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Numerical evaluation of Fourier transform of Daubechies scaling funct… (#921)
* Numerical evaluation of Fourier transform of Daubechies scaling functions. * Update example/calculate_fourier_transform_daubechies_constants.cpp Co-authored-by: Matt Borland <matt@mattborland.com> * Update example/fourier_transform_daubechies_ulp_plot.cpp Co-authored-by: Matt Borland <matt@mattborland.com> * Update include/boost/math/special_functions/fourier_transform_daubechies_scaling.hpp Co-authored-by: Matt Borland <matt@mattborland.com> * Update include/boost/math/special_functions/fourier_transform_daubechies_scaling.hpp Co-authored-by: Matt Borland <matt@mattborland.com> * Rename include file to reflect it implements both the scaling and wavelet. * Add performance to docs. * Update test/math_unit_test.hpp Co-authored-by: Matt Borland <matt@mattborland.com> * Add boost-no-inspect to files with non-ASCII characters. --------- Co-authored-by: Matt Borland <matt@mattborland.com>
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doc/graphs/fourier_transform_daubechies.png
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doc/graphs/fourier_transform_daubechies.png
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@@ -127,6 +127,48 @@ The 2 vanishing moment scaling function.
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[$../graphs/daubechies_8_scaling.svg]
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The 8 vanishing moment scaling function.
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Boost.Math also provides numerical evaluation of the Fourier transform of these functions.
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This is useful in sparse recovery problems where the measurements are taken in the Fourier basis.
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The usage is exhibited below:
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#include <boost/math/special_functions/fourier_transform_daubechies_scaling.hpp>
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using boost::math::fourier_transform_daubechies_scaling;
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// Evaluate the Fourier transform of the 4-vanishing moment Daubechies scaling function at ω=1.8:
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std::complex<float> hat_phi = fourier_transform_daubechies_scaling<float, 4>(1.8f);
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The Fourier transform convention is unitary with the sign of the imaginary unit being given in Daubechies Ten Lectures.
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In particular, this means that `fourier_transform_daubechies_scaling<float, p>(0.0)` returns 1/sqrt(2π).
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The implementation computes an infinite product of trigonometric polynomials as can be found from recursive application of the identity 𝓕[φ](ω) = m(ω/2)𝓕[φ](ω/2).
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This is neither particularly fast nor accurate, but there appears to be no literature on this extremely useful topic, and hence the naive method must suffice.
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[$../graphs/fourier_transform_daubechies.png]
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A benchmark can be found in `reporting/performance/fourier_transform_daubechies_performance.cpp`; the results on a ~2021 M1 Macbook pro are presented below:
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Run on (10 X 24.1212 MHz CPU s)
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CPU Caches:
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L1 Data 64 KiB (x10)
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L1 Instruction 128 KiB (x10)
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L2 Unified 4096 KiB (x5)
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Load Average: 1.33, 1.52, 1.62
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-----------------------------------------------------------
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Benchmark Time
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-----------------------------------------------------------
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FourierTransformDaubechiesScaling<double, 1> 70.3 ns
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FourierTransformDaubechiesScaling<double, 2> 330 ns
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FourierTransformDaubechiesScaling<double, 3> 335 ns
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FourierTransformDaubechiesScaling<double, 4> 364 ns
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FourierTransformDaubechiesScaling<double, 5> 386 ns
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FourierTransformDaubechiesScaling<double, 6> 436 ns
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FourierTransformDaubechiesScaling<double, 7> 447 ns
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FourierTransformDaubechiesScaling<double, 8> 473 ns
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FourierTransformDaubechiesScaling<double, 9> 503 ns
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FourierTransformDaubechiesScaling<double, 10> 554 ns
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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.
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[heading References]
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* Daubechies, Ingrid. ['Ten Lectures on Wavelets.] Vol. 61. Siam, 1992.
