of Diseased Livers: A 3D Refinement Approach

with A p ∈ {“obj”, “bkg”} can be used to represent the segmentation of P into object (“obj”) and background (“bkg”) voxels. Let N be the set of unordered neighboring pairs {p, q} in set P according to the used neighborhood relation. In our case, a 6-neighborhood relation is used to save memory. The cost of a given graph cut segmentation A is defined as 
$$ E(A) = B(A) + \lambda R(A) $$
where 
$$ R(A) = \sum\nolimits_{p^{\in P}} R_{p} (A_{p}) $$
takes region properties into account and 
$$ B(A) = \sum\nolimits_{p,q \in N} \;B_{p,q} \,\delta A_{p} \neq A_{q} $$
, with 
$$ \delta A_{p} \neq A_{q} $$
equals 1 if 
$$ A_{p} = A_{q} $$
and 0 if 
$$ A_{p} = A_{q} $$
, being boundary properties. The parameter λ with λ ≥ 0 allows to tradeoff the influence of both cost terms. Using the st cut algorithm, a partition A can be found which globally minimizes E(A).



Region term

The region term R(A) specifies the costs of assigning a voxel to a label based on its gray-value similarity to object and background regions. For this purpose, user defined seed regions are utilized. The region cost R p (·) for a given voxel p is defined for labels “obj” and “bkg” as negative log-likelihoods 
$$ R_{p} ("obj") = -ln(Pr(I_{p} |"obj")) $$
and 
$$ R_{p} ("bkg") = -ln(Pr(I_{p} |"bkg")) $$
with 
$$ Pr(I_{p} |"obj") = $$

$$ {e}^{-{\left({I}_p-{m}_{obj}\right)}^2/\left(2{\sigma}_{obj}^2\right)} $$
and 
$$ Pr({I_p} | "{\mathrm{bkg}}") = 1 - Pr(I_{p} | "{\mathrm{obj}}") $$
, respectively. From a object seed region placed inside the liver, the mean m obj and standard deviation σ obj are calculated. Clearly, in the above outlined approach, a simplification is made since liver gray-value appearance is usually not homogeneous. However, in combination with the other processing steps this simplification works quite well. Further, the specified object seeds are incorporated as hard constraints, and the boundary of the scene is used as background seeds.


Boundary term

The basic idea is to utilize a surfaceness measure as boundary term which is calculated in four steps:

1.

Gradient tensor calculation: First, to reduce the effect of unrelated structures on the gradient, the gray value range of the image is adapted:

Ĩ f = 
$$ \kappa \left({I}_{{}_{{}_f}}\right)=\left\{\begin{array}{lll} {v}_{low} & \mathrm{if}\;{I}_f<{t}_{low} \\ {v}_{high} & \mathrm{if}\;{I}_{{}_f}>{t}_{high} \\ {I}_{{}_f} & \mathrm{otherwise} \end{array}\right. $$
” src=”/wp-content/uploads/2016/09/A151032_1_En_22_Chapter_IEq13.gif”></SPAN> . Second, a gradient vector <SPAN id=IEq14 class=InlineEquation><IMG alt= is calculated for each voxel f on the with 
$$ \kappa $$
gray-value transformed data volume V by means of Gaussian derivatives with the kernel 
$$ {g}_{\sigma}=1/{\left(2\uppi {\sigma}^2\right)}^{\frac{3}{2}}{e}^{-\frac{x^2+{y}^2+{z}^2}{2{\sigma}^2}} $$
and standard deviation σ. The gradient tensor 
$$ \mathbf{S} = \nabla f \nabla f^{T} $$
is calculated for each voxel after gray-value transformation.

 

2.

Spatial non-linear filtering: To enhance weak edges and to reduce false responses, a spatial non-linear averaging of the gradient tensors is applied. The non-linear filter kernel consists of a Gaussian kernel which is modulated by the local gradient vector ∇f. Given a vector x that points from the center of the kernel to any neighboring voxel, the weight for this voxel is calculated as: 
$$ {h}_{\sigma^{\prime }, p}\left(\mathrm{x},\nabla f\right)=\left\{\begin{array}{lll} \frac{1}{N}{e}^{-\frac{r}{2{\sigma^{\prime}}^2}}{e}^{\frac{- \tan {\left(\phi \right)}^2}{2{\rho}^2}} & \mathrm{if}\;\phi \ne \frac{\uppi}{2}\\ 0 & \mathrm{if}\;\phi \ne \frac{\uppi}{2} \mbox{ and } r=0, \\ \frac{1}{N} & \mathrm{otherwise} \end{array}\right. $$
with 
$$ \mathrm{r} = \mathbf{x}^{T} \mathbf{x} $$
and 
$$ \phi = \frac{\pi}{2} \,- \mid $$
arccos
$$ (\nabla f^{T} \mathbf{x}/(|\nabla f ||\mathbf{x}|))| $$
. Parameter ρ determines the strength of orientedness, and σ′ determines the strength of punishment depending on the distance. N is a normalization factor that makes the kernel integrate to unity. The resulting structure tensor is denoted as W.

 

3.

Surfaceness measure calculation: Let 
$$ \mathbf{e}_{1_{\mathrm{W}(\mathrm{x})}} ,\; \mathbf{e}_{2_{\mathrm{W}(\mathrm{x})}} ,\; \mathbf{e}_{3_{\mathrm{W}(\mathrm{x})}} $$
be the eigenvectors and 
$$ \lambda_{1_{\mathrm{W}(\mathrm{x})}} \;\geq\; \lambda_{2_{\mathrm{W}(\mathrm{x})}} \;\geq\; \lambda_{3_{\mathrm{W}(\mathrm{x})}} $$
the corresponding eigenvalues of W(x) at position x. If x is located on a plane-like structure, we can observe that 
$$ \lambda_{1} \gg 0,\; \lambda_{2} \approx 0, \;\mathrm{and} \;\lambda_{3} \approx 0 $$
. Thus, we define the surfaceness measure as t(W(x)) = 
$$ \sqrt{\uplambda_{1_{\mathrm{W}\left(\mathrm{x}\right)}}-{\uplambda}_{2_{\mathrm{W}\left(\mathrm{x}\right)}}} $$
and the direction of the normal vector to the surface is given by 
$$ \mathbf{e}_{1_{\mathrm{W}(\mathrm{x})}} $$
.

 

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Sep 16, 2016 | Posted by in GENERAL RADIOLOGY | Comments Off on of Diseased Livers: A 3D Refinement Approach

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