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|---|---|---|---|
| 1 | // This file is part of INSTINCT, the INS Toolkit for Integrated | ||
| 2 | // Navigation Concepts and Training by the Institute of Navigation of | ||
| 3 | // the University of Stuttgart, Germany. | ||
| 4 | // | ||
| 5 | // This Source Code Form is subject to the terms of the Mozilla Public | ||
| 6 | // License, v. 2.0. If a copy of the MPL was not distributed with this | ||
| 7 | // file, You can obtain one at https://mozilla.org/MPL/2.0/. | ||
| 8 | |||
| 9 | /// @file Math.hpp | ||
| 10 | /// @brief Simple Math functions | ||
| 11 | /// @author T. Topp (topp@ins.uni-stuttgart.de) | ||
| 12 | /// @author N. Stahl (HiWi: Elliptical integral) | ||
| 13 | /// @date 2023-07-04 | ||
| 14 | |||
| 15 | #pragma once | ||
| 16 | |||
| 17 | #include <concepts> | ||
| 18 | #include <cstdint> | ||
| 19 | #include <optional> | ||
| 20 | #include <type_traits> | ||
| 21 | #include <Eigen/Core> | ||
| 22 | #include <gcem.hpp> | ||
| 23 | #include <fmt/format.h> | ||
| 24 | |||
| 25 | #include "util/Assert.h" | ||
| 26 | |||
| 27 | namespace NAV::math | ||
| 28 | { | ||
| 29 | |||
| 30 | /// @brief Calculates the factorial of an unsigned integer | ||
| 31 | /// @param[in] n Unsigned integer | ||
| 32 | /// @return The factorial of 'n' | ||
| 33 | uint64_t factorial(uint64_t n); | ||
| 34 | |||
| 35 | /// @brief Round the number to the specified amount of digits | ||
| 36 | /// @param[in] value Value to round | ||
| 37 | /// @param[in] digits Amount of digits | ||
| 38 | /// @return The rounded value | ||
| 39 | template<std::floating_point T> | ||
| 40 | 343402 | constexpr T round(const T& value, size_t digits) | |
| 41 | { | ||
| 42 | 343402 | auto factor = std::pow(10, digits); | |
| 43 | 343402 | return std::round(value * factor) / factor; | |
| 44 | } | ||
| 45 | |||
| 46 | /// @brief Round the number to the specified amount of significant digits | ||
| 47 | /// @param[in] value Value to round | ||
| 48 | /// @param[in] digits Amount of digits | ||
| 49 | /// @return The rounded value | ||
| 50 | template<std::floating_point T> | ||
| 51 | 40 | constexpr T roundSignificantDigits(T value, size_t digits) | |
| 52 | { | ||
| 53 |
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40 | if (value == 0.0) { return 0.0; } |
| 54 | // LOG_DEBUG("value = {:.13e} --> Round to {} digits", value, digits); | ||
| 55 | 39 | auto absVal = gcem::abs(value); | |
| 56 | 39 | auto log10 = static_cast<int32_t>(gcem::log10(absVal)); | |
| 57 |
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39 | auto exp = log10 + (log10 > 0 || (log10 == 0 && absVal >= 1.0)); |
| 58 | 39 | auto fac = static_cast<T>(digits) - static_cast<T>(exp); | |
| 59 | // LOG_DEBUG(" log10 = {}, exp = {}, fac = {}", log10, exp, fac); | ||
| 60 | 39 | auto factor = static_cast<T>(gcem::pow(10.0, fac)); | |
| 61 | // LOG_DEBUG(" factor = {:.0e} --> value * factor = {}", factor, value * factor); | ||
| 62 | // LOG_DEBUG(" round = {} --> ... / factor = {}", gcem::round(value * factor), gcem::round(value * factor) / factor); | ||
| 63 | 39 | return static_cast<T>(gcem::round(value * factor) / factor); | |
