System and method for performing wavelet and inverse wavelet transformations of digital data using semi-orthogonal wavelets

A method of decomposing a set of input coefficients into a first set of output coefficients and a second set of output coefficients by applying a first sequence of decomposition coefficients and a second sequence of decomposition coefficients to the set of input coefficients, the method comprising the steps of:

• deinterleaving the set of input coefficients into a first subset of the set of input coefficients and a second subset of the set of input coefficients;
• applying a first subsequence of the first sequence of coefficients to the first subset of the set of input coefficients to generate a first set of intermediate coefficients;
• applying a first subsequence of the second sequence of coefficients to the first subset of the set of input coefficients to generate a second set of intermediate coefficients;
• summing the first and second sets of intermediate coefficients to generate the first set of output coefficients;
• applying a second subsequence of the first sequence of coefficients to the second subset of the set of input coefficients to generate a third set of intermediate coefficients;
• applying a second subsequence of the second sequence of coefficients to the second subset of the set of input coefficients to generate a fourth set of intermediate coefficients;
• and
• summing the third and fourth sets of intermediate coefficients to generate the second set of output coefficients.

A reconstruction filter stage for reconstructing a first set of input coefficients and a second set of input coefficients into a set of output coefficients by applying a first sequence of coefficients and a second sequence of coefficients to the first and second sets of input coefficients, the reconstruction filter stage comprising:

• a first filter that applies a first subsequence of the first sequence of coefficients to the first set of input coefficients to generate a first set of intermediate coefficients;
• a second filter that applies a first subsequence of the second sequence of coefficients to the first set of input coefficients to generate a second set of intermediate coefficients;
• a first summer that sums the first and second sets of intermediate coefficients to generate a first subset of the set of output coefficients;
• a third filter that applies a second subsequence of the first sequence of coefficients to the second set of input coefficients to generate a third set of intermediate coefficients;
• a fourth filter that applies a second subsequence of the second sequence of coefficients to the second set of input coefficients to generate a fourth set of intermediate coefficients;
• a second summer that sums the third and fourth sets of intermediate coefficients to generate a second subset of the set of output coefficients;
• and
• an interleaver that interleaves the first and second subsets of the set of output coefficients to generate the set of output coefficients.

A method of reconstructing a first set of input coefficients and a second set of input coefficients into a set of output coefficients by applying a first sequence of coefficients and a second sequence of coefficients to the first and second sets of input coefficients, the method comprising the steps of:

• applying a first subsequence of the first sequence of coefficients to the first set of input coefficients to generate a first set of intermediate coefficients;
• applying a first subsequence of the second sequence of coefficients to the first set of input coefficients to generate a second set of intermediate coefficients;
• summing the first and second sets of intermediate coefficients to generate a first subset of the set of output coefficients;
• applying a second subsequence of the first sequence of coefficients to the second set of input coefficients to generate a third set of intermediate coefficients;
• applying a second subsequence of the second sequence of coefficients to the second set of input coefficients to generate a fourth set of intermediate coefficients;
• summing the third and fourth sets of intermediate coefficients to generate a second subset of the set of output coefficients;
• and
• interleaving the first and second subsets of the set of output coefficients to generate the set of output coefficients.

A pre-decomposition filter of a wavelet transform system for mapping a set of original data samples into a set of dual scaling function coefficients for decomposition by the wavelet transform system, the sets of original data samples and dual scaling function coefficients being related to a set of scaling function coefficients, to scaling functions that correspond to the set of scaling function coefficients, and to dual scaling functions that are dual to the scaling functions and correspond to the set of dual scaling function coefficients, the pre-decomposition filter comprising:

• a first filter stage that filters the set of original data samples to generate a first set of intermediate coefficients;
• a second filter stage that is parallel to the first filter and filters the set of original data samples to generate a second set of intermediate coefficients;
• and
• a summing stage that sums the first and second sets of intermediate data samples to generate the set of dual scaling function coefficients.

