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A new algorithm is developed for solving the maximum entropy (ME) image reconstruction problem. The problem is reduced to solving a system of ordinary differential equations with appropriate initial values. The choice of initial values closely relates to the satisfaction of constraints, and we show how initial values are determined. The algorithm does not involve any optimization method. Instead of searching in the (n + 1)-dimensional space as required for most ME algorithms, our approach relies on solving a one-dimensional search along a well-defined and easily mastered path. Moreover, an efficient algorithm is developed to handle the search. The computer reconstruction verifies the theory.