Wavelet-based multi-resolution representation has become a cutting-edge technology in the area of image data compression. Though the discrete wavelet transform is closely related to the perfect-reconstruction octave-band filter banks used in subband coding schemes, wavelets have provided very promising new ideas and insights for image data compression. Wavelet-based coding techniques have achieved competitive performance compared with other well-known image coding techniques. This thesis develops a general framework for wavelet-based lossy image coding. We discuss two application problems and develop the corresponding fast algorithms using wavelet-based techniques (e.g., optimum wavelet-packet bases selection, wavelet-domain soft-thresholding).