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Stereo Readings • Trucco & Verri, Chapter 7 – Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. – The rest is optional. Single image stereogram, by Niklas Een

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Page 1: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo

Readings• Trucco & Verri, Chapter 7

– Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. – The rest is optional.

Single image stereogram, by Niklas Een

Page 2: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,
Page 3: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Public Library, Stereoscopic Looking Room, Chicago, by Phillips, 1923

Page 4: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Teesta suspension bridge-Darjeeling, India

Page 5: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Woman getting eye exam during immigration procedure at Ellis Island, c. 1905 - 1920 , UCR Museum of Phography

Page 6: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Mark Twain at Pool Table", no date, UCR Museum of Photography

Page 7: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo

scene pointscene point

optical centeroptical center

image planeimage plane

Page 8: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo

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

• Requires – camera pose (calibration)– point correspondence

Page 9: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo correspondenceDetermine Pixel Correspondence

• Pairs of points that correspond to same scene point

Epipolar Constraint• Reduces correspondence problem to 1D search along conjugate

epipolar lines• Java demo: http://www.ai.sri.com/~luong/research/Meta3DViewer/EpipolarGeo.html

epipolar planeepipolar lineepipolar lineepipolar lineepipolar line

Page 10: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Fundamental matrix

• This epipolar geometry of two views is described by a Very Special 3x3 matrix , called the fundamental matrix

• maps (homogeneous) points in image 1 to lines in image 2!• The epipolar line (in image 2) of point p is:

• Epipolar constraint on corresponding points:

epipolar plane

epipolar lineepipolar lineepipolar lineepipolar line

0

(projection of ray)

Image 1 Image 2

Page 11: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Fundamental matrix – uncalibrated case

0

the Fundamental matrix

: intrinsics of camera 1 : intrinsics of camera 2

: rotation of image 2 w.r.t. camera 1

Page 12: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Cross-product as linear operator

Useful fact: Cross product with a vector t can be represented as multiplication with a (skew-symmetric) 3x3 matrix

Page 13: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Fundamental matrix – calibrated case

0

: ray through p in camera 1’s (and world) coordinate system

: ray through q in camera 2’s coordinate system

{

the Essential matrix

Page 14: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Properties of the Fundamental Matrix

• is the epipolar line associated with

• is the epipolar line associated with

• and

• is rank 2

• How many parameters does F have?14

T

Page 15: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Rectified case

Page 16: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo image rectification

Page 17: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo image rectification

• reproject image planes onto a commonplane parallel to the line between optical centers

• pixel motion is horizontal after this transformation• two homographies (3x3 transform), one for each

input image reprojection C. Loop and Z. Zhang. Computing Rectifying Homographies for

Stereo Vision. IEEE Conf. Computer Vision and Pattern Recognition, 1999.

Page 18: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Estimating F

• If we don’t know K1, K2, R, or t, can we estimate F for two images?

• Yes, given enough correspondences. We’ll see soon…

Page 19: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo Matching

Given a pixel in the left image, how to find its match?• Assume the photos have been rectified

Page 20: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Your basic stereo algorithm

For each epipolar line

For each pixel in the left image• compare with every pixel on same epipolar line in right image

• pick pixel with minimum match cost

Improvement: match windows• This should look familar...

Page 21: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Window size

• Smaller window+ –

• Larger window+ –

W = 3 W = 20

Effect of window size

Page 22: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo results

Ground truthScene

• Data from University of Tsukuba• Similar results on other images without ground truth

Page 23: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Results with window search

Window-based matching(best window size)

Ground truth

Page 24: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Better methods exist...

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

International Conference on Computer Vision, September 1999.

Ground truth

For the latest and greatest: http://www.middlebury.edu/stereo/

Page 25: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as energy minimization

What defines a good stereo correspondence?1. Match quality

– Want each pixel to find a good match in the other image

2. Smoothness– If two pixels are adjacent, they should (usually) move about

the same amount

Page 26: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as energy minimization

• Find disparity map d that minimizes an energy function

• Simple pixel / window matching

SSD distance between windows I(x, y) and J(x + d(x,y), y)

=

Page 27: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as energy minimization

I(x, y) J(x, y)

y = 141

C(x, y, d); the disparity space image (DSI)x

d

Page 28: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as energy minimization

y = 141

x

d

Simple pixel / window matching: choose the minimum of each column in the DSI independently:

Page 29: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as energy minimizationBetter objective function

{ {

match cost smoothness cost

Want each pixel to find a good match in the other image

Adjacent pixels should (usually) move about the

same amount

Page 30: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as energy minimization

match cost:

smoothness cost:

4-connected neighborhood

8-connected neighborhood

: set of neighboring pixels

Page 31: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Smoothness cost

“Potts model”

L1 distance

How do we choose V?

