Single View Geometry

Course web page:vision.cis.udel.edu/cv

April 7, 2003 Lecture 19

Announcements

• Readings...– From Hartley & Zisserman:

• Chapter 3.1-3.1.1, 3.4.4 describe basis of DLT algorithm that camera calibration lecture referred to

• Chapters 1-1.4 (skip 1.2.3), 2-2.2.1 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 (x1, x2, 0)T lie on

a single line called the line at infinity: l1 = (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 ¼1 = (0, 0, 0, 1)T at a 3-D ideal point

• A plane ¼ intersects ¼1 in a line which

is the 3-D line at infinity l1 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 withoriginal camera location