minkowski_add1 ( Region, StructElement : RegionMinkAdd : Iterations : )

Perform a Minkowski addition on a region.

minkowski_add1 dilates the input regions with a structuring element. By applying minkowski_add1 to a region, its boundary is smoothed. In the process, the area of the region is enlarged. Furthermore, disconnected regions may be merged. Such regions, however, remain logically distinct region. The Minkowski addition is a set-theoretic region operation. It is based on translations and union operations.

Let M (StructElement) and R (Region) be two regions, where M is the ``structuring element'' and R is the region to be precessed. Furthermore, let m be a point in M. Then the displacement vector v(m) = (dx,dy) is defined as the difference of the center of gravity of M and the vector v(m). Let t(v(m))(R) denote the translation of a region R by a vector v(m). Then

                            |  |
     minkowski_add1(R,M) := |  | t    (R)
                            \__/  v(m)
                           m in M

For each point m in M a translation of the region R is performed. The union of all these translations is the Minkowski addition of R with M. minkowski_add1 is similar to the operator dilation1, the difference being that in dilation1 the structuring element is mirrored at the origin. The position of StructElement is meaningless, since the displacement vectors are determined with respect to the center of gravity of M.

The parameter Iterations determines the number of iterations which are to be performed with the structuring element. The result of iteration n-1 is used as input for iteration n. From the above definition it follows that an empty region is generated in case of an empty structuring element.

Structuring elements (StructElement) can be generated with operators such as circle, rectangle1, rectangle2, ellipse, draw_region, store_polygon, store_coord, etc.


Attention

A Minkowski addition always results in the regions being enlarged. Closely spaced regions which may touch or overlap as a result of the dilation are still treated as two separate regions. If the desired behaviour is to merge them into one region, the operator union1 has to be called first.


Parameters

Region (input_object)
region(-array) -> object
Regions to be dilated.

StructElement (input_object)
region -> object
Structuring element.

RegionMinkAdd (output_object)
region(-array) -> object
Dilated regions.

Iterations (input_control)
integer -> integer
Number of iterations.
Default value: 1
Suggested values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 20, 30, 40, 50
Range of values: 1 <= Iterations
Minimum increment: 1
Recommended increment: 1


Complexity

Let F1 be the area of the input region, and F2 be the area of the structuring element. The the runtime complexity for one region is:

    O(sqrt(F1) * sqrt(F2) * Iterations) .


Result

minkowski_add1 returns TRUE if all parameters are correct. The behavior in case of empty or no input region can be set via:

  a) no region:    set_system(::'no_object_result',<RegionResult>:)
  b) empty region: set_system(::'empty_region_result',<RegionResult>:)
Otherwise, an exception is raised.


Possible Predecessors

threshold__, regiongrowing__, connection, union1, watersheds, class_ndim1__, circle, ellipse, rectangle1, rectangle2, draw_region, store_coord, struct_elements, store_filled_polygon


Possible Successors

reduce_domain, select_shape, area_center, connection


Alternatives

minkowski_add2, dilation1


See also

transpose, minkowski_sub1



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