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