This chapter provides information regarding camera calibration.
To achieve maximum accuracy of measurement for your camera setup, you have to calibrate it accordingly. Therefore, a camera model is determined, which describes the projection of a 3D world point into a (sub-)pixel in the image.
HALCON provides a wide range of operators to approach diverse tasks related to calibration, such as
describing and finding a calibration object (Calibration / Calibration Object),
the projection of points from the 3D scene onto the image plane and the other way around (Calibration / Projection, Calibration / Inverse Projection),
compensating perspective and radial distortions (Calibration / Rectification),
handling the camera parameters (Calibration / Camera Parameters),
performing a self-calibration (Calibration / Self-Calibration), and
calibrating different setups consisting of
one camera (Calibration / Monocular),
multiple cameras (Calibration / Binocular, Calibration / Multi-View), or
a camera in combination with a robot (Calibration / Hand-Eye).
This chapter gives guidance regarding the basic concept of retrieving the internal and external parameters of your camera. The following paragraphs state how to successfully calibrate a camera. In particular, they describe
the needed calibration object,
the individual steps to calibrate the cameras, including
how to prepare the calibration input data,
how to perform the actual calibration with
calibrate_cameras
how to check the success of the calibration,
the camera parameters,
additional information about the calibration process, including
how to obtain an appropriate calibration plate,
how to take a set of suitable images, and
which distortion model to use,
the available 3D camera models and how 3D points are transformed into the image coordinate system, and
limitations related to specific camera types.
For a successful calibration of your camera setup, at least one calibration object with accurately known metric properties is needed, e.g., a HALCON calibration plate. For the calibration, take a series of images of the calibration object in different positions and orientations. The success of the calibration highly depends on the quality of the calibration object and the images. So you might want to exercise special diligence during the acquisition of the calibration images. See the section “How to take a set of suitable images?” for further information.
A calibration plate is covered by multiple calibration marks, which are extracted in the calibration images in order to retrieve their coordinates. The orientation of the plate has to be known distinctively, hence, a finder pattern is also part of the imprint.
Your distributor can provide you with two different types of standard HALCON calibration plates:
As finder pattern, there are special groups of mark hexagons where some of
the marks contain dot-shaped holes (see create_caltab
camera_calibrationcalibrate_camerasgen_caltabWhen acquiring your calibration images, note that there are different recommendations on how to take them, depending on your used calibration plate (see section “How to take a set of suitable images?”).
Before calling a calibration operator (e.g., calibrate_cameras
Create a calibration data model with the operator
create_calib_data
Specify the camera type and the initial internal
camera parameters with the operator
set_calib_data_cam_param
Specify the description of all calibration objects
with the operator set_calib_data_calib_object
Collect observation data with the operators
find_calib_objectset_calib_data_observ_points
set_calib_dataUsing all the information stored within the calibration data model, the
actual calibration can be performed calling calibrate_cameras
The results can then be queried with the operator get_calib_data
After a successful calibration, the root mean square error (RMSE) of
the back projection of the optimization is returned in
Error
If only a single camera is calibrated, an ErrorError
For information about how to check the success of the calibration using a multi-view camera setup, see the respective section in the chapter Calibration / Multi-View.
Regarding camera parameters, you can distinguish between internal and external camera parameters.
These parameters describe the characteristics of the used camera,
especially the dimension of the sensor itself and the projection
properties of the used combination of lens, camera, and frame
grabber. Below is an overview of all available camera types and
their respective parameters CameraParam. In the list,
“projective cameras” refers to the property that the lens
performs a perspective projection on the object-side of the lens,
while “telecentric cameras” refers to the property that the lens
performs a telecentric projection on the object-side of the lens.
have 9 to 16 internal parameters depending on the camera type.
For reasons explained below, parameters that are marked with an * asterisk are fixed and not estimated by the algorithm.
