Some Properties of Interpolations Using Mathematical Morphology.

The problem of interpolation of images is defined as - given two images at time t = 0 and t = T, one must find the series of images for the intermediate time. This problem is not well posed, in the sense that without further constraints, there are many possible solutions. The solution is thus usually dictated by the choice of the constraints/assumptions, which in turn relies on the domain of application. In this article we follow the approach of obtaining a solution to the interpolation problem using the operators from Mathematical Morphology (MM). These operators have an advantage of preserving structures since the operators are defined on sets. In this work we explore the solutions obtained using MM, and provide several results along with proofs which corroborates the validity of the assumptions, provide links among existing methods and intuition about them. We also summarize few possible extensions and prospective problems of current interest.

Ranks for pairs of spatial fields via metric based on grayscale morphological distances.

Based on a set of morphological distances computed between the grayscale images (spatial fields) of similar size specifications, the ratios of selected morphological distances, and the ratios of areas of infima and suprema of grayscale images, a new metric to quantify the degree of similarity between the grayscale images is proposed. We denote the two spatial fields (grayscale images), respectively, with f(i) and f(j), and the infima and suprema of these spatial fields with (f(i)∧f(j)) and ( f(i)∨f(j)). The three morphology-based distances include: 1) dilation distance d( f(i),f(j)) ; 2) erosion distance e( f(i),f(j)); and 3) median-based distance MN ( f(i),f(j)) . By employing these parameters, which play vital role in construction of parameter-specific interaction matrices, we provide a metric to designate every possible pair of images that can be considered out of a database consisting of a huge number of images. We demonstrate the whole approach on: 1) synthetic spatial fields; 2) a set of 12 similar-sized grayscale images representing cloud-top temperatures of a specific region for 12 different time instants; and 3) four spatial elevation fields to rank possible pairs of images.