Numerical evaluation of Fourier transform of Daubechies scaling functions. [ci skip] [CI SKIP]

Author: NAThompson

doc/graphs/fourier_transform_daubechies.png

@@ -127,6 +127,23 @@ The 2 vanishing moment scaling function.
 [$../graphs/daubechies_8_scaling.svg]
 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 i 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.
+
+[$../graphs/fourier_transform_daubechies.png]
+
 [heading References]
 
 * Daubechies, Ingrid. ['Ten Lectures on Wavelets.] Vol. 61. Siam, 1992.

doc/sf/daubechies.qbk

@@ -0,0 +1,38 @@
+#include <utility>
+#include <boost/math/filters/daubechies.hpp>
+#include <boost/math/tools/polynomial.hpp>
+#include <boost/multiprecision/cpp_bin_float.hpp>
+#include <boost/math/constants/constants.hpp>
+
+using std::pow;
+using boost::multiprecision::cpp_bin_float_100;
+using boost::math::filters::daubechies_scaling_filter;
+using boost::math::tools::polynomial;
+using boost::math::constants::half;
+using boost::math::constants::root_two;
+
+template<typename Real, size_t N>
+std::vector<Real> get_constants() {
+   auto h = daubechies_scaling_filter<cpp_bin_float_100, N>();
+   auto p = polynomial<cpp_bin_float_100>(h.begin(), h.end());
+
+   auto q = polynomial({half<cpp_bin_float_100>(), half<cpp_bin_float_100>()});
+   q = pow(q, N);
+   auto l = p/q;
+   return l.data(); 
+}
+
+template<typename Real>
+void print_constants(std::vector<Real> const & l) {
+   std::cout << std::setprecision(std::numeric_limits<Real>::digits10 -10);
+   std::cout << "return std::array<Real, " << l.size() << ">{";
+   for (size_t i = 0; i < l.size() - 1; ++i) {
+       std::cout << "BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits, " << l[i]/root_two<Real>() << "), ";
+   }
+   std::cout << "BOOST_MATH_BIG_CONSTANT(Real, std::numeric_limits<Real>::digits, " << l.back()/root_two<Real>() << ")};\n";
+}
+
+int main() {
+   auto constants = get_constants<cpp_bin_float_100, 1>();
+   print_constants(constants);
+}

example/calculate_fourier_transform_daubechies_constants.cpp

@@ -0,0 +1,54 @@
+#include <boost/math/special_functions/fourier_transform_daubechies_scaling.hpp>
+#include <boost/math/tools/ulps_plot.hpp>
+
+using boost::math::fourier_transform_daubechies_scaling;
+using boost::math::tools::ulps_plot;
+
+template<int p>
+void real_part() {
+   auto phi_real_hi_acc = [](double omega) {
+       auto z = fourier_transform_daubechies_scaling<double, p>(omega);
+       return z.real();
+   };
+
+   auto phi_real_lo_acc = [](float omega) {
+      auto z = fourier_transform_daubechies_scaling<float, p>(omega);
+      return z.real();
+   };
+   auto plot = ulps_plot<decltype(phi_real_hi_acc), double, float>(phi_real_hi_acc, float(0.0), float(100.0), 20000);
+   plot.ulp_envelope(false);
+   plot.add_fn(phi_real_lo_acc);
+   plot.clip(100);
+   plot.title("Accuracy of 𝔑(𝓕[𝜙](ω)) with " + std::to_string(p) + " vanishing moments.");
+   plot.write("real_ft_daub_scaling_"  + std::to_string(p) + ".svg");
+ 
+}
+
+template<int p>
+void imaginary_part() {
+   auto phi_imag_hi_acc = [](double omega) {
+       auto z = fourier_transform_daubechies_scaling<double, p>(omega);
+       return z.imag();
+   };
+
+   auto phi_imag_lo_acc = [](float omega) {
+      auto z = fourier_transform_daubechies_scaling<float, p>(omega);
+      return z.imag();
+   };
+   auto plot = ulps_plot<decltype(phi_imag_hi_acc), double, float>(phi_imag_hi_acc, float(0.0), float(100.0), 20000);
+   plot.ulp_envelope(false);
+   plot.add_fn(phi_imag_lo_acc);
+   plot.clip(100);
+   plot.title("Accuracy of 𝕴(𝓕[𝜙](ω)) with " + std::to_string(p) + " vanishing moments.");
+   plot.write("imag_ft_daub_scaling_"  + std::to_string(p) + ".svg");
+ 
+}
+
+
+int main() {
+   real_part<3>();
+   imaginary_part<3>();
+   real_part<6>();
+   imaginary_part<6>();
+   return 0;
+}