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43
example/calculate_fourier_transform_daubechies_constants.cpp
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43
example/calculate_fourier_transform_daubechies_constants.cpp
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@@ -0,0 +1,43 @@
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// (C) Copyright Nick Thompson 2023.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#include <utility>
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#include <boost/math/filters/daubechies.hpp>
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#include <boost/math/tools/polynomial.hpp>
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#include <boost/multiprecision/cpp_bin_float.hpp>
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#include <boost/math/constants/constants.hpp>
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using std::pow;
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using boost::multiprecision::cpp_bin_float_100;
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using boost::math::filters::daubechies_scaling_filter;
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using boost::math::tools::polynomial;
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using boost::math::constants::half;
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using boost::math::constants::root_two;
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template<typename Real, size_t N>
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std::vector<Real> get_constants() {
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auto h = daubechies_scaling_filter<cpp_bin_float_100, N>();
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auto p = polynomial<cpp_bin_float_100>(h.begin(), h.end());
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auto q = polynomial({half<cpp_bin_float_100>(), half<cpp_bin_float_100>()});
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q = pow(q, N);
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auto l = p/q;
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return l.data();
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}
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template<typename Real>
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void print_constants(std::vector<Real> const & l) {
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std::cout << std::setprecision(std::numeric_limits<Real>::digits10 -10);
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std::cout << "return std::array<Real, " << l.size() << ">{";
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for (size_t i = 0; i < l.size() - 1; ++i) {
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std::cout << "BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits, " << l[i]/root_two<Real>() << "), ";
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}
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std::cout << "BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits, " << l.back()/root_two<Real>() << ")};\n";
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}
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int main() {
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auto constants = get_constants<cpp_bin_float_100, 1>();
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print_constants(constants);
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}
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60
example/fourier_transform_daubechies_ulp_plot.cpp
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60
example/fourier_transform_daubechies_ulp_plot.cpp
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@@ -0,0 +1,60 @@
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// boost-no-inspect
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// (C) Copyright Nick Thompson 2023.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#include <boost/math/special_functions/fourier_transform_daubechies.hpp>
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#include <boost/math/tools/ulps_plot.hpp>
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using boost::math::fourier_transform_daubechies_scaling;
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using boost::math::tools::ulps_plot;
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template<int p>
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void real_part() {
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auto phi_real_hi_acc = [](double omega) {
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auto z = fourier_transform_daubechies_scaling<double, p>(omega);
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return z.real();
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};
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auto phi_real_lo_acc = [](float omega) {
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auto z = fourier_transform_daubechies_scaling<float, p>(omega);
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return z.real();
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};
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auto plot = ulps_plot<decltype(phi_real_hi_acc), double, float>(phi_real_hi_acc, float(0.0), float(100.0), 20000);
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plot.ulp_envelope(false);
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plot.add_fn(phi_real_lo_acc);
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plot.clip(100);
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plot.title("Accuracy of 𝔑(𝓕[𝜙](ω)) with " + std::to_string(p) + " vanishing moments.");
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plot.write("real_ft_daub_scaling_" + std::to_string(p) + ".svg");
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}
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template<int p>
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void imaginary_part() {
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auto phi_imag_hi_acc = [](double omega) {
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auto z = fourier_transform_daubechies_scaling<double, p>(omega);
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return z.imag();
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};
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auto phi_imag_lo_acc = [](float omega) {
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auto z = fourier_transform_daubechies_scaling<float, p>(omega);
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return z.imag();
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};
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auto plot = ulps_plot<decltype(phi_imag_hi_acc), double, float>(phi_imag_hi_acc, float(0.0), float(100.0), 20000);
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plot.ulp_envelope(false);
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plot.add_fn(phi_imag_lo_acc);
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plot.clip(100);
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plot.title("Accuracy of 𝕴(𝓕[𝜙](ω)) with " + std::to_string(p) + " vanishing moments.");
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plot.write("imag_ft_daub_scaling_" + std::to_string(p) + ".svg");
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}
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int main() {
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real_part<3>();
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imaginary_part<3>();
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real_part<6>();
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imaginary_part<6>();
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return 0;
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}
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@@ -0,0 +1,248 @@
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// boost-no-inspect
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/*
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* Copyright Nick Thompson, Matt Borland, 2023
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* Use, modification and distribution are subject to the
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* Boost Software License, Version 1.0. (See accompanying file
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* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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*/
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#ifndef BOOST_MATH_SPECIAL_FOURIER_TRANSFORM_DAUBECHIES_HPP
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#define BOOST_MATH_SPECIAL_FOURIER_TRANSFORM_DAUBECHIES_HPP
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#include <array>
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#include <cmath>
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#include <complex>
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#include <iostream>
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#include <limits>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/tools/big_constant.hpp>
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#include <boost/math/tools/estrin.hpp>
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namespace boost::math {
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namespace detail {
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// See the Table 6.2 of Daubechies, Ten Lectures on Wavelets.
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// These constants are precisely those divided by 1/sqrt(2), because otherwise
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// we'd immediately just have to divide through by 1/sqrt(2).
