In this paper, the authors present a binary image compression scheme that can be used either for lossless or lossy compression requirements. This scheme contains five new contributions. The lossless component of the scheme partitions the input image into a number of non-overlapping rectangles using a new line-by-line method. The upper-left and the lower-right vertices of each rectangle are identified and the coordinates of which are efficiently encoded using three methods of representation and compression. The lossy component, on the other hand, provides higher compression through two techniques. 1) It reduces the number of rectangles from the input image using our mathematical regression models. These mathematical models guarantees image quality so that rectangular reduction should not produce visual distortion in the image. The mathematical models have been obtained through subjective tests and regression analysis on a large set of binary images. 2) Further compression gain is achieved through discarding isolated pixels and 1-pixel rectangles from the image. Simulation results show that the proposed schemes provide significant improvements over previously published work for both the lossy and the lossless components.