## 2D Transformations

List of Operators ↓

To specify a location in an image, we need a convention how to do so. Such a convention is set via a coordinate system. There are different coordinate systems used in HALCON. Here, we explain the ones used in 2D.

Pixels are discrete and to address them, we have a coordinate system using only integer values, the pixel coordinate system. For a higher accuracy that goes beyond the pixel grid, we need floating point coordinates, like e.g., . This leads to subpixel accurate coordinate systems. In HALCON, we have three different implementations of subpixel coordinate systems:

• Pixel Centered Coordinates, the HALCON Standard Subpixel Coordinate System

• Edge Centered Coordinates

• Polar Coordinates

Thereof the first two vary only in the coordinate origin, as visible in the figures below. Calibration makes it possible to map the image coordinates distances to real-world distances. For more information about these Calibrated Coordinates we refer to the `“Solution Guide III-C - 3D Vision”`.

### HALCON Standard Coordinate System

Pixel Accurate Coordinate System

The pixel coordinate system treats the image as a grid of discrete elements, the pixels. In HALCON, we put the origin in the middle of the upper left pixel. Now, we assign the pixel coordinates specifying its row and column like in a matrix.

Note that this implies for an image of size height width = pixels that the row coordinate values run from to and the column coordinate values from to , as visualized in the figure below.

Subpixel Accurate Coordinate System: Pixel Centered

The origin of this coordinate system is in the center of the upper left image pixel. Therewith this convention embeds the pixel coordinate system. The upper left image corner has the coordinates and for an image of size height width = pixels the bottom right corner has the coordinates (=, remember the coordinate values start at 0). It also implies that a pixel (k,l) covers the area of the rectangle , , , . This convention is called the standard coordinate system, or also Image Coordinate System.

This coordinate system is treated like a projection of the xy-plane of the 3D camera coordinate system. Accordingly, the axes have the assignment row: y coordinate, column: x coordinate.

 image/svg+xml (0,0) (0,5) (0,2) (6,0) (2,0) image/svg+xml (0.0,0.0) y x (1) (2)
Visualization of the HALCON standard pixel and subpixel Cartesian coordinate systems. The cross indicates the pixel in the bottom right image corner. Its center has the coordinates (in pixel coordinates (1)), (in standard subpixel coordinates (2)). The circle center has the coordinates .

### HALCON Non-Standard Cartesian Coordinate System

If we rotate an image around its origin by (=90 degrees), we want the two images with touching edges but not overlapping with each other. Also, scaling the image is not expected to result in negative image coordinates. For this, the origin has to be set in an image corner. This motivates the following coordinate system.

Subpixel Accurate Coordinate System: Edge Centered

For this coordinate system we set the origin in the upper left image corner. Thus the center of the upper left pixel has the coordinates and for an image of size height width = pixels the bottom right corner has the coordinates . A pixel covers the area of the rectangle , , , .

Visualization of the HALCON non-standard subpixel Cartesian coordinate system. The cross indicates the pixel in the bottom right image corner. Its center has the coordinates . The circle center has the coordinates .

For this coordinate system rotations are defined in the mathematically positive direction and thus counterclockwise. A rotation of (=90 degrees) maps the first axis (= x-axis) onto the second axis (= y-axis). Accordingly, the axes have the assignment row: x coordinate, column: y coordinate.

Visualization of a rotation () using the edge centered coordinate system.

### Operators Expecting Parameters in any Cartesian Coordinate System

The operator `affine_trans_point_2d` applies the transformation given by `HomMat2D` to the point coordinates. This means, `affine_trans_point_2d` works in both Cartesian Coordinate systems, as long you make sure that the point and the transformation are given in the same coordinate system.

The operators `angle_ll` and `angle_lx` may take the input points in pixel centered coordinates, but the returned angle is in the convention of rotations in a mathematically positive direction, thus counterclockwise, and with the horizontal axis as 0, like in the edge centered coordinate system.

### Operators Expecting Parameters in Different Coordinate Systems

In HALCON there is also the case that an operator expects its input in different coordinate systems. On the one hand, the object is expected in its usual coordinates, the standard coordinates. On the other hand, for the transformation matrix `HomMat2D`, the operator expects edge centered coordinates with their advantages regarding transformations described above. The operator converts the coordinates of the object from HALCON's standard coordinate system (with the origin in the center of the upper left pixel) to the edge centered coordinate system (with the origin in the upper left corner of the upper left pixel). After the transformation with `HomMat2D`, the result is converted back to the standard coordinate system.

