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Two-dimensional recursive filters are conveniently described in terms of two-dimensional z transforms. The designer of these filters faces two fundamental problems, their stability and their synthesis. Stability is determined by the location of the zero-valued region of the filter's denominator polynomial. A conjecture based on a one-dimensional stability theorem leads to a useful empirical stabilization procedure. Two-dimensional recursive filters can be synthesized to approximate large varieties of desired two-dimensional pulse responses. A conformal transformation yields two-dimensional recursive bandpass filters from appropriately specified one-dimensional filters.