Criminisi, Reid, Zisserman <Single View Metrology>

Criminisi, A., Reid, I., and Zisserman, A. 2000. Single View Metrology. Int. J. Comput. Vision 40, 2 (Nov. 2000), 123-148. DOI= http://dx.doi.org/10.1023/A:1026598000963

Single_View_Metrology.pdf

ref.
Constructing 3D Models from a Single View
3D Models from a Single View

Antonio Criminisi
http://research.microsoft.com/en-us/people/antcrim/
http://www.robots.ox.ac.uk/~criminis/

 

3D affine measurements
single perspective view
minimal geometric information - the vanishing line  & a vanishing point 
affine scene structure

first order error propagation analysis

(i) compute the distance between planes parellel to the reference plane (up to a common scale factor)
(ii) compute area and length ratios on any plane parallel to the reference plane
(iii) determine the camera's location

1. Introduction

perspective projection

reference plane
reference direction

three canonical types of measurement
(i) measurements of the distance between any of the planes which are parallel to the reference plane
(ii) measurements on these planes (and comparision of these measurements to those obtained on any parallel plane)
(iii) determining the camera's position in terms of the reference plane and direction

: independent of the camera's internal parammeters: focal length, aspect ratio, principal point, skew

http://en.wikipedia.org/wiki/Leon_Battista_Alberti

Maximum Likelihood estimates or measurements (when more than the minimum number of references are available)

1) planar homology to transfer measurements from one reference plane to another
2) analysing the uncertainty of the computted distances
3) statistical tests to validate the analytical uncertainty predictions

1) introduction
2) geometric derivations
3) algebraic representation
4) confidence intervals = a quantitative assessment of accuracy
5) applications

2. Geometry

central projection

vanishing point = projection of a point at infinity
: defined by any set of parallel lines on the plane
: Any parallel lines have the same vanishing point.

radial distortion

parallel projection

The vanishing line partitions all points in scene space. Any scene point which projects onto the vanishing line is at the same distance from the plane as the camera center; if it lies above the line it is farther from the plane, and if below the vanishing line, then it is closer to the plane than the camera center.

http://en.wikipedia.org/wiki/Projective_plane


2.1 Measurements Between Parallel Planes

Definition 1.
Two points X, X0 on separate planes (parallel to the reference plane) correspond if the line joining them is parallel to the reference direction.

The images of corresponding points and the vanishing point are collinear.

Theorem 1.
Given the vanishing line of a reference plane and the vanishing point for a reference direction; then distances from the reference plane parallel to the reference direction can be computed from their imaged end points up to a common scale factor. The scale factor can be determined from one known reference length.

cross-ratio
http://en.wikipedia.org/wiki/Cross-ratio
http://www.geom.uiuc.edu/docs/forum/photo/
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/OWENS/LECT13/node5.html
http://www.lems.brown.edu/vision/people/leymarie/Refs/Maths/Perspective.html
http://www.mathpages.com/home/kmath543/kmath543.htm

Although distances and ratios of distances are not preserved under projection, the cross ratio, defined as AC/BC ∙ BD/AD, is preserved. That is, AC/BC ∙ BD/AD = A′C′/B′C′ ∙ B′D′/A′D′. Encyclopædia Britannica, Inc.

Geometry and Analysis of Projective Spaces

By Charles Eugene Springer

Edition: illustrated

Published by Freeman, 1964

ISBN 0608309672, 9780608309675

299 pages

line-to-line homography

eqn(2)

1) Given a known reference distance
2) Compute the distance of the camera
3) Apply to a new pair of end points and compute and compute the distance

Definition 2.
A set of parallel planes are linked if it is possible to go from one plane to any other plane in the set through a chain of pairs of corresponding points (see also Definition 1).

Theorem 2.
Given a set of linked parallel planes; the distance between any pair of planes is sufficient to determine the absolute distance between any other pair; the link being provided by a chain of point correspondences between the set of planes.

2.2 Measurements on Parallel Planes

affine calibrated plane : vanishing line known

pencil of planes
http://en.wikipedia.org/wiki/Pencil_(mathematics)

The vanishing line is shared by the pencil of planes parallel to the reference plane.

http://en.wikipedia.org/wiki/Projective_map


planar homology

A map in the world between parallel planes induces a projective map in the image between images of points on the two planes. This imagemapis a planar homology (Springer, 1964), which is a plane projective transformation with five degrees of freedom, having a line of fixed points called the axis, and a distinct fixed point not on the axis known as the vertex. Planar homologies arise naturally in an image when two planes related by a perspectivity in three-dimensional space are imaged (Van Gool et al., 1998).

Viéville, T. and Lingrand, D. 1999. Using Specific Displacements to Analyze Motion without Calibration. Int. J. Comput. Vision 31, 1 (Feb. 1999), 5-29. DOI= http://dx.doi.org/10.1023/A:1008082308694

homology mapping


2.3 Determining the Camera Position

back-projection

homography

3. Algebraic Representation

affine coordinate system

projection matrix P
http://en.wikipedia.org/wiki/Projection_matrix

http://en.wikipedia.org/wiki/Homogeneous_coordinates

Columns 1, 2 and 4 of the projection matrix are the three columns of the reference plane to image homogrphy.

The vanishing line determines two of the eight d.o.f. of the homography.

Coordinate measurements within the planes depend on the first two and the fourth columns of P.

Affine measurements (e.g. area ratios) depend only on the fourth column of P.

3.1. Measurements Between Parallel Planes


Metric Calibration from Multiple Reference
error minimization algorithm


3.2. Measurements on Parallel Planes

homology


3.3. Determining Camera Position

http://en.wikipedia.org/wiki/Cramer%27s_rule

4. Uncertainty Analysis

4.1. Uncertainty on the P Matrix

The uncertainty in P is modeled as a 6-by-6 homogeneous covariance matrix.


4.2. Uncertainty on Measurements Between Planes

Maximum Likelihood Estimates
minimizing the sum of the Mahalanobis distances

constrained minimization problem
Lagrange multiplier method

http://en.wikipedia.org/wiki/Levenberg-Marquardt


4.3. Uncertainty on Camera Position


4.4. Example - Uncertainty on Measurements Between Planes


4.5. Monte Carlo Test