776 Computer Vision

Jan-Michael Frahm, Enrique DunnSpring 2012

From Previous Lecture• Homographies• Fundamental matrix• Normalized 8-point Algorithm• Essential Matrix

Plane Homography for Calibrated Cameras

• In the calibrated case o Two cameras P=K[I |0] and P’ = K’[R | t]o A plane π=(nT,d) T

• The homography is given by x’=Hx

H = K’(R – tnT/d)K-1

• For the plane at infinityH = K’RK-1

The Fundamental Matrix F

P0m0

L

l1

M

m1

M

P1

Hm0

Epipole e1

F = [e]xH = Fundamental Matrix

1 1 0Tm I 1 0I Fm 1 0 0Tm Fm

1 0Te F

The eight-point algorithmx = (u, v, 1)T, x’ = (u’, v’, 1)T

Minimize:

under the constraintF33

= 1

2

1

)( i

N

i

Ti xFx

Essential Matrix(Longuet-Higgins, 1981)

Epipolar constraint: Calibrated case

0)]([ xRtx RtExExT ][with0

X

x x’

The vectors x, t, and Rx’ are coplanar slide: S. Lazebnik

Essential Matrix

Epipolar constraint: Calibrated case

0)]([ xRtx RtExExT ][with0

X

x x’

The vectors x, t, and Rx’ are coplanar slide: S. Lazebnik

1det( ) 0, ( ) 02

T TE EE E trace EE E 'FKKE T

Cubic constraint

Today: Binocular stereo• Given a calibrated binocular stereo pair, fuse it to

produce a depth image

Where does the depth information come from?

Binocular stereo• Given a calibrated binocular stereo pair, fuse it to

produce a depth imageo Humans can do it

Stereograms: Invented by Sir Charles Wheatstone, 1838

Binocular stereo• Given a calibrated binocular stereo pair, fuse it to

produce a depth imageo Humans can do it

Autostereograms: www.magiceye.com

Binocular stereo• Given a calibrated binocular stereo pair, fuse it to

produce a depth imageo Humans can do it

Autostereograms: www.magiceye.com

Real-time stereo

• Used for robot navigation (and other tasks)o Software-based real-time stereo techniques

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

slide: R. Szeliski

Stereo image pair

slide: R. Szeliski

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

Anaglyphs

http://www.rainbowsymphony.com/freestuff.html

(Wikipedia for images)

slide: R. Szeliski

Stereo: epipolar geometry

• Match features along epipolar lines

viewing ray

epipolar plane

epipolar line

slide: R. Szeliski

Simplest Case: Parallel images

• Image planes of cameras are parallel to each other and to the baseline

• Camera centers are at same height

• Focal lengths are the same

slide: S. Lazebnik

Simplest Case: Parallel images

• Image planes of cameras are parallel to each other and to the baseline

• Camera centers are at same height

• Focal lengths are the same

• Then, epipolar lines fall along the horizontal scan lines of the images

slide: S. Lazebnik

Essential matrix for parallel images

RtExExT ][,0

0000

000][

TTRtE

R = I t = (T, 0, 0)

Epipolar constraint:

00

0][

xy

xz

yz

aaaa

aaa

t

x

x’

Essential matrix for parallel images

RtExExT ][,0

0000

000][

TTRtE

Epipolar constraint:

vTTvvTTvuv

u

TTvu

0

010

10000

0001

R = I t = (T, 0, 0)

The y-coordinates of corresponding points are the same!

t

x

x’

Depth from disparity

f

x x’

BaselineB

z

O O’

X

f

zfBxxdisparity

Disparity is inversely proportional to depth!

Depth Sampling Depth sampling for integer pixel disparity

Quadratic precision loss with depth!

Depth Sampling Depth sampling for wider baseline

Depth Sampling Depth sampling is in O(resolution6)

Stereo: epipolar geometry• for two images (or images with collinear camera

centers), can find epipolar lines• epipolar lines are the projection of the pencil of

planes passing through the centers

• Rectification: warping the input images (perspective transformation) so that epipolar lines are horizontal

slide: R. Szeliski

Rectification• Project each image onto same plane, which is

parallel to the epipole• Resample lines (and shear/stretch) to place lines

in correspondence, and minimize distortion

• [Loop and Zhang, CVPR’99]

slide: R. Szeliski

Rectification

BAD!

slide: R. Szeliski

Rectification

GOOD!

slide: R. Szeliski

Problem: Rectification for forward moving

cameras

• Required image can become very large (infinitely large) when the epipole is in the image

• Alternative rectifications are available using epipolar lines directly in the imageso Pollefeys et al. 1999, “A simple and efficient method for general

motion”, ICCV

Your basic stereo algorithm

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

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

slide: R. Szeliski

Finding correspondences

• apply feature matching criterion (e.g., correlation or Lucas-Kanade) at all pixels simultaneously

• search only over epipolar lines (many fewer candidate positions)

slide: R. Szeliski

Matching cost

disparity

Left Right

scanline

Correspondence search

• Slide a window along the right scanline and compare contents of that window with the reference window in the left image

