Convolution in (max,min)-algebra and its role in mathematical morphology

International audienceThe purpose of this theoretical paper is to study the convolution of two functions in the $(\max,\min)$-algebra. More precisely, a formal definition of morphological operators in $(\max,\min)$-algebra is introduced and their relevant properties from an algebraic viewpoint are stated and proved. Some previous works in mathematical morphology have already encountered this type of operators but a systematic study of them has not yet been undertaken in the morphological literature. It is shown in particular that their fundamental property is the equivalence with level set processing using Minkowski addition and subtraction. Some powerful results from nonlinear analysis can be straightforward related to the present $(\max, \min)$-operators. On the one hand, the theory of viscosity solutions of the Hamilton-Jacobi equation with Hamiltonians containing $u$ and $Du$ is summarized, in particular, the corresponding Hopf-Lax-Oleinik formulas are given. On the other hand, results on quasi-concavity preservation, Lipschitz approximation and conjugate/tranform related to $(\max,\min)$-convolutions are discussed. Links between $(\max,\min)$-convolutions and some previous approaches of unconventional morphology, in particular fuzzy morphology and viscous morphology, are fully reviewed. In addition, the interest of $(\max,\min)$-convolutions in Boolean random function characterization is considered. Links of $(\max,\min)$-morphology framework to geodesic dilation and erosion are also provided. We discuss two important conclusions. First, it is proven in the paper that $(\max,\min)$-openings are compatible with Matheron's axiomatic of Euclidean granulometries for functions with quasiconcave structuring functions. Second, it is also shown that the adjoint supmin convolution is the operator underlying the extension of Matheron's characterization of Boolean random closed sets to the case of Boolean random upper semicontinuous function. For all these reasons, we state that $(\max,\min)$-convolution provides the natural framework to generalize some key notions from Matheron's theory from sets to functions