Page 32: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Dynamic programming

Can minimize this independently per scanline using dynamic programming (DP)

: minimum cost of solution such that d(x,y) = d

Page 33: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Dynamic programming

Finds “smooth” path through DPI from left to right

y = 141

x

d

Page 34: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Dynamic Programming

Page 35: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Can we apply this trick in 2D as well?

No: dx,y-1 and dx-1,y may depend on different values of dx-1,y-1

Dynamic programming

Slide credit: D. Huttenlocher

Page 36: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Stereo as global optimizationExpressing this mathematically

1. Match quality– Want each pixel to find a good match in the other image

2. Smoothness– If two pixels are adjacent, they should (usually) move about

the same amount

We want to minimize sum of these two cost terms• This is a special type of cost function known as an

MRF (Markov Random Field)– Effective and fast algorithms have been recently developed:

» Graph cuts, belief propagation….» for more details (and code): http://vision.middlebury.edu/MRF/

Page 37: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

38

Middlebury Stereo Evaluationhttp://vision.middlebury.edu/stereo/

38

Page 38: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Depth from disparity

f

x x’

baseline

z

C C’

X

f

Page 39: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Real-time stereo

Used for robot navigation (and other tasks)• Several software-based real-time stereo techniques have

been developed (most based on simple discrete search)

Nomad robot searches for meteorites in Antarticahttp://www.frc.ri.cmu.edu/projects/meteorobot/index.html

Page 40: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

• Camera calibration errors• Poor image resolution• Occlusions• Violations of brightness constancy (specular reflections)• Large motions• Low-contrast image regions

Stereo reconstruction pipelineSteps

• Calibrate cameras• Rectify images• Compute disparity• Estimate depth

What will cause errors?

Page 41: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Active stereo with structured light

Project “structured” light patterns onto the object• simplifies the correspondence problem• can remove one of the cameras (replace with projector)

camera 1

projector

Microsoft’s Kinect

http://www.youtube.com/watch?v=7QrnwoO1-8A

Page 42: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Active stereo with structured light

Page 43: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Laser scanning

Optical triangulation• Project a single stripe of laser light• Scan it across the surface of the object• This is a very precise version of structured light scanning

Digital Michelangelo Projecthttp://graphics.stanford.edu/projects/mich/

Page 44: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Page 45: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Page 46: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Page 47: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Page 48: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Page 49: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Estimating F

• If we don’t know K1, K2, R, or t, can we estimate F for two images?

• Yes, given enough correspondences

Page 50: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

Estimating F – 8-point algorithm

• The fundamental matrix F is defined by

0Fxx'

for any pair of matches x and x’ in two images.

• Let x=(u,v,1)T and x’=(u’,v’,1)T,

333231

232221

131211

fff

fff

fff

F

each match gives a linear equation

0'''''' 333231232221131211 fvfuffvfvvfuvfufvufuu

Page 51: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

8-point algorithm

0

1´´´´´´

1´´´´´´

1´´´´´´

33

32

31

23

22

21

13

12

11

222222222222

111111111111

f

f

f

f

f

f

f

f

f

vuvvvvuuuvuu

vuvvvvuuuvuu

vuvvvvuuuvuu

nnnnnnnnnnnn

• In reality, instead of solving , we seek f to minimize , least eigenvector of .

0Af

Af AA

Page 52: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

8-point algorithm – Problem?• F should have rank 2• To enforce that F is of rank 2, F is replaced by F’ that

minimizes subject to the rank constraint. 'FF

• This is achieved by SVD. Let , where

, let

then is the solution.

VUF Σ

3

2

1

00

00

00

Σ

000

00

00

Σ' 2

1

VUF Σ''

Page 53: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

8-point algorithm% Build the constraint matrix A = [x2(1,:)‘.*x1(1,:)' x2(1,:)'.*x1(2,:)' x2(1,:)' ... x2(2,:)'.*x1(1,:)' x2(2,:)'.*x1(2,:)' x2(2,:)' ... x1(1,:)' x1(2,:)' ones(npts,1) ]; [U,D,V] = svd(A); % Extract fundamental matrix from the column of V % corresponding to the smallest singular value. F = reshape(V(:,9),3,3)'; % Enforce rank2 constraint [U,D,V] = svd(F); F = U*diag([D(1,1) D(2,2) 0])*V';

Page 54: Stereo Readings Trucco & Verri, Chapter 7 –Read through 7.1, 7.2.1, 7.2.2, 7.3.1, 7.3.2, 7.3.7 and 7.4, 7.4.1. –The rest is optional. Single image stereogram,

8-point algorithm

• Pros: it is linear, easy to implement and fast• Cons: susceptible to noise


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