## Name

`minkowski_add1` — Perform a Minkowski addition on a region.

## Signature

`minkowski_add1(Region, StructElement : RegionMinkAdd : Iterations : )`

## Description

`minkowski_add1` dilates the input regions with a structuring element. By applying `minkowski_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 (`StructElement`) and R (`Region`) 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_add1` is similar to the operator `dilation1`, the difference is 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 `gen_circle`, `gen_rectangle1`, `gen_rectangle2`, `gen_ellipse`, `draw_region`, `gen_region_polygon`, `gen_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 `union1` has to be called first.

## Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Automatically parallelized on tuple level.

## 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

Typical range of values: ```1 ≤ Iterations ``` (lin)

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. Then the runtime complexity for one region is:

## Result

`minkowski_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

`threshold`, `regiongrowing`, `connection`, `union1`, `watersheds`, `class_ndim_norm`, `gen_circle`, `gen_ellipse`, `gen_rectangle1`, `gen_rectangle2`, `draw_region`, `gen_region_points`, `gen_region_polygon_filled`

## Possible Successors

`reduce_domain`, `select_shape`, `area_center`, `connection`

## Alternatives

`minkowski_add2`, `dilation1`

`transpose_region`, `minkowski_sub1`