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We present a new method for lossless image compression that gives compression comparable to JPEG lossless mode with about five times the speed. Our method, called FELICS, is based on a novel use of two neighboring pixels for both prediction and error modeling. For coding we use single bits, adjusted binary codes, and Golomb or Rice codes. For the latter we present and analyze a provably good method for estimating the single coding parameter. (Note: This method is the foundation for the subsequently developed state-of-the-art methods now used for lossless image compression.)