Biomedical Engineering Reference
In-Depth Information
8.3.1 Initialization
Because level set models move using gradient descent, they seek
local solutions
,
and therefore the results are strongly dependent on the initialization, i.e., the
starting position of the surface. Thus, one controls the nature of the solution
by specifying an initial model from which the surface deformation process pro-
ceeds. We have implemented both computational (i.e. “semi-automated”) and
manual/interactive initialization schemes that may be combined to produce rea-
sonable initial estimates directly from the input data.
Linear filtering
: We can filter the input data with a low-pass filter (e.g. Gaussian
kernel) to blur the data and thereby reduce noise. This tends to distort
shapes, but the initialization need only be approximate.
Voxel classification
: We can classify pixels based on the filtered values of the
input data. For grayscale images, such as those used in this chapter, the
classification is equivalent to high and low thresholding operations. These
operations are usually accurate to only voxel resolution (see [12] for alter-
natives), but the deformation process will achieve subvoxel results.
Topological/logical operations
: This is the set of basic voxel operations that
takes into account position and connectivity. It includes unions or intersec-
tions of voxel sets to create better initializations. These logical operations
can also incorporate user-defined primitives. Topological operations consist
of connected-component analyses (e.g. flood fill) to remove small pieces or
holes from objects.
Morphological filtering
: This includes binary and grayscale morphological op-
erators on the initial voxel set. For the results in the chapter we imple-
ment openings and closings using
morphological propagators
[38,39] imple-
mented with level set surface models. This involves defining offset surfaces
of
φ
by expanding/contracting a surface according to the following PDE,
∂φ
∂
t
= ±|∇
φ
|
,
(8.5)
up to a certain time
t
. The value of
t
controls the offset distance from the
original surface of
φ
(
t
=
0). A dilation of size
α
,
D
α
, corresponds to the
solution of Eq. (8.5) at
t
=
α
using the positive sign, and likewise erosion,
E
α
,
uses the negative sign. One can now define a morphological opening operator
