Numerical evaluation of Fourier transform of Daubechies scaling funct…

…ions. [ci skip]

doc/sf/daubechies.qbk

@@ -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.
+
+
+
 [heading References]
 
 * Daubechies, Ingrid. ['Ten Lectures on Wavelets.] Vol. 61. Siam, 1992.

example/calculate_fourier_transform_daubechies_constants.cpp

@@ -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, 10>();
+   print_constants(constants);
+}

example/fourier_transform_daubechies_ulp_plot.cpp

@@ -0,0 +1,25 @@
+#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;
+
+int main() {
+
+   auto phi_real_hi_acc = [](double omega) {
+       auto z = fourier_transform_daubechies_scaling<double, 6>(omega);
+       return z.real();
+   };
+
+   auto phi_real_lo_acc = [](float omega) {
+      auto z = fourier_transform_daubechies_scaling<float, 6>(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 Fourier transform of 6 vanishing moment Daubechies scaling function");
+   plot.write("ft_daub_scaling_6.svg");
+   return 0;
+}

reporting/performance/fourier_transform_daubechies_scaling_performance.cpp

@@ -0,0 +1,33 @@
+//  (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_scaling.hpp>
+
+using boost::math::fourier_transform_daubechies_scaling;
+
+template<class Real>
+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, 3>(unif(mt)));
+    }
+}
+
+BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, float);
+BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, double);
+//BENCHMARK_TEMPLATE(FourierTransformDaubechiesScaling, long double);
+
+BENCHMARK_MAIN();

test/Jamfile.v2

@@ -1214,6 +1214,7 @@ 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 test/fourier_transform_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 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 ] ]

test/fourier_transform_daubechies_scaling_test.cpp

@@ -0,0 +1,124 @@
+/*
+ * 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/fourier_transform_daubechies_scaling.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::tools::summation_condition_number;
+using boost::math::constants::two_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() {
+   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);
+   }
+
+    for (Real omega = 10; omega < std::sqrt(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<int p>
+void test_quadrature() {
+   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);
+   }
+}
+
+// Tests Daubechies "Ten Lectures on Wavelets", equation 5.1.19:
+template<typename Real, int p>
+void test_ten_lectures_eq_5_1_19() {
+   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 < 50 && 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";
+       }
+   }
+}
+
+int main()
+{
+  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_quadrature<2>();
+  //test_quadrature<3>();
+  //test_quadrature<4>();
+  //test_quadrature<5>();
+
+  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>();
+
+  return boost::math::test::report_errors();
+}

test/math_unit_test.hpp

@@ -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 (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";