Projective Geometry

Projection

Vanishing lines m and n

Projective Plane (Extended Plane)

Projective Plane How??? Ordinary plane

Point Representation A point in the projective plane is represented as a ray in R 3

Projective Geometry

Homogeneous coordinates Homogeneous representation of 2D points and lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3  (0,0,0) T forms P 2 The point x lies on the line l if and only if Homogeneous coordinates Inhomogeneous coordinates but only 2DOF Note that scale is unimportant for incidence relation

Projective Geometry

Projective plane = S 2 with antipodal points identified Ordinary plane is unbound Projective plane is bound!

Projective Geometry

Pappus’ Theorem

Conic Section

Form of Conics

Transformation Projective: incidence, tangency Affine: plane at infinity, parallelism Similarity: absolute conics

Circular Point Circular points

Euclidean Transformation Any transformation of the projective plane which leaves the circular points fixed is a Euclidean transformation, and Any Euclidean transformation leaves the circular points fixed. A Euclidean transformation is of the form:

Euclidean Transformation

Calibration

Use circular point as a ruler…

Calibration

Today Cross ratio More on circular points and absolute conics Camera model and Zhang’s calibration Another calibration method

Transformation Let X and X’ be written in homogeneous coordinates, when X’=PX P is a projective transformation when….. P is an affine transformation when….. P is a similarity transformation when…..

Transformation Projective Affine Similarity Euclidean

Matrix Representation

Invariance Mathematician loves invariance ! Fixed point theorem Eigenvector

Cross Ratio Projective line P = (X,1) t Consider

Cross Ratio

Consider determinants: Rewritting So we have Consider

Cross Ratio How do we eliminate |T| and the coefficients The idea is to use the ratio. Consider and The remaining coefficients can be eliminated by using the fourth point

Pinhole Camera

3x4 projection matrix 3x3 intrinsic matrix Extrinsic matrix Principle point Skew factor

Pinhole Camera

Absolute Conic

Important: absolute conic is invariant to any rigid transformation We can writeand That is, and obtain

Absolute Conic Now consider the image of the absolute conic It is defined by

Typical Calibration 1.Estimate the camera projection matrix from correspondence between scene points and image points (Zhang p.12) 2.Recover intrinsic and extrinsic parameters

Typical Calibration P[3][4], B[3][3], b[3]

Calibration with IAC Can we calibrate without correspondence? (British Machine Vision)

Calibration with IAC

From Zhang’s, the image of the absolute conic is the conic Let’s assume that the model plane is on the X-Y plane of the world coordinate system, so we have:

Calibration with IAC Points on the model plane with t=0 form the line at infinity It is sufficient to consider model plane in homogeneous coordinates We know that the circular points I = (1,i,0,0) T and J = (1,-i,0,0) T must satisfy Let the image of I and J be denoted by

Calibration with IAC Consider the circle in the model plane with center (Ox,Oy,1) and radius r. This circle intersects the line at infinitywhen or Any circle (any center, any radius) intersects line at infinity in the two circular points The image of the circle should intersect the image of the line at infinity (vanishing line) in the image of the two circular points

Calibration with IAC