On implicit image differentiation and filtering

The main goal of this talk is to demonstrate advantages of using compact (implicit) finite differencing, filtering, and interpolating schemes for image processing applications. Finite difference schemes can be categorized as "explicit" and "implicit." Explicit schemes express the nodal derivatives as a weighted sum of the function nodal values. For example, f'i=(fi+1-fi-1)/2h is an explicit finite difference approximation of the first-order derivative. By comparison, compact (implicit) finite difference schemes equate a weighted sum of nodal derivatives to a weighted sum of the function nodal values. For instance, f'i-1+4f'i+f'i+1=3(fi+1-fi-1)/2h is an implicit (compact) scheme. Some implicit schemes correspond to Pade approximations and produce significantly more accurate approximations for the small scales to compare with explicit schemes having the same stencil widths. Some other implicit schemes are designed to deliver accurate approximations of function derivatives over a wide range of spatial scales. Compact (implicit) finite difference schemes, as well as implicit filtering and interpolating schemes, constitute advanced but standard tools for accurate numerical simulations of problems involving linear and nonlinear wave propagation phenomena.

In this talk, I show how Fourier-Pade-Galerkin approximations can be adapted for designing high-quality implicit finite difference schemes, establish a link between implicit schemes and standard explicit finite differences used for image gradient estimation, and demonstrate usefulness of implicit differencing and filtering schemes for various image processing tasks including image deblurring, feature detection, and sharpening.