Projective area scan cameras with regular lenses
'area_scan_division':
['area_scan_division', Focus, Kappa, Sx, Sy*, Cx,
Cy, ImageWidth, ImageHeight]
'area_scan_polynomial':
['area_scan_polynomial', Focus, K1, K2, K3, P1, P2,
Sx, Sy*, Cx, Cy, ImageWidth, ImageHeight]
Telecentric area scan cameras with regular lenses
'area_scan_telecentric_division':
['area_scan_telecentric_division', Magnification,
Kappa, Sx, Sy*, Cx, Cy, ImageWidth, ImageHeight]
'area_scan_telecentric_polynomial':
['area_scan_telecentric_polynomial', Magnification,
K1, K2, K3, P1, P2, Sx, Sy*, Cx, Cy, ImageWidth, ImageHeight]
Projective area scan cameras with tilt lenses
'area_scan_tilt_division':
['area_scan_tilt_division', Focus, Kappa,
ImagePlaneDist, Tilt, Rot, Sx, Sy*, Cx, Cy, ImageWidth, ImageHeight]
'area_scan_tilt_polynomial':
['area_scan_tilt_polynomial', Focus, K1, K2, K3, P1,
P2, ImagePlaneDist, Tilt, Rot, Sx, Sy*, Cx, Cy, ImageWidth,
ImageHeight]
'area_scan_tilt_image_side_telecentric_division':
['area_scan_tilt_image_side_telecentric_division',
Focus, Kappa, Tilt, Rot, Sx*, Sy*, Cx, Cy, ImageWidth, ImageHeight]
'area_scan_tilt_image_side_telecentric_polynomial':
['area_scan_tilt_image_side_telecentric_polynomial',
Focus, K1, K2, K3, P1, P2, Tilt, Rot, Sx*, Sy*, Cx, Cy,
ImageWidth, ImageHeight]
Telecentric area scan cameras with tilt lenses
'area_scan_tilt_bilateral_telecentric_division':
['area_scan_tilt_bilateral_telecentric_division',
Magnification, Kappa, Tilt, Rot, Sx*, Sy*, Cx, Cy,
ImageWidth, ImageHeight]
'area_scan_tilt_bilateral_telecentric_polynomial':
['area_scan_tilt_bilateral_telecentric_polynomial',
Magnification, K1, K2, K3, P1, P2, Tilt, Rot, Sx*, Sy*, Cx,
Cy, ImageWidth, ImageHeight]
'area_scan_tilt_object_side_telecentric_division':
['area_scan_tilt_object_side_telecentric_division',
Magnification, Kappa, ImagePlaneDist, Tilt, Rot, Sx, Sy*,
Cx, Cy, ImageWidth, ImageHeight]
'area_scan_tilt_object_side_telecentric_polynomial':
['area_scan_tilt_object_side_telecentric_polynomial',
Magnification, K1, K2, K3, P1, P2, ImagePlaneDist, Tilt,
Rot, Sx, Sy*, Cx, Cy, ImageWidth, ImageHeight]
Projective area scan cameras with hypercentric lenses
'area_scan_hypercentric_division':
['area_scan_hypercentric_division', Focus, Kappa, Sx, Sy*,
Cx, Cy, ImageWidth, ImageHeight]
'area_scan_hypercentric_polynomial':
['area_scan_hypercentric_polynomial', Focus, K1, K2, K3, P1,
P2, Sx, Sy*, Cx, Cy, ImageWidth, ImageHeight]
Type of the camera, as listed above.
Focal length of the lens (only for lenses that perform a perspective projection on the object side of the lens).
The initial value is the nominal focal length of the used lens, e.g., 0.008m.
Magnification of the lens (only for lenses that perform a telecentric projection on the object side of the lens).
The initial value is the nominal magnification of the used telecentric lens (the image size divided by the object size), e.g., 0.2.
Distortion coefficient to model the radial lens distortions (only for the division model).
Use 0.0 as initial value.
Distortion coefficients to model the radial and decentering lens distortions (only for the polynomial model).
Use 0.0 as initial value for all five coefficients.
Distance of the exit pupil of the lens to the image plane. The exit pupil is the (virtual) image of the aperture stop (typically the diaphragm), as viewed from the image side of the lens. Typical values are in the order of a few centimeters to very large values if the lens is close to being image-side telecentric.
The tilt angle describes the angle by which the optical axis is tilted with respect to the normal of the sensor plane (corresponds to a rotation around the x-axis). The rotation angle describes the rotation around the optical axis (z-axis). For a rotation the optical axis gets tilted vertically down with respect to the camera housing, corresponds to the optical axis being tilted horizontally to the left (direction of view along the z-axis), corresponds to the optical axis being tilted vertically up, and corresponds to the optical axis being tilted horizontally to the right.
These parameters are only used if a tilt lens is part of the camera setup.ImagePlaneDist.
describes the orientation of the tilt axis in
relation to the x-axis of the sensor and has to be applied first.
describes the actual tilt of
the lens. ImagePlaneDist is the distance of the
exit pupil of the lens to the image plane.
