minkowski_sub1minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1 (Operator)

Name

minkowski_sub1minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1 — Erode a region.

Signature

Description

minkowski_sub1minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1 computes the Minkowski subtraction of the input regions with a structuring element. By applying minkowski_sub1minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1 to a region, its boundary gets smoothed. In the process, the area of the region is reduced. Furthermore, connected regions may be split. Such regions, however, remain logically one region. The Minkowski subtraction is a set-theoretic region operation. It uses the intersection operation.

Let M (StructElementStructElementStructElementStructElementstructElement) and R (RegionRegionRegionRegionregion) 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 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_sub1(R,M) := |  | t    (R)
                             |  |  v(m)
                            m in M

For each point m in M a translation of the region R is performed. The intersection of all these translations is the Minkowski subtraction of R with M. minkowski_sub1minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1 is similar to the operator erosion1erosion1erosion1Erosion1Erosion1, the difference is that in erosion1erosion1erosion1Erosion1Erosion1 the structuring element is mirrored at the origin. The position of StructElementStructElementStructElementStructElementstructElement is meaningless, since the displacement vectors are determined with respect to the center of gravity of M.

The parameter IterationsIterationsIterationsIterationsiterations 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 the maximum region is generated in case of an empty structuring element.

Structuring elements (StructElementStructElementStructElementStructElementstructElement) can be generated with operators such as gen_circlegen_circlegen_circleGenCircleGenCircle, gen_rectangle1gen_rectangle1gen_rectangle1GenRectangle1GenRectangle1, gen_rectangle2gen_rectangle2gen_rectangle2GenRectangle2GenRectangle2, gen_ellipsegen_ellipsegen_ellipseGenEllipseGenEllipse, draw_regiondraw_regiondraw_regionDrawRegionDrawRegion, gen_region_polygongen_region_polygongen_region_polygonGenRegionPolygonGenRegionPolygon, gen_region_pointsgen_region_pointsgen_region_pointsGenRegionPointsGenRegionPoints, etc.

Parallelization

• 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:

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

Result

minkowski_sub1minkowski_sub1minkowski_sub1MinkowskiSub1MinkowskiSub1 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

thresholdthresholdthresholdThresholdThreshold, regiongrowingregiongrowingregiongrowingRegiongrowingRegiongrowing, connectionconnectionconnectionConnectionConnection, union1union1union1Union1Union1, watershedswatershedswatershedsWatershedsWatersheds, class_ndim_normclass_ndim_normclass_ndim_normClassNdimNormClassNdimNorm, gen_circlegen_circlegen_circleGenCircleGenCircle, gen_ellipsegen_ellipsegen_ellipseGenEllipseGenEllipse, gen_rectangle1gen_rectangle1gen_rectangle1GenRectangle1GenRectangle1, gen_rectangle2gen_rectangle2gen_rectangle2GenRectangle2GenRectangle2, draw_regiondraw_regiondraw_regionDrawRegionDrawRegion, gen_region_pointsgen_region_pointsgen_region_pointsGenRegionPointsGenRegionPoints, gen_struct_elementsgen_struct_elementsgen_struct_elementsGenStructElementsGenStructElements, gen_region_polygon_filledgen_region_polygon_filledgen_region_polygon_filledGenRegionPolygonFilledGenRegionPolygonFilled

Possible Successors

reduce_domainreduce_domainreduce_domainReduceDomainReduceDomain, select_shapeselect_shapeselect_shapeSelectShapeSelectShape, area_centerarea_centerarea_centerAreaCenterAreaCenter, connectionconnectionconnectionConnectionConnection

Alternatives

minkowski_sub2minkowski_sub2minkowski_sub2MinkowskiSub2MinkowskiSub2, erosion1erosion1erosion1Erosion1Erosion1

See also

transpose_regiontranspose_regiontranspose_regionTransposeRegionTransposeRegion

Module

Foundation