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Daubechies Wavelets and Scaling Functions

Synopsis
```#include <boost/math/special_functions/daubechies_scaling.hpp>

namespace boost::math {

template<class Real, int p>
class daubechies_scaling {
public:
daubechies_scaling(int grid_refinements = -1);

inline Real operator()(Real x) const;

inline Real prime(Real x) const;

inline Real double_prime(Real x) const;

std::pair<Real, Real> support() const;

int64_t bytes() const;
};

template<class Real, int p, int order>
std::vector<Real> dyadic_grid(int64_t j_max);

#include <boost/math/special_functions/daubechies_wavelet.hpp>
template<class Real, int p>
class daubechies_wavelet {
public:
daubechies_wavelet(int grid_refinements = -1);

inline Real operator()(Real x) const;

inline Real prime(Real x) const;

inline Real double_prime(Real x) const;

std::pair<Real, Real> support() const;

int64_t bytes() const;
};

} // namespaces
```

Daubechies wavelets and scaling functions are a family of compactly supported functions indexed by an integer p which have p vanishing moments and an associated filter of length 2p. They are used in signal denoising, Galerkin methods for PDEs, and compression.

The canonical reference on these functions is Daubechies' monograph Ten Lectures on Wavelets, whose notational conventions we attempt to follow here.

A basic usage is as follows:

```auto phi = boost::math::daubechies_scaling<double, 8>();
double y = phi(0.38);
double dydx = phi.prime(0.38);

auto psi = boost::math::daubechies_wavelet<double, 8>();
y = psi(0.38);
```

Note that the constructor call is expensive, as it must assemble a dyadic grid--values of pφ at dyadic rationals, i.e., numbers of the form n/2j. You should only instantiate this class once in the duration of a program. The class is pimpl'd and all its member functions are threadsafe, so it can be copied cheaply and shared between threads. The default number of grid refinements is chosen so that the relative error is controlled to ~2-3 ULPs away from the right-hand side of the support, where superexponential growth of the condition number of function evaluation makes this impossible. However, controlling relative error of Daubechies wavelets and scaling functions is much more difficult than controlling absolute error, and the memory consumption is much higher in relative mode. The memory consumption of the class can be queried via

```int64_t mem = phi.bytes();
```

and if this is deemed unacceptably large, the user may choose to control absolute error via calling the constructor with the `grid_refinements` parameter set to -2, so

```auto phi = boost::math::daubechies_scaling<double, 8>(-2);
```

gives a scaling function which keeps the absolute error bounded by roughly the double precision unit roundoff.

If context precludes the ability to reuse the class throughout the program, it makes sense to reduce the accuracy even further. This can be done by specifying the grid refinements, for example,

```auto phi = boost::math::daubechies_scaling<double, 8>(12);
```

creates a Daubechies scaling function interpolated from a dyadic grid computed down to depth j = 12. The call to the constructor is exponential time in the number of grid refinements, and the call operator, `.prime`, and `.double_prime` are constant time.

Note that the only reason that this is a class, rather than a free function is that the dyadic grids would make the Boost source download extremely large. Hence, it may make sense to precompute the dyadic grid and dump it in a `.cpp` file; this can be achieved via

```using boost::multiprecision::float128;
int grid_refinements = 12;
constexpr const derivative = 0;
constexpr const p = 8;
std::vector<float128> v = boost::math::dyadic_grid<float128, p, derivative>(grid_refinements);
```

Note that quad precision is the most accurate precision provided, for both the dyadic grid and for the scaling function. 1ULP accuracy can only be achieved for float and double precision, in well-conditioned regions.

Derivatives are only available if the wavelet and scaling function has sufficient smoothness. The compiler will gladly inform you of your error if you try to call `.prime` on 2φ, which is not differentiable, but be aware that smoothness increases with the number of vanishing moments.

The axioms of a multiresolution analysis ensure that integer shifts of the scaling functions are elements of the multiresolution analysis; a side effect is that the supports of the (unshifted) wavelet and scaling functions are arbitrary. For this reason, we have provided `.support()` so that you can check our conventions:

```auto [a, b] = phi.support();
```

For definiteness though, for the scaling function, the support is always [0, 2p - 1], and the support of the wavelet is [ -p + 1, p].

The 2 vanishing moment scaling function.

The 8 vanishing moment scaling function.

Boost.Math also provides numerical evaluation of the Fourier transform of these functions. This is useful in sparse recovery problems where the measurements are taken in the Fourier basis. The usage is exhibited below:

```#include <boost/math/special_functions/fourier_transform_daubechies_scaling.hpp>
using boost::math::fourier_transform_daubechies_scaling;
// Evaluate the Fourier transform of the 4-vanishing moment Daubechies scaling function at ω=1.8:
std::complex<float> hat_phi = fourier_transform_daubechies_scaling<float, 4>(1.8f);
```

The Fourier transform convention is unitary with the sign of the imaginary unit being given in Daubechies Ten Lectures. In particular, this means that ```fourier_transform_daubechies_scaling<float, p>(0.0)``` returns 1/sqrt(2π).

The implementation computes an infinite product of trigonometric polynomials as can be found from recursive application of the identity 𝓕[φ](ω) = m(ω/2)𝓕[φ](ω/2). This is neither particularly fast nor accurate, but there appears to be no literature on this extremely useful topic, and hence the naive method must suffice.

A benchmark can be found in `reporting/performance/fourier_transform_daubechies_performance.cpp`; the results on a ~2021 M1 Macbook pro are presented below:

Run on (10 X 24.1212 MHz CPU s) CPU Caches: L1 Data 64 KiB (x10) L1 Instruction 128 KiB (x10) L2 Unified 4096 KiB (x5) Load Average: 1.33, 1.52, 1.62 ----------------------------------------------------------- Benchmark Time ----------------------------------------------------------- FourierTransformDaubechiesScaling<double, 1> 70.3 ns FourierTransformDaubechiesScaling<double, 2> 330 ns FourierTransformDaubechiesScaling<double, 3> 335 ns FourierTransformDaubechiesScaling<double, 4> 364 ns FourierTransformDaubechiesScaling<double, 5> 386 ns FourierTransformDaubechiesScaling<double, 6> 436 ns FourierTransformDaubechiesScaling<double, 7> 447 ns FourierTransformDaubechiesScaling<double, 8> 473 ns FourierTransformDaubechiesScaling<double, 9> 503 ns FourierTransformDaubechiesScaling<double, 10> 554 ns

Due to the low accuracy of this method, `float` precision is arg-promoted to `double`, and hence takes just as long as `double` precision to execute.

References

• Daubechies, Ingrid. Ten Lectures on Wavelets. Vol. 61. Siam, 1992.
• Mallat, Stephane. A Wavelet Tour of Signal Processing: the sparse way Academic press, 2008.
• Thompson, Nicholas, Maddock, John et al. Towards 1ULP Evaluation of Daubechies Wavelets https://arxiv.org/ftp/arxiv/papers/2005/2005.05424.pdf