An approach to color-morphology based on Einstein addition and Loewner order

In this article, a new method to process color images via mathematical morphology is presented. Precisely, each pixel of an image contained in the rgb-space is converted into a 2x2-matrix representing a point in a color bi-cone. The supremum and infimum needed for dilation and erosion, respectively, are calculated with respect to the Loewner order. Since the result can be outside the bi-cone, a map is proposed to ensure the algebraic closure in the bi-cone. A new addition and subtraction based on Einstein's Relativity Theory is used to define morphological operation such as top-hats, gradients, and morphological Laplacian. Since the addition and subtraction is defined in the unit ball, a suitable mapping between those two spaces is constructed. A comparison with the component-wise approach and the full ordering using lexicographical cascades approach demonstrate the feasibility and capabilities of the proposed approach.