Computer Aided Detection of Bone Metastases in the Thoracolumbar Spine - Spinal Imaging and Image Analysis - page 107

o /
o

t ¼ a *

Þdjr/jþb * * ð x ð x Þjð x Þjr/jþcr ð x Þr/ ð 19 Þ

here is the

curvature, and * is the speed function. We used the ITK implementation of the level
set algorithm in our system [ 35 ].
In the fast marching level set, a Gaussian gradient convolution is

ʱ , ʲ , ʳ are weighting parameters for each term, δ is the step size, ʺ

first applied to
the image as the speed function. Then a sigmoid function is applied to remap the
speed image. The sigmoid function is designed so that the propagation speed of the
front is low when it is close to high image gradients and moves rather fast in the low
gradient areas. The sigmoid function can be written as

1

þ

S ð I Þ ¼ ð Max Min Þ Min ð 20 Þ

ðÞ

I b

e 1 þ

a

where I is the intensity of the input pixel, Min and Max are the range for output,
a de nes the center. a and b control the

shape of sigmoid function and the function of the speed image. The determination
of a and b is based on the pixel statistics in the region [ 35 ]. The speed image for fast
marching level set can be written as

nes the width of the Sigmoid, and b de

* f ð x Þ ¼ S ð G ð I ð x ÞÞÞ ð 21 Þ

where I(x) is the image intensity, G(.) is the Gaussian gradient operator, and S(.) is
the sigmoid function.
The Laplacian level set de nes the speed term based on second derivative features

in the image. The speed term is calculated as the Laplacian of the image values. The
goal is to attract the evolving level set surface to local zero-crossings in the Laplacian
image. In our implementation, the image is

first convolved with a few iterations of
gradient anisotropic diffusion. Gradient anisotropic diffusion has the attribute to
reduce the noise and texture and meanwhile preserve the edge. After the anisotropic
diffusion and the Laplacian filter, the speed image for Laplacian level set is

* L ð x Þ ¼ r

2

ð A ð I ð x ÞÞÞ ð 22 Þ

2

where I is the image intensity, A(.) is the anisotropic diffusion and r is the

Laplacian operator.
Segmentation results from the watershed and graph cut algorithms are used as
the initialization for the level set algorithm. After the level set algorithm has run, a
smooth 3D surface is computed for each detection. The level set results for sclerotic
and lytic metastases are shown in Fig. 6 e, i respectively.