minkowski_add1minkowski_add1MinkowskiAdd1MinkowskiAdd1minkowski_add1 (Operator)

Name

minkowski_add1minkowski_add1MinkowskiAdd1MinkowskiAdd1minkowski_add1 — Perform a Minkowski addition on a region.

Signature

Description

minkowski_add1minkowski_add1MinkowskiAdd1MinkowskiAdd1MinkowskiAdd1minkowski_add1 dilates the input regions with a structuring element. By applying minkowski_add1minkowski_add1MinkowskiAdd1MinkowskiAdd1MinkowskiAdd1minkowski_add1 to a region, its boundary gets 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 (StructElementStructElementStructElementStructElementstructElementstruct_element) and R (RegionRegionRegionRegionregionregion) be two regions, where M is the structuring element and R is the region to be processed. Furthermore, let m be a point in M. Then the displacement vector is defined as the difference of the center of gravity of M and the vector . Let denote the translation of a region R by a vector . Then

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_add1minkowski_add1MinkowskiAdd1MinkowskiAdd1MinkowskiAdd1minkowski_add1 is similar to the operator dilation1dilation1Dilation1Dilation1Dilation1dilation1, the difference is that in dilation1dilation1Dilation1Dilation1Dilation1dilation1 the structuring element is mirrored at the origin. The position of StructElementStructElementStructElementStructElementstructElementstruct_element is meaningless, since the displacement vectors are determined with respect to the center of gravity of M.

The parameter IterationsIterationsIterationsIterationsiterationsiterations 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 (StructElementStructElementStructElementStructElementstructElementstruct_element) can be generated with operators such as gen_circlegen_circleGenCircleGenCircleGenCirclegen_circle, gen_rectangle1gen_rectangle1GenRectangle1GenRectangle1GenRectangle1gen_rectangle1, gen_rectangle2gen_rectangle2GenRectangle2GenRectangle2GenRectangle2gen_rectangle2, gen_ellipsegen_ellipseGenEllipseGenEllipseGenEllipsegen_ellipse, draw_regiondraw_regionDrawRegionDrawRegionDrawRegiondraw_region, gen_region_polygongen_region_polygonGenRegionPolygonGenRegionPolygonGenRegionPolygongen_region_polygon, gen_region_pointsgen_region_pointsGenRegionPointsGenRegionPointsGenRegionPointsgen_region_points, etc.

Attention

A Minkowski addition always results in enlarged regions. Closely spaced regions which may touch or overlap as a result of the dilation are still treated as two separate regions. If the desired behavior is to merge them into one region, the operator union1union1Union1Union1Union1union1 has to be called first.

Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Multithreading scope: global (may be called from any thread).
• Automatically parallelized on tuple level.

Parameters

Complexity

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

Result

minkowski_add1minkowski_add1MinkowskiAdd1MinkowskiAdd1MinkowskiAdd1minkowski_add1 returns 2 (H_MSG_TRUE) if all parameters are correct. The behavior in case of empty or no input region can be set via:

• no region: set_system('no_object_result',<RegionResult>)

• empty region: set_system('empty_region_result',<RegionResult>)

Otherwise, an exception is raised.

Possible Predecessors

thresholdthresholdThresholdThresholdThresholdthreshold, regiongrowingregiongrowingRegiongrowingRegiongrowingRegiongrowingregiongrowing, connectionconnectionConnectionConnectionConnectionconnection, union1union1Union1Union1Union1union1, watershedswatershedsWatershedsWatershedsWatershedswatersheds, class_ndim_normclass_ndim_normClassNdimNormClassNdimNormClassNdimNormclass_ndim_norm, gen_circlegen_circleGenCircleGenCircleGenCirclegen_circle, gen_ellipsegen_ellipseGenEllipseGenEllipseGenEllipsegen_ellipse, gen_rectangle1gen_rectangle1GenRectangle1GenRectangle1GenRectangle1gen_rectangle1, gen_rectangle2gen_rectangle2GenRectangle2GenRectangle2GenRectangle2gen_rectangle2, draw_regiondraw_regionDrawRegionDrawRegionDrawRegiondraw_region, gen_region_pointsgen_region_pointsGenRegionPointsGenRegionPointsGenRegionPointsgen_region_points, gen_region_polygon_filledgen_region_polygon_filledGenRegionPolygonFilledGenRegionPolygonFilledGenRegionPolygonFilledgen_region_polygon_filled

Possible Successors

reduce_domainreduce_domainReduceDomainReduceDomainReduceDomainreduce_domain, select_shapeselect_shapeSelectShapeSelectShapeSelectShapeselect_shape, area_centerarea_centerAreaCenterAreaCenterAreaCenterarea_center, connectionconnectionConnectionConnectionConnectionconnection

Alternatives

minkowski_add2minkowski_add2MinkowskiAdd2MinkowskiAdd2MinkowskiAdd2minkowski_add2, dilation1dilation1Dilation1Dilation1Dilation1dilation1

See also

transpose_regiontranspose_regionTransposeRegionTransposeRegionTransposeRegiontranspose_region, minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1MinkowskiSub1minkowski_sub1

Module

Foundation