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 {\cal P}_b of a parent Hilbert space {\cal H}( , but all that is available to the observer is its image P_{a} f , the projection of f onto a known linear subspace {\cal P}_a (also in \cal H ). 1) Find necessary and sufficient conditions under which f is uniquely determined by P_{a} f and 2) find necessary and sufficient conditions for the stable linear reconstruction of f from P_{a} 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 {\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 of f employing 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.