Periodic Reporting for period 4 - OPREP (Operator Based Representations for Geometry Processing)
Reporting period: 2021-07-01 to 2022-12-31
Geometric data is prevalent in many areas of science and technology. From the surface of the brain to the intricate shapes of free-form architecture, complex geometric structures arise in many fields, and problems such as analysis, processing and synthesis of geometric data are of great importance.
One major challenge in tackling such problems is choosing an adequate discrete representation of the geometric data. Traditionally, surface geometric data is treated as an irregularly sampled signal in three-dimensional space, yielding a representation as either a point cloud, or a polygonal mesh. Further analysis and manipulation are done directly on this discrete representation, resulting in algorithms which are often combinatorial, leading to difficult numerical optimization problems.
The goal of this research was to explore a fundamentally different approach of representing geometric data through the space of scalar functions defined on it, and representing geometric operations as algebraic manipulations of linear operators acting on such functions.
We investigated the basic theory behind such a representation, addressing questions such as: what are the best function spaces to work with? Which operators can be consistently discretized, leading to discrete theorems analogous to continuous ones? How should multi-scale processing of geometric data be treated in this novel representation?
To validate our approach, we explored how this representation can be leveraged for devising efficient solutions to difficult real-world geometry processing problems, such as numerical simulation of intricate phenomena on curved surfaces, surface correspondence and quadrangular remeshing.
One major challenge in tackling such problems is choosing an adequate discrete representation of the geometric data. Traditionally, surface geometric data is treated as an irregularly sampled signal in three-dimensional space, yielding a representation as either a point cloud, or a polygonal mesh. Further analysis and manipulation are done directly on this discrete representation, resulting in algorithms which are often combinatorial, leading to difficult numerical optimization problems.
The goal of this research was to explore a fundamentally different approach of representing geometric data through the space of scalar functions defined on it, and representing geometric operations as algebraic manipulations of linear operators acting on such functions.
We investigated the basic theory behind such a representation, addressing questions such as: what are the best function spaces to work with? Which operators can be consistently discretized, leading to discrete theorems analogous to continuous ones? How should multi-scale processing of geometric data be treated in this novel representation?
To validate our approach, we explored how this representation can be leveraged for devising efficient solutions to difficult real-world geometry processing problems, such as numerical simulation of intricate phenomena on curved surfaces, surface correspondence and quadrangular remeshing.
The work addressed various theoretical properties and practical applications of the operator based representations.
We have applied it for applications such as: computing the correspondence between two very different shapes, which can potentially be used, e.g. for medical imaging (for computing the correspondence between the surface of different organs obtained by MRI or CT scans, such as the surface of the brain or the placenta), or for paleontology (for computing the correspondence between bones or teeth of different species of monkeys).
We have further showed how to simulate intricate phenomena on curved surfaces (such as fluid flow), which can potentially be used for applications such as weather prediction.
We have worked with multi-resolution representations, both in the context of shape correspondence, and for making general linear solves on surfaces more efficient.
On the theoretical side, we developed discrete representations of different geometric operators (such as the Levi-Civita covariant derivative), and consistency conditions (such as vector field integrability).
We have applied it for applications such as: computing the correspondence between two very different shapes, which can potentially be used, e.g. for medical imaging (for computing the correspondence between the surface of different organs obtained by MRI or CT scans, such as the surface of the brain or the placenta), or for paleontology (for computing the correspondence between bones or teeth of different species of monkeys).
We have further showed how to simulate intricate phenomena on curved surfaces (such as fluid flow), which can potentially be used for applications such as weather prediction.
We have worked with multi-resolution representations, both in the context of shape correspondence, and for making general linear solves on surfaces more efficient.
On the theoretical side, we developed discrete representations of different geometric operators (such as the Levi-Civita covariant derivative), and consistency conditions (such as vector field integrability).
All the papers that we published have shown clear advancement over the state of the art in applications such as shape correspondence, numerical simulation, meshing and architectural geometry.