ALGORITHMS FOR PROCESSING CONNECTED COMPONENTS WITH GRAY VALUES - Fundamentals of Three-Dimensional Digital Image Processing - page 188

6

ALGORITHMS FOR PROCESSING
CONNECTED COMPONENTS WITH GRAY
VA LUES

In this chapter we present algorithms processing a connected component in
which each voxel has an arbitrary gray value (= a gray-tone connected compo-
nent). We assume here that a gray value (= a density value) is positive inside
a connected component and that a gray value outside a connected component
is zero unless stated otherwise. Therefore, the image is a gray-tone image with
a background.
A gray-tone connected component has the same topological features and
general shape features as a binary connected component. Furthermore, it has
additional density value features. The important characteristics of an algo-
rithm processing a gray-tone connected component are a combination of those
two types of information. The notice about the description of algorithms in the
beginning of the previous chapter also concerns the description of algorithms
in this chapter.

6.1 Distance transformation of a gray-tone image

6.1.1 Definition

The definition of the distance transformation (DT) of a gray-tone image is
immediately derived from the extension of a 2D gray-tone image.
Definition 6.1 (DT of a gray-tone image). A process to calculate the
following image

f ijk }
with a background is called a gray weighted distance transformation ( GWDT ).

G = { g ijk } from an input 3D gray-tone image F = {

F { f ijk }→ G { g ijk }

GWDT : = =

min {| ; π represents a path from a 0-voxel of

an image

π |

F = { f ijk } toavoxel( i, j, k ) ,

represents the sum of density values of
voxels on a path π in an input image ,
if f ijk > 0 (( i, j, k ) is a voxel of a connected component) ,
0 , if f ijk = 0 (( i, j, k ) is on the background)

and | π |

g ijk = (6.1)