Noise Filtering Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 172

be arranged according to their size and the evaluation of the standard deviation
can be started from larger templates. Then, the first template having a standard
deviation less than the threshold may be the optimal template. If no template is
found having a standard deviation less than the threshold, the template having
the minimum standard deviation is selected as the optimal template. Through this
procedure, the computational time spent searching for the optimal template is
substantially reduced compared to searching for the templates in arbitrary order.
The filtering algorithm can be applied iteratively so that more noise reduction
can be achieved.

6.5 ANISOTROPIC DIFFUSION FILTERING

Perona and Malik [12] first proposed a nonlinear anisotropic smoothing filter for
removal of background noise in images. It uses local gradients to control the
anisotropy of the filter. A comprehensive review of anisotropic filter theory can
be found in [17].
The smoothing operation is assumed to be a diffusive process that is suppressed
or stopped at boundaries by selecting appropriate spatial diffusion strengths. In
particular, depending on the values assumed by diffusion strength, the filter is able
to realize intraregion smoothing in preference to smoothing across boundaries. In
other words, the nonlinear anisotropic diffusion equation is:

∂
∂ t

It x ivct x x t (6.11) (,) = [(,) ⋅ ∇ I (, ]

represents the spatial
coordinate, and the variable t in our discrete implementation corresponds to iteration
step n. The function I(

The diffusion strength is controlled by c( x , t). The vector x

, t)
assumes a constant value for linear isotropic diffusion. In that case, the diffused
image is derived from isotropic application of the Laplacian operator to the image.
But the price of eliminating the noise with linear diffusion is blurring of the edges.
This results in their detection and localization being difficult.
In order to preserve the edges, the diffusion must be reduced or even blocked
when close to a discontinuity. The diffusion function c(

x , t) is the image intensity. The diffusion function c( x

, t) can be chosen to be
a function of gradient magnitude evaluated on image intensity I(

x

x , t):

2

− ∇

| It

K

(, | x

cx , ( e

2

(6.12) ) =

2

Figure 6.2 shows the monotonic decrease of the diffusion coefficient c( x , t) with

increasing gradient I. A more effective view of the relationship between parameter

K and image gradient

∇

∇ I is obtained by defining the flow function φ ( ∇ I) as the

product c I.

The parameter K is the diffusion constant, and it is chosen in order to preserve
edge strength at the object boundary and to reduce the noise contribution. The

⋅∇