Computer Vision : CISC 4/689 CREDITS Rasmussen, UBC (Jim Little), Seitz (U. of Wash.), Camps (Penn....

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 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

Computer Vision : CISC 4/689 CREDITS Rasmussen, UBC (Jim Little), Seitz (U. of Wash.), Camps (Penn....

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