These angles are typically roughly known based on the considerations that led to the use of the tilt lens or can be read off from the mechanism by which the lens is tilted.
Scale factors. They correspond to the horizontal and vertical distance between two neighboring cells on the sensor. Since in most cases the image signal is sampled line-synchronously, is determined by the dimension of the sensor and does not need to be estimated by the calibration process.
The initial values depend on the dimensions of the used chip of the camera. See the technical specification of your camera for the actual values. Attention: These values increase if the image is subsampled!
As projective cameras are described through the pinhole camera
model, it is impossible to determine
Focus
For telecentric lenses, it is
impossible to determine Magnification,
, and simultaneously. Therefore,
the algorithm will keep fixed.
For image-side telecentric tilt lenses (see chapter “Basics”,
section “Camera Model and Parameters” in the
“Solution Guide III-C 3D Vision” for an overview of different
types of tilt lenses), it is impossible
to determine Focus
For bilateral telecentric tilt lenses,
it is impossible to determine Magnification,
, , and the tilt parameters
and simultaneously. Therefore,
additionally to , the algorithm will keep
fixed.
Column () and row () coordinate of the principal point of the image (center of the radial distortion).
Use the half image width and height as initial values. Attention: These values decrease if the image is subsampled!
Width and height of the sampled image. Attention: These values decrease if the image is subsampled!
have 12 or 16 internal parameters depending on the camera type.
For reasons explained below, parameters that are marked with an * asterisk are fixed and not estimated by the algorithm.
Projective line scan cameras with regular lenses
'line_scan_division':
['line_scan_division', Focus, Kappa, Sx*, Sy*, Cx,
Cy, ImageWidth, ImageHeight, Vx, Vy, Vz]
'line_scan_polynomial':
['line_scan_polynomial', Focus, K1, K2, K3, P1, P2,
Sx*, Sy*, Cx, Cy, ImageWidth, ImageHeight, Vx, Vy, Vz]
Telecentric line scan cameras with regular lenses
'line_scan_telecentric_division':
['line_scan_telecentric_division', Magnification,
Kappa, Sx*, Sy*, Cx, Cy, ImageWidth, ImageHeight, Vx, Vy, Vz*]
'line_scan_telecentric_polynomial':
['line_scan_telecentric_polynomail', Magnification,
K1, K2, K3, P1, P2, Sx*, Sy*, Cx, Cy, ImageWidth,
ImageHeight, Vx, Vy, Vz*]
Type of the camera, as listed above.
Focal length of the lens (only for lenses that perform a perspective projection on the object side of the lens).
The initial value is the nominal focal length of the used lens, e.g., 0.008m.
Magnification of the lens (only for lenses that perform a telecentric projection on the object side of the lens).
The initial value is the nominal magnification of the used telecentric lens (the image size divided by the object size), e.g., 0.2.
Distortion coefficient of the division model to model the radial lens distortions.
Use 0.0 as initial value.
Distortion coefficients to model the radial and decentering lens distortions (only for the polynomial model).
Use 0.0 as initial value for all five coefficients.
Scale factor. Corresponds to the horizontal distance between two
neighboring cells on the sensor. Note that Focus or
Magnification, respectively, and cannot
be determined simultaneously. Therefore, is
kept fixed in the calibration. The initial value for
can be taken from the technical specifications
of the camera. Attention: This value increases if the image is
subsampled!
Scale factor. During the calibration, it appears only in the form . Consequently, and cannot be determined simultaneously. Therefore, in the calibration, is kept fixed. describes the distance of the image center point from the sensor line in meters. The initial value for can be taken from the technical specifications of the camera. Attention: This value increases if the image is subsampled!
Column coordinate of the image center point (center of the radial distortions). Use half of the image width as the initial value for . Attention: This value decreases if the image is subsampled!
Distance of the image center point (center of the radial distortions) from the sensor line in scanlines. The initial value can normally be set to 0.
Width and height of the sampled image. Attention: These values decrease if the image is subsampled!
X-, Y-, and Z-component of the motion vector.
