# Grayscale morphology

**Written by Stanley R.Sternberg**

Date of Publication: 22 October 2006

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**INTRODUCTION**

### Extending the principles of mathematical morphology from two to three dimen- sions reveals a surprising landscape of beauty and utility. Continuous tone images are equated to 3dimensional surfaces whose topology is modified by sliding geometric structures. The grayscale morphology is easily visualized in the continu- ous domain of Euclidean 3-space, conveying a clear impression of the grayscale morphological transformations. Complications which are introduced by the details of the digital approximations required for computer image processing are high- lighted, but not considered in depth. The appendix deals with the issues of the digital umbra shading and shadowing algorithm.

### Historically, morphological grayscale image processing originally treated the gray level functions as sets or piles of binary cross sections (thresholds). Processing was based on a stereological approach where cross sections are transformed individually or in pairs [7]. The transition into a topological description of a function by its watersheds is due to Lantuejoul [4]. Meyer [5] developed grayscale image contrast descriptors based on differences of openings of the functions by three dimensional structuring elements composed of disks. Rosenfeld [6] proposed a generalization of connectivity to functions, from which he derived a grayscale thinning algorithm in collaboration with Dyer [9]. Goetcharian has contributed several original notions such as the lower skeletons [l] and relations of the fuzzy logic concepts to grayscale morphological processing [2]. Recently, Serra [7] has introduced formal notions of grayscale homotopy. Lantuejoul and Serra [4] have recently examined the relation- ships between the morphological filters and the classical linear filters, such as convolutions.

**GRAYSCALE MORPHOLOGY**

### The principles of mathematical morphology are applicable to sets in Euclidean or digital spaces without regard to their dimension. The concepts introduced in the previous paper dealt with sets in Euclidean 2-space, representing silhouette pictures or binary images. Here we consider the morphology of sets in Euclidean 3-space. The binary images of Euclidean 2-space are seen here as flat cutouts of the horizontal X, Y plane, henceforth referred to the binary plane. Other sets we will consider in Euclidean 3-space have solid volumes. Of particular interest are the umbrae, solid sets which extend unbroken indefinitely downward in the negative 2’ direction. The details of an umbra’s third dimension can be determined by a single parameter, the height z of the umbra at coordinates (x, y) of the binary plane.

*This publication was originally done on Science Direct. Read the original publication.*

*This publication was originally done on Science Direct. Read the original publication.*