3D statistical shape models of human bones : their construction using a finite element registration algorithm, formulation on Hilbert spaces, and application to medical image analysis

Statistical shape models have become a widely used tool in computer vision and medical image analysis. They are constructed from a representative set of example shapes and represent the normal shape variations of a class of objects,
in our case of human bones. The foundation of statistical shape models is the concept of correspondence. In order to draw meaningful statistical conclusions and to build a generative model from the example shapes, we should compare and relate only corresponding parts of the shape. The task of establishing correspondence between shapes and images is known as the registration problem and is one of the fundamental problems of computer vision. To approximate a solution of the registration problem for our bone shapes, we propose a new registration algorithm, which is formulated as a continuous minimization problem, whose solution is sought with a state of the art finite element method.
Once the shapes have been brought into correspondence, a statistical shape model can be built. We present a formulation of the shape model on general Hilbert
spaces, which incorporates all associated models which can be constructed in a similar way, like models of shape, color, intensity, deformations etc. Which of
these models is used depends only on the choice of the Hilbert space. Because this includes the choice between continuously defined models and models based on
any kind of discretization method, we can easily integrate the statistical model into our registration method and its finite element discretization. This inclusion of class-specific prior knowledge into makes the registration more
robust against outliers and damaged data sets.
Finally, we show how the statistical models can be applied to a number of practical problems from medical image analysis and surgery planning, like the fitting of the model to novel shapes or images, the design of optimized medical implants or the automatic repositioning of fractured bones.