SuperEllipseTrace3D

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the medial axis of each vessel in the vasculature. Another key advantage when using the superellipsoid model is that its local nature allows us to adapt vessel intensity estimates across different regions of the vasculature.
 
the medial axis of each vessel in the vasculature. Another key advantage when using the superellipsoid model is that its local nature allows us to adapt vessel intensity estimates across different regions of the vasculature.
  
 +
'''Vascular local shape modelling using superellipsoid '''
  
 
[[Image:se_model.png |thumb|800px|'''Figure1:''' Shows the boundary of a superellispoid while the shape parameter <math> \epsilon_{1} </math> is varied between 1.0 and 0.25 and  <math> \epsilon_{2} </math> is set to 1.|center]]
 
[[Image:se_model.png |thumb|800px|'''Figure1:''' Shows the boundary of a superellispoid while the shape parameter <math> \epsilon_{1} </math> is varied between 1.0 and 0.25 and  <math> \epsilon_{2} </math> is set to 1.|center]]
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in early computer vision literature [2]. Superellipsoids are expressed implicitly as:
 
in early computer vision literature [2]. Superellipsoids are expressed implicitly as:
  
 +
[[Image:seWorldGeometry.png |thumb|400px|'''Figure2:''' |right]]
 
<math>
 
<math>
 
\left(\left|x\right|^{\frac{2}{\epsilon_{2}}}+\left|y\right|^{\frac{2}{\epsilon_{2}}}\right)^{\frac{\epsilon_{2}}{\epsilon_{1}}}+\left|z\right|^{\frac{2}{\epsilon_{1}}}=1,
 
\left(\left|x\right|^{\frac{2}{\epsilon_{2}}}+\left|y\right|^{\frac{2}{\epsilon_{2}}}\right)^{\frac{\epsilon_{2}}{\epsilon_{1}}}+\left|z\right|^{\frac{2}{\epsilon_{1}}}=1,
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the shape parameters, first note that with <math> \epsilon_{1}=\epsilon_{2}=1 </math> equation 1 is simply an ellipsoid. With <math>\epsilon_{1}\leq </math> and <math>\epsilon_{2}= 1</math> the superellipsoid becomes a cylindroid, i.e., the implicit surface bounds a convex region, where for constant <math>z</math>, the 2-D level curves are
 
the shape parameters, first note that with <math> \epsilon_{1}=\epsilon_{2}=1 </math> equation 1 is simply an ellipsoid. With <math>\epsilon_{1}\leq </math> and <math>\epsilon_{2}= 1</math> the superellipsoid becomes a cylindroid, i.e., the implicit surface bounds a convex region, where for constant <math>z</math>, the 2-D level curves are
 
elliptical. Setting <math>\epsilon_{1},\epsilon_{2}</math> to other values produces a variety of shapes including cuboids and pinched cuboids [2]. Since the goal in this work is to find an approximation to an elliptical cylinder, we will restrict our focus to cylindroidal forms. This means that <math>\epsilon_{2}</math> in equation is set to unity.
 
elliptical. Setting <math>\epsilon_{1},\epsilon_{2}</math> to other values produces a variety of shapes including cuboids and pinched cuboids [2]. Since the goal in this work is to find an approximation to an elliptical cylinder, we will restrict our focus to cylindroidal forms. This means that <math>\epsilon_{2}</math> in equation is set to unity.
 +
  
 
Starting with a set of points in the set
 
Starting with a set of points in the set
Line 51: Line 54:
  
  
−
'''Vessel Model Formulation'''
+
'''Robust Vascular Intensity model'''
 +
A class of robust estimators known as ''M-estimates'' can be defined as any solution <math>T_{n}</math> to:
  
