projective geometry ece 847: digital image processing stan birchfield clemson university
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Projective geometry
ECE 847:Digital Image Processing
Stan BirchfieldClemson University
Lines
• A line in 2D is described by two parameters:
• But vertical lines?• Only two parameters are sufficient,
but requires nonlinear formulation:
slope y-intercept
almost
^
Lines• A better parameterization can represent all lines:
• Here the line is represented by 3 parameters:
• But nonzero scalar multiple does not change the equation:
• So we have only 2 degrees of freedom• To make this work, we have to introduce a non-intuitive
definition:
• I.e., the vector u and its scalar multiple are the same
Lines
• While we are at it, let us put the point into a vector, too:
• Which leads to the beautiful expression:
• Nonzero scalar multiple also does not change the point:
• So we introduce an analogous non-intuitive definition:
Example
• Ques: What does the vector [4, 6, 2]T represent?
• Ans: It depends.
– If the vector is a 2D point, then the point is(4/2, 6/2) = (2, 3) -- divide by 3rd coordinate
– If the vector is a 2D line, then the line is4x + 6y + 2 = 0, or 2x + 3y + 1 = 0
• Points and lines are represented in the same way. Context determines which.
Lines• Ques.: Is the point p on the line u?• Ans: Check whether pTu = 0
• Ques.: Which line passes through two points p1 and p2?
• Ans.: Compute u = p1 x p2
• Ques.: Which points lies at the intersection of two lines?
• Ans.: Compute p = u1 x u2
Euclidean transformation
• 2D Euclidean transformation:
• is more conveniently represented as
• Again, we use 3 numbers to represent 2D point(These are homogeneous coordinates)
Perspective projection
• Nonlinear perspective projection
• can be replaced by linear equation
• where (x,y,w)T are homogeneous coordinates of (u,v,1)T:
Recap
• Homogeneous coordinates of 2D point (x,y)T are p=(wx,wy,w)T where w ≠ 0
• We have seen three reasons for homogeneous coordinates:– simple representation of points and lines, no
special cases– simple representation of Euclidean
transformation– simple representation of perspective
projection
Q & A
• Questions:– Is there a unifying theory to explain
homogeneous coordinates?– How can they be extended to 3D?– Are they useful for anything else?
• Answers:– Projective geometry– Useful for planar warping, 3D
reconstruction, image mosaicking, camera calibration, etc.
Euclidean Projective
• Start with 2D Euclidean point (x,y)
• To convert to Projective,– Append 1 to the coordinates: p=(x,y,1)– Declare equivalence class: p=p, ≠0
• To convert back to Euclidean,– Divide by last coordinate:
(u, v, w) (u/w, v/w) x=u/w, y=v/w
Ideal points
• What if last coordinate is zero?(u,v,0)
• Cannot divide by zero• Projective plane contains more
points than the Euclidean plane:– All Euclidean planes, plus– Points at infinity (a.k.a. ideal points)
• All ideal points lie on ideal line: (0, 0, 1)
Are ideal points special?
• In pure projective geometry, there is no distinction between real points and ideal points– Transformations will often convert one to
another– We will freely make use of this, and often
ignore the distinction
• However, distinction is necessary to convert back to Euclidean– Distinction will be made when we need to
interpret results
Geometries
• Every geometry has– transformations– invariants
Stratification of geometries
• Euclidean
• similarity
• affine
• projective
Ray space
Unit hemisphere
Augmented affine plane
Homogeneous coordinates
Image formation
• 3D world point is (X,Y,Z,W)T
• 2D image point is (x,y,w)T
• Therefore, perspective projection is a 3x4 matrix P
Perspective projection
• Camera calibration matrix K
Homography
• Simple case is projection from plane to plane– Can be either world plane to image plane, or– image plane to another image plane, or– world plane to another world plane, – etc.
• 3x3 matrix is a projective transformation• Called a homography
Euclidean homography
• Needs K
Essential and fundamental matrices
Relationship b/w FM and H
• Fundamental matrix and homography
How to compute homography
• Direct Linear Transform
Normalization
• Important
Stratification
Euclidean ProjectiveAffineSimilarity
ideal lineabsolute pointsone length
allow scale allow parallelprojection
allow perspectiveprojection
Conics
Circular points
• aka absolute points
Absolute conic
Image of the absolute conic
Calibration using intersecting circles
Calibration using squares
Calibration using vanishing points
Calibration algorithm
Tsai’s calibration routine
Radial alignment constraint
What can you do with a calibrated camera
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