In the previous sections we have surveyed the considerable and rapidly expanding body of work on deformable models in medical image analysis. The survey has revealed several issues that are relevant to the continued development of the deformable model approach. This section summarizes the key issues and indicates some promising research directions.
Interactive (semiautomatic) algorithms and fully automatic algorithms represent two alternative approaches to computerized medical image analysis. Certainly automatic interpretation of medical images is a desirable, albeit very difficult, long-term goal, since it can potentially increase the speed, accuracy, consistency, and reproducibility of the analysis. However, the interactive or semiautomatic methodology is likely to remain dominant in practice for some time to come, especially in applications where erroneous interpretations are unacceptable. Consequently, the most immediately successful deformable model based techniques will likely be those that drastically decrease the labor intensiveness of medical image processing tasks through partial automation and significantly increase their reproducibility, while still allowing for interactive guidance or editing by the medical expert. Although fully automatic techniques based on deformable models will likely not reach their full potential for some time to come, they can be of immediate value in specific application domains such as the segmentation of healthy tissue surrounding a pathology for enhanced visualization.
Ideally a deformable model should be capable of representing a broad range of shapes and be useful in a wide array of medical applications. Generality is the basis of deformable model formulations with local shape parameters such as snakes. Alternatively, highly specific, "hand-crafted" or constrained deformable models appear to be useful in applications such as tracking the nonrigid motion of the heart (Section 3.5), automatically matching and labeling structures in the brain from 3D MR images (Section 3.4), or segmenting very noisy images such as echocardiograms. Certainly attempts to completely automate the processing of medical images would require a high degree of application and model specificity. A promising direction for future study appears to be techniques for learning "tailored" models from simple general-purpose models. The work of Cootes et al.  may be viewed as an example of such a strategy.
4.3 Compactness vs Geometric Coverage vs Topological Flexibility
A geometric model of shape may be evaluated based on the parsimony of its formulation, its representational power, and its topological flexibility. Generally, parameterized models offer the greatest parsimony, free-form (spline) models feature the broadest coverage, and implicit models have the greatest topological flexibility. Deformable models have been developed based on each of these geometric classes. Increasingly, researchers are turning to the development of hybrid deformable models that combine complementary features. For objects with a simple, fixed topology and without significant protrusions, parameterized models coupled with local (spline) and/or global deformations schemes appear to provide a good compactness-descriptiveness trade-off [24,112,136,148]. On the other hand, the segmentation and modeling of complex, multipart objects such as arterial or bronchial "tree" structures, or topologically complex structures such as vertebrae, is a difficult task with these types of models. Polygon-based or particle-based deformable modeling schemes seem promising in segmenting and reconstructing such structures. Polygon-based models may be compacted by removing and "retiling" [40,57,144] polygons in regions of low shape variation, or by replacing a region of polygons with a single, high-order finite element or spline patch . A possible research direction is to develop alternative models that blend or combine descriptive primitive elements (rather than simple particles), such as flexible cylinders, into a global structure.
The earliest deformable models were curves and surfaces. Anatomic structures in the human body, however, are either solid or thick-walled. To support the expanding role of medical images into tasks such as surgical planning and simulation, and the functional modeling of structures such as bones, muscles, skin, or arterial blood flow, may require volumetric or solid deformable models rather than surface models. For example, the planning of facial reconstructive surgery requires the extraction and reconstruction of the skin, muscles, and bones from 3D images using accurate solid models. It also requires the ability to simulate the movement and interactions of these structures in response to forces, the ability to move, cut, and fuse pieces of the model in a realistic fashion, and the ability to stimulate the simulated muscles of the model to predict the effect of the surgery. Several researchers have begun to explore the use of volumetric or solid deformable models of the human face and head for computer graphics applications [44, 79] and for medical applications [39,54,55,74, 114,151,152], particularly reconstructive surgery simulation, and there is much room for further research. Researchers have also begun to use volumetric deformable models to more accurately track and analyze LV motion [31,42,110,158].
Ideally it should be possible to measure and control the accuracy of a deformable model. The most common accuracy control mechanisms are the global or local subdivision of model basis functions , or the repositioning of model points to increase their density in regions of the data exhibiting rapid shape variations . Other mechanisms that warrant further research are the local control and adaptation of model continuity, parameter evolution (including the rate and scheduling of the evolution), and the automation of all accuracy control mechanisms. The parametric formulation of a deformable model should not only yield an accurate description of the object; it should also provide quantitative information about the object in an intuitive, convenient form. That is, the model parameters should be useful for operations such as measuring, matching, modification, rendering, and higher-level analysis or geometric reasoning. This "parameter descriptiveness'' criterion may be achieved in a postprocessing step by adapting or optimizing the parameterization to more efficiently or more descriptively match the data. However, it is preferable to incorporate the descriptive parameterization directly into the model formulation. An example of this strategy is the deformable model of Park et al. .
Ideally, a deformable model should be insensitive to initial conditions and noisy data. Deformable models are able to exploit multiple image attributes and high level or global information to increase the robustness of shape recovery. For example, many snakes models now incorporate region-based image features as well as the traditional edge-based features (Section 3.1). Strategies worthy of further research include the incorporation of shape constraints into the deformable model that are derived from low-level image processing operations such as thinning, medial axis transforms , or mathematical morphology. A classical approach to improve the robustness of model fitting is the use of multiscale image preprocessing techniques [73,138], perhaps coupled with a multiresolution deformable model . A multiresolution technique that merits further research in the context of deformable models, is the use of wavelet bases  for deformations [148,149]. A deform-able model should be able to easily incorporate added constraints and any other a priori anatomic knowledge of object shape and motion. Section 3.3 reviewed several of the most promising techniques to incorporate a priori knowledge. For example, for LV motion tracking, a promising research direction is the incorporation of biomechanical properties of the heart and the inclusion of the temporal periodic characteristics of the heart motion. Future directions include modeling schemes that incorporate reasoning and recognition mechanisms using techniques from artificial intelligence such as rule-based systems or neural networks.
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