A method in a wavelet transform system for mapping a set of original data samples into a set of dual scaling function coefficients for decomposition by the wavelet transform system, the sets of original data samples and dual scaling function coefficients being related to a set of scaling function coefficients, to scaling functions that correspond to the set of scaling function coefficients, and to dual scaling functions that are dual to the scaling functions and correspond to the set of dual scaling function coefficients, the method comprising the steps of:

• filtering the set of original data samples to generate a first set of intermediate coefficients;
• filtering the set of original data samples to generate a second set of intermediate coefficients;
• and
• summing the first and second sets of intermediate data samples to generate the set of dual scaling function coefficients.

A reconstruction filter stage for reconstructing a set of dual scaling function coefficients and a set of dual wavelet coefficients at a lower resolution level into a set of dual scaling function coefficients at a next higher resolution level in an inverse wavelet transform, the inverse wavelet transform using semi-orthogonal standard wavelets as the basic wavelets in the inverse wavelet transform and using related standard scaling functions as the basic scaling functions in the inverse wavelet transform, the standard scaling functions at the lower resolution level being related to the standard scaling functions at the next higher resolution level by a first finite sequence of reconstruction coefficients in a first two-scale relation, the semi-orthogonal standard wavelets at the lower resolution level being related to the standard scaling functions at the next higher resolution level by a second finite sequence of reconstruction coefficients in a second two-scale relation, the reconstruction filter stage comprising:

• a first mapping filter substage to apply a sequence of mapping coefficients to the set of dual scaling function coefficients at the lower resolution level to generate a set of standard scaling function coefficients at the lower resolution level;
• a second mapping filter substage to apply the sequence of mapping coefficients to the set of dual wavelet coefficients at the lower resolution level to generate a set of formatted wavelet coefficients at the lower resolution level;
• and
• a reconstruction filter substage to reconstruct the sets of standard scaling function and formatted wavelet coefficients at the lower resolution level into the set of dual scaling function coefficients at the next higher resolution level by applying a third finite sequence of reconstruction coefficients to the set of standard scaling function coefficients and a fourth finite sequence of reconstruction coefficients to the set of formatted wavelet coefficients, the third finite sequence of reconstruction coefficients being derived from the second finite sequence of reconstruction coefficients and the fourth finite sequence of reconstruction coefficients being derived from the first finite sequence of reconstruction coefficients.

A method-of reconstructing a set of dual scaling function coefficients and a set of dual wavelet coefficients at a lower resolution level into a set of dual scaling function coefficients at a next higher resolution level in an inverse wavelet transform, the method using semi-orthogonal standard wavelets as the basic wavelets in the inverse wavelet transform and using related standard scaling functions as the basic scaling functions in the inverse wavelet transform, the standard scaling functions at the lower resolution level being related to the standard scaling functions at the next higher resolution level by a first finite sequence of reconstruction coefficients in a first two-scale relation, the semi-orthogonal standard wavelets at the lower resolution level being related to the standard scaling functions at the next higher resolution level by a second finite sequence of reconstruction coefficients in a second two-scale relation, the method comprising the steps:

• applying a sequence of mapping coefficients to the set of dual scaling function coefficients at the lower resolution level to generate a set of standard scaling function coefficients at the lower resolution level;
• applying the sequence of mapping coefficients to the set of dual wavelet coefficients at the lower resolution level to generate a set of formatted wavelet coefficients at the lower resolution level;
• and
• reconstructing the sets of standard scaling function and formatted wavelet coefficients at the lower resolution level into the set of dual scaling function coefficients at the next higher resolution level by applying a third finite sequence of reconstruction coefficients to the set of standard scaling function coefficients and a fourth finite sequence of reconstruction coefficients to the set of formatted wavelet coefficients, the third finite sequence of reconstruction coefficients being derived from the second finite sequence of reconstruction coefficients and the fourth finite sequence of reconstruction coefficients being derived from the first finite sequence of reconstruction coefficients.