Generalized Image Restoration by the Method of Alternating Orthogonal Projections

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 original f is a vector known a priori to belong to a linear subspace Pb of a parent Hilbert space H (, but all that is available to the observer is its image P₀ f, the projection of f onto a known linear subspace Pₐ (also in H). 1) Find necessary and sufficient conditions under which f is uniquely determined by P₀ f and 2) find necessary and sufficient conditions for the stable linear reconstruction of f from P₀ f in 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) f is uniquely determined by P₀ iff P₁ and the orthogonal complement of P₀ have only the zero vector in common. b) The reconstruction problem is completely posed iff the angle between P₁ and the orthogonal complement of P₀, 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 of f employing only the operations of projection onto P₁ and projection onto the orthogonal complement of P₀ These operations define the necessary instrumentation.