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// These numbers agree with Table 6.2, but are generated via example/calculate_fourier_transform_daubechies_constants.cpp
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template <typename Real, unsigned N> constexpr std::array<Real, N> ft_daubechies_scaling_polynomial_coefficients() {
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static_assert(N >= 1 && N <= 10, "Scaling function only implemented for 1-10 vanishing moments.");
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if constexpr (N == 1) {
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return std::array<Real, 1>{static_cast<Real>(1)};
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}
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if constexpr (N == 2) {
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return {BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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1.36602540378443864676372317075293618347140262690519031402790348972596650842632007803393058),
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.366025403784438646763723170752936183471402626905190314027903489725966508441952115116994061)};
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}
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if constexpr (N == 3) {
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return std::array<Real, 3>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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1.88186883113665472301331643028468183320710177910151845853383427363197699204347143889269703),
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-1.08113883008418966599944677221635926685977756966260841342875242639629721931484516409937898),
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
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0.199269998947534942986130341931677433652675790561089954894918152764320227250084833874126086)};
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}
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if constexpr (N == 4) {
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return std::array<Real, 4>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
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2.60642742441038678619616138456320274846457112268350230103083547418823666924354637907021821),
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
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-2.33814397690691624172277875654682595239896411009843420976312905955518655953831321619717516),
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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0.851612467139421235087502761217605775743179492713667860409024360383174560120738199344383827),
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
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-0.119895914642891779560885389233982571808786505298735951676730775016224669960397338539830347)};
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}
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if constexpr (N == 5) {
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return std::array<Real, 5>{
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
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3.62270372133693372237431371824382790538377237674943454540758419371854887218301659611796287),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-4.45042192340421529271926241961545172940077367856833333571968270791760393243895360839974479),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
2.41430351179889241160444590912469777504146155873489898274561148139247721271772284677196254),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-0.662064156756696785656360678859372223233256033099757083735935493062448802216759690564503751),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
0.0754788470250859443968634711062982722087957761837568913024225258690266500301041274151679859)};
|
||||
}
|
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if constexpr (N == 6) {
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return std::array<Real, 6>{
|
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BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
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5.04775782409284533508504459282823265081102702143912881539214595513121059428213452194161891),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-7.90242489414953082292172067801361411066690749603940036372954720647258482521355701761199),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
5.69062231972011992229557724635729642828799628244009852056657089766265949751788181912632318),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-2.29591465417352749013350971621495843275025605194376564457120763045109729714936982561585742),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
0.508712486289373262241383448555327418882885930043157873517278143590549199629822225076344289),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-0.0487530817792802065667748935122839545647456859392192011752401594607371693280512344274717466)};
|
||||
}
|
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if constexpr (N == 7) {
|
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return std::array<Real, 7>{
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
7.0463635677199166580912954330590360004554457287730448872409828895500755049108034478397642),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-13.4339028220058085795120274851204982381087988043552711869584397724404274044947626280185946),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
12.0571882966390397563079887516068140052534768286900467252199152570563053103366694003818755),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-6.39124482303930285525880162640679389779540687632321120940980371544051534690730897661850842),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
2.07674879424918331569327229402057948161936796436510457676789758815816492768386639712643599),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-0.387167532162867697386347232520843525988806810788254462365009860280979111139408537312553398),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
0.0320145185998394020646198653617061745647219696385406695044576133973761206215673170563538)};
|
||||
}
|
||||
if constexpr (N == 8) {
|
||||
return std::array<Real, 8>{
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
9.85031962984351656604584909868313752909650830419035084214249929687665775818153930511533915),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-22.1667494032601530437943449172929277733925779301673358406203340024653233856852379126537395),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
23.8272728452144265698978643079553442578633838793866258585693705776047828901217069807060715),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-15.6065825916019064469551268429136774427686552695820632173344334583910793479437661751737998),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
6.63923943761238270605338141020386331691362835005178161341935720370310013774320917891051914),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-1.81462830704498058848677549516134095104668450780318379608495409574150643627578462439190617),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
0.292393958692487086036895445298600849998803161432207979583488595754566344585039785927586499),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-0.0212655694557728487977430067729997866644059875083834396749941173411979591559303697954912042)};
|
||||
}
|
||||
if constexpr (N == 9) {
|
||||
return std::array<Real, 9>{
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
13.7856894948673536752299497816200874595462540239049618127984616645562437295073582057283235),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-35.79362367743347676734569335180426263053917566987500206688713345532850076082533131311371),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
44.8271517576868325408174336351944130389504383168376658969692365144162452669941793147313),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-34.9081281226625998193992072777004811412863069972654446089639166067029872995118090115016879),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
18.2858070519930071738884732413420775324549836290768317032298177553411077249931094333824682),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-6.53714271572640296907117142447372145396492988681610221640307755553450246302777187366825001),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
1.5454286423270706293059630490222623728433659436325762803842722481655127844136128434034519),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-0.219427682644567750633335191213222483839627852234602683427115193605056655384931679751929029),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
0.0142452515927832872075875380128473058349984927391158822994546286919376896668596927857450578)};
|
||||
}
|
||||
if constexpr (N == 10) {
|
||||
return std::array<Real, 10>{
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
19.3111846872275854185286532829110292444580572106276740012656292351880418629976266671349603),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-56.8572892818288577904562616825768121532988312416110097001327598719988644787442373891037268),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
81.3040184941182201969442916535886223134891624078921290339772790298979750863332417443823932),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-73.3067370305702272426402835488383512315892354877130132060680994033122368453226804355121917),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
45.5029913577892585869595005785056707790215969761054467083138479721524945862678794713356742),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-20.0048938122958245128650205249242185678760602333821352917865992073643758821417211689052482),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
6.18674372398711325312495154772282340531430890354257911422818567803548535981484584999007723),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-1.29022235346655645559407302793903682217361613280994725979138999393113139183198020070701239),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
0.16380852384056875506684562409582514726612462486206657238854671180228210790016298829595125),
|
||||
BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits,
|
||||
-0.00960430880128020906860390254555211461150702751378997239464015046967050703218076318595987803)};
|
||||
}
|
||||
}
|
||||
|
||||
} // namespace detail
|
||||
|
||||
/*
|
||||
* Given ω∈ℝ, computes a numerical approximation to 𝓕[𝜙](ω),
|
||||
* where 𝜙 is the Daubechies scaling function.
