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

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 $\Omega$ in euclidean space is given by the following relations:

${\begin{cases}x^{2}+y^{2}+z^{2}=0\\t=0\end{cases}}$

where $\textstyle x,y,z$ and $\textstyle t$ are expressed in homogeneous coordinates. It is easily shown that $\Omega$ is invariant to euclidean transformations. From an algebraic perspective every circle in 3D space intersects $\Pi _{\infty }$ in two complex points, and these two points lay on the absolute conic. The position of $\Omega$ on $\Pi _{\infty }$ can be reconstructed from three of these circles. The resulting equations would be:

$X^{T}QX={\begin{bmatrix}x&y&z\end{bmatrix}}{\begin{bmatrix}a_{1}&a_{4}&a_{5}\\a_{4}&a_{2}&a_{6}\\a_{5}&a_{6}&a_{3}\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=0$

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

$Q=P^{T}{\begin{bmatrix}\lambda _{1}&&\\&\lambda _{2}&\\&&\lambda _{3}\end{bmatrix}}P$

where $P$ is an orthogonal upper triangular matrix. By doing the variable change $X'=PX$ we get:

$X^{T}QX=(X')^{T}{\begin{bmatrix}\lambda _{1}&&\\&\lambda _{2}&\\&&\lambda _{3}\end{bmatrix}}X'$

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 ($\Pi _{\infty }$) intersect $\Omega$ in exactly two points. Another useful property for upgrading the projective reconstruction to euclidean up to scale because it angles. The angle $\textstyle \theta$ between two lines is given by the following equation:

$\cos \theta ={d_{1}^{T}\Omega d_{2} \over {\sqrt {(d_{1}^{T}\Omega d_{1})(d_{2}^{T}\Omega d_{2})}}}$

where $\textstyle d_{1}$ and $\textstyle d_{2}$ 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 $\Omega$ has only complex points.

Image of the absolute conic

The projection of the absolute conic onto an image also generates a conic $\omega$ that is invariant to rigid transformations. If the camera parameters are known, the projection $\omega$ of $\Omega$ 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 $c$ represents the optical center of the camera, while $q_{1}$ and $q_{2}$ represent the projection of $r_{1}$ and $r_{2}$ through the optical center. Laguerre formula states that the angle $\theta$ between $cr_{1}$ and $cr_{2}$ is proportional to the cross-ratio $\lbrace r_{l},r_{2};i,j\rbrace$. For the projection $\pi$ of $\Pi _{\infty }$ onto the image the cross-ratio remains constant as $\Omega$ is invariant under $\pi$. This means that $\theta$ is also proportional to $\lbrace p_{l},p_{2};\pi (i),\pi (j)\rbrace$ and the camera calibration can be recovered from $\omega$.

The points on $\Pi _{\infty }$ can be expressed as $X=(d^{T},0)^{T}$ and are mapped on the image plane by a projection matrix $P=KR[I|t]$, where K is the calibration matrix, R is a rotation matrix and t is a translation vector. By replacing the representation of $X$ we obtain

$x\simeq PX=KR[I|t](d^{T},0)^{T}=KRd$

where $x$ represents the projection of the point $X$ in the image and $\simeq$ is equality up to a scale factor. This means that $KR$ represents the planar homography $H$ between $\Pi _{\infty }$ and the camera plane. Under homography a conic $A$ is mapped as $H^{-T}AH^{-1}$. Taking into account that $\Omega =I$ on $\Pi _{\infty }$ we get

$\textstyle \omega =(KR)^{-T}I(KR)^{-1}=K^{-T}RR^{-1}K-^{1}=K^{-T}K^{-1}$

This means that if $\omega$ is known it can be decomposed in an unique upper triangular matrix by using the Cholesky decomposition retrieving $K^{-1}$.

Dual image of the absolute conic

The dual image of the absolute conic (DIAC) $\omega ^{*}$ 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.

$\textstyle \omega ^{*}=\omega ^{-1}=KK^{T}$

Examples

Camera calibration with the absolute conic

The absolute conic provides a useful mathematical tool for retrieving the calibration matrix. The position of $\omega$ 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:

${\begin{cases}\omega _{1}^{*}=P_{1}\Omega ^{*}P_{1}^{T}\\\omega _{k}^{*}=P_{k}\Omega ^{*}P_{k}^{T}\end{cases}}$

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 $F$ and the two epipoles $e_{1}$ and $e_{2}$ from the point matches with the epipole transformation

$\tau ^{'}={a\tau +b \over c\tau +d}$

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

• if we parameterise the lines that go through the epipole $e_{1}$ with the point of intersection with $\Pi _{\infty }$ we get:

$y_{1}=e_{1}\times (1,\tau ,0)$.

• we put the condition $y_{1}^{T}Ky_{1}=0$ so that the line is tangent to the absolute conic which extends to:

$\textstyle \alpha _{1}\tau ^{2}+\beta _{1}\tau +\gamma _{1}=0$. (1)

• since $y_{1}=e_{1}\times (1,\tau ,0)$ maps to $y_{2}=e_{2}\times (1,\tau ,0)$. Because under epipolar transformation tangents keep their properties we have:

$\textstyle y_{2}Ky^{T}=0$

which expands to:

$\textstyle \alpha _{2}\tau ^{2}+\beta _{2}\tau +\gamma _{2}=0$. (2)

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

${\alpha _{1} \over \alpha _{2}}={\beta _{1} \over \beta _{2}}={\gamma _{1} \over \gamma _{2}}$.

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.

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.

External links

• Advanced Vision homepage