 +
<math>
 +
\sum_{i = 1}^{n}\psi(x_{i};T_{n})=0.
 +
</math>
 +
 +
 +
The notation <math>T_{n}</math> refers to the dependency of the
 +
estimate on a finite i.i.d. sample, <math>\{X_{1},X_{2},...,X_{n}\}</math>,
 +
i.e., <math>T_{n}=T_{n}(X_{1},X_{2},...,X_{n})</math>.
 +
 +
When estimating the background, the main challenge is to deal with
 +
the fact that the image data invariably contains vessel,
 +
background clutter, and uncorrelated noise. Taken as a whole, the
 +
resulting image data is a mixture of various components. Finding
 +
the single component corresponding to the background region can be
 +
difficult.
 +
 +
This is especially true since the actual background voxels may not
 +
be a majority population in the mixture. This often happens when a
 +
small vessel is adjacent to a very large one. Hence, the median
 +
may no longer be a good choice. The real problem in this case is
 +
that when estimating the background one must deal with
 +
pseudo-outliers, i.e., data that appears as an outlier to
 +
the current process but might be an inlier to another process. For
 +
example an adjacent vessel appears as an outlier when estimating
 +
the local background intensity. However, the data is actually an
 +
inlier to the adjacent vessel.
 +
 +
To deal with these outliers, we propose the \textit{skipped
 +
median} which simply restricts the support of the signum function,
 +
thereby excluding very large residuals from the intensity
 +
estimate. The corresponding <math>psi</math>-function is given by:
 +
 +
<math>
 +
\psi_{med(s)}=\text{sgn}(r) \times
 +
1_{[-s,s]}(r)=\left\{\begin{array}{r}\text{sgn}(r),|r|\leq
 +
s\\0,|r|>s\end{array}\right..
 +
</math>
 +
 +
 +
''' Generalized likelihood ratio test'''
 
When estimating the superellipsoid model parameters we will assume
 
When estimating the superellipsoid model parameters we will assume
 
that a vessel segment consists of two piece-wise constant regions
 
that a vessel segment consists of two piece-wise constant regions
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boundary using a gradient search. The following two subsections will detail each step.
 
boundary using a gradient search. The following two subsections will detail each step.
  
 +
'''Traversing Vessels Using the
 +
Superellipsoid '''
 +
 +
[[Image:seTraversal.png |thumb|400px|'''Figure3: Illustrating vessel traversal. Figure shows a sequence of
 +
superellipsoidal fits denoted <math>k</math>, <math>k+1</math>, and <math>k+2</math> superimposed
 +
on a vessel. Note that the traversal is directed using the local
 +
pose to determine the next step in the traversal.''' |left]]
  
−
[[Image:seWorldGeometry.png |thumb|300px|'''Figure2:''' |center]]
 
−
[[Image:seTraversal.png |thumb|300px|'''Figure3:''' |center]]
 
 
[[Image:SeTraceExample.png |thumb|1200px|'''Figure4:''' |center]]
 
[[Image:SeTraceExample.png |thumb|1200px|'''Figure4:''' |center]]

Revision as of 15:49, 11 May 2009

This page presents automated methods for robust modeling and analysis of 3-D vascular images using a \textit{cylindroidal superellipsoid} to model localized segments of vasculature. The proposed vessel model has an explicit, low-order parameterization, allowing for joint estimation of boundary and centerline information, thereby approximating the medial axis of tubular structures. Further, this explicit parameterization provides a geometric framework for traversing vessels in a directed manner. Topological information like branch point location and connectivity can also be detected as a side effect. An M-estimators provide robust region-based statistics that are used to drive the superellipsoid toward a vessel boundary. The proposed methodology behaves quite well across scale-space, shows a high degree of insensitivity to adjacent structures and implicitly handles branching.


Contents


Local vascular modelling using Superellipsoid

The superellipsoid provides the basis for a simple shape model that describes localized vessel segments. It has an explicit low-order parameterization relating directly to the local pose of a vessel segment. Using this geometric information, it is straightforward to traverse entire vascular networks.Also, since the superellipsoid model furnishes joint estimates of the boundary and centerpoint of a vessel, the resulting traversal approximates the medial axis of each vessel in the vasculature. Another key advantage when using the superellipsoid model is that its local nature allows us to adapt vessel intensity estimates across different regions of the vasculature.

Also, since the superellipsoid model furnishes joint estimates of the boundary and centerpoint of a vessel, the resulting traversal approximates the medial axis of each vessel in the vasculature. Another key advantage when using the superellipsoid model is that its local nature allows us to adapt vessel intensity estimates across different regions of the vasculature.

Vascular local shape modelling using superellipsoid

Figure1: Shows the boundary of a superellispoid while the shape parameter ε1 is varied between 1.0 and 0.25 and ε2 is set to 1.


Superellipsoids are geometric modeling primitives first introduced in early computer vision literature [2]. Superellipsoids are expressed implicitly as:

Figure2:

\left(\left|x\right|^{\frac{2}{\epsilon_{2}}}+\left|y\right|^{\frac{2}{\epsilon_{2}}}\right)^{\frac{\epsilon_{2}}{\epsilon_{1}}}+\left|z\right|^{\frac{2}{\epsilon_{1}}}=1,

where the parameters \epsilon=\left(\epsilon_{1},\epsilon_{2}\right) control the shape of the superellipsoid. To understand the role of the shape parameters, first note that with ε1 = ε2 = 1 equation 1 is simply an ellipsoid. With \epsilon_{1}\leq and ε2 = 1 the superellipsoid becomes a cylindroid, i.e., the implicit surface bounds a convex region, where for constant z, the 2-D level curves are elliptical. Setting ε1,ε2 to other values produces a variety of shapes including cuboids and pinched cuboids [2]. Since the goal in this work is to find an approximation to an elliptical cylinder, we will restrict our focus to cylindroidal forms. This means that ε2 in equation is set to unity.


