LOCAL PROCESSING OF 3D IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 59

Let us denote by f p an output value of an element filter O p at a voxel
( i, j, k )( p = 1 , 2 ,...,n ). Then, the following examples show the output of an
omnidirectionalized filter f g .

n n

(1) f q = f p , (or f q = f p /n ) . (3.35)

p =1 p =1

n n

f p 2 . (2) f q = f p ,f q = (3.36)

p =1 p =1

(3) f q = order statistics of {| f p |} . (3.37)

(max {| f p |} , min {| f p |} , median of {| f p |} , etc . )

3.3.6 1D difference filters and their combinations

The simplest form of a difference filter is one in which voxels used for calcula-
tion are arranged along a line segment. This we call a simple difference filter .
Examples are shown below:

1st order difference
LDF 1 [ p, q, r ]: F = { f ijk }→ G = { g ijk } ,

g ijk = f i−p,j−q,k−r − f i + p,j + q,k + r (3.38)

2nd order difference
LDF 2 [ p, q, r ]: F = { f ijk }→ G = { g ijk } ,

g ijk = f i−p,j−q,k−r + f i + p,j + q,k + r − 2 f ijk (3.39)

1st order difference - rotationary type
LDF rot1 [ r, θ, φ ]: F = { f ijk }→ G = { g ijk } ,

g ijk = f 1 ( r, θ, φ ) − f 2 ( r, θ, φ ) (3.40)

2nd order difference - rotationary type
LDF rot2 [ r, θ, φ ]: F = { f ijk }→ G = { g ijk } .

g ijk = f 1 ( r, θ, φ )+ f 2 ( r, θ, φ ) − 2 f ijk (3.41)

In the rotationary type above, two voxels ( i ±

r cos θ ) are selected at each voxel ( i, j, k ), so that they are located separately
by r from the voxel ( i, j, k ) and symmetrically with respect to ( i, j, k )inthe
directional angles φ (Fig. 3.3). Then, f 1 ( r, θ, φ )and f 2 ( r, θ, φ )intheabove
rotationary type represent input density values at these voxels.

± r cos θ sin φ, j ± r sin φ sin θ, k