The initial values for the x-, y-, and z-component of the motion vector depend on the image acquisition setup. Assuming a camera that looks perpendicularly onto a conveyor belt and that is rotated around its optical axis such that the sensor line is perpendicular to the conveyor belt, i.e., the y-axis of the camera coordinate system is parallel to the conveyor belt, use the initial values . The initial value for can then be determined, e.g., from a line scan image of an object with known size (e.g., calibration plate, ruler):
With
If, compared to the above setup, the camera is rotated 30 degrees around its optical axis, i.e., around the z-axis of the camera coordinate system, the above determined initial values must be changed as follows:
If, compared to the first setup, the camera is rotated -20 degrees around the x-axis of the camera coordinate system, the following initial values result:
The quality of the initial values for , , and are crucial for the success of the whole calibration. If they are not precise enough, the calibration may fail.
Note that for telecentric line scan cameras, the value of has no influence on the image position of 3D points and therefore cannot be determined. Consequently, is not optimized and left at its initial value for telecentric line scan cameras. Therefore, the initial value of should be set to 0. For setups with multiple telecentric line scan cameras that share a common motion vector (for a detailed explanation, see Calibration / Multi-View), however, can be determined based on the camera poses. Therefore, in this case is optimized.
Note that the term focal length is not quite correct and would be appropriate only for an infinite object distance. To simplify matters, always the term focal length is used even if the image distance is meant.
For all operators that use camera parameters as input the respective parameter values are checked as to whether they fulfill the following restrictions:
For some operators the restrictions differ slightly. In particular, for operators that do not support line scan cameras the following restriction applies:
The following 6 parameters describe the 3D pose, i.e., the position and orientation of the world coordinate system relative to the camera coordinate system. The x- and y-axis of the camera coordinate system are parallel to the column and row axes of the image, while the z-axis is perpendicular to the image plane. For line scan cameras, the pose of the world coordinate system refers to the camera coordinate system of the first image line.
Translation along the x-axis of the camera coordinate system.
Translation along the y-axis of the camera coordinate system.
Translation along the z-axis of the camera coordinate system.
Rotation around the x-axis of the camera coordinate system.
Rotation around the y-axis of the camera coordinate system.
Rotation around the z-axis of the camera coordinate system.
The pose tuple contains one more element, which is the
representation type of the pose. It codes the combination of the
parameters OrderOfTransformOrderOfRotationViewOfTransformcreate_pose
When using a standard HALCON calibration plate, the world coordinate system is defined by the coordinate system of the calibration plate. See the section “Calibration Object” above for further information.
If a HALCON calibration plate is used, you can use the operator
find_calib_objectfind_caltabfind_marks_and_pose
HALCON calibration plates use meters as unit. The camera parameters use corresponding units. Of course, calibration can be done using different units, but in this case the related parameters have to be adapted. Here, we list the HALCON default units for the different camera parameters:
| Parameter | Unit | |
|---|---|---|
| External | RotX, RotY, RotZ | , , |
| TransX, TransY, TransZ | , , | |
| Internal | Cx, Cy | , |
| Focus | ||
| ImagePlaneDist | ||
| ImageWidth, ImageHeight | , | |
| K1, K2, K3 | , , | |
| Kappa | ||
| P1, P2 | , | |
| Magnification | - (scalar) | |
| Sx, Sy | , | |
| Tilt, Rot | , | |
| Vx, Vy, Vz | , , |
The use of calibrate_cameras
You can obtain high-precision calibration plates in various sizes
and materials from your local distributor. These calibration
plates come with associated description files and can be easily
extracted with find_calib_object
It is also possible to use any arbitrary object for
calibration. The only requirement is that the object has
characteristic points that can be robustly detected in the image
and that the 3D world position of these points is known with high
accuracy. See the “Solution Guide III-C 3D Vision” for details.
Self-printed calibration objects are usually not accurate enough for high-precision applications.
With the combination of lens (fixed focus
setting!), camera, and frame grabber to be calibrated, a set of
images of the calibration plate must be taken (see
open_framegrabbergrab_image
Your local distributor can provide you with two different types of
standard HALCON calibration plates: Calibration plates with hexagonally
arranged marks (see create_caltabgen_caltab
The parameters and hints listed below should be considered when taking the calibration images. For a successful calibration, the setup and the used set of images should have certain qualities. These qualities may vary for the specific task and demand. In order to give guidance, values and hints suitable for a basic monocular camera setup are mentioned.
Aperture
The aperture of the camera must not be changed during the acquisition of the images. If the aperture is changed after the calibration, the camera must be calibrated anew.
Camera pose
The position of the camera must not be changed during the image acquisition.
Focus
The calibration images should be sharply focused, i.e., transitions between objects should be clearly delimited. The focus, respectively the focal length, must not be changed during the image acquisition.