|
||||
* Fast and accurate evaluation of these function seems to me to be a rather involved research project,
|
||||
* which I have not endeavored to complete.
|
||||
* In particular, recovering ~1ULP evaluation is not possible using the techniques
|
||||
* employed here-you should use this with the understanding it is good enough for almost
|
||||
* all uses with empirical data, but probably doesn't recover enough accuracy
|
||||
* for pure mathematical uses (other than graphing-in which case it's fine).
|
||||
* The implementation uses an infinite product of trigonometric polynomials.
|
||||
* See Daubechies, 10 Lectures on Wavelets, equation 5.1.17, 5.1.18.
|
||||
* It uses the factorization of m₀ shown in Corollary 5.5.4 and equation 5.5.5.
|
||||
* See more discusion near equation 6.1.1,
|
||||
* as well as efficiency gains from equation 7.1.4.
|
||||
*/
|
||||
template <class Real, unsigned p> std::complex<Real> fourier_transform_daubechies_scaling(Real omega) {
|
||||
// This arg promotion is kinda sad, but IMO the accuracy is not good enough in
|
||||
// float precision using this method. Requesting a better algorithm!
|
||||
if constexpr (std::is_same_v<Real, float>) {
|
||||
return static_cast<std::complex<float>>(fourier_transform_daubechies_scaling<double, p>(static_cast<double>(omega)));
|
||||
}
|
||||
using boost::math::constants::one_div_root_two_pi;
|
||||
using std::abs;
|
||||
using std::exp;
|
||||
using std::norm;
|
||||
using std::pow;
|
||||
using std::sqrt;
|
||||
using std::cbrt;
|
||||
// Equation 7.1.4 of 10 Lectures on Wavelets is singular at ω=0:
|
||||
if (omega == 0) {
|
||||
return std::complex<Real>(one_div_root_two_pi<Real>(), 0);
|
||||
}
|
||||
// For whatever reason, this starts returning NaNs rather than zero for |ω|≫1.
|
||||
// But we know that this function decays rather quickly with |ω|,
|
||||
// and hence it is "numerically zero", even if in actuality the function does not have compact support.
|
||||
// Now, should we probably do a fairly involved, exhaustive calculation to see where exactly we should set this threshold
|
||||
// and store them in a table? .... yes.
|
||||
if (abs(omega) >= sqrt(std::numeric_limits<Real>::max())) {
|
||||
return std::complex<Real>(0, 0);
|
||||
}
|
||||
auto const constexpr lxi = detail::ft_daubechies_scaling_polynomial_coefficients<Real, p>();
|
||||
auto xi = -omega / 2;
|
||||
std::complex<Real> phi{one_div_root_two_pi<Real>(), 0};
|
||||
do {
|
||||
std::complex<Real> arg{0, xi};
|
||||
auto z = exp(arg);
|
||||
phi *= boost::math::tools::evaluate_polynomial_estrin(lxi, z);
|
||||
xi /= 2;
|
||||
} while (abs(xi) > std::numeric_limits<Real>::epsilon());
|
||||
std::complex<Real> arg{0, omega};
|
||||
// There is no std::expm1 for complex numbers.
|
||||
// We may therefore be leaving accuracy gains on the table for small |ω|:
|
||||
std::complex<Real> prefactor = (Real(1) - exp(-arg))/arg;
|
||||
return phi * static_cast<std::complex<Real>>(pow(prefactor, p));
|
||||
}
|
||||
|
||||
template <class Real, unsigned p> std::complex<Real> fourier_transform_daubechies_wavelet(Real omega) {
|
||||
// See Daubechies, 10 Lectures on Wavelets, page 193, unlabelled equation in Theorem 6.3.6:
|
||||
// 𝓕[ψ](ω) = -exp(-iω/2)m₀(ω/2 + π)^{*}𝓕[𝜙](ω/2)
|
||||
if constexpr (std::is_same_v<Real, float>) {
|
||||
return static_cast<std::complex<float>>(fourier_transform_daubechies_wavelet<double, p>(static_cast<double>(omega)));
|
||||
}
|
||||
|
||||
using std::exp;
|
||||
using std::pow;
|
||||
auto Fphi = fourier_transform_daubechies_scaling<Real, p>(omega/2);
|
||||
auto phase = -exp(std::complex<Real>(0, -omega/2));
|
||||
// See Section 6.4 for the sign convention on the argument,
|
||||
// as well as Table 6.2:
|
||||
auto z = phase; // strange coincidence.
