Computer graphics and signal processing have been jointly developed in the last decades. From one side, several tools from the signal processing can be directly extended on non-Euclidean domain thanks to the spectral geometry processing. From the other side, new devices and techniques for the acquisition of signals provide a huge quantity of information defined on complex non-Euclidean geometries.
The increased opportunity to precisely scan and acquire the geometry of real 3D objects gives rise to several new solutions in different applications such as fabrication and quality control. One application, with a huge impact in real life, is anomaly detection and analysis in the medical scenario. Our proposal aims to exploit the spectral geometry processing tools for this precise goal.
Recently, in medical centres and clinics, a huge diffusion of devices for 3D acquisition gave rise to a dramatical enhancement of the availability of digital representations of anatomical shapes, such as the external surfaces of organs, muscles, tissues and bones that compose the human body.
The geometry of these objects already contains a lot of information and is interesting by itself, some deformations of the surface of specific organs are indeed correlated with the occurrence of a disease. At the same time, these surfaces are usually enriched with additional numerical evaluations that give rise to a collection of signals defined on the surface such as thickness, curvatures and other clinical annotations.
The main idea of our project is to analyze signals defined on the considered surface through standard tools from signal processing in order to detect and classify the anomalies.
We believe that in the medical context, and in particular for organs analysis and disease detection, signal processing on the non-Euclidean domain can give rise to innovative and useful techniques and methods.
As we outlined in the previous paragraphs, this proposal can be placed in the context of functional analysis on non-Euclidean manifolds. More precisely, the manifolds considered this project are the external surfaces of bones and organs of the human body.
The functional analysis on non-Euclidean manifolds we will exploit is mainly based on the functional maps framework introduced in 2012.
The functional maps framework has been extended in several subsequent works.
Among many others, we authored at least three papers that can be considered as three of the more recent advances of this technique.
In 2016, I published "Functional Maps for Brain Classification on Spectral Domain", this work indicates the direction to this project. Unfortunately, no further explorations have been performed after this first attempt. One of the main contributions of this paper from 2016 is that it shows how the well-defined framework of Functional maps in the context of computer graphics gives rise to high-quality performances in the medical scenario such as the anomaly detection in the analysis of brain surfaces.
In that paper not all the properties of the functional maps frameworks have been explored. For instance, no transfer of information between shapes has been performed in that work exploiting the functional space connection provided by the method. Moreover, in the last years, several new refinement techniques have been proposed for the functional maps and none of them has been used in the medical context. We authored some of these works and we know this topic very well. We believe that novel contributions could be achieved by exploring other properties of the functional maps and exploiting the recent advancements proposed.
Another novelty and an additional concrete idea at the base of this project is to adopt the shape difference operator proposed in 2013 in the context of the analysis of surfaces for medical tasks. The shape difference operator which is defined over the functional map could be used to localize and better understand the deformations that occur between a pair of shapes.
This tool has been deeply explored in recent works ("On the Stability of Functional Maps and Shape Difference Operators", "Functional Characterization of Deformation Fields"). We believe that this tool is now mature enough to be exploited in medical applications similar to what we did for the functional maps a few years ago.
As described above, we would like to perform an additional subproject trying to enrich a given class of shapes (i.e. a specific organ) with a parametric model that can be easily deformed to register all the shapes in this class. Thanks to this tool, we will be able to provide a common template to all the considered shapes. With a common template, all the shapes will be in 1:1 correspondence enabling us to perform a wide quantity of analysis. This method is related to the existing literature about parametrization of anatomical surfaces but will propose a complete change of perspective in the paradigm of the analysis. The existing methods indeed perform the analysis on the 2-dimensional domain where the given surfaces are mapped while thanks to the common 3D template provided by our method our analysis of the shapes will be directly performed on the 3D domain. This new opportunity will allow us to consider all the geometric structures of the data in our process without losing some of the information in the mapping between the 3D and the 2D. This approach is completely new in this context. We claim that this part of the project has great potential and it can have a huge impact on real applications.
Finally, applying data-driven techniques to all the three previously described directions could improve the quality of the obtained results and increase the novelty of the contributions achieved by this project.