We present lossless and lossy image compression algorithms, based on biorthogonal wavelets, which provide high computational speed and excellent compression performance. A specific pair of spline biorthogonal wavelets are chosen, having dyadic rational filter coefficients; convolutions with these filters can be performed by using only arithmetic bit-shifting and integer addition operations. For lossless compression, we have investigated a new reversible embedded wavelet transform. For lossy compression, we have constructed two-dimensional reconstruction filter masks for carrying out bit-shifting to minimize data access operations when performing the inverse wavelet transform.