|
||||
//auto z = exp(std::complex<Real>(0, -omega/2 - boost::math::constants::pi<Real>()));
|
||||
auto constexpr lxi = detail::ft_daubechies_scaling_polynomial_coefficients<Real, p>();
|
||||
auto m0 = std::complex<Real>(pow((Real(1) + z)/Real(2), p))*boost::math::tools::evaluate_polynomial_estrin(lxi, z);
|
||||
return Fphi*std::conj(m0)*phase;
|
||||
}
|
||||
|
||||
} // namespace boost::math
|
||||
#endif
|
||||
@@ -0,0 +1,51 @@
|
||||
// (C) Copyright Nick Thompson 2023.
|
||||
// Use, modification and distribution are subject to the
|
||||
// Boost Software License, Version 1.0. (See accompanying file
|
||||
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
|
||||
|
||||
#include <random>
|
||||
#include <array>
|
||||
#include <vector>
|
||||
#include <iostream>
|
||||
#include <benchmark/benchmark.h>
|
||||
#include <boost/math/special_functions/fourier_transform_daubechies.hpp>
|
||||
|
||||
using boost::math::fourier_transform_daubechies_scaling;
|
||||
|
||||
template<class Real, size_t p>
|
||||
void FourierTransformDaubechiesScaling(benchmark::State& state)
|
||||
{
|
||||
std::random_device rd;
|
||||
auto seed = rd();
|
||||
std::mt19937_64 mt(seed);
|
||||
std::uniform_real_distribution<Real> unif(0, 10);
|
||||
|
||||
for (auto _ : state)
|
||||
{
|
||||
benchmark::DoNotOptimize(fourier_transform_daubechies_scaling<Real, p>(unif(mt)));
|
||||
}
|
||||
}
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 1);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 2);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 3);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 4);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 5);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 6);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 7);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 8);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 9);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float, 10);
|
||||
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 1);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 2);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 3);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 4);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 5);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 6);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 7);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 8);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 9);
|
||||
BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double, 10);
|
||||
|
||||
|
||||
BENCHMARK_MAIN();
|
||||
@@ -1220,6 +1220,7 @@ test-suite interpolators :
|
||||
[ run gegenbauer_test.cpp : : : [ requires cxx11_auto_declarations cxx11_constexpr cxx11_smart_ptr cxx11_defaulted_functions ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] ]
|
||||
[ run daubechies_scaling_test.cpp : : : <toolset>gcc-mingw:<cxxflags>-Wa,-mbig-obj <debug-symbols>off <toolset>msvc:<cxxflags>/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : <build>no ] [ check-target-builds ../config//is_cygwin_run "Cygwin CI run" : <build>no ] ]
|
||||
[ run daubechies_wavelet_test.cpp : : : <toolset>gcc-mingw:<cxxflags>-Wa,-mbig-obj <debug-symbols>off <toolset>msvc:<cxxflags>/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : <build>no ] [ check-target-builds ../config//is_cygwin_run "Cygwin CI run" : <build>no ] ]
|
||||
[ run fourier_transform_daubechies_test.cpp : : : <toolset>gcc-mingw:<cxxflags>-Wa,-mbig-obj <debug-symbols>off <toolset>msvc:<cxxflags>/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : <build>no ] [ check-target-builds ../config//is_cygwin_run "Cygwin CI run" : <build>no ] ]
|
||||
[ run wavelet_transform_test.cpp : : : <toolset>msvc:<cxxflags>/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] [ check-target-builds ../config//is_ci_sanitizer_run "Sanitizer CI run" : <build>no ] ]
|
||||
[ run agm_test.cpp : : : <toolset>msvc:<cxxflags>/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] ]
