My name is Samy Blusseau and I started a Tenure track at the Centre for Mathematical Morphology (CMM) of Mines ParisTech in March. I had already done a one year postdoc at the CMM since February 2017, jointly with the LTCI lab of Télécom ParisTech, under the supervision of Jesus Angulo, Santiago Velasco-Forero, Isabelle Bloch and Yann Gousseau.

During that project, I focused on extending mathematical morphology to positive semidefinite matrices – in 2 and 3 dimensions. These objects usually model local structural information that can be integrated to infer larger structures. For example, structure tensors in 2D images or diffusion tensors in 3D diffusion MRI can help reconstruct elongated connected components such as vessels and white matter fibers

in the brain. This topic led me to start developing a morphological method for signal processing on weighted graphs, which is at the crossroads between mathematical morphology, graphs theory and tropical algebra [1]. The illustration below shows an image of vessels (left), its corresponding anisotropy image (centre) computed from structure tensors (the brighter, the more anisotropic), and the result of an opening (right) based on the just mentioned method.

Currently, my scientific project is partly concerned with the study of tropical algebra, in relation with graphs, mathematical morphology and non linearity in deep learning. Indeed, like many approaches in image processing and computer vision, mathematical morphology seems jeopardized by deep learning methods, which have already solved many practical problems where massive data are available. Yet, along with tropical algebra, mathematical morphology actually offers a fresh and original view to address deep learning issues.

Tropical algebra is a branch of non linear mathematics in which the sum of classical linear algebra is replaced by the max (or min) and the usual product is sometimes replaced by the sum (in that case we refer to “max-plus” or “min-plus” algebra). It is strongly linked to mathematical morphology since “linear” transformations in the tropical sense are often morphological operators [2]. For example most dilations of images can be represented by a max-plus product of a matrix by a the image seen as a vector.

Another asset of tropical algebra is that it can provide geometrical insights into non linear processes such as the non-linear activation functions in deep neural networks. For example the so called *maxout* units introduced by Goodfellow [3] is actually a tropical polynomial. Like in classical algebra, such polynomial is associated to a convex polytope, called Newton polytope. As shown recently [4], the number of linear regions defined by a maxout unit (which somehow measures its representation capacity) is exactly the number of vertices of its Newton polytope. Similar approaches could help understand existing deep architectures but also new ones based on units like the *morphological perceptron*, which still needs to be investigated in light of the progress in modern machine learning.

[1] S. Blusseau, S. Velasco-Forero, J. Angulo, I. Bloch, *Tropical and morphological operators for signals on graphs,* International Confernce on Image Processing (ICIP) 2018 (accepted for publication).

[2] P. Maragos, *Chapter Two – Representations for morphological image operators and analogies with linear operators*, vol. 177 of Advances in Imaging and Electron Physics, pp. 45 – 187. Elsevier, 2013.

[3] I. J. Goodfellow, D. Warde-Farley, M. Mirza, A. Courville and Y. Bengio. *Maxout networks*. arXiv preprint arXiv:1302.4389 (2013).

[4] V. Charisopoulos and P. Maragos. *Morphological Perceptrons: Geometry and Training Algorithms*. In International Symposium on Mathematical Morphology (ISMM) 2017, pp. 3–15.

Congratulations on your tenure track. While tenure track(TT)’s are rare in France, back in 2014/15, when I was switching from pure math to more applied areas (eventually medical imaging and a bit of computer vision), I gave a talk at CMM at Fontainebleau and met Jesus Angulo there. Is there any other TT’s like this that’re available there or in other universities in or around Paris? Thanks in advance!