Regularized multi-structural shape modeling of the knee complex based on deep functional maps

Abstract

The incorporation of a-priori knowledge on the shape of anatomical structures and their variation through Statistical Shape Models (SSMs) has shown to be very effective in guiding highly uncertain image segmentation problems. In this paper, we construct multiple-structure SSMs of purely geometric nature, that describe the relationship between adjacent anatomical components through Canonical Correlation Analysis. Shape inference is then conducted based on a regularization term on the shape likelihood providing more reliable structure representations. A fundamental prerequisite for performing statistical shape analysis on a set of objects is the identification of corresponding points on their associated surfaces. We address the correspondence problem using the recently proposed Functional Maps framework, which is a generalization of point-to-point correspondence to manifolds. Additionally, we show that, by incorporating techniques from the deep learning theory into this framework, we can further enhance the ability of SSMs to better capture the shape variation in a given dataset. The efficiency of our approach is illustrated through the creation of 3D models of the human knee complex in two application scenarios: incomplete or noisy shape reconstruction and missing structure estimation.

Keywords: Deep functional maps; Knee modeling; Missing structure estimation; Multi-structure statistical shape model; Shape correspondence; Shape reconstruction.