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[3][4], B[3][3], b[3]
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
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- What is projective listening
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- Youtube
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- Theory of structures
- Geometry in the real world
- Perspective geometry of aerial photography
- Camera pinhole model
- Basic geometry terms
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- Essential questions for geometry
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- Inductive reasoning in geometry
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