Daubechies D4 wavelet transform (D4 denotes four coefficients)

I have to confess up front that the comment here does not even come
close to describing wavelet algorithms and the Daubechies D4
algorithm in particular. I don't think that it can be described in
anything less than a journal article or perhaps a book. I even have
to apologize for the notation I use to describe the algorithm, which
is barely adequate. But explaining the correct notation would take
a fair amount of space as well. This comment really represents some
notes that I wrote up as I implemented the code. If you are
unfamiliar with wavelets I suggest that you look at the bearcave.com
web pages and at the wavelet literature. I have yet to see a really
good reference on wavelets for the software developer. The best
book I can recommend is *Ripples in Mathematics* by Jensen and
Cour-Harbo.

All wavelet algorithms have two components, a wavelet function and a scaling function. These are sometime also referred to as high pass and low pass filters respectively.

The wavelet function is passed two or more samples and calculates a wavelet coefficient. In the case of the Haar wavelet this is

coef_{i}= odd_{i}- even_{i}or coef_{i}= 0.5 * (odd_{i}- even_{i})

depending on the version of the Haar algorithm used.

The scaling function produces a smoother version of the original data. In the case of the Haar wavelet algorithm this is an average of two adjacent elements.

The Daubechies D4 wavelet algorithm also has a wavelet
and a scaling function. The coefficients for the
scaling function are denoted as h_{i} and the
wavelet coefficients are g_{i}.

Mathematicians like to talk about wavelets in terms of a wavelet algorithm applied to an infinite data set. In this case one step of the forward transform can be expressed as the infinite matrix of wavelet coefficients represented below multiplied by the infinite signal vector.

a_{i}= ...h0,h1,h2,h3, 0, 0, 0, 0, 0, 0, 0, ... s_{i}c_{i}= ...g0,g1,g2,g3, 0, 0, 0, 0, 0, 0, 0, ... s_{i+1}a_{i+1}= ...0, 0, h0,h1,h2,h3, 0, 0, 0, 0, 0, ... s_{i+2}c_{i+1}= ...0, 0, g0,g1,g2,g3, 0, 0, 0, 0, 0, ... s_{i+3}a_{i+2}= ...0, 0, 0, 0, h0,h1,h2,h3, 0, 0, 0, ... s_{i+4}c_{i+2}= ...0, 0, 0, 0, g0,g1,g2,g3, 0, 0, 0, ... s_{i+5}a_{i+3}= ...0, 0, 0, 0, 0, 0, h0,h1,h2,h3, 0, ... s_{i+6}c_{i+3}= ...0, 0, 0, 0, 0, 0, g0,g1,g2,g3, 0, ... s_{i+7}

The dot product (inner product) of the infinite vector and
a row of the matrix produces either a smoother version of the
signal (a_{i}) or a wavelet coefficient (c_{i}).

In an ordered wavelet transform, the smoothed (a_{i}) are
stored in the first half of an *n* element array region. The
wavelet coefficients (c_{i}) are stored in the second half
the *n* element region. The algorithm is recursive. The
smoothed values become the input to the next step.

The transpose of the forward transform matrix above is used to calculate an inverse transform step. Here the dot product is formed from the result of the forward transform and an inverse transform matrix row.

s_{i}= ...h2,g2,h0,g0, 0, 0, 0, 0, 0, 0, 0, ... a_{i}s_{i+1}= ...h3,g3,h1,g1, 0, 0, 0, 0, 0, 0, 0, ... c_{i}s_{i+2}= ...0, 0, h2,g2,h0,g0, 0, 0, 0, 0, 0, ... a_{i+1}s_{i+3}= ...0, 0, h3,g3,h1,g1, 0, 0, 0, 0, 0, ... c_{i+1}s_{i+4}= ...0, 0, 0, 0, h2,g2,h0,g0, 0, 0, 0, ... a_{i+2}s_{i+5}= ...0, 0, 0, 0, h3,g3,h1,g1, 0, 0, 0, ... c_{i+2}s_{i+6}= ...0, 0, 0, 0, 0, 0, h2,g2,h0,g0, 0, ... a_{i+3}s_{i+7}= ...0, 0, 0, 0, 0, 0, h3,g3,h1,g1, 0, ... c_{i+3}

Using a standard dot product is grossly inefficient since most of the operands are zero. In practice the wavelet coefficient values are moved along the signal vector and a four element dot product is calculated. Expressed in terms of arrays, for the forward transform this would be:

a_{i}= s[i]*h0 + s[i+1]*h1 + s[i+2]*h2 + s[i+3]*h3 c_{i}= s[i]*g0 + s[i+1]*g1 + s[i+2]*g2 + s[i+3]*g3

This works fine if we have an infinite data set, since we don't have to worry about shifting the coefficients "off the end" of the signal.

