fit_ellipse_contour_xld — Approximate XLD contours by ellipses or elliptic arcs.
fit_ellipse_contour_xld(Contours : : Algorithm, MaxNumPoints, MaxClosureDist, ClippingEndPoints, VossTabSize, Iterations, ClippingFactor : Row, Column, Phi, Radius1, Radius2, StartPhi, EndPhi, PointOrder)
fit_ellipse_contour_xld approximates the XLD contours
Contours by elliptic arcs or closed ellipses. It does not
perform a segmentation of the input contours. Thus, one has to make sure
that each contour corresponds to one and only one elliptic structure.
The operator returns for each contour the center
Column), the orientation of the main
Phi, the length of the larger half axis
and the length of the smaller half axis
Radius2 of the underlying
ellipse. In addition to that, the angle corresponding to the start point
and the end point
and the point order along the boundary
is returned for elliptic arcs. These parameters are set to
0, , and 'positive' for closed ellipses.
The algorithm used for the fitting of ellipses can be selected via
This approach minimizes the algebraic distance between the contour points and the resulting ellipse. The constraint guarantees that the resulting polynomial characterizes an ellipse (instead of a hyperbola or a parabola).
Similar to 'fitzgibbon'. Here the contour points are weighted to decrease the impact of outliers based on the approach of Huber (see below).
Similar to 'fitzgibbon'. Here the contour points are weighted and outliers are ignored based on the approach of Tukey (see below).
This approach minimizes the geometric distance between the contour points and the resulting ellipse. The distance measure is statistically optimal, but takes more computational time, because the computation is an iterative process. This option is recommended if the contour points are considerably distorted by noise.
Similar to 'geometric'. Here the contour points are weighted to decrease the impact of outliers based on the approach of Huber (see below).
Similar to 'geometric'. Here the contour points are weighted and outliers are ignored based on the approach of Tukey (see below).
Each input contour is transformed in an affine standard position.
Based on the moments of the transformed contour (that is of the
enclosed image region) the standard circular segment is chosen
whose standard position matches best with the standard position
of the contour. The ellipse corresponding to the standard
position of the selected circular segment is re-transformed
based on the affine transformation which produced the standard position
of the contour resulting in the ellipse matching the original contour.
VossTabSize standard circular segments are used for
this computation. To speed up the process the corresponding moments
and other data is stored in a table which is created during the
first call (with a specific value for
Each point P on an ellipse satisfies the constraint that the sum of distances to the focal points equals twice the length of the larger half axis a. In this approach, the deviation is minimized for all contour points by a least squares optimization.
Similar to 'focpoints'. Here a weighted least squares optimization is done to decrease the impact of outliers based on the approach of Huber (see below).
Similar to 'focpoints'. Here a weighted least squares optimization is done and outliers are ignored based on the approach of Tukey (see below).
For '*huber' and '*tukey' a robust error statistics is used
to estimate the standard deviation of the distances from the contour points
without outliers from the approximating ellipse. The parameter
ClippingFactor (a scaling factor for the standard deviation)
controls the amount of outliers: the smaller the value chosen for
ClippingFactor the more points are classified as outliers.
Every contour point is individually weighted and contributes according to
its weight in the fitting process. The bundle of robust weighting and
fitting can be iterated. The overall number of iterations is
Iterations. This parameter is of relevance for all
algorithms except the two based on the geometric distance,
i.e., 'geohuber' and 'geotukey', where this parameter
is ignored. If the algorithm does
not find a fitting ellipse within this number of iterations, it fits a line;
in this case
Radius2 is zero.
In the Tukey algorithm outliers are removed, whereas in the Huber algorithm
outliers are only damped, or more precisely they are weighted linearly.
Without any robust weighting the squares of the distances are taken as
error values in the optimization, i.e., a least square formulation.
In practice, the approach of Tukey is recommended.
To reduce the computational load, the fitting of ellipses can be
restricted to a subset of the contour points: If a value other than
-1 is assigned to
MaxNumPoints, only up to
MaxNumPoints points - uniformly distributed over the
contour - are used.
For elliptic arcs, the points on the ellipse closest to the
start points and end points of the original contours are chosen
as start and end points. The corresponding angles referring to the
main axis of the ellipse are returned in
EndPhi, see also
Contours, for which the distance between their start points and
their end points is less or equal
are considered to be closed. Thus, they are approximated by
ellipses instead of elliptic arcs. The 'focpoints' algorithms
determine the initial parameters for the optimization for closed contours in
a faster and less accurate way than for open contours.
As a result, in some cases more
Iterations may be necessary when applied to closed contours.
Due to artifacts in the pre-processing the start and end points
of a contour might be faulty. Therefore, it is possible to
ClippingEndPoints points at the beginning and at the
end of a contour from the fitting of ellipses. However, they are still
used for the determination of
The minimum necessary number of contour points for fitting an ellipse is five. Therefore, it is required that the number of contour points is at least .
Algorithm for the fitting of ellipses.
Default value: 'fitzgibbon'
List of values: 'fhuber', 'fitzgibbon', 'focpoints', 'fphuber', 'fptukey', 'ftukey', 'geohuber', 'geometric', 'geotukey', 'voss'
Maximum number of contour points used for the computation (-1 for all points).
Default value: -1
MaxNumPoints >= 5
Maximum distance between the end points of a contour to be considered as 'closed'.
Default value: 0.0
MaxClosureDist >= 0.0
Number of points at the beginning and at the end of the contours to be ignored for the fitting.
Default value: 0
ClippingEndPoints >= 0
Number of circular segments used for the Voss approach.
Default value: 200
VossTabSize >= 25 && VossTabSize <= 5000
Maximum number of iterations for the robust weighted fitting.
Default value: 3
Iterations >= 0
Clipping factor for the elimination of outliers (typical: 1.0 for '*huber' and 2.0 for '*tukey').
Default value: 2.0
Suggested values: 1.0, 1.5, 2.0, 2.5, 3.0
ClippingFactor > 0
Row coordinate of the center of the ellipse.
Column coordinate of the center of the ellipse.
Orientation of the main axis [rad].
Length of the larger half axis.
Length of the smaller half axis.
Angle of the start point [rad].
Angle of the end point [rad].
point order along the boundary.
List of values: 'negative', 'positive'
read_image (Image, 'caltab') find_caltab (Image, CalPlate, 'caltab_800mm.descr', 3, 112, 5) reduce_domain (Image, CalPlate, ImageReduced) edges_sub_pix (ImageReduced, Edges, 'lanser2', 0.5, 20, 40) select_contours_xld (Edges, EdgesClosed, 'closed', 0, 2.0, 0, 0) select_contours_xld (EdgesClosed, EdgesMarks, 'length', 20, 80, 0, 0) fit_ellipse_contour_xld (EdgesMarks, 'fitzgibbon', -1, 2, 0, 200, 3, 2.0, \ Row, Column, Phi, Radius1, Radius2, StartPhi, \ EndPhi, PointOrder) gen_ellipse_contour_xld (EllMarks, Row, Column, Phi, Radius1, Radius2, \ StartPhi, EndPhi, PointOrder, 1.5) length_xld(EllMarks,Length)
fit_ellipse_contour_xld returns TRUE if all parameter values
are correct, and ellipses could be fitted to the input contours.
If the input is empty the behavior can be set via
If necessary, an exception is raised.
If the parameter
ClippingFactor is chosen too small, i.e.,
all points are classified as outliers, the error 3264 is raised.