# lec4 - Announcements Image Formation Cameras(cont Computer...

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1 CS252A, Fall 2010 Computer Vision I Image Formation, Cameras (cont.) Computer Vision I CSE 252A Lecture 4 CS252A, Fall 2010 Computer Vision I Announcements Assignment 0: “ Getting Started with Matlab ” is due today Read Chapters 1 & 2 of Forsyth & Ponce CS252A, Fall 2010 Computer Vision I The course Part 1: The physics of imaging Part 2: Early vision Part 3: Reconstruction Part 4: Recognition CS252A, Fall 2010 Computer Vision I Pinhole Camera: Perspective projection • Abstract camera model - box with a small hole in it Forsyth&Ponce CS252A, Fall 2010 Computer Vision I Geometric properties of projection 3-D points map to points 3-D lines map to lines Planes map to whole image or half-plane Polygons map to polygons Important point to note: Angles & distances not preserved, nor are inequalities of angles & distances. Degenerate cases: line through focal point project to point plane through focal point projects to a line CS252A, Fall 2010 Computer Vision I Equation of Perspective Projection Cartesian coordinates: We have, by similar triangles, that (x, y, z) -> (f’ x/z, f’ y/z, f’) Establishing an image plane coordinate system at C’ aligned with i and j, we get

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2 CS252A, Fall 2010 Computer Vision I A Digression Projective Geometry and Homogenous Coordinates CS252A, Fall 2010 Computer Vision I Homogenous coordinates Our usual coordinate system is called a Euclidean or affine coordinate system Rotations, translations and projection in Homogenous coordinates can be expressed linearly as matrix multiplies Euclidean World 3D Homogenous World 3D Homogenous Image 2D Euclidean World 2D Convert Convert Projection CS252A, Fall 2010 Computer Vision I What is the intersection of two lines in a plane? A Point CS252A, Fall 2010 Computer Vision I Do two lines in the plane always intersect at a point? No, Parallel lines don’t meet at a point. CS252A, Fall 2010 Computer Vision I Can the perspective image of two parallel lines meet at a point? YES CS252A, Fall 2010 Computer Vision I Projective geometry provides an elegant means for handling these different situations in a unified way and homogenous coordinates are a way to represent entities (points & lines) in projective spaces.
3 CS252A, Fall 2010 Computer Vision I Projective Geometry Axioms of Projective Plane 1. Every two distinct points define a line 2. Every two distinct lines define a point (intersect at a point) 3. There exists three points, A,B,C such that C does not lie on the line defined by A and B. Different than Euclidean (affine) geometry Projective plane is “bigger” than affine plane – includes “line at infinity” Projective Plane Affine Plane = + Line at Infinity CS252A, Fall 2010 Computer Vision I Homogenous coordinates A way to represent points in a projective space • Use three numbers to represent a point on a projective plane Why? The projective plane has to be bigger than the Cartesian plane.

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