Master’s Paper [ROUGH DRAFT]: 2 Dimensional Min-Filters with Polygons

Computing min-filters is an important operation in image processing. Min-filters are the basic building blocks for translation invariant morphological operators, which are used to perform operations such as noise suppression, image smoothing, contrast enhancement, and edge detection. Min-filters are a special case of min-convolutions, which are used for a variety of applications, including signal processing, and combinatorial optimization. The most widely used approach for computing min-filters decomposes the filtering element as a Minkowsky sum of smaller filters. However this approach can be expensive for large filtering elements[1]. There is an algorithm to perform the min-filter efficiently, but it only applies to axis parallel rectangles [2]. We present an efficient algorithm to compute the min-filter when the filtering element is a polygon for certain polygons. Because the polygon will be given as a non digital geometric object, we relax the conditions by allowing the algorithm to choose a valid digitization of the polygon for each placement. With this relaxation, we present an algorithm to compute the min-filter by arbitrary rectangles and isosceles right triangles in logarithmic time per pixel, in the size of the polygon. We also show how to compute the min-filter by arbitrary triangles in time O ( logA sinγ ) per pixel, where A is the area of the triangle, and γ is the size of the smallest angle in the triangle.