BMF is an image compression format produced by a program of the same name, written by Dmitry Shkarin. The first version was apparently released in 1998.. It offers lossless and lossy (called "near-lossless") compression modes. There was a "version 2" released in 2009. A cryptic remark by Shkarin that "BMF v.2.0 does not exist, there are only separate codecs for it" may indicate that version 2 is something comparable to a container format for image files; however, the format is proprietary, so this cannot be verified.
BMF was never in wide use, and is best described as a hobbyist format.