# vector_to_proj_hom_mat2d (Operator)

## Name

`vector_to_proj_hom_mat2d` — Compute a projective transformation matrix using given point correspondences.

## Signature

`vector_to_proj_hom_mat2d( : : Px, Py, Qx, Qy, Method, CovXX1, CovYY1, CovXY1, CovXX2, CovYY2, CovXY2 : HomMat2D, Covariance)`

## Description

`vector_to_proj_hom_mat2d` determines the homogeneous projective transformation matrix `HomMat2D` that optimally fulfills the following equations given by at least 4 point correspondences If fewer than 4 pairs of points (`Px`,`Py`), (`Qx`,`Qy`) are given, there exists no unique solution, if exactly 4 pairs are supplied the matrix `HomMat2D` transforms them in exactly the desired way, and if there are more than 4 point pairs given, `vector_to_proj_hom_mat2d` seeks to minimize the transformation error. To achieve such a minimization, several different algorithms are available. The algorithm to use can be chosen using the parameter `Method`. `Method`='dlt' uses a fast and simple, but also rather inaccurate error estimation algorithm while `Method`='normalized_dlt' offers a good compromise between speed and accuracy. Finally, `Method`='gold_standard' performs a mathematically optimal but slower optimization.

If 'gold_standard' is used and the input points have been obtained from an operator like `points_foerstner`, which provides a covariance matrix for each of the points, which specifies the accuracy of the points, this can be taken into account by using the input parameters `CovYY1`, `CovXX1`, `CovXY1` for the points in the first image and `CovYY2`, `CovXX2`, `CovXY2` for the points in the second image. The covariances are symmetric 2×2 matrices. `CovXX1`/`CovXX2` and `CovYY1`/`CovYY2` are a list of diagonal entries while `CovXY1`/`CovXY2` contains the non-diagonal entries which appear twice in a symmetric matrix. If a different `Method` than 'gold_standard' is used or the covariances are unknown the covariance parameters can be left empty.

In contrast to `hom_vector_to_proj_hom_mat2d`, points at infinity cannot be used to determine the transformation in `vector_to_proj_hom_mat2d`. If this is necessary, `hom_vector_to_proj_hom_mat2d` must be used. If the correspondence between the points has not been determined, `proj_match_points_ransac` should be used to determine the correspondence as well as the transformation.

If the points to transform are specified in standard image coordinates, their row coordinates must be passed in `Px` and their column coordinates in `Py`. This is necessary to obtain a right-handed coordinate system for the image. In particular, this assures that rotations are performed in the correct direction. Note that the (x,y) order of the matrices quite naturally corresponds to the usual (row,column) order for coordinates in the image.

## Attention

It should be noted that homogeneous transformation matrices refer to a general right-handed mathematical coordinate system. If a homogeneous transformation matrix is used to transform images, regions, XLD contours, or any other data that has been extracted from images, the row coordinates of the transformation must be passed in the x coordinates, while the column coordinates must be passed in the y coordinates. Consequently, the order of passing row and column coordinates follows the usual order (`Row`,`Column`). This convention is essential to obtain a right-handed coordinate system for the transformation of iconic data, and consequently to ensure in particular that rotations are performed in the correct mathematical direction.

Furthermore, it should be noted that if a homogeneous transformation matrix is used to transform images, regions, XLD contours, or any other data that has been extracted from images, it is assumed that the origin of the coordinate system of the homogeneous transformation matrix lies in the upper left corner of a pixel. The image processing operators that return point coordinates, however, assume a coordinate system in which the origin lies in the center of a pixel. Therefore, to obtain a consistent homogeneous transformation matrix, 0.5 must be added to the point coordinates before computing the transformation.

## Execution Information

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

## Parameters

`Px` (input_control)  point.x-array `→` (real / integer)

Input points in image 1 (row coordinate).

`Py` (input_control)  point.y-array `→` (real / integer)

Input points in image 1 (column coordinate).

`Qx` (input_control)  point.x-array `→` (real)

Input points in image 2 (row coordinate).

`Qy` (input_control)  point.y-array `→` (real)

Input points in image 2 (column coordinate).

`Method` (input_control)  string `→` (string)

Estimation algorithm.

Default value: 'normalized_dlt'

List of values: 'dlt', 'gold_standard', 'normalized_dlt'

`CovXX1` (input_control)  real-array `→` (real)

Row coordinate variance of the points in image 1.

Default value: []

`CovYY1` (input_control)  real-array `→` (real)

Column coordinate variance of the points in image 1.

Default value: []

`CovXY1` (input_control)  real-array `→` (real)

Covariance of the points in image 1.

Default value: []

`CovXX2` (input_control)  real-array `→` (real)

Row coordinate variance of the points in image 2.

Default value: []

`CovYY2` (input_control)  real-array `→` (real)

Column coordinate variance of the points in image 2.

Default value: []

`CovXY2` (input_control)  real-array `→` (real)

Covariance of the points in image 2.

Default value: []

`HomMat2D` (output_control)  hom_mat2d `→` (real)

Homogeneous projective transformation matrix.

`Covariance` (output_control)  real-array `→` (real)

9×9 covariance matrix of the projective transformation matrix.

## Possible Predecessors

`proj_match_points_ransac`, `proj_match_points_ransac_guided`, `points_foerstner`, `points_harris`

## Possible Successors

`projective_trans_image`, `projective_trans_image_size`, `projective_trans_region`, `projective_trans_contour_xld`, `projective_trans_point_2d`, `projective_trans_pixel`

## Alternatives

`hom_vector_to_proj_hom_mat2d`, `proj_match_points_ransac`, `proj_match_points_ransac_guided`

## 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.

Calibration