This thesis develops two new techniques for grey-scale fingerprint image compression, quadtree decomposition with multifractal analysis (QDMA) and wavelet packet with multifractal analysis (WPMA). Considering the limitations of the human vision system, the QDMA and WPMA use a wavelet transform guided by a multifractal measure to obtain the best reconstructed image in terms of a higher peak signal to noise ratio (PSNR) at the lowest bit rate. The fingerprint images and the corresponding wavelet coefficients are considered to be an approximation of strange attractors and can be analyzed by their multifractality. Wavelets can not only provide the grouping of subband information and the highest compression for optimum bit allocation (quantization), but also an optimum synthesis (combination of subbands) by the inverse wavelet transform to achieve the highest image quality. Using the QDMA technique, the compression ratio can reach 13.95:1 with 28.06 dB PSNR, while the compression ratio of WPMA can exceed 17.7:1 with PSNR up to 28.57 dB. The QDMA and WPMA techniques can make the reconstructed image with good quality for identification or as evidence in court cases. (Abstract shortened by UMI.)