear diffusion. International Journal of Computer Vision, 28(2):155-174, July 1998

• Effect of window size

(Seitz)

http://www.cs.cmu.edu/~ph/869/papers/kanade-okutomi.pdf

http://www.cs.cmu.edu/~ph/869/papers/kanade-okutomi.pdf

http://www.cs.washington.edu/education/courses/cse590ss/01wi/notes/papers/ScharsteinSzeliskiIJCV98.pdf


Correspondence Using Window-based matching

SSD error

disparity

Left Right

scanline


Sum of Squared (Pixel) DifferencesLeft Right

Lw Rw

LI RI

),(),(

2

2222

)],(),([),,(

:disparity offunction a as differenceintensity themeasurescost SSD The

},|,{),(

:function window thedefine We

pixels. of windowsby ingcorrespond are and

yxWvuRLr

mmmmm

RL

m

vduIvuIdyxC

yvyxuxvuyxW

mmww

LwRw

),( LL yx ),( LL ydx

m

m


Image Normalization• Even when the cameras are identical models, there can be differences in gain and sensitivity.• The cameras do not see exactly the same surfaces, so their overall light levels can differ.• For these reasons and more, it is a good idea to normalize the pixels in each window:

pixel Normalized ),(

),(ˆ

magnitude Window )],([

pixel Average ),(

),(

),(),(

2

),(

),(),(),(

1

yxW

yxWvuyxW

yxWvuyxW

m

mm

m

m

II

IyxIyxI

vuII

vuII


Stereo results

Ground truthScene

– Data from University of Tsukuba

(Seitz)


Results with window correlation

Window-based matching(best window size)

Ground truth

(Seitz)


Results with better method

State of the art methodBoykov et al., Fast Approximate Energy Minimization via Graph Cuts,

International Conference on Computer Vision, September 1999.

Ground truth

(Seitz)

http://www.cs.cornell.edu/rdz/Papers/BVZ-iccv99.pdf

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