• Matching cost: SSD or normalized correlation

slide: S. Lazebnik

Left Right

scanline

Correspondence search

SSDslide: S. Lazebnik

Left Right

scanline

Correspondence search

Norm. corrslide: S. Lazebnik

Neighborhood size• Smaller neighborhood: more details• Larger neighborhood: fewer isolated mistakes

• w = 3 w = 20

slide: R. Szeliski

Matching criteria• Raw pixel values (correlation)• Band-pass filtered images [Jones & Malik 92]• “Corner” like features [Zhang, …]• Edges [many people…]• Gradients [Seitz 89; Scharstein 94]• Rank statistics [Zabih & Woodfill 94]• Intervals [Birchfield and Tomasi 96]• Overview of matching metrics and their

performance:o H. Hirschmüller and D. Scharstein, “Evaluation of Stereo Matching

Costs on Images with Radiometric Differences”, PAMI 2008

slide: R. Szeliski

Adaptive Weighting• Boundary Preserving• More Costly

Failures of correspondence search

Textureless surfaces Occlusions, repetition

Non-Lambertian surfaces, specularitiesslide: S. Lazebnik

Stereo: certainty modeling• Compute certainty map from correlations

• input depth map certainty map

slide: R. Szeliski

Results with window search

Window-based matching Ground truth

Data

slide: S. Lazebnik

Better methods exist...

Graph cuts Ground truth

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

Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 2001

slide: S. Lazebnik

How can we improve window-based

matching?• The similarity constraint is local (each reference

window is matched independently)• Need to enforce non-local correspondence

constraints

slide: S. Lazebnik

Non-local constraints• Uniqueness

o For any point in one image, there should be at most one matching point in the other image

slide: S. Lazebnik

Non-local constraints• Uniqueness

o For any point in one image, there should be at most one matching point in the other image

• Orderingo Corresponding points should be in the same order in both views

slide: S. Lazebnik

Non-local constraints• Uniqueness

o For any point in one image, there should be at most one matching point in the other image

• Orderingo Corresponding points should be in the same order in both views

Ordering constraint doesn’t holdslide: S. Lazebnik

Non-local constraints• Uniqueness

o For any point in one image, there should be at most one matching point in the other image

• Orderingo Corresponding points should be in the same order in both views

• Smoothnesso We expect disparity values to change slowly (for the most part)

slide: S. Lazebnik

Scanline stereo• Try to coherently match pixels on the entire

scanline• Different scanlines are still optimized

independentlyLeft image Right image

slide: S. Lazebnik

“Shortest paths” for scan-line stereo

Left image

Right image

Can be implemented with dynamic programmingOhta & Kanade ’85, Cox et al. ‘96

leftS

rightS

correspondence

q

p

Left

occlu

sion

t

Rightocclusion

s

occlC

II

corrC

Slide credit: Y. Boykov

Coherent stereo on 2D grid• Scanline stereo generates streaking artifacts

• Can’t use dynamic programming to find spatially coherent disparities/ correspondences on a 2D grid

slide: S. Lazebnik

Stereo matching as energy minimizationI1 I2 D

• Energy functions of this form can be minimized using graph cuts

Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 2001

W1(i ) W2(i+D(i )) D(i )

jii

jDiDiDiWiWDE,neighbors

221 )()())(()()(

data term smoothness term

slide: S. Lazebnik

Active stereo with structured light

• Project “structured” light patterns onto the objecto Simplifies the correspondence problemo Allows us to use only one camera

camera

projector

L. Zhang, B. Curless, and S. M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi-pass Dynamic Programming. 3DPVT 2002

slide: S. Lazebnik

Active stereo with structured light

L. Zhang, B. Curless, and S. M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi-pass Dynamic Programming. 3DPVT 2002

slide: S. Lazebnik

Active stereo with structured light

http://en.wikipedia.org/wiki/Structured-light_3D_scannerslide: S. Lazebnik

Kinect: Structured infrared light

http://bbzippo.wordpress.com/2010/11/28/kinect-in-infrared/ slide: S. Lazebnik

Laser scanning

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

Digital Michelangelo ProjectLevoy et al.

http://graphics.stanford.edu/projects/mich/

Source: S. Seitz

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Source: S. Seitz

Laser scanned models

The Digital Michelangelo Project, Levoy et al. Source: S. Seitz

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Source: S. Seitz

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Source: S. Seitz

Laser scanned models

The Digital Michelangelo Project, Levoy et al.

Source: S. Seitz

1.0 mm resolution (56 million triangles)

Aligning range images• A single range scan is not sufficient to describe a

complex surface• Need techniques to register multiple range

images

B. Curless and M. Levoy, A Volumetric Method for Building Complex Models from Range Images, SIGGRAPH 1996

Aligning range images• A single range scan is not sufficient to describe a

complex surface• Need techniques to register multiple range

images

• … which brings us to multi-view stereo