29 April 2008 Birkbeck College, U. Lond on 1 Geometric Model Acquisition Steve Maybank School of Computer Science and Information Systems Birkbeck College London, WC1E 7HX Edited version of the slides for the VVG Summer School, held at the University of Bath 21 September 2007
Geometric Model Acquisition. Steve Maybank School of Computer Science and Information Systems Birkbeck College London, WC1E 7HX Edited version of the slides for the VVG Summer School, held at the University of Bath 21 September 2007. Geometric Model Acquisition. - PowerPoint PPT Presentation
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29 April 2008 Birkbeck College, U. London 1
Geometric Model Acquisition
Steve MaybankSchool of Computer Science and Information
SystemsBirkbeck College
London, WC1E 7HX
Edited version of the slides for the VVG Summer School, held at the University of Bath
21 September 2007
29 April 2008 Birkbeck College, U. London 2
Geometric Model Acquisition
Aim: make a 3D model of a scene from two or more images taken from different viewpoints.
Why is it possible: the image differences depend in part on the shapes of the objects in the scene.
29 April 2008 Birkbeck College, U. London 3
Two Images of the Same Scene
http://vasc.ri.cmu.edu/idb/images/stereo/fruitSOURCE "University of Illinois, Bill Hoff“DESCRIPTION "Fruit on table, digitized from 35mm."
29 April 2008 Birkbeck College, U. London 4
Two Images of a Point in R3
• p
c1 •
c2•
q1 •
q2 •
Epipolar plane: <c1,c2,x>
Image 1
Image 2
objectpoint
opticalcentre
opticalcentre
29 April 2008 Birkbeck College, U. London 5
Corresponding Points Points in different images
correspond,qq ~,
qq ~ if they are projections of the same
scene point p. In projective coordinates, projection is a matrix application,
qpM
qMp~~
29 April 2008 Birkbeck College, U. London 6
Method for Finding Corresponding Points
oods.neighbourh levelgrey similar have ,~ i.e. large, is
~
~.~.~.
~.~,
ncorrelatio theif ~ thatassumed isIt .~, of oodsneighbourh levelgrey be ~,Let
Description of the relative positions of the cameras.
Equations involving the measurements, the scene points and the relative positions of the cameras.
Statistical description of the errors in the measurements.
29 April 2008 Birkbeck College, U. London 9
Pinhole Camera
Light tight box
Small hole (optical centre)
Viewingscreen(image)
Object
Light rays
Central perspective projection model for imageformation (Brunelleschi, 15th C.).
29 April 2008 Birkbeck College, U. London 10
Camera Coordinate Frame
(X,Y,Z)(0,0,0)(0,0,-f)
Origin (0,0,0) at the pin hole.Focal length of the camera = f.Axes of image coordinate frame are parallel to X, Y axes
of the CCF.Image point = (-Xf/Z, -Yf/Z)
•
•
• Z
YXx
y
29 April 2008 Birkbeck College, U. London 11
Mathematical Version of the Camera Coordinate Frame
(X,Y,Z)
(0,0,0)
(0,0,f)
Origin (0,0,0) at the pin hole.Focal length of the camera = f.The image is in front of the pin hole!Image point = (Xf/Z, Yf/Z). The minus signs have gone.
•
•
• Z
yxX
Y Image plane
•
29 April 2008 Birkbeck College, U. London 12
Relative Position of the Cameras
Y X~
ZX Z~
Y~
The relative position of the cameras is describedby an orthogonal matrix R and a translation vector t.
R, t
29 April 2008 Birkbeck College, U. London 13
Transformation of Coordinates
YX~
ZX
Z~
Y~
If a point p has coordinates (X,Y,Z)T in the first CCF,then in the second CCF the same point p has coordinates
R, t
● p
t
Z
Y
X
R
29 April 2008 Birkbeck College, U. London 14
Properties of Orthogonal Matrices
angle.rotation the
for one and axis for the two:freedom of degrees threehave rotations The
100
0cossin
0sincos
:matrixrotation a of Example
.1det ifonly and ifrotation a represents matrix orthogonal The
.1det that followsIt
.
100
010
001
ifonly and if orthogonal is matrix 3x3 The
R
RR
R
IRRR T
29 April 2008 Birkbeck College, U. London 15
Projection Ray
Tfyx ,,point Image
Y
X
Z
●
0,,, :ray Projection Tfyx
Any scene point projecting to (x, y, f)T is on the projection ray.
CCF
29 April 2008 Birkbeck College, U. London 16
Projection Rays of Corresponding Points 1
●
The projection rays of corresponding points intersect ata scene point. Geometric model acquisition is based onthis single constraint.
For an extreme example, see http://www.wisdom.weizmann.ac.il/~vision/VideoAnalysis/Demos/Traj2Traj/hall.htm
29 April 2008 Birkbeck College, U. London 17
Projection Rays of Corresponding Points 2
Tfyx ,, Tfyx ~
,~,~~ ●
The equations of the projectionrays are known, but they hold indifferent coordinate systems.
f
y
x
f
y
x
~~
~
29 April 2008 Birkbeck College, U. London 18
Transformation of Coordinates
by CCF second in thegiven is ,,ray The Tfyx
t
f
y
x
R
such that ~, numbers realexist therethus
.~~
~
~
f
y
x
t
f
y
x
R
,~
,~,~ith equation w theofproduct scalar theby taking eliminated are ~, numbersunknown TheT