A Survey of Mathematical Structures for Extending 2D Neurogeometry to 3D Image Processing

In the Big Data landscape, learning algorithms are often “black-boxes” and as such, hard to interpret. We need new constructive models, to eventually feed the Big Data framework. The emerging field of Neurogeometry provides inspiration for models in medical computer vision. Neurogeometry models the neuronal architecture of the visual cortex through Differential Geometry. First, Neurogeometry can explain visual phenomena like human perceptual completion. And second, it provides efficient algorithms for computer vision. Examples of applications are image completion (in-painting) and crossing-preserving smoothing. In medical computer vision, Neurogeometry is less known. One reason is that one often deals with 3D images, whereas Neurogeometry is essentially 2D (our retina is 2D). Moreover, the generalization to 3D is not mathematically straight-forward. This article presents the theoretical framework of a 3D-Neurogeometry inspired by the 2D case. The aim is to provide a “theoretical toolbox” and inspiration for new models in 3D medical computer vision.