Field of view coverage and orientation
Within the set of calibration images, every part of the field of view should be covered by the plate at least once. The calibration plate may also fill the entire image. The orientation of the plate should vary within the set of images.
Tilt angles
The set of calibration images should also contain images with tilted calibration plates. Thereby the plate should be tilted in different directions at an angle of about 30-45°. Note, that if the recommended angle cannot be realized due to, e.g., limited depth of field, you should at least tilt your plate as steeply as possible with your setup.
Number of images/ calibration plate poses
Plate with hexagonally arranged marks: At least 6 images (Not all, but at least 4 of them with tilted calibration plates).
Plate with rectangularly arranged marks: At least 15 images.
Nevertheless, you must make sure that the poses of the calibration plates also fulfill the other requirements.
Inverted acquisition of the calibration plate
The calibration marks must not be acquired inverted. This can, e.g., happen if a calibration plate made of glass is acquired from its backside or if a line scan camera is not moving downwards with respect to the image coordinate system (i.e., is negative).
Pattern coverage
How much of the calibration pattern must at least be contained in the images depends on the used plate.
Plate with hexagonally arranged marks: At least one finder pattern needs to be visible. If at least two finder patterns are visible in the image, it is possible to detect whether the calibration plate is mirrored or not. In a mirrored case, a suitable error will be returned.
Plate with rectangularly arranged marks: Plate needs to be completely visible, as the finder pattern is the frame surrounding the point marks.
Nevertheless, of course, the more of the calibration pattern is visible to the camera and the more of the field of view is filled by the calibration plate, the better.
Mark diameter
The marks of the calibration plates should have a diameter of at least 20 pixels in each image. This requirement is essential for a successful calibration.
Contrast
The contrast between the light and dark areas of the calibration plate should be at least 100 gray values (regarding byte images).
Overexposure
To avoid overexposed images, make sure that gray values of the light parts of the calibration plate do not exceed 240 (regarding byte images), especially not in the neighborhood of the calibration marks.
Homogeneity
The calibration plate should be illuminated homogeneously and reflections should be avoided. As a rule of thumb, the range of gray values of the light parts of the plate should not exceed 45 (regarding byte images).
Image format
Calibration images should be saved in an uncompressed format. Compression artifacts which occur, e.g., when using JPG format and high compression rates need to be avoided.
Preprocessing
Calibration images should not be preprocessed. If image properties like contrast or focus are insufficient (see above), the issues need to be resolved by adjusting the camera setup instead of processing the images ahead of the calibration.
Two distortion models can be used: The division model and the polynomial model. The division model uses one parameter to model the radial distortions while the polynomial model uses five parameters to model radial and decentering distortions (see the sections “Camera parameters” and “The Used 3D camera model”).
The advantages of the division model are that the distortions can be applied faster, especially the inverse distortions, i.e., if world coordinates are projected into the image plane. Furthermore, if only few calibration images are used or if the field of view is not covered sufficiently, the division model typically yields more stable results than the polynomial model. The main advantage of the polynomial model is that it can model the distortions more accurately because it uses higher order terms to model the radial distortions and because it also models the decentering distortions. Note that the polynomial model cannot be inverted analytically. Therefore, the inverse distortions must be calculated iteratively, which is slower than the calculation of the inverse distortions with the (analytically invertible) division model.
Typically, the division model should be used for the calibration. If the accuracy of the calibration is not high enough, the polynomial model can be used. Note, however, that the calibration sequence used for the polynomial model must provide an even better coverage of the area in which measurements will later be performed. The distortions may be modeled inaccurately outside of the area that was covered by the calibration plate. This holds for the image border as well as for areas inside the field of view that were not covered by the calibration plate.
In general, camera calibration means the exact determination of the parameters that model the (optical) projection of any 3D world point into a (sub-)pixel (r,c) in the image. This is important if the original 3D pose of an object must be computed from the image (e.g., for measuring industrial parts). The appropriate projection model depends on the camera type used in your setup.
For the modeling of this projection process, which is determined by the used combination of camera, lens, and frame grabber, HALCON provides the following 3D camera models:
The combination of an area scan camera with a lens that effects a perspective projection on the object side of the lens and that may show radial and decentering distortions. The lens may be a tilt lens, i.e., the optical axis of the lens may be tilted with respect to the camera's sensor (this is sometimes called a Scheimpflug lens). Since hypercentric lenses also perform a perspective projection, cameras with hypercentric lenses are pinhole cameras. The models for regular (i.e., non-tilt) pinhole and image-side telecentric lenses are identical. In contrast, the models for pinhole and image-side telecentric tilt lenses differ substantially, as described below.
