LOCAL PROCESSING OF 3D IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 50

simplicity of its calculation. The decomposition to a serial composition of
three 1D UFs is also possible as in a 2D case.

Uniform weight linear filter UF [ C ]: F = { f ijk }→ G = { g ijk }

g ijk = c

· f ijk . (3.4)

represents the sum over all voxels in the neighborhood

N (( i, j, k ))

We assume c = 1 in the sequel, if not described otherwise.
In order to describe explicitly that a weight W of the size of the neigh-

borhood P × Q × R voxels is employed, we use the notation

W P × Q × R ]: F { f ijk }→ G { g ijk } .

Linear filter LF [ = = (3.5)

The fast algorithm of the recursive type is available in the same way as the
2D UF [Preston79].
A weight matrix derived based on the probability density of Gaussian
distribution is frequently employed to smooth an input image in practical
applications. This type of filter is called a Gaussian filter .

Remark 3.2 (Gaussian distribution). The probability density function of
the 3D Gaussian distribution (normal distribution) p ( x 1 ,x 2 ,x 3 )isgivenas
follows

) t Σ − 1 ( p ( x 1 ,x 2 ,x 3 )=(2 π ) − 3 / 2 | Σ | − 1 / 2 exp {− ( x − µ x − µ ) / 2 }

=( x 1 ,x 2 ,x 3 ) t x

=( µ 1 ,µ 2 ,µ 3 ) t = mean vector µ

σ 11 σ 12 σ 13
σ 21 σ 22 σ 23
σ 31 σ 32 σ 33

= covariance matrix

Σ = (3.6)

To derive a weight matrix, we assume that the origin is located at the
center voxel of the neighborhood. Therefore, we assume the mean vector as
( 0 , 0 , 0 ). An arbitrarily selected positive definite matrix can be given as a
covariance matrix

. A scale factor may be neglected. Thus we determine
each element of the weight matrix by the equation.

Σ

) t Σ − 1 ( exp {− ( x − µ x − µ ) } (3.7)

3.2.2 Median filter and order statistics filter

The median filter is a filter that outputs at each voxel ( i, j, k ) the median of
density values of an input image in the neighborhood of ( i, j, k ). Formally it
is defined as follows.