Adaptive processes often attempt to minimize the mean square error (MSE) to filter a partially observed digital signal. While mathematically tractable, the MSE criterion often causes oversmoothing of the filtered signal. In this paper, we propose using maximum entropy (ME) as the optimization criterion to avoid the oversmoothing of signals. This criterion is motivated by the fact that ME methods make no assumptions regarding the unobserved data, aside from explicitly stated ones. The maximum entropy Kalman filter presented in this paper employs ME as its optimization criterion to explicitly identify the appropriate parameters of the standard Kalman filter, for the purpose of image compression and reconstruction.