Lecture 1: 3D Computer Vision - 2D and 1D Projective Geometry

Key Topics:

• Difference between Euclidean and Projective Geometry
• Cartesian 2D Coordinates vs. Homogeneous Coordinates
• Representing Points, Lines, Conics in 2D Projective Space
• Duality Relationship
• Application of 2D Transformation Hierarchies

Geometry Overview

Euclidean Geometry

• Described using Cartesian coordinates.
• Uses the axiomatic methods, related tools like compasses and straight edges.
• Represents real-world 3D scenes as Euclidean geometry.

Projective Geometry

• Described using Homogeneous coordinates.
• Deals with projective transformations (projection of 3D world onto a 2D image).
• Real-world scenes are seen through projective transformations affecting geometrical properties.

Homogeneous Coordinates

• Represent points in the projective space using three values (kX, kY, k).
• Point at infinity represented by setting k = 0.
• Homogeneous coordinates can be converted to Cartesian by dividing by the third coordinate (k).

Relation Between Euclidean and Projective Geometry

• Euclidean Space (E) is represented by Cartesian Coordinates in R^N.
• Projective Space (P) is represented by Homogeneous Coordinates in R^(N+1).
• Example: 2D Euclidean space to 3D homogeneous projective space.

Projective Transformations in Life

• Our eyes see the 3D world as a 2D image through projective transformations.
• Properties like angles, distances, and ratios are not preserved (e.g., parallel lines meeting at a point, circles turning into ellipses).

Straightness Property

• A key property preserved in projective transformations.
• Straight lines remain straight after transformation.
• Basis for defining projective mappings.

Analytical vs. Synthetic Geometry

• Euclidean Geometry uses synthetic methods (compass, ruler).
• Projective Geometry uses analytical methods (algebra, linear algebra).

Projective Mapping

• Defines a transformation from one projective space to another while preserving straight lines.

Points and Lines in Projective Space

Representation

• Point in Homogeneous Coordinate: Kx, Ky, K
• Line in Homogeneous Coordinate: Coefficients Ax + By + Cz = 0
• Example transformations provided in lecture.

Intersections and Duality

• Intersecting two lines results in a point (using cross product).
• Intersecting two points results in a line (using cross product).
• Dual Relationship: Points can represent lines and vice versa (interchangeable in incidence relationships).

Lines at Infinity

• Represents directions of lines in the plane.
• Parallel lines intersect at infinity.

Use of Linear Algebra

• Homogeneous coordinates and linear algebra help describe projective geometry formally.
• Uses dot products, cross products to describe incidences.

Conclusion

• Review and understand differences/relationships between Euclidean and Projective Geometry.
• Use of Homogeneous coordinates is crucial to understanding projective transformations.
• Recommended readings: Textbooks by Richard Hartley and Andrew Zigermann (Chapter 2), Mahi's Invitation to 3D Vision (Chapter 2).