M.Sc. Dissertation Proposal

Vision and Motivation

The problem of reconstructing the human anatomical structures (e.g., bones, liver, etc.) from MRI/CT datasets can be stated as follows: Given a point set concerning an anatomical structure’s surface, find a mesh that interpolates its points. This challenge assumes that the segmentation of MRI/CT datasets took place before. Thus, this challenge involves creating two geometric structures from segmentation:
(1) a set of points that samples each anatomical structure in the MRI dataset; and
(2) a piecewise linear (PL) surface or mesh from the point set of each anatomical structure.
Obtaining the point set of each anatomical structure is straightforward as it can be extracted from the multi-organ segmentation of the input dataset. Finding a mesh for each human organ is the objective of the research work behind this M.Sc. dissertation proposal.

State-of-the-Art

It is worth noting that there is currently no universal surface reconstruction algorithm available in the literature — whether it be a classical or a machine learning-based algorithm [2024-Leitao]. This means that both types of algorithms suffer from similar limitations when it comes to surface reconstruction. While the screened Poisson algorithm [2013-Kazhdan] is an exception in that it guarantees the manifoldness of the output mesh, other algorithms do not have this capability. However, the Poisson algorithm is not perfect and may suffer from shape drifting, meaning that the generated surface may not accurately reflect the original surface shape.

Research Methodology

Our innovative solution is based on the concept of Lie groups. A Lie group is a structure that possesses both algebraic and topological properties. The algebraic properties are derived from groups, while the topological properties are derived from smooth manifolds. For instance, following a geodesic curve in the topological space of the manifold is equivalent to performing a symbolic operation on the homogeneous matrix that represents a point on the manifold. We build a Lie algebra by linearizing a Lie group, which is accomplished by expanding the group combinatorial operator around the coordinates of the group elements at any given group element. This linearization of the Lie group creates a new set of elements that is its Lie algebra. A Lie algebra is a linear vector space, so the group operation is reduced to vector addition; it is this local linearization that makes it possible to analyze non-linear properties using simple linear tools. Our approach aims to linearize manifolds that correspond to anatomical structures. To achieve this, we have developed an algorithm that takes a point set of a particular anatomical structure as input and produces a PL surface as output. Our algorithm comprises four steps. In the first step, we generate a kD-tree to find the k-nearest neighbors of each point of the anatomical structure, which is required in the second step. In the second step, we determine the minimum spanning tree (MST) using Prim's algorithm. The MST acts as the PL surface spine. In the third step, based on the MST, we create the Lie group of each input point, which is similar to the concept of pixel kernel in image processing. However, since we are working on a 2D manifold in the 3D space, we only consider surrounding points for which there are shortest and geodesic paths to its central point. In the fourth step, we find geodesic paths from every single input point to any other point of the input point set, from which its triangulation follows.

References

[2013-Kazhdan] M. Kazhdan and H. Hoppe (2013): Screened Poisson Surface Reconstruction. ACM Transactions on Graphics, Vol. 32, No. 3, Article 29, pp. 1-29, June 2013.
https://doi.org/10.1145/2487228.2487237
[2024-Leitao] Leitão, G. & Gomes, A. (2022). PCR Cocktail – a mesh-growing algorithm for triangulating point clouds without using angle bounds. Computers & Graphics, 2022 (submitted for publication).
http://www.di.ubi.pt/~agomes/mistral/2022-Leitao.pdf