Geometry 2 A taste of projective geometry Introduction

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Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann

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:

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 •

Summary of last lecture •

Epipolar constraints, the Essential matrix • epipolar plane Baseline

Epipolar constraints, the Essential matrix • epipolar plane Baseline

Camera matrix •

Camera matrix •

The uncalibrated case: the Fundamental matrix •

The uncalibrated case: the Fundamental matrix •

The Fundamental matrix •

The Fundamental matrix •

Geometry • Geometry – Greek: earth measurement • Geometry concerns with shape, size, relative

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

Projective geometry • How does a plane looks after projection? How does perspective distorts geometry?

Plane perspective Pencil of rays

Plane perspective Pencil of rays

Plane perspective Pencil of rays

Plane perspective Pencil of rays

Projective transformation

Projective transformation

Projective transformation • How these change from Eucleadian geometry? • • • Point, line,

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 coordinates •

Projective line •

Projective line •

Intersection and incidence •

Intersection and incidence •

Ideal points •

Ideal points •

Line at infinity

Line at infinity

Line at infinity •

Line at infinity •

Homography •

Homography •

Homography •

Homography •

Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism Similarity Preserves angles, parallelism Affine

Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism Similarity Preserves angles, parallelism Affine Preserves parallelism Homography Preserves cross ratio

Camera rotation •

Camera rotation •

Planar scene •

Planar scene •

Summary Homography Perspective (calibrated) Perspective (uncalibrated) Orthographic One-to-one (group) Concentric epipolar lines Parallel epipolar

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

Recovering epipolar constraints •

Recovering epipolar constraints •

Interest points (Harris) •

Interest points (Harris) •

Descriptor: SIFT (Scale invariant feature transform) •

Descriptor: SIFT (Scale invariant feature transform) •

SIFT matches

SIFT matches

RANSAC •

RANSAC •

Epipolar lines

Epipolar lines

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