Probabilistic Segmentation of the Knee Joint from X-ray Images

A probabilistic method is proposed for segmentation of the knee joint. A likelihood function is formulated that explicitly models overlapping object appearance. Priors on global appearance and geometry (including shape) are learned from example images. Markov chain Monte Carlo methods are used to obtain samples from a posterior distribution over model parameters from which expectations can be estimated. The result is a probabilistic segmen- tation that quantifies uncertainty so that measurements such as joint space can be made with associated uncertainty. Joint space area and mean point-to-contour distance are used for evaluation. The aim of this paper is to outline a probabilistic, model-based segmentation method for the knee joint from x-ray images and to make explicit the uncertainty in the segmentation so obtained. The method explicitly handles the possible overlapping of femur and tibia and their appearance models. Such cases are not handled in methods based on active contours (1), active shape models or active appearance models (2), for example. Segmentation of objects is often only an intermediate result. Consider for example medical image analysis tasks which involve measuring the size of anatomical structures. Most standard segmentation algorithms result in a single solution without any information as to the confidence in this solution. No information about uncertainty is propagated to the subsequent size estimation step. Furthermore, anatomical structures almost inevitably overlap. In medical applications, especially, it is desirable to have an indication of the certainty of a measurement and to cope with structures that overlap or are in close proximity. The performance of the proposed method is evaluated by applying it to the segmentation of the knee joint to enable the measurement of joint space, an important biomarker for the assessment of osteoarthritis (3). 2 Modelling Knee Radiographs The general task of segmenting modelled objects from an image can be described in a Bayesian framework as that of inferring the conditional distribution P (M|I, I, S) which is the probability of the model parameters, M, given a test image I, a set of training images, I, and their annotations, S. Each object can be described by its geometry and appearance in the image and, more specifically, in terms of shape parameters, S, geometric (non-shape) parameters, G, global appearance parameters, Ag, and local appearance parameters, Al. Using Bayes' rule and assuming P (I) is fixed: P (M|I) ∝ p(I|M)P (M )= p(I|G, S, Ag ,A l)P (G)P (S)P (Ag)P (Al) (1)