|
||||
[ run rsqrt_test.cpp : : : <toolset>msvc:<cxxflags>/bigobj [ requires cxx17_if_constexpr cxx17_std_apply ] [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] ]
|
||||
|
||||
220
test/fourier_transform_daubechies_test.cpp
Normal file
220
test/fourier_transform_daubechies_test.cpp
Normal file
@@ -0,0 +1,220 @@
|
||||
// boost-no-inspect
|
||||
/*
|
||||
* Copyright Nick Thompson, 2023
|
||||
* Use, modification and distribution are subject to the
|
||||
* Boost Software License, Version 1.0. (See accompanying file
|
||||
* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
|
||||
*/
|
||||
|
||||
#include "math_unit_test.hpp"
|
||||
#include <numeric>
|
||||
#include <utility>
|
||||
#include <iomanip>
|
||||
#include <iostream>
|
||||
#include <random>
|
||||
#include <boost/math/tools/condition_numbers.hpp>
|
||||
#include <boost/math/constants/constants.hpp>
|
||||
#include <boost/math/quadrature/trapezoidal.hpp>
|
||||
#include <boost/math/special_functions/daubechies_scaling.hpp>
|
||||
#include <boost/math/special_functions/daubechies_wavelet.hpp>
|
||||
#include <boost/math/special_functions/fourier_transform_daubechies.hpp>
|
||||
|
||||
#ifdef BOOST_HAS_FLOAT128
|
||||
#include <boost/multiprecision/float128.hpp>
|
||||
using boost::multiprecision::float128;
|
||||
#endif
|
||||
|
||||
using boost::math::fourier_transform_daubechies_scaling;
|
||||
using boost::math::fourier_transform_daubechies_wavelet;
|
||||
using boost::math::tools::summation_condition_number;
|
||||
using boost::math::constants::two_pi;
|
||||
using boost::math::constants::pi;
|
||||
using boost::math::constants::one_div_root_two_pi;
|
||||
using boost::math::quadrature::trapezoidal;
|
||||
// 𝓕[φ](-ω) = 𝓕[φ](ω)*
|
||||
template<typename Real, int p>
|
||||
void test_evaluation_symmetry() {
|
||||
std::cout << "Testing evaluation symmetry on the " << p << " vanishing moment scaling function.\n";
|
||||
auto phi = fourier_transform_daubechies_scaling<Real, p>(0.0);
|
||||
CHECK_ULP_CLOSE(one_div_root_two_pi<Real>(), phi.real(), 3);
|
||||
CHECK_ULP_CLOSE(static_cast<Real>(0), phi.imag(), 3);
|
||||
|
||||
Real domega = Real(1)/128;
|
||||
for (Real omega = domega; omega < 10; omega += domega) {
|
||||
auto phi1 = fourier_transform_daubechies_scaling<Real, p>(-omega);
|
||||
auto phi2 = fourier_transform_daubechies_scaling<Real, p>(omega);
|
||||
CHECK_ULP_CLOSE(phi1.real(), phi2.real(), 3);
|
||||
CHECK_ULP_CLOSE(phi1.imag(), -phi2.imag(), 3);
|
||||
|
||||
auto psi1 = fourier_transform_daubechies_wavelet<Real, p>(-omega);
|
||||
auto psi2 = fourier_transform_daubechies_wavelet<Real, p>(omega);
|
||||
CHECK_ULP_CLOSE(psi1.real(), psi2.real(), 3);
|
||||
CHECK_ULP_CLOSE(psi1.imag(), -psi2.imag(), 3);
|
||||
}
|
||||
|
||||
for (Real omega = 10; omega < std::cbrt(std::numeric_limits<Real>::max()); omega *= 10) {
|
||||
auto phi1 = fourier_transform_daubechies_scaling<Real, p>(-omega);
|
||||
auto phi2 = fourier_transform_daubechies_scaling<Real, p>(omega);
|
||||
CHECK_ULP_CLOSE(phi1.real(), phi2.real(), 3);
|
||||
CHECK_ULP_CLOSE(phi1.imag(), -phi2.imag(), 3);
|
||||
}
|
||||
return;
|
||||
}
|
||||
|
||||
template<typename Real, int p>
|
||||
void test_roots() {
|
||||
std::cout << "Testing roots on the " << p << " vanishing moment scaling function.\n";
|
||||
for (long n = 1; n < 100; ++n) {
|
||||
// All arguments of the form 2πn are roots of the complex function:
|
||||
// See Daubechies, 10 Lectures on Wavelets, Section 6.2:
|
||||
Real omega = n*two_pi<Real>();
|
||||
Real residual = std::norm(fourier_transform_daubechies_scaling<Real, p>(omega));
|
||||
CHECK_LE(residual, std::numeric_limits<Real>::epsilon()*std::numeric_limits<Real>::epsilon());
|
||||
}
|
||||
std::cout << "Testing roots on the " << p << " vanishing moment wavelet.\n";
|
||||
// If ωₙ is a root of the 𝓕[𝜙], then 2ωₙ is a root of 𝓕[ψ].
|
||||
// In addition, m₀(π) = 0, and m₀ is 2π periodic.
