Description of the Shape Constrained Deformable Model
     In this method, we use an initial model that is placed approximately to the position of the organ we want to reconstruct. This model is modified using a generic model and the information from the medical images of the patient. The segmentation process is formulated as the minimization of a cost functions associated to each point of the shape:

• A shape memory force that keeps the shape of the generic model.
• Image interest force that tends towards vertices to move into high edge density.

Initial shape and generic shape
     As described earlier, our method uses two shapes: the initial shape and the generic shape. The initial shape is the shape at the first iteration in the computing process. We can use an initial shape which is different from the generic shape. The only restriction is that initial shape and generic shape must have the same topology. Most of cases, initial shape is identical to generic shape. Sometimes for specific images as CT images, it's very helpful to define an initial shape. On these images, bones appear with two contours very close to each other. During the reconstruction process, some vertices move to the internal contour and some other to the external contour. In that case, we define an initial shape big enough to contain the entire organ to be reconstructed. In this way, vertices only move to the external contour. A generic shape is used to compute the shape memory force which will be described later.

The deformable model
     Our model is represented by points that define the object contours. The surface of the object is defined by a triangular mesh where each vertex knows its neighboring vertices. Thus, reconstruction algorithms such as the labeling tool described earlier can directly be used in a first step to build the generic model. The vertices of the mesh triangles correspond with voxels in the volume being segmented.

Cost functions
     These cost functions can be defined as a set of forces applied to each vertex. As described earlier, there are two types of force, shape memory force and image interest force. All these forces are explained in next two sections.

Shape memory force
     The elastic properties of an object can be defined as its ability to come back in its original shape once deformed. It corresponds to a force, called shape memory force that is deduced from the generic shape and the current shape. It is different from the elastic model that uses spring to link each neighboring point and it offers better convergence. A triangular mesh defines the surface of the object and each vertex knows its neighborhoods. For each vertex, we use an approximate coordinate system calculated with neighboring vertices. The computing of the shape memory force is made into two steps described as follow:

• During initialization, we calculate for each vertex the local coordinate system with the generic model. In this local coordinate system, we compute the coordinates of the vertex. This is the position of the local null deformation of the vertex relative to its neighbors.
• For each iteration, we calculate for each vertex the local coordinate system relative to its neighbors. This local coordinate system changes at each iteration because the shape is modified during the fitting process. In this local coordinate system, we know the current position of the vertex and position of the local null deformation. This position has been computed in the initialization step. Between these two positions, we apply a force that will move the current position to the position of the local null deformation. The local shape tends towards to be similar of the local shape of the generic model.

• G is the isobarycenter of the neighbors of point P.
• Z is the unit vector normal to the approximate neighbors plane. X and Y are two vectors normal to Z. (X,Y,Z) is an orthonormal coordinate system.
• Pcurrent is the current position of the vertex. This position is defined by three values xc, yc and zc.
• Pideal is the position of the local null deformation ( Figure 2).

     The length of the vector Pideal is proportional to the size of the generic model. It means that if we apply a global scaling on the generic model, the direction of Pideal for all vertices will be invariant and the variation of the length will be equal to the variation of the model size. In that way, a coefficient can be added to Pideal in the previous equation. With this coefficient, we can add a force that inflates or deflates the deformable model during the fitting process. This force can be used on a closed contour to force the contour to expand in the absence of external influences.

Image interest force
     External forces attract the deformable contour to interesting features, such as object boundaries, in an image. Any force expression that accomplishes this attraction can be considered for use. In our case, the computation of external forces is defined in two steps:

• At the initialization, we apply a 3D Sobel operator on the entire voxmap. The gradient map is then stored in a cache. Using a memory cache is a good strategy because we intensively use the value of the gradient for each vertex in the next process. The input voxmap can be represented as gray level data as well as color.
• During the process, we find the closest position of the gradient value which is greater or equal to the threshold parameter. This position is considered as ideal position. The threshold parameter is controlled by the user. In order to limit time, the research for the ideal position is restricted along the normal of the surface around the vertex. This normal of the surface has already been computed during the computation of the internal force. This normal is the Z vector in the local coordinate system described earlier. A parameter defines the size of the neighbor. This parameter can be edited by the user. We put a force to move the vertex to this ideal position.

• Pi1Pi2 is the segment on which the search of the ideal position is done.
• Ni is the normal to the surface
• Si is the vertex and Mi is the position of the ideal position ( Figure 3).

The parameter K is the coefficient of the external energy. This parameter can be modified by the user. Since we are considering noisy data, we may perform a smoothing of external forces by taking the average over a neighborhood. The expression of the smoothed force applied on Pi is:

Nt(Pi) is the neighbor vertices of the vertex Pi. It can be defined as the set of points connected to Pi by a topological path of length lesser than or equal to t. The parameter t can be modified by the user. If this parameter is equal to zero, the smoothing is not used in the external force computation.

Energy minimization
     For each vertex, we can compute the total energy which is the sum of internal and external energy. We then use minimization of the total energy using variable metric methods in multidimensions [15]. The minimization process is made in several iterations. Between each iteration, the user can intervene to modify the deformable model.

 Implementation
     The software which implements this method has been developed on SGI machines. The user interface is composed of a main window and a dialog to set the parameters of the fitting process. The main window shows the voxmap with the deformable model. The user can modify the model in two ways. He can make some global transformation as translation, rotation or scaling. He can also modify a part of the model by moving a set of vertices at the surface. This modification can be done on the initial shape but also during the fitting process. The following figure shows a snap shot of the user interface ( Figure 4).