Geometry 2 A taste of projective geometry Introduction
- Slides: 32
Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann Institute of Science
Material covered • Pinhole camera model, perspective projection • Two view geometry, general case: • Epipolar geometry, the essential matrix • Camera calibration, the fundamental matrix • Two view geometry, degenerate cases • Homography (planes, camera rotation) • A taste of projective geometry • Stereo vision: 3 D reconstruction from two views • Multi-view geometry, reconstruction through factorization
Summary of last lecture •
Epipolar constraints, the Essential matrix • epipolar plane Baseline
Camera matrix •
The uncalibrated case: the Fundamental matrix •
The Fundamental matrix •
Geometry • Geometry – Greek: earth measurement • Geometry concerns with shape, size, relative positions, and properties of spaces • Euclidean geometry: • • • Point, line, plane Incidence Continuity Order, “between” Parallelism Congruence = invariance: angles, lengths, areas are preserved under rigid transformations
Projective geometry • How does a plane looks after projection? How does perspective distorts geometry?
Plane perspective Pencil of rays
Plane perspective Pencil of rays
Projective transformation
Projective transformation • How these change from Eucleadian geometry? • • • Point, line, plane Incidence Continuity Order, “between” Parallelism Congruence • Under projective transformation • A (straight) line transforms to a line and a conic to a conic • But order and parallelism are not preserved • Likewise, angles, lengths and areas are not preserved
Projective coordinates •
Projective line •
Intersection and incidence •
Ideal points •
Line at infinity
Line at infinity •
Homography •
Homography •
Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism Similarity Preserves angles, parallelism Affine Preserves parallelism Homography Preserves cross ratio
Camera rotation •
Planar scene •
Summary Homography Perspective (calibrated) Perspective (uncalibrated) Orthographic One-to-one (group) Concentric epipolar lines Parallel epipolar lines Form Properties DOFs 8(5) 8(7) 4 Eqs/pnt 2 1 1 1 Minimal configuration 4 5+ (8, linear) 7+ (8, linear) 4 Depth No Yes, up to scale Yes, projective Affine structure (third view required for Euclidean structure)
Recovering epipolar constraints
Recovering epipolar constraints •
Interest points (Harris) •
Descriptor: SIFT (Scale invariant feature transform) •
SIFT matches
RANSAC •
Epipolar lines