Self-similar intermediate asymptotics for nonlinear degenerate parabolic free-boundary problems that occur in image processing

In the boundary layers around the edges of images, basic nonlinear paraboli c equations for image intensity used in image processing assume a special d egenerate asymptotic form. An asymptotic self-similar solution to this dege nerate equation is obtained in an explicit form. The solution reveals a sub stantially nonlinear effect-the formation of sharp steps at the edges of th e images, leading to edge enhancement. Positions of the steps and the time shift parameter cannot be determined by direct construction of a self-simil ar solution; they depend on the initial condition of the pre-self-similar s olution. The free-boundary problem is formulated describing the image inten sity evolution in the boundary layer.