Starting with a set of points in the set X=\left\{\mathbf{x}|S(\mathbf{x};\epsilon)=1\right\} (ε is fixed), a transformation given by T(\mathbf{x})=\mathbf{R}({\phi})\mathbf{D}({\sigma})\mathbf{x}+{\mu} can be performed. This transformation preserves orthogonality of the local coordinate frame while allowing for anisotropic scaling. The action of T(\mathbf{x}) is to first scale \mathbf{x} by \mathbf{D}({\sigma}), which is a 3\times 3 matrix with scale parameters {\sigma}=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right) on the diagonal. Then, a rotation is performed via the 3\times 3 matrix \mathbf{R}({\phi}) with three degrees of freedom that is parameterized by φ. Finally, a shift via {\mu}=\left(\mu_{x},\mu_{y},\mu_{z}\right)^{T} is performed. This transformation leads to a new superellipsoid defined by the set having 10 degrees of freedom in 3D, X'=\left\{\mathbf{x}|S(T^{-1}(\mathbf{x});\epsilon)=1\right\}, where, T^{-1}(\mathbf{x})=\mathbf{D}({\sigma})^{-1}\mathbf{R}({\phi})^{T}(\mathbf{x}-{\mu}).


Robust Vascular Intensity model A class of robust estimators known as M-estimates can be defined as any solution Tn to:

\sum_{i = 1}^{n}\psi(x_{i};T_{n})=0.


The notation Tn refers to the dependency of the estimate on a finite i.i.d. sample, {X1,X2,...,Xn}, i.e., Tn = Tn(X1,X2,...,Xn).

When estimating the background, the main challenge is to deal with the fact that the image data invariably contains vessel, background clutter, and uncorrelated noise. Taken as a whole, the resulting image data is a mixture of various components. Finding the single component corresponding to the background region can be difficult.

This is especially true since the actual background voxels may not be a majority population in the mixture. This often happens when a small vessel is adjacent to a very large one. Hence, the median may no longer be a good choice. The real problem in this case is that when estimating the background one must deal with pseudo-outliers, i.e., data that appears as an outlier to the current process but might be an inlier to another process. For example an adjacent vessel appears as an outlier when estimating the local background intensity. However, the data is actually an inlier to the adjacent vessel.

To deal with these outliers, we propose the \textit{skipped median} which simply restricts the support of the signum function, thereby excluding very large residuals from the intensity estimate. The corresponding psi-function is given by:

\psi_{med(s)}=\text{sgn}(r) \times 1_{[-s,s]}(r)=\left\{\begin{array}{r}\text{sgn}(r),|r|\leq s\\0,|r|>s\end{array}\right..


Generalized likelihood ratio test When estimating the superellipsoid model parameters we will assume that a vessel segment consists of two piece-wise constant regions with intensities denoted IF for foreground and IB for background and that the vessel boundary separating these regions is well-modeled by a superellipsoid. The local vessel intensities are modeled using a homogeneous grayscale appearance model as

V(\mathbf{x};{\theta})= I_{B}-(I_{B}-I_{F})\times 1_{\left[S(\mathbf{x},{\beta})<1\right]},

where {\theta}=\left(I_{F},I_{B},{\beta}\right) is the full parameter set of the superellipsoid vessel model. This includes the intensity parameters \left(I_{F},I_{B}\right) that describe the foreground and background intensities respectively, as well as the underlying parameters of the superellipsoid shape model, i.e., S(\mathbf{x},{\beta}). The notation 1_{\left[\cdot\right]} denotes an indicator function that evaluates to one on the set specified in the subscript and is zero elsewhere.


The strategy for maximizing equation is to alternate between estimating the intensity parameters and then updating the boundary using a gradient search. The following two subsections will detail each step.

Traversing Vessels Using the Superellipsoid

Figure3: Illustrating vessel traversal. Figure shows a sequence of superellipsoidal fits denoted k, k + 1, and k + 2 superimposed on a vessel. Note that the traversal is directed using the local pose to determine the next step in the traversal.
Figure4:
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