| 64 | } | ||
| 65 | |||
| 66 | /// @brief Interprets the input integer with certain amount of Bits as Output type. Takes care of sign extension | ||
| 67 | /// @tparam Out Output type | ||
| 68 | /// @tparam Bits Size of the input data | ||
| 69 | /// @tparam In Input data type (needs to be bigger than the amount of Bits) | ||
| 70 | /// @param[in] in Number as two's complement, with the sign bit (+ or -) occupying the MSB | ||
| 71 | /// @return Output type | ||
| 72 | template<std::integral Out, size_t Bits, std::integral In> | ||
| 73 | 3099 | constexpr Out interpretAs(In in) | |
| 74 | { | ||
| 75 | static_assert(Bits < sizeof(In) * 8); | ||
| 76 | static_assert(Bits < sizeof(Out) * 8); | ||
| 77 | |||
| 78 | 3099 | constexpr size_t N = sizeof(Out) * 8 - Bits; | |
| 79 | 3099 | return static_cast<Out>(static_cast<Out>((in & static_cast<In>(gcem::pow(2, Bits) - 1)) << N) >> N); | |
| 80 | } | ||
| 81 | |||
| 82 | /// @brief Calculates the skew symmetric matrix of the given vector. | ||
| 83 | /// This is needed to perform the cross product with a scalar product operation | ||
| 84 | /// @tparam Derived Derived Eigen Type | ||
| 85 | /// @param[in] a The vector | ||
| 86 | /// @return Skew symmetric matrix | ||
| 87 | /// @note See Groves (2013) equation (2.50) | ||
| 88 | template<typename Derived> | ||
| 89 | 3013918 | Eigen::Matrix<typename Derived::Scalar, 3, 3> skewSymmetricMatrix(const Eigen::MatrixBase<Derived>& a) | |
| 90 | { | ||
| 91 |
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3013918 | INS_ASSERT_USER_ERROR(a.cols() == 1, "Given Eigen Object must be a vector"); |
| 92 |
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3013245 | INS_ASSERT_USER_ERROR(a.rows() == 3, "Given Vector must have 3 Rows"); |
| 93 | |||
| 94 | 3013867 | Eigen::Matrix<typename Derived::Scalar, 3, 3> skewMat; | |
| 95 |
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3005786 | skewMat << 0, -a(2), a(1), |
| 96 |
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3014844 | a(2), 0, -a(0), |
| 97 |
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3015993 | -a(1), a(0), 0; |
| 98 | |||
| 99 | 3015366 | return skewMat; | |
| 100 | } | ||
| 101 | |||
| 102 | /// @brief Calculates the square of a skew symmetric matrix of the given vector. | ||
| 103 | /// @tparam Derived Derived Eigen Type | ||
| 104 | /// @param[in] a The vector | ||
| 105 | /// @return Square of skew symmetric matrix | ||
| 106 | template<typename Derived> | ||
| 107 | Eigen::Matrix<typename Derived::Scalar, 3, 3> skewSymmetricMatrixSquared(const Eigen::MatrixBase<Derived>& a) | ||
| 108 | { | ||
| 109 | INS_ASSERT_USER_ERROR(a.cols() == 1, "Given Eigen Object must be a vector"); | ||
| 110 | INS_ASSERT_USER_ERROR(a.rows() == 3, "Given Vector must have 3 Rows"); | ||
| 111 | |||
| 112 | Eigen::Matrix<typename Derived::Scalar, 3, 3> skewMat2; | ||
| 113 | skewMat2 << std::pow(a(2), 2) + std::pow(a(1), 2), a(0) * a(1), a(0) * a(2), | ||
| 114 | a(0) * a(1), std::pow(a(2), 2) + std::pow(a(0), 2), a(1) * a(2), | ||
| 115 | a(0) * a(2), a(1) * a(2), std::pow(a(0), 2) + std::pow(a(1), 2); | ||
| 116 | |||
| 117 | return skewMat2; | ||
