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===3D reconstruction with a hand held camera===
===3D reconstruction with a hand held camera===


[[File:3dcastle.png|thumb|center|upright=2.0|A representation of the absolute conic and its relation with the epipolar geometry.]]
[[File:3dcastle.png|thumb|left|upright=2.0|A representation of the reconstructed Arenberg Castle (shaded on the left and textured on the right) from a video taken with an off the shelf handheld video camera.]]
[[File:3dreconstruction.png|thumb|center|A representation of the absolute conic and its relation with the epipolar geometry.]]
[[File:3dreconstruction.png|thumb|right|The result of the camera calibration from a series of frames of the Arenberg Castle.]]
[[File:jaintempleview.png|thumb|center|A representation of the absolute conic and its relation with the epipolar geometry.]]
[[File:jaintempleview.png|thumb|center|Three views of the Jain Temple.]]
[[File:3djaintemple.png|thumb|center|A representation of the absolute conic and its relation with the epipolar geometry.]]
[[File:3djaintemple.png|thumb|center|A perspective of the reconstructed Jain Temple by using self-calibration.]]


== References ==
== References ==

Revision as of 07:11, 9 February 2012

The Absolute Conic (AC) is a concept used in computer vision which is discussed in CVonline [1]


Introduction

In computer vision the Absolute Conic (AC) is a concept used in geometric camera calibration and represents a special curve that is invariant to rigid transformation. The relative position of this conic to the movement and rotation of a camera does not change and Euclidean structure is determined by its location on the plane at infinity.

A concept tightly correlated with the absolute conic is the image of the absolute conic or IAC. The IAC represents the projection of the absolute conic on the images taken by a camera. It has been shown that the IAC is invariant in between images in respect to translation and rotations and is determined only by the camera intrinsic parameters. The concept of the IAC can be more easily understood by making an analogy with the impression that the moon is following you when traveling on the train. By imposing various constraints in between images, like assuming all the camera parameters are constant through the scene, all the intrinsic camera parameters can be recovered directly from point correspondences in between images without the need for a specific calibration phase by using a predefined pattern.

Theory

Representation of the absolute conic

The representation of the absolute conic in euclidean space is given by the following relations:

where and are expressed in homogeneous coordinates. It is easily shown that is invariant to euclidean transformations. From an algebraic perspective every circle in 3D space intersects in two complex points, and these two points lay on the absolute conic. The position of on can be reconstructed from three of these circles. The resulting equations would be:

Since is symmetric is can be transformed by using the Cholesky decomposition by symmetric indefinite factorization resulting in:

where is an orthogonal upper triangular matrix. By doing the variable change we get:

and since we work in homogeneous coordinates, by doing a re scaling on each axis we would get the system of equations defined above.

The absolute conic has several useful properties for upgrading projective geometry to metric up to scale. One property is that the projections of all circles on the plane at infinity () intersect in exactly two points. Another useful property for upgrading the projective reconstruction to euclidean up to scale because it angles. The angle between two lines is given by the following equation:

[2]

where and are intersection points with the plane at infinity of the two lines. The absolute conic is to be viewed as a mathematical tool as it has no real representation. Even though the coefficients equation that defines it are real has only complex points.

Image of the absolute conic

The projection of the absolute conic onto an image also generates a conic that is invariant to rigid transformations. If the camera parameters are known, the projection of on the camera image can be determined. The reverse is also true, and the image of the absolute conic is determined by the camera intrinsic parameters.

flow.
A representation of the absolute conic and its projection on the image plane. In the image represents the optical center of the camera, while and represent the projection of and through the optical center. Laguerre formula states that the angle between and is proportional to the cross-ratio . For the projection of onto the image the cross-ratio remains constant as is invariant under . This means that is also proportional to and the camera calibration can be recovered from .


The points on can be expressed as and are mapped on the image plane by a projection matrix , where K is the calibration matrix, R is a rotation matrix and t is a translation vector. By replacing the representation of we obtain



where represents the projection of the point in the image and is equality up to a scale factor. This means that represents the planar homography between and the camera plane. Under homography a conic is mapped as . Taking into account that on we get


This means that if is known it can be decomposed in an unique upper triangular matrix by using the Cholesky decomposition retrieving .