These operators are

A matrix representing a transformation in pixel centered coordinates can be converted to represent the same transformation (e.g., a rotation around the same point) written in edge centered coordinates, e.g., through

hom_mat2d_translate(HomMat2D, 0.5, 0.5, HomMat2DTmp) hom_mat2d_translate_local(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)

Note, the operators beginning with `projective_` mentioned above use a projective transformation matrix. These transformation matrices can, e.g., be obtained from a 3D camera pose. Doing so, the matrix used is written in a projection of the xy-plane within the 3D coordinate system. Accordingly, the axes have the assignment row: y coordinate, column: x coordinate and therewith the coordinates need to be converted.

### Shape-based Matching and Correlation-based Matching

Results from Shape-based matching, like e.g., `find_shape_model`, are given in edge centered coordinates. This means that with these results one can create a transformation `HomMat2D` directly applicable for e.g., `affine_trans_contour_xld` and the other operators listed in the paragraph above, entitled ```Operators Expecting Parameters in Different Coordinate Systems```.

To display the results found by shape-based matching, namely from the operators

we highly recommend the usage of the procedure `dev_display_shape_matching_results`.

In the following images we give an example how a displayed match may look when using the transformation matrix in the correct and the erroneous coordinate system, respectively. For the latter one, shown in image (3), the transition matrix is given in pixel centered coordinates as well and therefore the match shown by `affine_trans_contour_xld` is off by 0.5 pixels. Note, this effect is only visible when a rotation is involved.

 image/svg+xml image/svg+xml image/svg+xml (1) (2) (3)
The original image of the paperclip (1), a part of the match where the inputs are given in the correct coordinates (2), and a match with inputs given in wrong coordinates.

The results from `find_ncc_model` and `find_ncc_models` are given in edge centered coordinates as well. Therefore, also with these results one can create a transformation `HomMat2D` directly applicable for, e.g., `affine_trans_contour_xld` and the other operators listed in the paragraph above, entitled `Operators Expecting Parameters in Different Coordinate Systems`.

To display the results found by correlation-based matching, we highly recommend the usage of the procedure `dev_display_ncc_matching_results`.

### Non-Cartesian Coordinate Systems

Subpixel Accurate Coordinate System: Polar Coordinates

In polar coordinates, points are defined through a distance and an angle. The distance is called the radial coordinate and is given in relation to the fix point, the pole. The angular coordinate is given with respect to a defined axis, the polar axis. In HALCON, the pole is specified by and the polar axis is the horizontal axis. The angular coordinate is given in radians.

After a transformation with `polar_trans_image_ext`, the upper left pixel in the output image always corresponds to the point in the input image that is specified by `RadiusStart` and `AngleStart`. Analogously, the lower right pixel in the output image corresponds to the point in the input image that is specified by `RadiusEnd` and `AngleEnd`. In the usual mode (`AngleStart` < `AngleEnd` and `RadiusStart` < `RadiusEnd`), the polar transformation is performed in the mathematically positive orientation (counterclockwise). Furthermore, points with smaller radius lie in the upper part of the output image. By suitably exchanging the values of these parameters (e.g., `AngleStart` > `AngleEnd` or `RadiusStart` > `RadiusEnd`), any desired orientation of the output image can be achieved.

 image/svg+xml α 2 2 r α 1 1 r image/svg+xml 2 r α 1 ) , ( + + + + 2 r α 2 ) , ( 1 r α 1 ) , ( 1 r α 2 ) , ( image/svg+xml α 2 2 r α 1 1 r (1) (2) (3)
As an example, we show an annular arc defined by its pole (`Row`,`Column`) (+), the polar axis (...), two angular coordinates `AngleStart` (), `AngleEnd` () and two radial coordinates `RadiusStart` (), `RadiusEnd` (). (1) The original image and the parameters defining the annular arc. (2) The annular arc, shown in a figure where the polar coordinates form an equidistant grid obtained by `polar_trans_image_ext`. (3) The annular arc in the representation of the original image. The Cartesian coordinates have been obtained through `polar_trans_image_inv` on image (2). The origin is in the center of the pixel in the upper left corner.

