Single View Geometry Course web page: vision.cis.udel.edu/cv April 7, 2003  Lecture 19

Announcements Readings... –From Hartley & Zisserman: Chapter , describe basis of DLT algorithm that camera calibration lecture referred to Chapters (skip 1.2.3), are background for today and Wednesday –Criminisi et al.’s “Single View Metrology” for Wednesday But first, conclusion of calibration lecture

Outline Homogeneous representation of lines, planes Vanishing points and lines Single view metrology

Homogeneous Representation of 2-D Lines A line in a plane is specified by the equation ax + by + c = 0 In vector form, l = (a, b, c) T This is equivalent to k(a, b, c) T for non- zero k, as with homogeneous point coordinates (0, 0, 0) T is undefined for lines

Homogeneous Lines: Results Point on line –A 2-D homogeneous point x = (x, y, 1) T is on the line l = (a, b, c) T only when ax + by + c = 0 –We can write this as a dot product: l ¢ x = 0 Intersection of lines –We want a point x that is on both lines l and l’. This would imply that l ¢ x = l’ ¢ x = 0 –Because the cross product is orthogonal to both multiplicands, x = l £ l’ satisfies this requirement and thus defines the point of intersection Line joining points –This is the dual of line intersection, so l = x £ x’

The Intersection of Parallel Lines Consider two parallel lines l = (a, b, c) T and l’ = (a, b, c’) T The intersection of these two lines is given by l £ l’ = (b, {a, 0) T This is not a finite point on the plane, but rather an ideal point, or a point at infinity For example, the lines x = 1 and x = 2 are l = (-1, 0, 1) T and l’ = (-1, 0, 2) T, respectively, and their intersection is (0, 1, 0) T –This is the point at infinity in the direction of the Y -axis

Line at Infinity All ideal points (x 1, x 2, 0) T lie on a single line called the line at infinity: l 1 = (0, 0, 1) T This can be thought of as the set of all directions in the plane

Homogeneous Representation of Planes A plane in 3-D is specified by the equation ax + by + cz + d = 0 In vector form, ¼ = (a, b, c, d) T Analogous to lines, a 3-D homogeneous point x = (x, y, z, 1) T is on the plane ¼ only when ¼ ¢ x = 0 More results: –The intersection of 2 planes is a line –The intersection of 3 planes is a point –By duality, a plane is the join of 3 non-collinear points

More 3-D Analogues 3-D lines have various representations –E.g., 4 x 4 Plucker matrices allow straightforward interaction with 3-D homogeneous points and planes Parallel 3-D lines intersect on the plane at infinity ¼ 1 = (0, 0, 0, 1) T at a 3-D ideal point A plane ¼ intersects ¼ 1 in a line which is the 3-D line at infinity l 1 of ¼

Vanishing Points & Lines Vanishing point: Finite image projection of ideal point Vanishing line: Image projection of plane’s line at infinity from Hartley & Zisserman

Basic method: Detect edges, identify parallel segments, find intersection point But...because of image noise, etc., lines do not intersect at a unique point Computing a Vanishing Point from an Image from Hartley & Zisserman

Vanishing Point Estimation Idea: Fit lines independently, then choose point closest to all of them –Not optimal Better approach: Pick vanishing point location which results in best overall fit to lines –E.g., Levenberg-Marquardt minimization of SSD between endpoints of measured line segments and lines radiating from vanishing point from Hartley & Zisserman

Vanishing Line Estimation Compute vanishing points for sets of parallel lines in plane (or parallel planes) Then fit line to vanishing points from Hartley & Zisserman

Vanishing Points of Lines Parallel to Plane are on Same Vanishing Line from Hartley & Zisserman

Single View Metrology Definition: Obtaining information on scene struct- ure (e.g., lengths, areas) from a single image Idea: Use constraints imposed by parallel lines, planes to get measurements up to scale (Criminisi et al., 1999) from Criminisi et al.

Metrology Applications: Forensic Science from Criminisi et al. Knowing the height of the phone booth, can we determine the height of the person?

Metrology Applications: Virtual Modeling from Criminisi et al. Original image Synthesized view Synthesized view with original camera location