Model based approach for Improved 3-D Segmentation of Aggregated-Nuclei in Confocal

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Introduction

This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821)

Nicolas Roussel1,James. A. Tyrell2, Qin Shen3, Sally Temple3, Badrinath Roysam1

1Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 121802Massachusetts General Hospital, Boston, MA 02114

3Albany Medical Center, Albany, NY 12208

We present a study on a new modeling based approach to cell segmentation and its application to cell detection in 3D space. As a model, we use an ellipsoid to which three types of deformation can be applied, namely translation, rotation and scaling. The geometric and intensity parameters of the models are optimized independently at each iteration step. The purpose of this approach is to derive a virtual “fitting” Pseudo-Force that drives the ellipsoid to the apparent boundaries of the cell.

In an environment where a number of cells can coexist and partially overlap, it is also necessary to model the inter-object constraints that limit the extent to which overlap is possible. We propose to implement this constraint as an “interaction” force that will counteract the “fitting” Force and prevent two objects from occupying the same space.

Appearance model

Geometric descriptor

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Background

AmplitudeIndicator Function

Foreground

Gradient search which requires the derivative of L with respect to the deformation of the ellipsoid

R(C)

In the case where multiple object are to occupy the same field of view and possibly demonstrate partial overlap pattern, it can be useful to incorporate a non overlap constraint that limits the extent to which overlap is possible. To that end, we are going to introduce an additional set of “interaction” pseudo-forces created by neighboring objects that will counteract the “fitting” Force and prevent two objects from occupying the same space.

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We define as B the ellipsoid Surface and R its volume

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R

As an appearance model we assume uniform intensity IB and IF inside and outside the ellipsoid boundaries respectively.

2D illustration of the appearance model

Model Description

Image slice Iso-surface 3D representation

Neighboring inside outside function

Validation

Model Fitting

Clustering

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Edge detection Channel:

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Our modeling approach is based on a common template is used for all objects in the field of view. In 3D space, this geometric template is defined over a 3D mesh, each facet defined by its center Xe() and normal Ne() . Essentially, every object in the field of view is considered a deformed version of this template. Actual object structure if thus captured by the deformation T(X,β). This formalism allows for the use of a wide range of shape families.

A parameterized object surface can be implicitly formulated in term of an inside/outside function that allows us to define the volume occupied by a given shape:

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If S R

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Where R is the region of space occupied by the object, C its surface and the domain. For a given template Su(X) and deformation parameter T(X,β), the object is defined by:

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In this paper, our focus is on the segmentation of brain cells. On a first level of approximation, those cells can be considered to be of ellipsoidal shape. As a result and under the formalism described above, they can be modeled as a unit sphere Su(X) defined as:

2 2 2( ) 1uS x y z x

to which the transformation T(x, β)=R(Φ)D(σ)x+μ is applied. The Matrix D(σ) is a 3x3 matrix with positive scale parameters on the diagonal, the rotation R(Φ) matrix and a translation factor.

Pixel appearance within a geometric model can be defined in term of a homogeneity estimator ki() for the pixels within R and ko() within Ω\(RUC) . For instance, in the case where we assume uniform intensity distribution within and outside the object model, the homogeneity estimator could be defined as:

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Edge homogeneity estimator kb() can be defined for the pixel on the object boundaries from a separate edge detection channel Ib(). In snake literature, commonly used edge channel called external energy, include:

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Fitting the region based active contour model as defined in the previous section can be seen as an optimization process whereas the quantity J(β) is to be minimized with respect the geometric parameters β. For the sake of clarity, the objective function can be seen as the weighted sum from a level based factor Ja(β) and an edge based factor Je(β) . The weighting coefficients can be used to balance the appearance and edge energy terms.

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Model based segmentation Framework

Problem illustration

Interaction model

Model Fitting

Segmentation Framework

Illustration of our fitting framework: Ellipsoid model for each cell are overlaid on the image projection (left) and iso-surface (right). Using this modeling approach, it is feasible to detect specific objects based on their assumed shape and appearance model.

Illustration of the segmentation problem. In the field of view, a number of ellipsoidally shaped cells coexist and partially overlap. The presence of cell aggregates combined with the inconsistency of their appearance model makes for a challenging segmentation task

References

J. Tyrrell, V. Mahadevan, R. Tong, B. Roysam, E. Brown, and R. Jain, \3-dmodel-based complexity analysis of tumor microvasculature from in vivomultiphoton confocal images," J. of Microvascular Research, vol. 70, pp.165{178, 2005.

S. Jehan-Besson, M. Barlaud, and G. Aubert, \Dreams: Deformable regionsdriven by an eulerian accurate minimization method for image and videosegmentation: Application to face detection in color video sequences."[Online]. Available: citeseer.ist.psu.edu/558842.html

Harris, J. W. and Stocker, H. "Ellipsoid." §4.10.1 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 111, 1998.

Conclusion

Results

Using a model based approach to 3D cell segmentation has proven successful in a number of test cases. Overall the results are encouraging and suggest that such an approach would prove useful in dealing with situation involving dense cell population with irregular appearance.

One of the main advantage of this algorithm is its modularity as it can be easily generalized using alternative template and interaction models in application involving the segmentation of different anatomical structure.

Template

Deformation Model

Interaction Model

Modular object representation

Seeding

Validation

Clustering

Highly parallelizable Segmentation scheme