Maximally stable extremal regions

In computer vision, maximally stable extremal regions (MSER) are used as a method of blob detection in images. This technique was proposed by Matas et al. to find correspondences between image elements from two images with different viewpoints. This method of extracting a comprehensive number of corresponding image elements contributes to the wide-baseline matching, and it has led to better stereo matching and object recognition algorithms. Image is a mapping . Extremal regions are well defined on images if: is totally ordered (total, antisymmetric and transitive binary relations exist). An adjacency relation is defined. We will denote that two points are adjacent as . Region is a contiguous (aka connected) subset of . (For each there is a sequence such as .) Note that under this definition the region can contain "holes" (for example, a ring-shaped region is connected, but its internal circle is not the part of ). (Outer) region boundary , which means the boundary of is the set of pixels adjacent to at least one pixel of but not belonging to . Again, in case of regions with "holes", the region boundary is not obliged to be connected subset of (a ring has inner bound and outer bound which do not intersect). Extremal region is a region such that either for all (maximum intensity region) or for all (minimum intensity region). As far as is totally ordered, we can reformulate these conditions as for maximum intensity region and for minimum intensity region, respectively. In this form we can use a notion of a threshold intensity value which separates the region and its boundary. Maximally stable extremal region Let an extremal region such as all points on it have an intensity smaller than . Note for all positive . Extremal region is maximally stable if and only if has a local minimum at . (Here denotes cardinality). is here a parameter of the method. The equation checks for regions that remain stable over a certain number of thresholds. If a region is not significantly larger than a region , region is taken as a maximally stable region.