I sometimes joke that I left my infinite data set in my other bear suit. The only problem with the algorithm described so far is that we don't have an infinite signal. The signal is finite. In fact not only must the signal be finite, but it must have a power of two number of elements.

If i=N-1, the i+2 and i+3 elements will be beyond the end of the array. There are a number of methods for handling the wavelet edge problem. This version of the algorithm acts like the data is periodic, where the data at the start of the signal wraps around to the end.

This algorithm uses a temporary array. A Lifting Scheme version of
the Daubechies D4 algorithm does not require a temporary. The
matrix discussion above is based on material from *Ripples in
Mathematics*, by Jensen and Cour-Harbo. Any error are mine.

**Author**: Ian Kaplan

**Use**: You may use this software for any purpose as long
as I cannot be held liable for the result. Please credit me
with authorship if use use this source code.

This comment is formatted for the doxygen documentation generator

*/ class Daubechies { private: /** forward transform scaling coefficients */ double h0, h1, h2, h3; /** forward transform wave coefficients */ double g0, g1, g2, g3; double Ih0, Ih1, Ih2, Ih3; double Ig0, Ig1, Ig2, Ig3; /** Forward Daubechies D4 transform */ void transform( double* a, const int n ) { if (n >= 4) { int i, j; const int half = n >> 1; double* tmp = new double[n]; for (i = 0, j = 0; j < n-3; j += 2, i++) { tmp[i] = a[j]*h0 + a[j+1]*h1 + a[j+2]*h2 + a[j+3]*h3; tmp[i+half] = a[j]*g0 + a[j+1]*g1 + a[j+2]*g2 + a[j+3]*g3; } tmp[i] = a[n-2]*h0 + a[n-1]*h1 + a[0]*h2 + a[1]*h3; tmp[i+half] = a[n-2]*g0 + a[n-1]*g1 + a[0]*g2 + a[1]*g3; for (i = 0; i < n; i++) { a[i] = tmp[i]; } delete [] tmp; } } /** Inverse Daubechies D4 transform */ void invTransform( double* a, const int n ) { if (n >= 4) { int i, j; const int half = n >> 1; const int halfPls1 = half + 1; double* tmp = new double[n]; // last smooth val last coef. first smooth first coef tmp[0] = a[half-1]*Ih0 + a[n-1]*Ih1 + a[0]*Ih2 + a[half]*Ih3; tmp[1] = a[half-1]*Ig0 + a[n-1]*Ig1 + a[0]*Ig2 + a[half]*Ig3; for (i = 0, j = 2; i < half-1; i++) { // smooth val coef. val smooth val coef. val tmp[j++] = a[i]*Ih0 + a[i+half]*Ih1 + a[i+1]*Ih2 + a[i+halfPls1]*Ih3; tmp[j++] = a[i]*Ig0 + a[i+half]*Ig1 + a[i+1]*Ig2 + a[i+halfPls1]*Ig3; } for (i = 0; i < n; i++) { a[i] = tmp[i]; } delete [] tmp; } } public: Daubechies() { const double sqrt_3 = sqrt( 3 ); const double denom = 4 * sqrt( 2 ); // // forward transform scaling (smoothing) coefficients // h0 = (1 + sqrt_3)/denom; h1 = (3 + sqrt_3)/denom; h2 = (3 - sqrt_3)/denom; h3 = (1 - sqrt_3)/denom; // // forward transform wavelet coefficients // g0 = h3; g1 = -h2; g2 = h1; g3 = -h0; Ih0 = h2; Ih1 = g2; // h1 Ih2 = h0; Ih3 = g0; // h3 Ig0 = h3; Ig1 = g3; // -h0 Ig2 = h1; Ig3 = g1; // -h2 } void daubTrans( double* ts, int N ) { int n; for (n = N; n >= 4; n >>= 1) { transform( ts, n ); } } void invDaubTrans( double* coef, int N ) { int n; for (n = 4; n <= N; n <<= 1) { invTransform( coef, n ); } } }; // Daubechies