Operators

# vector_to_fundamental_matrix (Operator)

## Name

vector_to_fundamental_matrix — Compute the fundamental matrix given a set of image point correspondences and reconstruct 3D points.

## Signature

vector_to_fundamental_matrix( : : Rows1, Cols1, Rows2, Cols2, CovRR1, CovRC1, CovCC1, CovRR2, CovRC2, CovCC2, Method : FMatrix, CovFMat, Error, X, Y, Z, W, CovXYZW)

## Description

For a stereo configuration with unknown camera parameters the geometric relation between the two images is defined by the fundamental matrix. The operator vector_to_fundamental_matrix determines the fundamental matrix FMatrix from given point correspondences (Rows1,Cols1), (Rows2,Cols2), that fulfill the epipolar constraint:

```              T
/ Cols2 \             / Cols1 \
| Rows2 | * FMatrix * | Rows1 | = 0  .
\   1   /             \   1   /
```

Note the column/row ordering in the point coordinates: since the fundamental matrix encodes the projective relation between two stereo images embedded in 3D space, the x/y notation must be compliant with the camera coordinate system. Therefore, (x,y) coordinates correspond to (column,row) pairs.

For a general relative orientation of the two cameras the minimum number of required point correspondences is eight. Then, Method is chosen to be 'normalized_dlt' or 'gold_standard'. If left and right camera are identical and the relative orientation between them is a pure translation then choose Method equal to 'trans_normalized_dlt' or 'trans_gold_standard'. In this special case the minimum number of correspondences is only two. The typical application of the motion beeing a pure translation is that of a single fixed camera looking onto a moving conveyor belt.

The fundamental matrix is determined by minimizing a cost function. To minimize the respective error different algorithms are available, and the user can choose between the direct-linear-transformation ('normalized_dlt') and the gold-standard-algorithm ('gold_standard'). Like the motion case, the algorithm can be selected with the parameter Method. For Method = 'normalized_dlt' or 'trans_normalized_dlt', a linear algorithm minimizes an algebraic error based on the above epipolar constraint. This algorithm offers a good compromise between speed and accuracy. For Method = 'gold_standard' or 'trans_gold_standard', a mathematically optimal, but slower optimization is used, which minimizes the geometric backprojection error of reconstructed projective 3D points. In this case, in addition to the fundamental matrix its covariance matrix CovFMat is output, along with the projective coordinates (X,Y,Z,W) of the reconstructed points and their covariances CovXYZW. Let n be the number of points. Then the concatenated covariances are stored in a 16xn tuple.

If an optimal gold-standard-algorithm is chosen the covariances of the image points (CovRR1, CovRC1, CovCC1, CovRR2, CovRC2, CovCC2) can be incorporated in the computation. They can be provided for example by the operator points_foerstner. If the point covariances are unknown, which is the default, empty tuples are input. In this case the optimization algorithm internally assumes uniform and equal covariances for all points.

The value Error indicates the overall quality of the optimization procedure and is the mean euclidian distance in pixels between the points and their corresponding epipolar lines.

If the correspondence between the points are not known, match_fundamental_matrix_ransac should be used instead.

## Parallelization

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Processed without parallelization.

## Parameters

Rows1 (input_control)  number-array (real / integer)

Input points in image 1 (row coordinate).

Restriction: length(Rows1) >= 8 || length(Rows1) >= 2

Cols1 (input_control)  number-array (real / integer)

Input points in image 1 (column coordinate).

Restriction: length(Cols1) == length(Rows1)

Rows2 (input_control)  number-array (real / integer)

Input points in image 2 (row coordinate).

Restriction: length(Rows2) == length(Rows1)

Cols2 (input_control)  number-array (real / integer)

Input points in image 2 (column coordinate).

Restriction: length(Cols2) == length(Rows1)

CovRR1 (input_control)  number-array (real / integer)

Row coordinate variance of the points in image 1.

Default value: []

CovRC1 (input_control)  number-array (real / integer)

Covariance of the points in image 1.

Default value: []

CovCC1 (input_control)  number-array (real / integer)

Column coordinate variance of the points in image 1.

Default value: []

CovRR2 (input_control)  number-array (real / integer)

Row coordinate variance of the points in image 2.

Default value: []

CovRC2 (input_control)  number-array (real / integer)

Covariance of the points in image 2.

Default value: []

CovCC2 (input_control)  number-array (real / integer)

Column coordinate variance of the points in image 2.

Default value: []

Method (input_control)  string (string)

Estimation algorithm.

Default value: 'normalized_dlt'

List of values: 'gold_standard', 'normalized_dlt', 'trans_gold_standard', 'trans_normalized_dlt'

FMatrix (output_control)  hom_mat2d (real)

Computed fundamental matrix.

CovFMat (output_control)  real-array (real)

9x9 covariance matrix of the fundamental matrix.

Error (output_control)  real (real)

Root-Mean-Square of the epipolar distance error.

X (output_control)  real-array (real)

X coordinates of the reconstructed points in projective 3D space.

Y (output_control)  real-array (real)

Y coordinates of the reconstructed points in projective 3D space.

Z (output_control)  real-array (real)

Z coordinates of the reconstructed points in projective 3D space.

W (output_control)  real-array (real)

W coordinates of the reconstructed points in projective 3D space.

CovXYZW (output_control)  real-array (real)

Covariance matrices of the reconstructed 3D points.

## References

Richard Hartley, Andrew Zisserman: “Multiple View Geometry in Computer Vision”; Cambridge University Press, Cambridge; 2000.
Olivier Faugeras, Quang-Tuan Luong: “The Geometry of Multiple Images: The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications”; MIT Press, Cambridge, MA; 2001.

## Module

3D Metrology

 Operators