Mathematical morphology skeleton image processing makes many partial skeleton image planes from an original binary image. And the original binary image can be reconstructed without any distortion by summing the first partial skeleton image plane and each dilated partial skeleton image planes using the same structuring element. Especially compression effects of Elias coding to the morphological globally minimal skeleton(GMS) image, is better than that of PCX and Huffman coding. And then this paper proposes mathematical morphological GMS image processing which can be applied to a binary image transmitting for facimile and big size(bigger than $64{ imes}64$ size) bitmap fonts storing in a memory.