Smoothing and edge detection by time-varying coupled nonlinear diffusion equations

In this paper, we develop new methods for de-noising and edge detection in images by the solution of nonlinear diffusion partial differential equations. Many previous methods in this area obtain a de-noising u of the noisy image I as the solution of an equation of the form partial derivative (t)u = L(g(\del upsilon\), delu, u - I), when g controls the speed of the diffusion and defines the edge map. The usual choice for g(s) is (1 + ks(2))(-1) and the function upsilon is always some smoothing of u. Previous choices include upsilon = u, upsilon = G(sigma) * u, and upsilon = G sigma * I. Numerical results indicate that the choice of upsilon plays a very important role in the quality of the images obtained. Notice that all these choices involve an isotropic smoothing of u, which sometimes fails to presence important corners and junctions, and this may also fail to resolve small features which are closely grouped together. This paper obtains u as the solution of a nonlinear diffusion equation which depends on u. The equation can be obtained as the energy descent equation for the total variation of upsilon penalized by the mean squared error between u and upsilon. The parameters in this energy descent equation are regarded as functions of time rather than constants, to allow for a reduction in the amount of smoothing as time progresses. Numerical tests indicate that our new method is faster and able to resolve small details and junctions better than standard methods. (C) 2001 Academic Press.