Segmentation of 3D Tubular Tree Structures in Medical Images ...

28 Chapter 2. Extraction of Tubular Structures
The fitting term [70] is obtained as the mean flow through the circle and depends on the
radius r: T (x, r) = 1 ∫ 2Π
2rΠ α=0 〈V (α, r), D(α)〉 dα ∗ . V (α, r) represents the GVFs vector
at the circle point and D(α) defines a normal vector on the circle pointing towards its
center. For the circle fitting, the TDF response steadily increases with increasing radius
until the circle touches an actual edge/surface of the object. Increasing r further results in
a drop of T (x, r) as the magnitude of the vectors drop off. Thus, the fitting is performed
for increasing radii as long as the fitting term increases. The best fit provides the tubelikeliness
T (x).
In this way a tube-likelness measure and a radius estimate are obtained for each potential
centerline point.
2.3.3 Adaption to Varying Background Conditions
In the last sections we showed how the GVF field can be utilized for detection of tubular
structures using conventional central or offset medialness functions. The basic requirement
was an initial vector field F n (x) where at edge points the vectors have a larger magnitude
than the noise regions and where the vectors point toward the center of the tubular object.
Depending on the expected background conditions, such an initial vector field can be
obtained from the gray value image by computing image gradient vectors, using F = ∇I σ
for structures surrounded by darker tissue (e.g. angiography images) or F = −∇I σ for
structures surrounded by brighter tissue (e.g. airways), or from a gradient magnitude map
using F = −∇|∇I σ | in case of tubular structures that may be surrounded by arbitrary
backgrounds. The resulting vectors fields for these three cases are are shown in Fig. 2.5.
As can be seen from the results, in all cases the resulting vector fields show the characteristic
properties necessary for the tube detection approaches. In contrast to conventional
Gaussian scale space based TDFs, the gradient information can be obtained on a small
scale that only accounts for image noise. The ability to cope with structures surrounded
by arbitrary step edges is an issue that can not be addresses by conventional Gaussian
scale space methods, and represents an advantage of our approach.
The tubular structures found in medical datasets may show a large variability of contrast.
To account for this variability, a normalization of the initial gradient information
can be used, treating all gradients with a minimum magnitude as definite edges:
F n (x) = F (x)/|F (x)| ∗ (min(|F (x)|, F max )/F max ) where x = (x, y, z). This normalization
has the effect that the response for tubular structures primarily depends on the shape and
∗ The integral is approximated in practice by computing the sum over 32 discrete circle points.