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
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
surfNorm.cpp
lights.txt
Single_View_Metrology.pdf