A snake for model-based segmentation of biomedical images1
Introduction
In the field of medicine images are a very important source of knowledge for the diagnosis and evaluation of pathological processes. A computational procedure that automatically identifies, isolates and also presents the 3D view of different organs can be very helpful to the clinician. On the other hand, a 3D finite element analysis of bony structures in areas such as bone remodeling, fracture prediction and prosthesis design has become an important tool for research (Keyak et al., 1990). However, the custom design of a model is a hard task that implies various steps: determination of the 3D bone geometry, generation of the elements and determination of its mechanical properties. A very important step is the geometrical reconstruction, since the success of later studies depends on its precision. This reconstruction has been approached using sequences of images corresponding to the transversal slices of the anatomical structure under study obtained by means of computational tomography (CT). A 3D model is generally constructed by stacking the 2D segmented contours. Normally, the contours are obtained using simple segmentation techniques which then need to be corrected (Kang et al., 1993; Steele et al., 1994). Our objective is to develop a completely automatic method for carrying out this segmentation, freeing the specialist from manual intervention, thus saving time and obtaining a greater objectivity in the segmentation.
Biomedical images are characterized by their low signal to noise ratio. This is due to the image formation methods and the process of acquiring and digitizing them for computational segmentation. Classical methods fail to provide satisfactory results and it is necessary to introduce a priori knowledge of the domain. For instance, Dhawan and Juvvadi (1990)proposed an automatic knowledge based system for the analysis of three dimensional anatomical images. They employed specific domain knowledge represented by means of production rules. However, this system is aimed more at labeling of the structures than at their correct determination. If the final objective is the analysis of the structure using finite elements, the precise localization of the structure boundaries is necessary. Ayache et al. (1990)approached the problem of segmenting sets of 3D images by applying deformable models and taking into account the previously extracted contours of the adjacent slice. Deformable models have shown their effectiveness in the segmentation of anatomical structures due to their ability of exploiting a mixed bottom-up (image data) and top-down (a priori knowledge about the location, shape and size of the structures) control of the segmentation. An exhaustive summary of the application of deformable models to medical images can be found in the work by McInerney and Terzopoulos (1996).
Our aim is the accurate segmentation of the proximal tibia for the construction of a solid model that is as close as possible to the real bone. This model will later be used for structural analysis using finite elements. A drawing identifying tibia, fibula, cortical bone and trabecular bone can be seen in Fig. 1. Fig. 2 shows two CT images of tibia and fibula at higher definition. The width and the high density of the cortical bone and also the clear separation between tibia and fibula, Fig. 1(b) and Fig. 2(b), make the segmentation of the tibia in the distal part fairly easy. As we approach the proximal part of the tibia, the thickness of the cortical bone decreases and the distance from the fibula is reduced. When knee injuries are considered, problems of bone loss, cortical bone narrowness, malformations, etc., can be found. This translates into: poorly defined external contours of the bones, little separation between the internal and external contour of the cortical bone and the proximity between contours of different structures (tibia and fibula, tibia and femur). Fig. 1(a) and Fig. 2(a) illustrate two cases with several of such problems. These considerations imply several problems in the application of classical snakes. Some of these are: (1) existence of noise and/or textures, in the background or inside the structures (trabecular structure of the bone), (2) the structures whose contours we want to determine may be close to other structures, and (3) the initial approximation is located inside the structure which presents double contours (the cortical bone). In these situations the contour can be trapped by spurious edge points. This makes the final result very sensitive to the initial conditions. For a correct determination of the contour, the adjustment to the external part of the cortical bone is of paramount importance. When sequences of slices are being processed, the contour in the previous slice can be used as the initial deformation state. Serious approximation problems arise when injuries or deformations appear. In this case the initial contour can be located close to the internal contour of the cortical bone, and become trapped by it.
In this work, new terms for internal and external energy are introduced in order to solve such problems. A simplified method for the minimization of the energy function, whose performance is based on the proximity in shape and scale between the initial deformable model and the real contour is also introduced. In Section 2, the background theory of classical snakes is introduced; in 3 New internal energy term, 4 Modification of the snake's external potential, we introduce the new energy terms. In Section 5, the minimization process is described. Finally, the results presented in Section 6prove the validity of our approach.
Section snippets
Classical snake model
A snake, introduced by Kass et al. (1988), is an elastic curve, that from an initial state tries to adjust to the most significant features of the scene. It is deformed due to external forces that attract it towards salient features of the image, and internal forces which try to preserve the condition of smoothness in the shape of the curve. A final solution is given by the minimum total energy of the snake, which is the result of the equationwhere Eint and E
New internal energy term
In order to numerically compute a minimal energy solution it is necessary to discretize the derivatives in the expression of the internal energy. This leads to the expressionwhere the vi's are the vertices of the discrete version of the snake. In the absence of important external forces, the movements of the snake try to minimize its internal energy. Taking the partial derivative of Eq. (4)and equating it to zero, the following expression is
Modification of the snake's external potential
One of the problems in deformable contour based methods that must be addressed is that of a bad initialization that may cause the snake to fall into a local minimum. When trying to determine the external contour of a bone, these problems occur if the initial model is positioned inside the cortical bone, or when this model is located in the external part, but closer to the contour of another structure.
In an attempt to mitigate such problems of bad initialization, we propose a new external energy
A simplified method for the minimization of the energy
The solution to the problem of detecting the contour is found in the minimization of the snakes' total energy function. In order to solve variational problems in vision, Amini et al. (1990)proposed the use of dynamic programming owing to its characteristics of global optimality of the solution, numerical stability and the possibility of imposing strong constraints on the behaviour of the structure. However, dynamic programming methods are slow and require a large amount of memory. Geiger et al.
Results
In order to illustrate the behaviour of the system, several sequences of contiguous transverse CT scan proximal tibia slices were processed. These were obtained in vivo from different patients. The final objective was the 3D reconstruction of the tibia geometry. The first step of the analysis concerns the identification of the most significant bone structure (i.e., the tibia) in the slice in which it is best defined.
The effect of the new internal energy term can be observed in Fig. 6. Fig. 6(a)
Conclusions
Segmenting structures in medical images and reconstructing a compact geometric representation of these structures is difficult due to well known reasons. Deformable models seem to be very accurate for such images because of their ability to manage image data and a priori knowledge.
Our proposed snake implements new definitions of internal and external energy in order to solve problems in the classical formulation which affect the segmentation of images obtained from CT bone scans.
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This work was supported by Xunta de Galicia under Grant XUGA20603B96.