We compare a variety of lossless image compression methods on a large sample of astronomical images and show how the compression ratios and speeds of the algorithms are affected by the amount of noise in the images. In the ideal case where the image pixel values have a random Gaussian distribution, the equivalent number of uncompressible noise bits per pixel is given by Nbits =log2(sigma * sqrt(12)) and the lossless compression ratio is given by R = BITPIX / Nbits + K where BITPIX is the bit length of the pixel values and K is a measure of the efficiency of the compression algorithm. We perform image compression tests on a large sample of integer astronomical CCD images using the GZIP compression program and using a newer FITS tiled-image compression method that currently supports 4 compression algorithms: Rice, Hcompress, PLIO, and GZIP. Overall, the Rice compression algorithm strikes the best balance of compression and computational efficiency; it is 2--3 times faster and produces about 1.4 times greater compression than GZIP. The Rice algorithm produces 75%--90% (depending on the amount of noise in the image) as much compression as an ideal algorithm with K = 0. The image compression and uncompression utility programs used in this study (called fpack and funpack) are publicly available from the HEASARC web site. A simple command-line interface may be used to compress or uncompress any FITS image file.
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