Basic geometric concepts to understand

1
Basic geometric concepts to understand

• Affine, Euclidean geometries (inhomogeneous
  coordinates)
• projective geometry (homogeneous coordinates)
• plane at infinity affine geometry

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Intuitive introduction
Naturally everything starts from the known vector
space
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• Vector space to affine isomorph, one-to-one
• vector to Euclidean as an enrichment scalar
  prod.
• affine to projective as an extension add ideal
  elements

Pts, lines, parallelism
Angle, distances, circles
Pts at infinity
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Relation between Pn (homo) and Rn (in-homo)
Rn --gt Pn, extension, embedded in
Pn --gt Rn, restriction,
P2 and R2
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Examples of projective spaces

• Projective plane P2
• Projective line P1
• Projective space P3

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Projective plane
Space of homogeneous coordinates (x,y,t)
Pts are elements of P2
Pts are elements of P2
Pts at infinity (x,y,0), the line at infinity
4 pts determine a projective basis
3 ref. Pts 1 unit pt to fix the scales for
ref. pts
Relation with R2, (x,y,0), line at inf., (0,0,0)
is not a pt
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Lines
Linear combination of two algebraically
independent pts
Operator is span or join
Line equation
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Point/line duality

• Point coordinate, column vector
• A line is a set of linearly dependent points
• Two points define a line

• Line coordinate, row vector
• A point is a set of linearly dependent lines
• Two lines define a point

• What is the line equation of two given points?
• line (a,b,c) has been always homogeneous
  since high school!

9
Given 2 points x1 and x2 (in homogeneous
coordinates), the line connecting x1 and x2 is
given by
Given 2 lines l1 and l2, the intersection point x
is given by
NB cross-product is purely a notational device
here.
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Conics
Conics a curve described by a second-degree
equation

• 33 symmetric matrix
• 5 d.o.f
• 5 pts determine a conic

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Projective line
Homogeneous pair (x1,x2)
Finite pts Infinite pts how many? A basis
by 3 pts Fundamental inv cross-ratio
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Projective space P3

• Pts, elements of P3
• Relation with R3, plane at inf.
• planes linear comb of 3 pts
• Basis by 4 (ref pts) 1 pts (unit)

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planes
In practice, take SVD
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Key points

• Homo. Coordinates are not unique
• 0 represents no projective pt
• finite points embedded in proj. Space (relation
  between R and P)
• pts at inf. (x,0) missing pts, directions
• hyper-plane (co-dim 1)
• dualily between u and x,

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Introduction to transformation
2D general Euclidean transformation
2D general affine transformation
2D general projective transformation
Colinearity Cross-ratio
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Projective transformation
collineation homography
Consider all functions
All linear transformations are represented by
matrices A
Note linear but in homogeneous coordinates!
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How to compute transformatins and canonical
projective coordinates?
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Geometric modeling of a camera
How to relate a 3D point X (in oxyz) to a 2D
point in pixels (u,v)?
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Camera coordinate frame
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Image coordinate frame
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5 intrinsic parameters

• Focal length in horizontal/vertical pixels (2)
• (or focal length in pixels aspect ratio)
• the principal point (2)
• the skew (1)

one rough example 135 film
In practice, for most of CCD cameras

• alpha u alpha v i.e. aspect ratio1
• alpha 90 i.e. skew s0
• (u0,v0) the middle of the image
• only focal length in pixels?

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World (object) coordinate frame
Xw
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6 extrinsic parameters
World coordinate frame extrinsic parameters
Relation between the abstract algebraic and
geometric models is in the intrinsic/extrinsic
parameters!
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Finally, we have a map from a space pt (X,Y,Z)
to a pixel (u,v) by
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What does the calibration give us?
It turns the camera into an angular/direction
sensor!
Normalised coordinates
Direction vector
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Camera calibration
Given
from image processing or by hand ?

• Estimate C
• decompose C into intrinsic/extrinsic

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Decomposition

• analytical by equating K(R,t)P

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Pose estimation calibration of only extrinsic
parameters

• Given
• Estimate R and t

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3-point algebraic method
3 reference points 3 beacons

• First convert pixels u into normalized points x
  by knowing the intrinsic parameters
• Write down the fundamental equation
• Solve this algebraic system to get the point
  distances first
• Compute a 3D transformation

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3D transformation estimation
given 3 corresponding 3D points

• Compute the centroids as the origin
• Compute the scale
• (compute the rotation by quaternion)
• Compute the rotation axis
• Compute the rotation angle

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Linear pose estimation from 4 coplanar points

• Vector based (or affine geometry) method

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Midterm statistics
059 7 6069 12 7079 17 8089 8 9099
5 100 2
Total 71.80392157 16.30953047 Q1
14.98039216 5.82920302 Q2 12.03921569
6.141533308 Q3 14.56862745
4.817696138 Q4 12.35294118
7.638909685 Q5 14.90196078
7.105645367 Q6 7.254901961 4.511510334