 # Projective Geometry Projective Geometry Projective Geometry Projective Geometry

• Slides: 60 Projective Geometry Projective Geometry Projective Geometry Projective Geometry Projective Geometry Projection 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 2 D points and lines The point x lies on the line l if and only if Note that scale is unimportant for incidence relation equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 (0, 0, 0)T forms P 2 Homogeneous coordinates Inhomogeneous coordinates but only 2 DOF Projective Geometry Projective Geometry Projective plane = S 2 with antipodal points identified Ordinary plane is unbound Projective plane is bound! Projective Geometry Projective Geometry Pappus’ Theorem Pappus’ Theorem Pappus’ Theorem Conic Section Conic Section Conic Section Conic Section Conic Section Conic Section Conic Section Conic Section Form of Conics Transformation • Projective • Affine • Similarity : incidence, tangency : plane at infinity, parallelism : 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 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 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 Pinhole Camera Skew factor Principle point Extrinsic matrix 3 x 4 projection matrix 3 x 3 intrinsic matrix Pinhole Camera Absolute Conic Absolute Conic Absolute Conic Important: absolute conic is invariant to any rigid transformation That is, We can write and obtain and 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, B, b Calibration with IAC Can we calibrate without correspondence? (British Machine Vision) Calibration with IAC 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 It is sufficient to consider model plane in homogeneous coordinates Points on the model plane with t=0 form the line at infinity 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 infinity when 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 Calibration with IAC