int QccWAVWaveletRedundantDWT2D(const QccMatrix input_matrix, QccMatrix
*output_matrices, int num_rows, int num_cols, int num_scales, const QccWAVWavelet
*wavelet);
int QccWAVWaveletInverseRedundantDWT2D(const QccMatrix *input_matrices,
QccMatrix output_matrix, int num_rows, int num_cols, int num_scales, const
QccWAVWavelet *wavelet);
int QccWAVWaveletRedundantDWT2DSubsample(const
QccMatrix *input_matrices, QccMatrix output_matrix, int num_rows, int
num_cols, int num_scales, int subsample_pattern_row, int subsample_pattern_col,
const QccWAVWavelet *wavelet);
QccMatrix *QccWAVWaveletRedundantDWT2DAlloc(int
num_rows, int num_cols, int num_scales);
void QccWAVWaveletRedundantDWT2DFree(QccMatrix
*rdwt, int num_rows, int num_scales);
During each level or scale of decomposition, a baseband, a horizontal, a vertical, and a diagonal subband are produced. The baseband subband is then recursively decomposed. Consequently, the 2D RDWT produces num_scales * 3 highpass subbands and one final baseband subband, for a total of num_scales * 3 + 1 subbands. These subbands are returned in output_matrices which is an array of num_scales * 3 + 1 matrices each of which having size num_rows rows and num_cols columns. The first entry in the output_matrices array is the baseband matrix; subsequent entries contain highpass bands of decreasing scale (increasing spatial resolution) in the order horizontal, vertical, and diagonal. Sufficient storage space for output_matrices must be allocated prior to calling QccWAVWaveletRedundantDWT2D().
wavelet can be either a filter-bank or lifting-scheme implementation. In either case, the 2D RDWT is implemented as a separable transform in that each row is decomposed in a fashion similar to the 1D RDWT implemented by QccWAVWaveletRedundantDWT1D(3) followed by a similar 1D decomposition along each column.
QccWAVWaveletInverseRedundantDWT2D() performs the inverse RDWT of input_matrices. Sufficient space for output_matrix must be allocated prior to calling QccWAVWaveletInverseRedundantDWT2D().
QccWAVWaveletRedundantDWT2DAlloc() allocates num_scales * 3 + 1 subband matrices, each of size num_rows by num_cols, returning the allocated subbands. QccWAVWaveletRedundantDWT2DFree() frees the matrices allocated via QccWAVWaveletRedundantDWT2DAlloc().
The RDWT produces an oversampled wavelet transform; that is, an overcomplete expansion of the original signal. However, the coefficients of the usual critically sampled DWT are amongst the RDWT coefficients. More accurately, there exist 4 ^ num_scales sets of RDWT coefficients that are identical to a critically sampled, dyadic DWT. Each one of these 4 ^ num_scales DWTs differs from the others in the subsampling phase choice (one can choose even or odd for both rows and columns) at each level of the transform. QccWAVWaveletRedundantDWT2DSubsample() subsamples the coefficients output from QccWAVWaveletRedundantDWT2D() to obtain the coefficients for a critically sampled DWT. Together, subsample_pattern_row and subsample_pattern_col indicate which of the 4 ^ num_scales DWT-coefficient sets to choose; the resulting critically sampled DWT then consists of the same coefficients that would have been produced by a call to QccWAVWaveletDWT2D(3) with these particular subsample_pattern_row and subsample_pattern_col values. subsample_pattern_row and subsample_pattern_col both can take on values 0 to (2 ^ num_levels) - 1 (see QccWAVWaveletDWT2D(3) ). num_scales gives the number of levels of decomposition that exist in input_matrices, which is assumed to be an array of num_scales * 3 + 1 matrices as produced by QccWAVWaveletRedundantDWT2D(). The output of QccWAVWaveletRedundantDWT2DSubsample() is a matrix of nested subbands as is present in the QccWAVSubbandPyramid(3) structure. Sufficient space for output_matrix must be allocated prior to calling QccWAVWaveletInverseRedundantDWT2DSubsample().
QccWAVWaveletInverseRedundantDWT2D() is the "proper" procedure for inverting the RDWT. That is, QccWAVWaveletInverseRedundantDWT2D() performs inverse filtering as well as weighted "averaging" of the transform redundancies to properly reconstruct the original signal from the RDWT coefficients. A "quick and dirty" reconstruction is possible, however, by subsampling and then inverting using the critically sampled inverse DWT, i.e., by calling QccWAVWaveletRedundantDWT2DSubsample() and then QccWAVWaveletInverseDWT2D(). Both these approaches to inverting the RDWT will produce the same results on unaltered RDWT coefficients; however, should the RDWT coefficients be processed in some fashion, say, through quantization, then QccWAVWaveletInverseRedundantDWT2D() should be used to invert the transform. Additionally, this subsampling approach to inverting the RDWT works only for subsample_pattern = 0, which is the only subsampling pattern whose phase choices match those assumed by QccWAVWaveletInverseDWT2D().
Shensa points out that the Mallat and a trous implementations are very closely related; in fact, it is possible to obtain the coefficients for the Mallat algorithm by subsampling, or decimating, the coefficients resulting from the algorithm a trous, except possibly at the matrix boundaries when symmetric extension is used. This observation leads to an "alternative" implementation of the algorithm a trous, and it is this implementation that is used here in QccWAVWaveletRedundantDWT2D(). This alternative implementation of the algorithm a trous eliminates the above-mentioned symmetric-boundary inconsistencies; in addition, the alternative implementation permits the direct use of lifting instead of filter banks for improved computation efficiency.
M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, "A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform," in Wavelets: Time-Frequency Methods and Phase Space, Berlin: Springer-Verlag, pp. 286-297, 1989.
P. Dutilleux, "An Implementation of the Algorithm A Trous to Compute the Wavelet Transform," in Wavelets: Time-Frequency Methods and Phase Space, Berlin: Springer-Verlag, pp. 298-304, 1989.
M. J. Shensa, "The Discrete Wavelet Transform: Wedding the A Trous and Mallat Algorithms," IEEE Trans. Signal Processing, vol. 40, no. 10, pp. 2464-2482, Oct. 1998.