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Algorithms for reconstruction of a two-dimensional binary pattern Z from its row-projection vector A and column-projection vector B have been developed by Chang. The projection set (A,B) is said to be unique, nonunique, or inconsistent if it determines one binary pattern, more than one binary pattern, or no binary pattern. A binary patern Z is said to be ambiguous if there exists another pattern with the same projections, otherwise it is unambiguous. Two characterization questions of binary patterns and their projections are posed. First, given Z what is the necessary and sufficient condition for Z to be ambiguous, or unambiguous? Second, given (A,B), what is the necessary and sufficient condition for (A,B) to be unique, nonunique, or inconsistent? These two combinational questions are discussed and efficient algorithms to answer them are derived from some ideas of Ryser.