The combination of an area scan camera with a lens that is telecentric on the object-side of the lens, i.e., that effects a parallel projection on the object-side of the lens, and that may show radial and decentering distortions. The lens may be a tilt lens. The models for regular (i.e., non-tilt) bilateral and object-side telecentric lenses are identical. In contrast, the models for bilateral and object-side telecentric tilt lenses differ substantially, as described below.
The combination of a line scan camera with a lens that effects a perspective projection and that may show radial distortions. Tilt lenses are currently not supported for line scan cameras.
The combination of a line scan camera with a lens that effects a telecentric projection and that may show radial distortions. Tilt lenses are currently not supported for line scan cameras.
To transform a 3D point which is given in world coordinates, into a 2D point , which is given in pixel coordinates, a chain of transformations is needed:
| 3D world point | |
| Transformed into camera coordinate system | |
| Projected into image plane (2D point, still in metric coordinates) | |
| Lens distortion applied | |
| If tilted lens is used, the point is projected on a tilted image plane. In this case the distorted point only lies on a virtual image plane of a system without tilt. | |
| Pixel coordinates |
The following paragraphs describe these steps in more detail for area scan
cameras and subsequently for line scan cameras.
For a even more detailed description of the different 3D camera models as
well as some explanatory diagrams please refer to the chapter “Basics”,
section “Camera Model and Parameters” in the
“Solution Guide III-C 3D Vision”.
The point is transformed from world
into camera coordinates (points as homogeneous vectors, compare
affine_trans_point_3d
with and being
the rotation and translation matrices (refer to
the chapter “Basics”, section “3D Transformations and Poses`” in the
“Solution Guide III-C 3D Vision” for detailed information).
If the underlying camera model is an area scan pinhole camera, the projection of into the image plane is described by the following equation: where . For cameras with hypercentric lenses, the following equation holds instead:
If an area scan telecentric camera is used, the corresponding equation is: where .
For all types of cameras, the lens distortions can be modeled either by the division model or by the polynomial model.
The division model uses one parameter Kappa
The following equations transform the distorted image plane coordinates into undistorted image plane coordinates if the division model is used:
These equations can be inverted analytically, which leads to the following equations that transform undistorted coordinates into distorted coordinates:
The polynomial model uses three parameters () to model the radial distortions and two parameters () to model the decentering distortions.
The following equations transform the distorted image plane coordinates into undistorted image plane coordinates if the polynomial model is used: with
These equations cannot be inverted analytically. Therefore, distorted image plane coordinates must be calculated from undistorted image plane coordinates numerically.
If the camera lens is a tilt lens, the tilt of the lens with respect to the image plane is described by the rotation angle and the tilt angle .
In this step you have to further distinguish between different types of tilt
lenses as described below. See chapter “Basics”, section “Camera Model and
Parameters” in the “Solution Guide III-C 3D Vision” for an
overview of different types of tilt lenses.
For projective tilt lenses and object-side
telecentric tilt lenses (which perform a perspective projection on
the image side of the lens) the projection of into the point , which lies
in the tilted image plane, is described by a projective 2D
transformation, i.e., by the homogeneous 3×3
matrix (see
projective_trans_point_2d
where is the additional coordinate from the projective transformation of a homogeneous point. where and with and .
For image-side telecentric tilt lenses and bilateral telecentric tilt lenses (which perform a parallel projection on the image side of the lens), the projection onto the tilted image plane is described by a linear 2D transformation, i.e., by a 2×2 matrix: where is defined as above for projective lenses.
Finally, the point (or if a tilt lens is present) is transformed from the image plane coordinate system into the image coordinate system (the pixel coordinate system):
For line scan cameras, also the relative motion between the camera and the object must be modeled. In HALCON, the following assumptions for this motion are made:
The camera moves with constant velocity along a straight line.
The orientation of the camera is constant.
The motion is equal for all images.
The motion is described by the motion vector that must be given in [meter/row] in the camera coordinate system. The motion vector describes the motion of the camera, assuming a fixed object. In fact, this is equivalent to the assumption of a fixed camera with the object traveling along .