Polar coordinates are used by the following operators:

### Images with a reduced domain, regions, and models

In the part before we spoke about coordinates of images. When it comes to the location of the origin of the coordinate system used, images with reduced domains, regions, and models are treated differently than images.

Images with a reduced domain and regions

Both images with a reduced domain and regions keep the coordinate system of the image from which they were created. This means, they inherit the origin and the points keep the coordinate values they had in the original image.

Models

Models, on the other side, can have a local coordinate system. E.g., models obtained over `create_shape_model` have their origin in the barycenter point of the ROI they are created from. For further information see the `“Solution Guide II-B - Matching”`.

### Calibrated Coordinates

While working with pixel units, we can not extract any information about real-world distances directly. When a camera is calibrated, it is possible to rectify the images. In this case one can assign world coordinates to the image. For further information we refer to the `“Solution Guide III-C - 3D Vision”`.

#### List of Operators

`affine_trans_pixel`
Apply an arbitrary affine 2D transformation to pixel coordinates.
`affine_trans_point_2d`
Apply an arbitrary affine 2D transformation to points.
`deserialize_hom_mat2d`
Deserialize a serialized homogeneous 2D transformation matrix.
`hom_mat2d_compose`
Multiply two homogeneous 2D transformation matrices.
`hom_mat2d_determinant`
Compute the determinant of a homogeneous 2D transformation matrix.
`hom_mat2d_identity`
Generate the homogeneous transformation matrix of the identical 2D transformation.
`hom_mat2d_invert`
Invert a homogeneous 2D transformation matrix.
`hom_mat2d_reflect`
Add a reflection to a homogeneous 2D transformation matrix.
`hom_mat2d_reflect_local`
Add a reflection to a homogeneous 2D transformation matrix.
`hom_mat2d_rotate`
Add a rotation to a homogeneous 2D transformation matrix.
`hom_mat2d_rotate_local`
Add a rotation to a homogeneous 2D transformation matrix.
`hom_mat2d_scale`
Add a scaling to a homogeneous 2D transformation matrix.
`hom_mat2d_scale_local`
Add a scaling to a homogeneous 2D transformation matrix.
`hom_mat2d_slant`
Add a slant to a homogeneous 2D transformation matrix.
`hom_mat2d_slant_local`
Add a slant to a homogeneous 2D transformation matrix.
`hom_mat2d_to_affine_par`
Compute the affine transformation parameters from a homogeneous 2D transformation matrix.
`hom_mat2d_translate`
Add a translation to a homogeneous 2D transformation matrix.
`hom_mat2d_translate_local`
Add a translation to a homogeneous 2D transformation matrix.
`hom_mat2d_transpose`
Transpose a homogeneous 2D transformation matrix.
`hom_mat3d_project`
Project an affine 3D transformation matrix to a 2D projective transformation matrix.
`hom_vector_to_proj_hom_mat2d`
Compute a homogeneous transformation matrix using given point correspondences.
`point_line_to_hom_mat2d`
Approximate an affine transformation from point-to-line correspondences.
`projective_trans_pixel`
Project pixel coordinates using a homogeneous projective transformation matrix.
`projective_trans_point_2d`
Project a homogeneous 2D point using a projective transformation matrix.
`serialize_hom_mat2d`
Serialize a homogeneous 2D transformation matrix.
`vector_angle_to_rigid`
Compute a rigid affine transformation from points and angles.
`vector_field_to_hom_mat2d`
Approximate an affine map from a displacement vector field.
`vector_to_aniso`
Approximate an anisotropic similarity transformation from point correspondences.
`vector_to_hom_mat2d`
Approximate an affine transformation from point correspondences.
`vector_to_proj_hom_mat2d`
Compute a projective transformation matrix using given point correspondences.
`vector_to_proj_hom_mat2d_distortion`
Compute a projective transformation matrix and the radial distortion coefficient using given image point correspondences.
`vector_to_rigid`
Approximate a rigid affine transformation from point correspondences.
`vector_to_similarity`
Approximate an similarity transformation from point correspondences.