Mathematical morphology (MM) is a theory of lattice operators concerned with their algebraic representation by the combination of canonical operators. These are the so-called erosions, dilations, anti-erosions and anti-dilations, that are, respectively, the operators which commute with meet, commute with join, anti-commute with meet and anti-commute with join. Representation theorems give general sufficient conditions for the minimal representational of operators in complete lattices by the join and meet of canonical operators. These general theorems can be instantiated to specific lattices, such as Boolean lattices and lattices of real-valued functions, to design operators to solve problems in signal and image processing. This design depends on properly calibrating the parameters of canonical operators, what can be performed based on prior information about the problem being solved or by learning from data. In a learning context, it is necessary to develop, implement and test algorithms that can efficiently learn the parameters from data without losing the algebraic representation of MM. This can be achieved by proving representation theorems that lead to parametric representations of lattice operators that can be leveraged to develop more efficient learning algorithms.

Projects can involve deducing representation theorems for lattice operators based on mathematical morphology or developing and implementing algorithms based on existing representation theorems with applications in signal and image processing. The research topics can range from abstract lattice algebra to applying machine learning methods, combining theory and application.

References:

Marcondes, D.; Barrera, J. Discrete Morphological Neural Networks. SIAM Journal on Image Sciences. 2024

Marcondes, D., Feldman, M., Barrera, J. An Algorithm to Train Unrestricted Sequential Discrete Morphological Neural Networks. In: Brunetti, S., Frosini, A., Rinaldi, S. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2024. Lecture Notes in Computer Science, vol 14605. Springer, Cham, 2024

Marcondes, D., Barrera, J. The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators. In: Brunetti, S., Frosini, A., Rinaldi, S. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2024. Lecture Notes in Computer Science, vol 14605. Springer, Cham, 2024