Download presentation

Presentation is loading. Please wait.

Published byErnest Vibert Modified about 1 year ago

1
Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann Institute of Science

2
Summery of last lecture

3
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

4
Camera matrix

5
The uncalibrated case: the Fundamental matrix

6
The Fundamental matrix

7
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

8
Projective geometry How does a plane looks after projection? How does perspective distorts geometry?

9
Plane perspective Pencil of rays

10
Plane perspective Pencil of rays

11
Projective transformation

12
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

13
Projective coordinates

14
Projective line

15
Intersection and incidence

16
Ideal points

17
Line at infinity

18

19
Homography

20

21
Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism SimilarityPreserves angles, parallelism AffinePreserves parallelism Homography Preserves cross ratio

22
Camera rotation

23
Planar scene

24
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)

25
Recovering epipolar constraints

26

27
Interest points (Harris)

28
Descriptor: SIFT ( Scale invariant feature transform)

29
SIFT matches

30
RANSAC

31
Epipolar lines

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google