Dual image of the absolute conic

The dual image of the absolute conic (DIAC) represents the inverse of the IAC. The form of the DIAC proves more useful in the computations for retrieving the position of the absolute conic and recovering the camera calibration.

Examples

Camera calibration with the absolute conic

The absolute conic provides a useful mathematical tool for retrieving the calibration matrix. The position of is retrieved by determining the point homography or the projection matrices of multiple views. The projection matrices are recovered by using various algorithms for point matching in between views like the Iterative Closest Point algorithm. Under the assumption of constant intrinsic parameters in between view this yields a system of equations similar to the following:

This does not usually have an unique solution which so multiple constraints have to be enforced. The most common are constant aspect ratio and pixel shape. Also due to the noise associated with point matching algorithms numerical methods must be employed for solving the system. The output is an approximation of the real intrinsic camera parameters as a result of an error minimization function.


Epipolar geometry and the absolute conic

flow.
A representation of the absolute conic and its relation with the epipolar geometry.

One way to retrieve several of the intrinsic camera parameters is to correlate the absolute conic with the epipolar geometry of two views. After computing the fundamental matrix and the two epipoles and from the point matches with the epipole transformation[3]


By taking into account only the epipolar lines that are tangent to the conic we can set a contraint on that will render three equation in five unknowns for the intrinsic camera parameters. [4]

  • if we parameterise the lines that go through the epipole with the point of intersection with we get:
[4].


  • we put the condition so that the line is tangent to the absolute conic[4] which extends to:
. (1)
  • since maps to . Because under epipolar transformation tangents keep their properties[4] we have:


which expands to:

. (2)
  • As equation (1) and equation (2) have the same coefficients we get the following equalities, known as the Kruppa equations:
.

Critical motion sequences

It has been observed that certain motion sequences are problematic for self-calibration. It appears that for specific transformations of the camera the reconstruction is not unique, as at least two solutions that satisfy all the constraints exist. The classes of CMS depend on the constraints that are enforced like constant camera parameters or variable focus. A list of the critical motion sequences for the before mentioned constraints is given in the tables below.

Critical motion sequence for constant but unknown intrinsic parameters
Movement Ambiguity
Pure translation affine transformation (5DOF)
Pure rotation arbitrary position for plane at infinity (3DOF)
Orbital motion projective distortion along rotation axis (2DOF)
Planar motion scaling axis perpendicular to plane (1DOF)
Critical motion sequence for variable focal lengths
Movement Ambiguity
Pure rotation arbitrary position for plane at infinity (3DOF)
Forward motion projective distortion along optical axis (2DOF)
Translation with rotation around optical axis scaling optical axis (1DOF)
Hyperbolic and/or elliptic motion one extra solution


Applications

3D reconstruction with a hand held camera

File:3dcastle.png
A representation of the reconstructed Arenberg Castle (shaded on the left and textured on the right) from a video taken with an off the shelf handheld video camera.
File:3dreconstruction.png
The result of the camera calibration from a series of frames of the Arenberg Castle.
File:Jaintempleview.png
Three views of the Jain Temple.
File:3djaintemple.png
A perspective of the reconstructed Jain Temple by using self-calibration.

References

  1. ^ R. B. Fisher, "CVonline: an overview", Int. Assoc. of Pat. Recog. Newsletter, 27(2), April 2005.
  2. ^ Bogusław Cyganek, J. Paul Siebert, An introduction to 3D computer vision techniques and algorithms, Volume 10, 384-385
  3. ^ Faugeras, O.D., Luong, Q.-T. and Maybank, S.J., Camera self-calibration: theory and experiments, Proc. 2nd ECCV, 321-334, Springer-Verlag, 1992.
  4. ^ a b c d S.D. Hippisley-Cox, J. Porrill, Auto-calibration — Kruppa's equations and the intrinsic parameters of a camera

Bibliography

  • Sturm, Peter (1997). "Critical Motion Sequences for Monocular Self-Calibration and Uncalibrated Euclidean Reconstruction". Proc. 1997 Conference on Computer Vision and Pattern Recognition (2): 1100–1105.