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The Kalman filtering method is extended to two dimensions. The resulting computational load is found to be excessive. Two new approximations are then introduced. One, called the strip processor, updates a line segment at a time; the other, called the reduced update Kalman filter, is a scalar processor. The reduced update Kalman filter is shown to be optimum in that it minimizes the post update mean-square error (mse) under the constraint of updating only the nearby previously processed neighbors. The resulting filter is a general two-dimensional recursive filter.