Particle-Based Shape Modeling for Arbitrary Regions-of-Interest

TL;DR

This work addresses the need for flexible, scalable ROI definitions in statistical shape modeling (SSM) by extending particle-based shape modeling (PSM) with mesh-field free-form constraints and a quadratic penalty optimization. The proposed approach formulates an unconstrained objective $F$ that penalizes constraint violations via $g_{i,m}^+({\bf p})$, enabling efficient, linear-in-particle optimization with Gauss-Seidel updates. Free-form constraints leverage surface distance and gradient fields ${\bf M}^d_i({\bf p})$ and ${\bf M}^g_i({\bf p})$ attached to meshes, allowing arbitrary ROI delineation beyond geometric primitives. A graphical interface supports cutting planes and FFCs, with constraint propagation across a population through image-registered deformations. Evaluations on synthetic ellipsoids, CT femurs, and left atria demonstrate that the method achieves intended ROI isolation, yields meaningful mode variation, and improves ROI flexibility without reprocessing data, offering practical benefits for ROI-focused shape analysis.

Abstract

Statistical Shape Modeling (SSM) is a quantitative method for analyzing morphological variations in anatomical structures. These analyses often necessitate building models on targeted anatomical regions of interest to focus on specific morphological features. We propose an extension to \particle-based shape modeling (PSM), a widely used SSM framework, to allow shape modeling to arbitrary regions of interest. Existing methods to define regions of interest are computationally expensive and have topological limitations. To address these shortcomings, we use mesh fields to define free-form constraints, which allow for delimiting arbitrary regions of interest on shape surfaces. Furthermore, we add a quadratic penalty method to the model optimization to enable computationally efficient enforcement of any combination of cutting-plane and free-form constraints. We demonstrate the effectiveness of this method on a challenging synthetic dataset and two medical datasets.

Figures (5)

• Figure 1: (a) Constrained particle distribution on a sphere, where yellow illustrates the feasible region of the constrained area where particles are allowed to be distributed, the gray is the infeasible region where if particles were to be there, they would be violating the constraint. (b) Distance field ${\bf M}^d_i({\bf p})$ of signed geodesic distances to the surface of every mesh vertex. (c) Gradient field ${\bf M}^g_i({\bf p})$ on the mesh surface at every mesh vertex represented using white arrows and the blue surface as the feasible region.
• Figure 2: (a) The constraint panel shows the constraints that have been defined and the tools to define the constraints. (b) Cutting-plane constraints are defined by ctrl-clicking 3 points on the shape surface. (c) Constraints can be flipped or applied to all other shapes via the right-click menu. (d) FFCs are defined with a painting tool with different brush sizes and options to customize included and excluded areas. We show how the painting of excluded areas of different sizes applied to a segmentation of a left atrium.
• Figure 3: (a) Sample ellipsoids from the dataset with feasible regions in yellow and restricted regions in grey. (b) The first three modes of variation in the dataset, which show the variation in corresponding major axes.
• Figure 4: (a) Example of defined constraints. The feasible region is shown in yellow and the constrained region in grey. (b) The first two modes of variation in the dataset. Notice that particle s are excluded from the lesser trochanter.
• Figure 5: (a) Example of defined constraints where the left atrium (in yellow) segmentations have the pulmonary veins excluded (in grey). (b) The first three modes of variation in the dataset. Notice how the pulmonary vein areas remain hollow.