LOCAL PROCESSING OF 3D IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 47

3

LOCAL PROCESSING OF 3D IMAGES

In this chapter we discuss the basics of 3D local operations using density
values in small subareas of an input image. A smoothing filter, a difference
filter, and features of the curvature of a surface are explained in detail.

3.1 Classification of local operations

3.1.1 General form

The general form of a local process was given in Section 2.4.2 as shown below.
Definition 3.1 (Local processing, local operation). The output density
g ijk

is obtained by the equation,

g ijk = φ ijk ( { f pqr ;( p, q, r ) ∈N pqr (( i, j, k )) } ) (3.1)

where

N pqr (( i, j, k )) ≡{ ( i + p, j + q, k + r ); ( p, q, r ) ∈S ijk } , (3.2)

S ijk × ×

is an appropriate subset of the set of all integer triads I I I.

N pqr (( i, j, k )) is called the neighborhood
of ( i, j, k ). This means that a density value at a voxel ( i, j, k ) of an output
image is calculated by a function φ ijk

The subarea of the image space

using the density values of an input

image N pqr (( i, j, k )). We call the function

φ ijk a local function . In the most general case, both the neighborhood and
the local function may be changeable according to their location on an image.
In the explanation in this topic, however, it is assumed that they are the
same over the whole of an image to be processed, unless claimed otherwise
( position-invariant processing ).
Remark 3.1. Local processes are sometimes called filtering and mask pro-
cessing . More specific names such as

F = { f ijk } in the neighborhood

operator are
also used for these processes. For example, terms such as edge detection filter,
difference filter, Gaussian filter, and smoothing filter are often used.

∗∗∗ filter and ∗∗∗