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 original <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> is a vector known a priori to belong to a linear subspace <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_b</tex> of a parent Hilbert space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal H}(</tex> , but all that is available to the observer is its image <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{a} f</tex> , the projection of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> onto a known linear subspace <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_a</tex> (also in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\cal H</tex> ). 1) Find necessary and sufficient conditions under which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> is uniquely determined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{a} f</tex> and 2) find necessary and sufficient conditions for the stable linear reconstruction of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{a} f</tex> 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) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> is uniquely determined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_{a}</tex> iff <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_{b}</tex> and the orthogonal complement of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{ \cal P}_{a}</tex> have only the zero vector in common. b) The reconstruction problem is completely posed iff the angle between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_{b}</tex> and the orthogonal complement of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_{a}</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> employing only the operations of projection onto <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_{b}</tex> and projection onto the orthogonal complement of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal P}_{a}</tex> These operations define the necessary instrumentation.

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