int QccWAVWaveletDWT1DInt(QccVectorInt signal,
int signal_length, int signal_origin, int subsample_pattern, int num_scales,
const QccWAVWavelet *wavelet);
int QccWAVWaveletInverseDWT1DInt(QccVectorInt
signal, int signal_length, int signal_origin, int subsample_pattern, int
num_scales, const QccWAVWavelet *wavelet);
Essentially, QccWAVWaveletDWT1DInt() calls QccWAVWaveletAnalysis1DInt(3) for each level of decomposition; QccWAVWaveletAnalysis1DInt(3) in turn calls QccWAVLiftingAnalysisInt(3) . signal_origin indicates the sample index at which signal starts. wavelet must indicate an integer-valued lifting scheme (see QccWAVLiftingSchemeInteger(3) ).
QccWAVWaveletInverseDWT1DInt() performs the inverse DWT of signal which is assumed to have been produced by QccWAVWaveletDWT1DInt(). num_scales gives the number of levels of decomposition that exist in signal. Essentially, QccWAVWaveletInverseDWT1DInt() calls QccWAVWaveletSynthesis1DInt(3) for each level of synthesis; QccWAVWaveletSynthesis1DInt(3) in turn calls QccWAVFilterBankSynthesisInt(3) . signal_origin indicates the sample index at which signal starts.
subsample_pattern indicates the even- or odd-phase subsampling to be used at each level of decomposition. In most applications, even subsampling at all levels is desired, in which case subsample_pattern should be set to zero. In more general settings, when some mixture of even- and odd-phase subsampling is desired, subsample_pattern can be an integer between 0 and 2^num_levels - 1. In this integer, the jth bit (where j = 1 is the least-significant bit) indicates whether the jth level of decomposition employs even or odd subsampling (0 = even, 1 = odd). For example, if subsample_pattern is 5, then the first and third decompositions use odd-phase subsampling, while all others use even subsampling.
In traditional floating-point lifting, the prediction and update steps are generally followed by a single application of scaling by a constant in order to produce the usual unitary normalization. This scaling step is somewhat problematic for integer-valued lifting since the scaling constant is usually not an integer. In applications wherein unitary scaling is not required (e.g., in some applications that process each subband completely independently), the scaling step is simply dropped in order to implement an integer-valued version of the transform. Alternatively, one can append three additional lifting steps to implement the scaling; these additional lifting steps can then be rendered integer-valued via appropriate rounding (e.g., Xiong et al.) making the transforms approximately normalized. This latter approach of scaling via additional lifting steps is employed in the integer-valued lifting schemes implemented in QccPack.
A. R. Calderbank, I. Daubechies, W. Sweldens, B.-L. Yeo, "Lossless Image Compression Using Integer to Integer Wavelet Transforms", in Proceedings of the International Conference on Image Processing, Lausanne, Switzerland, pp. 596-599, September 1997.
Z. Xiong, X. Wu, S. Cheng, J. Hua, "Lossy-to-Lossless Compression of Medical Volumetric Data Using Three-Dimensional Integer Wavelet Transforms," IEEE Transactions on Medical Imaging, vol. 22, pp. 459-470, March 2003.
I. Daubechies and W. Sweldens, "Factoring Wavelet Transforms Into Lifting Steps," J. Fourier Anal. Appl., vol. 4, no. 3, pp. 245-267, 1998.