Signal compression can be obtained by wavelet transformation of integer input data followed by quantification and coding. As the quantification is usually lossy, the whole compression/decompression scheme is lossy too. We define a critical wavelet coefficient quantification, i.e., the coarsest quantification that allows perfect reconstruction. This is demonstrated for the Haar transform and for arbitrarily smooth wavelet transforms derived from it. The new integer wavelet transform allows implementation of multiresolution subband compression schemes, in which the decompressed data are gradually refined, retaining the option of perfect reconstruction.