| 118 | } | ||
| 119 | |||
| 120 | /// @brief Calculates the secant of a value (sec(x) = csc(pi/2 - x) = 1 / cos(x)) | ||
| 121 | template<std::floating_point T> | ||
| 122 | ✗ | T sec(const T& x) | |
| 123 | { | ||
| 124 | ✗ | return 1.0 / std::cos(x); | |
| 125 | } | ||
| 126 | |||
| 127 | /// @brief Calculates the cosecant of a value (csc(x) = sec(pi/2 - x) = 1 / sin(x)) | ||
| 128 | template<std::floating_point T> | ||
| 129 | 31113 | T csc(const T& x) | |
| 130 | { | ||
| 131 | 31113 | return 1.0 / std::sin(x); | |
| 132 | } | ||
| 133 | |||
| 134 | /// @brief Returns the sign of the given value | ||
| 135 | /// @param[in] val Value to get the sign from | ||
| 136 | /// @return Sign of the given value | ||
| 137 | template<typename T> | ||
| 138 | 1343753 | int sgn(const T& val) | |
| 139 | { | ||
| 140 | 1343753 | return (T(0) < val) - (val < T(0)); | |
| 141 | } | ||
| 142 | |||
| 143 | /// @brief Find (L^T D L)-decomposition of Q-matrix via outer product method | ||
| 144 | /// @param[in] Qmatrix Symmetric positive definite matrix to be factored | ||
| 145 | /// @return L - Factor matrix (strict lower triangular) | ||
| 146 | /// @return D - Vector with entries of the diagonal matrix | ||
| 147 | /// @note See \cite deJonge1996 de Jonge 1996, Algorithm FMFAC5 | ||
| 148 | /// @attention Consider using NAV::math::LtDLdecomp_choleskyFact() because it is faster by up to a factor 10 | ||
| 149 | template<typename Derived> | ||
| 150 | std::optional<std::pair<Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime>, | ||
| 151 | Eigen::Vector<typename Derived::Scalar, Derived::RowsAtCompileTime>>> | ||
| 152 | LtDLdecomp_outerProduct(const Eigen::MatrixBase<Derived>& Qmatrix) | ||
| 153 | { | ||
| 154 | using Eigen::seq; | ||
| 155 | |||
| 156 | auto n = Qmatrix.rows(); | ||
| 157 | Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime> Q = Qmatrix.template triangularView<Eigen::Lower>(); | ||
| 158 | Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime> L; | ||
| 159 | Eigen::Vector<typename Derived::Scalar, Derived::RowsAtCompileTime> D; | ||
| 160 | |||
| 161 | if constexpr (Derived::RowsAtCompileTime == Eigen::Dynamic) | ||
| 162 | { | ||
| 163 | L = Eigen::Matrix<typename Derived::Scalar, Eigen::Dynamic, Eigen::Dynamic>::Zero(n, n); | ||
| 164 | D.setZero(n); | ||
| 165 | } | ||
| 166 | else | ||
| 167 | { | ||
| 168 | L = Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime>::Zero(); | ||
| 169 | D.setZero(); | ||
| 170 | } | ||
| 171 | |||
| 172 | for (Eigen::Index i = n - 1; i >= 0; i--) | ||
| 173 | { | ||
| 174 | D(i) = Q(i, i); | ||
| 175 | if (Q(i, i) <= 0.0) { return {}; } | ||
| 176 | L(i, seq(0, i)) = Q(i, seq(0, i)) / std::sqrt(Q(i, i)); | ||
| 177 | |||
| 178 | for (Eigen::Index j = 0; j <= i - 1; j++) | ||
| 179 | { | ||
| 180 | Q(j, seq(0, j)) -= L(i, seq(0, j)) * L(i, j); | ||
| 181 | } | ||
| 182 | L(i, seq(0, i)) /= L(i, i); | ||
| 183 | } | ||
| 184 | |||
| 185 | return std::make_pair(L, D); | ||
| 186 | } | ||
| 187 | |||
| 188 | /// @brief Find (L^T D L)-decomposition of Q-matrix via a backward Cholesky factorization in a bordering method formulation | ||
