Projective geometry ECE 847: Digital Image Processing Stan Birchfield Clemson University

Lines A line in 2D is described by two parameters: But vertical lines? almost A line in 2D is described by two parameters: But vertical lines? Only two parameters are sufficient, but requires nonlinear formulation: ^ slope y-intercept

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 is 4x + 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: Answers: 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=ap, a≠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

Stratification Euclidean Similarity Affine Projective allow parallel projection allow perspective projection allow scale Euclidean Similarity Affine Projective one length absolute points ideal line

Cross ratio

Ray space

Unit hemisphere

Augmented affine plane ∞ ∞ ∞ ℓ∞ ∞ The line at infinity ℓ∞ is “beyond infinity”

Intersection of parallel lines y=mx+b (where m is the same) intersect at (1, m, 0) (1,m,0) ∞ ∞ ∞ ℓ∞ ∞ Note: Antipodal points are identified

Representing line at infinity (1,m1,0) ∞ ∞ ∞ ℓ∞ (1,m2,0) ∞ Cross product of two points at infinity yields ℓ∞ = (0,0,1)

The strange world beyond infinity The line at infinity ℓ∞ = (0,0,1) ax+by+c=0 This means 1 = 0 !

Line transformations If point transforms according to p’ = Ap How does line transform? u’ = A-Tu

Conics Take picture of circle  ellipse No distinction between types of conic sections in projective geometry

3D Projective Points and planes Plucker coordinates for lines

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