On detecting all saddle points in 2D images

Although spatial critical points (saddle points and extrema-minima and maxima) are mathematically well-defined, it is non-trivial to detect them on an arbitrary discrete grid. Discretising a continuous method as well as a straightforward discrete neighbourhood based method do not guarantee to return all critical points. Although not all image analysis tasks require the right amount of critical point in mutual relations, it is obviously an advantage to know that all critical points are found. Furthermore, some methods do require the right amount of saddle points in relation to the extrema. The Enter number is an invariant stating explicitly the relation of the number of types of critical points. Using this, one is sure to find the right number of critical points. It is defined on a discrete lattice, so one only has to use the right grid. This appears to be a hexagonal one where each point has six neighbours. An easy way is given to use the hexagonal based critical point detection in a rectangular grid, which is commonly used in computer vision and image analysis tasks. (C) 2004 Elsevier B.V. All rights reserved.