We present a class of nonlinear adaptive image restoration filters which may be steered to preserve sharp edges and contrasts in the restorations. From a theoretical point of view we discuss the associated variational problems and prove existence of solutions in certain Sobolev spaces W1,p or in a BV -space. The degree of regularity of the solution may be understood as a mathematical explanation of the heuristic properties of the designed filters. Key words. adaptive image restoration filter, image restoration, variational methods AMS subject classifications. 49J52, 49N15, 94A17 PII. S0363012997324338