|
||||
// Recalling 𝓕[ψ](ω) = exp(iω/2)m₀(ω/2 + π)^{*}𝓕[𝜙](ω/2)*phase,
|
||||
// ω=4nπ are also roots:
|
||||
for (long n = 0; n < 100; ++n) {
|
||||
Real omega = 4*n*pi<Real>();
|
||||
Real residual = std::norm(fourier_transform_daubechies_wavelet<Real, p>(omega));
|
||||
CHECK_LE(residual, std::numeric_limits<Real>::epsilon()*std::numeric_limits<Real>::epsilon());
|
||||
}
|
||||
}
|
||||
|
||||
template<int p>
|
||||
void test_scaling_quadrature() {
|
||||
std::cout << "Testing numerical quadrature of the scaling function with " << p << " vanishing moments matches numerical evaluation.\n";
|
||||
auto phi = boost::math::daubechies_scaling<double, p>();
|
||||
auto [tmin, tmax] = phi.support();
|
||||
double domega = 1/8.0;
|
||||
for (double omega = domega; omega < 10; omega += domega) {
|
||||
// I suspect the quadrature is less accurate than special function evaluation, so this is just a sanity check:
|
||||
auto f = [&](double t) {
|
||||
return phi(t)*std::exp(std::complex<double>(0, -omega*t))*one_div_root_two_pi<double>();
|
||||
};
|
||||
auto expected = trapezoidal(f, tmin, tmax, 2*std::numeric_limits<double>::epsilon());
|
||||
auto computed = fourier_transform_daubechies_scaling<float, p>(static_cast<float>(omega));
|
||||
CHECK_MOLLIFIED_CLOSE(static_cast<float>(expected.real()), computed.real(), 1e-4);
|
||||
CHECK_MOLLIFIED_CLOSE(static_cast<float>(expected.imag()), computed.imag(), 1e-4);
|
||||
}
|
||||
}
|
||||
|
||||
template<int p>
|
||||
void test_wavelet_quadrature() {
|
||||
std::cout << "Testing numerical quadrature of the wavelet with " << p << " vanishing moments matches numerical evaluation.\n";
|
||||
auto psi = boost::math::daubechies_wavelet<double, p>();
|
||||
auto [tmin, tmax] = psi.support();
|
||||
double domega = 1/8.0;
|
||||
// There is a root at at ω=0, so skip this one because we can't recover the phase of a root.
|
||||
for (double omega = 2*domega; omega < 10; omega += domega) {
|
||||
// I suspect the quadrature is less accurate than special function evaluation, so this is just a sanity check:
|
||||
auto f = [&](double t) {
|
||||
return psi(t)*std::exp(std::complex<double>(0, -omega*t))*one_div_root_two_pi<double>();
|
||||
};
|
||||
auto expected = trapezoidal(f, tmin, tmax, std::numeric_limits<double>::epsilon());
|
||||
auto computed = fourier_transform_daubechies_wavelet<double, p>(omega);
|
||||
if(!CHECK_ABSOLUTE_ERROR(std::abs(expected), std::abs(computed), 1e-9)) {
|
||||
std::cerr << " |𝓕[ψ](" << omega << ")| is incorrect.\n";
|
||||
}
|
||||
// Again, lots of evidence that the quadrature is less accurate than what we've implemented.
|
||||
// Graph of the quadrature phase is super janky; graph of the implementation phase is pretty good.
|
||||
if(!CHECK_ABSOLUTE_ERROR(std::arg(expected), std::arg(computed), 1e-2)) {
|
||||
std::cerr << " arg(𝓕[ψ](" << omega << ")) is incorrect.\n";
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// Tests Daubechies "Ten Lectures on Wavelets", equation 5.1.19:
|
||||
template<typename Real, int p>
|
||||
void test_ten_lectures_eq_5_1_19() {
|
||||
std::cout << "Testing Ten Lectures equation 5.1.19 on " << p << " vanishing moments.\n";
|
||||
Real domega = Real(1)/Real(16);
|
||||
for (Real omega = 0; omega < 1; omega += domega) {
|
||||
Real term = std::norm(fourier_transform_daubechies_scaling<Real, p>(omega));
|
||||
auto sum = summation_condition_number<Real>(term);
|
||||
int64_t l = 1;
|
||||
while (l < 10000 && term > 2*std::numeric_limits<Real>::epsilon()) {
|
||||
Real tpl = std::norm(fourier_transform_daubechies_scaling<Real, p>(omega + two_pi<Real>()*l));
|
||||
Real tml = std::norm(fourier_transform_daubechies_scaling<Real, p>(omega - two_pi<Real>()*l));
|
||||
|
||||
sum += tpl;
|
||||
sum += tml;
|
||||
Real term = tpl + tml;
|
||||
++l;
|
||||
}
|
||||
// With arg promotion, I can get this to 13 ULPS:
|
||||
if (!CHECK_ULP_CLOSE(1/two_pi<Real>(), sum.sum(), 125)) {
