Generalized Image Restoration by the Method of Alternating Orthogonal Projections

Abstract

We adopt a view that suggests that many problems of image restoration are probably geometric in character and admit the following initial linear formulation: The originalfis a vector known a priori to belong to a linear subspace{\cal P}_bof a parent Hilbert space{\cal H}(, but all that is available to the observer is its imageP_{a} f, the projection offonto a known linear subspace{\cal P}_a(also in\cal H). 1) Find necessary and sufficient conditions under whichfis uniquely determined byP_{a} fand 2) find necessary and sufficient conditions for the stable linear reconstruction offfromP_{a} fin the face of noise. (In the later case, the reconstruction problem is said to be completely posed.) The answers torn out to be remarkably simple. a)fis uniquely determined by{\cal P}_{a}iff{\cal P}_{b}and the orthogonal complement of{ \cal P}_{a}have only the zero vector in common. b) The reconstruction problem is completely posed iff the angle between{\cal P}_{b}and the orthogonal complement of{\cal P}_{a}, is greater than zero. (All angles lie in the first quadrant.) c) In the absence of noise, there exists in both cases a) and b) an effective recursive algorithm for the recovery offemploying only the operations of projection onto{\cal P}_{b}and projection onto the orthogonal complement of{\cal P}_{a}These operations define the necessary instrumentation.