The camera coordinate system of line scan cameras is defined as follows: The origin of the coordinate system is the center of projection (for pinhole cameras) or the center of distortion (for telecentric cameras), respectively. The z-axis is identical to the optical axis and directed so that the visible points have positive z coordinates. The y-axis is perpendicular to the sensor line and to the z-axis. It is directed so that the motion vector has a positive y-component. The x-axis is perpendicular to the y- and z-axis, so that the x-, y-, and z-axis form a right-handed coordinate system.
As the camera moves over the object during the image acquisition,
also the camera coordinate system moves relatively to the object,
i.e., each image line has been imaged from a different
position. This means there would be an individual pose for each
image line. To make things easier, in HALCON all transformations
from world coordinates into camera coordinates and vice versa are
based on the pose of the first image line only. The motion is
taken into account during the projection of the point
into the image. Consequently, only
the pose of the first image line is computed by the operator
find_calib_objectcalibrate_cameras
For line scan cameras, the transformation from world to camera coordinates () works in the same way. Therefore, you can also apply transformation step 1 as described for area scan cameras above.
For line scan pinhole cameras, the projection of the point that is given in the camera coordinate system into (sub-)pixel coordinates in the image is modeled as follows:
Assuming the following set of equations must be solved for , , and : were and are the undistortion functions that are described above for area scan cameras and .
For line scan telecentric cameras, the following set of equations must be solved for and : with , and as defined above. Note that neither nor influence the projection for telecentric cameras.
The above formulas already include the compensation for image distortions.
Finally, the point is transformed into the image coordinate system, i.e., the pixel coordinate system:
For pinhole cameras, if the calibration plates are parallel
to each other in all images (in particular, if they all lie in the
same plane), it is impossible to determine FocusFocus
For telecentric lenses, the distance of the calibration plate from the camera cannot be determined. Therefore, the z-component of the resulting calibration plate pose is set to 1 m in the calibration results. Furthermore, as described previously, for telecentric line scan cameras, cannot be determined and is left at its initial value, except for multi-camera setups that have a common motion vector, in which case can be determined.
For tilt lenses, the greater the lens distortion is, the
more accurately the tilt can be determined. For lenses with small
distortions, the tilt cannot be determined robustly. Therefore, the
optimized tilt parameters may differ significantly from the nominal
tilt parameters of the setup. If this is the case, please check
ErrorError
For perspective tilt lenses and object-side
telecentric tilt lenses, the image plane distance can only be
determined uniquely if the tilt is not 0 degrees. The smaller the tilt, the
less accurately the image plane distance can be determined.
Therefore, the optimized image plane distance may differ
significantly from the nominal image plane distance of the setup.
If this is the case, please check ErrorError
For perspective tilt lenses and object-side telecentric tilt lenses that are tilted around the horizontal or vertical axis, i.e., for which the rotation angle is 0, 90, 180, or 270 degrees, the tilt angle , scale factor , the focal length (for perspective tilt lenses) or the magnification (for object-side telecentric tilt lenses), and the distance of the tilted image plane from the perspective projection center cannot be determined uniquely. In this case, should be excluded from the optimization by calling
set_calib_data(CalibDataID, 'camera', 'general', 'excluded_settings', 'sx')
Additionally, note that for tilt lenses it is only possible to determine and simultaneously. This is an implementation choice that makes the optimization numerically more robust. Consequently, the parameters and are excluded simultaneously from the optimization by calling
set_calib_data(CalibDataID, 'camera', 'general', 'excluded_settings', 'tilt')
Pinhole cameras with tilt lenses with large focal lengths have nearly telecentric projection characteristics. Therefore, as described before, and the tilt parameters and are correlated and can only be determined imprecisely simultaneously. In this case, it is again advisable to exclude from the optimization.
For telecentric lenses, there are always two possible poses of a calibration
plate for a single image. Therefore, it is not possible to
decide which one of the two poses is actually present in the image.
This ambiguity also effects the tilt parameters and
of a telecentric tilt lens. Consequently, depending
on the initial parameters for and
the camera calibration may return the alternative parameters instead
of the nominal ones. If this is the case, please check
ErrorError
For line scan cameras with the polynomial distortion model (for cameras with perspective as well as telecentric lenses), the parameters and are highly correlated with other parameters in the camera model. Therefore, they typically cannot be determined reliably and should be excluded from the calibration by calling
set_calib_data(CalibDataID, 'camera', 'general', 'excluded_settings', 'poly_tan_2')