Transcript Slide 1
Model based approach for Improved 3-D
Segmentation of Aggregated-Nuclei in Confocal
Image Stacks
Nicolas Roussel1,James. A. Tyrell2, Qin Shen3, Sally Temple3, Badrinath Roysam1
1Department
Bio-Med
of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180
2Massachusetts General Hospital, Boston, MA 02114
3Albany Medical Center, Albany, NY 12208
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)
L2
S2 S3
S1
L3
Enviro-Civil
S4
Validating
TestBEDs
R2
Fundamental
L1 Science
R1
Introduction
Problem illustration
S5
R3
Results
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.
Model Description
Geometric descriptor
C
z
Image slice
Iso-surface 3D representation
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
We define as B the ellipsoid
Surface and R its volume
Model based segmentation Framework
R
Model Fitting
y
x
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.
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.
1
1
J ki X , dV
ko X , dV kb X dS
V
RC Vi
\ R C o
C
Gradient search which requires the derivative of L with respect
to the deformation of the ellipsoid
i2 o2 dX
J a
Fa
ki X , ko X ,
NdS
Vi Vo d
C
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:
Fe
S (x) 1, x C
If S (x) 1, x R
S (x) 1, x \( R C )
S (x; ) S u (T 1(x; ))
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:
S (x) x y z 1
2
2
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.
J e
dX
ki
dS .
C
d
Interaction model
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:
u
Je
Ja
2
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.
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” pseudoforces created by neighboring objects that will counteract the “fitting” Force
and prevent two objects from occupying the same space.
Lk ( X ) Max 1 S X 1
ki ki
Interraction force from Neighbooring function
dX
Fint Lk X
NdS
d
C
Seeds
Image Channel:
Image:
I ()
Fe
2D illustration of the appearance model
Fa
Amplitude
Indicator Function
Edge
detection
Channel:
S1e , S2e ,
J e
, Sme
J a
I e ()
Fint Lk X
C
dX
NdS
d
Interaction Model:
Foreground
Clustering
k X , 1 V I ( X ) 2
i
i
i
2
k
X
,
1
V
I
(
X
)
o
o
o
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:
Ib X G X * I X
2
Deformation Model
Validation
Interaction Model
Clustering
Sk1 , Sk2 ,
, Skn
Highly parallelizable
Segmentation scheme
J. Tyrrell, V. Mahadevan, R. Tong, B. Roysam, E. Brown, and R. Jain, \3-d
model-based complexity analysis of tumor microvasculature from in vivo
multiphoton confocal images," J. of Microvascular Research, vol. 70, pp.
165{178, 2005.
Validation
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:
Seeding
References
Model Fitting
V (x, ) o (o i ) 1S (x; )1
Background
Template
Modular object
representation
Segmentation Framework
As an appearance model we assume uniform
intensity IB and IF inside and outside the ellipsoid
boundaries respectively.
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.
Neighboring inside outside function
Appearance model
R(C)
Conclusion
Lk ( X ) Max 1 S X 1
k
ki i
S. Jehan-Besson, M. Barlaud, and G. Aubert, \Dreams: Deformable regions
driven by an eulerian accurate minimization method for image and video
segmentation: 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.