This paper discusses lossless encoding methods of compressing two-dimensional black and white images for the efficient communication of such images. All the models concerned are adaptive in the sense that we assume no prior knowledge of statistical properties of any input and that the image is serially scanned and encoded in one pass. In the adaptive strategy, each picture element (pel) is encoded based on the time-variant coding parameters, which are empirically estimated from the already encoded pels in the present image. Since the coding parameters depend on the conditioning neighboring pels, modeling involves a pel-by-pel determination of conditioning neighboring pels. Two approximate solutions of how to select the conditioning pels are given, both of which are based on Rissanen's MDL criterion. Although no rigorous theory to account for the performance has been established, the experimental study demonstrates that one of them is superior in compression performance to any adaptive model utilizing a fixed number of conditioning pels. In this paper, an efficient implementation of models by means of an arithmetic code is also presented.