|
||||
std::cerr << " Failure with occurs on " << p << " vanishing moments.\n";
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// Tests Daubechies "Ten Lectures on Wavelets", equation 5.1.38:
|
||||
template<typename Real, int p>
|
||||
void test_ten_lectures_eq_5_1_38() {
|
||||
std::cout << "Testing Ten Lectures equation 5.1.38 on " << p << " vanishing moments.\n";
|
||||
Real domega = Real(1)/Real(16);
|
||||
for (Real omega = 0; omega < 1; omega += domega) {
|
||||
Real phi_omega_sq = std::norm(fourier_transform_daubechies_scaling<Real, p>(omega));
|
||||
Real psi_omega_sq = std::norm(fourier_transform_daubechies_wavelet<Real, p>(omega));
|
||||
Real phi_half_omega_sq = std::norm(fourier_transform_daubechies_scaling<Real, p>(omega/2));
|
||||
|
||||
if (!CHECK_ULP_CLOSE(phi_half_omega_sq, phi_omega_sq + psi_omega_sq, 125)) {
|
||||
std::cerr << " Failure with occurs on " << p << " vanishing moments at omega = " << omega << "\n";
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
int main()
|
||||
{
|
||||
test_roots<float, 1>();
|
||||
test_roots<float, 2>();
|
||||
test_roots<float, 3>();
|
||||
test_roots<float, 4>();
|
||||
test_roots<float, 5>();
|
||||
test_roots<float, 6>();
|
||||
test_roots<float, 7>();
|
||||
test_roots<float, 8>();
|
||||
test_roots<float, 9>();
|
||||
test_roots<float, 10>();
|
||||
|
||||
test_evaluation_symmetry<float, 1>();
|
||||
test_evaluation_symmetry<float, 2>();
|
||||
test_evaluation_symmetry<float, 3>();
|
||||
test_evaluation_symmetry<float, 4>();
|
||||
test_evaluation_symmetry<float, 5>();
|
||||
test_evaluation_symmetry<float, 6>();
|
||||
test_evaluation_symmetry<float, 7>();
|
||||
test_evaluation_symmetry<float, 8>();
|
||||
test_evaluation_symmetry<float, 9>();
|
||||
test_evaluation_symmetry<float, 10>();
|
||||
|
||||
// Slow tests:
|
||||
test_scaling_quadrature<9>();
|
||||
test_scaling_quadrature<10>();
|
||||
|
||||
// This one converges really slowly:
|
||||
//test_ten_lectures_eq_5_1_19<float, 1>();
|
||||
test_ten_lectures_eq_5_1_19<float, 2>();
|
||||
test_ten_lectures_eq_5_1_19<float, 3>();
|
||||
test_ten_lectures_eq_5_1_19<float, 4>();
|
||||
test_ten_lectures_eq_5_1_19<float, 5>();
|
||||
test_ten_lectures_eq_5_1_19<float, 6>();
|
||||
test_ten_lectures_eq_5_1_19<float, 7>();
|
||||
test_ten_lectures_eq_5_1_19<float, 8>();
|
||||
test_ten_lectures_eq_5_1_19<float, 9>();
|
||||
test_ten_lectures_eq_5_1_19<float, 10>();
|
||||
|
||||
test_ten_lectures_eq_5_1_38<float, 3>();
|
||||
test_ten_lectures_eq_5_1_38<float, 4>();
|
||||
test_ten_lectures_eq_5_1_38<float, 5>();
|
||||
test_ten_lectures_eq_5_1_38<float, 6>();
|
||||
|
||||
test_wavelet_quadrature<9>();
|
||||
|
||||
return boost::math::test::report_errors();
|
||||
}
|
||||
@@ -90,7 +90,11 @@ bool check_ulp_close(PreciseReal expected1, Real computed, size_t ulps, std::str
|
||||
using boost::math::lltrunc;
|
||||
// Of course integers can be expected values, and they are exact:
|
||||
if (!std::is_integral<PreciseReal>::value) {
|
||||
BOOST_MATH_ASSERT_MSG(!isnan(expected1), "Expected value cannot be a nan.");
|
||||
if (boost::math::isnan(expected1)) {
|
||||
std::ostringstream oss;
|
||||
oss << "Error in CHECK_ULP_CLOSE: Expected value cannot be a nan. Callsite: " << filename << ":" << function << ":" << line << ".";
|
||||
throw std::domain_error(oss.str());
|
||||
}
|
||||
if (sizeof(PreciseReal) < sizeof(Real)) {
|
||||
std::ostringstream err;
|
||||
err << "\n\tThe expected number must be computed in higher (or equal) precision than the number being tested.\n";
|
||||
@@ -100,7 +104,7 @@ bool check_ulp_close(PreciseReal expected1, Real computed, size_t ulps, std::str
|
||||
}
|
||||
}
|
||||
|
||||
if (isnan(computed))
|
||||
if (boost::math::isnan(computed))
|
||||
{
|
||||
std::ios_base::fmtflags f( std::cerr.flags() );
|
||||
std::cerr << std::setprecision(3);
|
||||
|
||||
Reference in New Issue
Block a user