Singular points of scalar images in any dimensions are classified by a topo
logical number. This number takes integer values and can efficiently be com
puted as a surface integral on any closed hypersurface surrounding a given
point. A nonzero value of the topological number indicates that in the corr
esponding point the gradient field vanishes, so the point is singular. The
value of the topological number classifies the singularity and extends the
notion of local minima and maxima in one-dimensional signals to the higher
dimensional scalar images. Topological numbers are preserved along the drif
t of nondegenerate singular points induced by any smooth image deformation.
When interactions such as annihilations, creations or scatter of singular
points occurs upon a smooth image deformation, the total topological number
remains the same.
Our analysis based on an integral method and thus is a nonperturbative exte
nsion of the order-by-order approach using sets of differential invariants
for studying singular points.
Examples of typical singularities in one- and two-dimensional images are pr
esented and their evolution induced by isotropic linear diffusion of the im
age is studied.