# Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann Institute of Science.

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

Summery of last lecture

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: 3D reconstruction from two views Multi-view geometry, reconstruction through factorization

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

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

Homography

Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism SimilarityPreserves angles, parallelism AffinePreserves parallelism Homography Preserves cross ratio

Camera rotation

Planar scene

Summary HomographyPerspective (calibrated) Perspective (uncalibrated) Orthographic Form PropertiesOne-to-one (group) Concentric epipolar lines Parallel epipolar lines DOFs 8(5) 8(7)4 Eqs/pnt 2111 Minimal configuration 45+ (8,linear)7+ (8,linear)4 DepthNoYes, up to scale Yes, projective structure Affine structure (third view required for Euclidean structure)

Recovering epipolar constraints

Interest points (Harris)

Descriptor: SIFT ( Scale invariant feature transform)

SIFT matches

RANSAC

Epipolar lines

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