| 189 | /// @param[in] Q Symmetric positive definite matrix to be factored | ||
| 190 | /// @return L - Factor matrix (strict lower triangular) | ||
| 191 | /// @return D - Vector with entries of the diagonal matrix | ||
| 192 | /// @note See \cite deJonge1996 de Jonge 1996, Algorithm FMFAC6 | ||
| 193 | template<typename Derived> | ||
| 194 | std::optional<std::pair<Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime>, | ||
| 195 | Eigen::Vector<typename Derived::Scalar, Derived::RowsAtCompileTime>>> | ||
| 196 | LtDLdecomp_choleskyFact(const Eigen::MatrixBase<Derived>& Q) | ||
| 197 | { | ||
| 198 | using Eigen::seq; | ||
| 199 | |||
| 200 | auto n = Q.rows(); | ||
| 201 | typename Derived::PlainObject L = Q.template triangularView<Eigen::Lower>(); | ||
| 202 | Eigen::Vector<typename Derived::Scalar, Derived::RowsAtCompileTime> D; | ||
| 203 | double cmin = 1; | ||
| 204 | |||
| 205 | if constexpr (Derived::RowsAtCompileTime == Eigen::Dynamic) { D.setZero(n); } | ||
| 206 | else { D.setZero(); } | ||
| 207 | |||
| 208 | for (Eigen::Index j = n - 1; j >= 0; j--) | ||
| 209 | { | ||
| 210 | for (Eigen::Index i = n - 1; i >= j + 1; i--) | ||
| 211 | { | ||
| 212 | L(i, j) = (L(i, j) - L(seq(i + 1, n - 1), j).dot(L(seq(i + 1, n - 1), i))) / L(i, i); | ||
| 213 | } | ||
| 214 | double t = L(j, j) - L(seq(j + 1, n - 1), j).dot(L(seq(j + 1, n - 1), j)); | ||
| 215 | if (t <= 0.0) { return {}; } | ||
| 216 | double c = t / L(j, j); | ||
| 217 | cmin = std::min(c, cmin); | ||
| 218 | L(j, j) = std::sqrt(t); | ||
| 219 | } | ||
| 220 | for (Eigen::Index i = 0; i < n; i++) | ||
| 221 | { | ||
| 222 | L.row(i).leftCols(i) /= L(i, i); | ||
| 223 | D(i) = std::pow(L(i, i), 2.0); | ||
| 224 | L(i, i) = 1; | ||
| 225 | } | ||
| 226 | |||
| 227 | return std::make_pair(L, D); | ||
| 228 | } | ||
| 229 | |||
| 230 | /// @brief Calculates the squared norm of the vector and matrix | ||
| 231 | /// | ||
| 232 | /// \anchor eq-squaredNorm \f{equation}{ \label{eq:eq-squaredNorm} | ||
| 233 | /// ||\mathbf{\dots}||^2_{\mathbf{Q}} = (\dots)^T \mathbf{Q}^{-1} (\dots) | ||
| 234 | /// \f} | ||
| 235 | /// @param a Vector | ||
| 236 | /// @param Q Covariance matrix of the vector | ||
| 237 | /// @return Squared norm | ||
| 238 | template<typename DerivedA, typename DerivedQ> | ||
| 239 | typename DerivedA::Scalar squaredNormVectorMatrix(const Eigen::MatrixBase<DerivedA>& a, const Eigen::MatrixBase<DerivedQ>& Q) | ||
| 240 | { | ||
| 241 | static_assert(DerivedA::ColsAtCompileTime == Eigen::Dynamic || DerivedA::ColsAtCompileTime == 1); | ||
| 242 | INS_ASSERT_USER_ERROR(a.cols() == 1, "Parameter 'a' has to be a vector"); | ||
| 243 | INS_ASSERT_USER_ERROR(a.rows() == Q.rows(), "Parameter 'a' and 'Q' need to have same size"); | ||
| 244 | INS_ASSERT_USER_ERROR(Q.cols() == Q.rows(), "Parameter 'Q' needs to be quadratic"); | ||
| 245 | |||
| 246 | return a.transpose() * Q.inverse() * a; | ||
| 247 | } | ||
| 248 | |||
| 249 | /// @brief Calculates the cumulative distribution function (CDF) of the standard normal distribution | ||
| 250 | /// | ||
| 251 | /// \anchor eq-normalDistCDF \f{equation}{ \label{eq:eq-normalDistCDF} | ||
| 252 | /// \Phi(x) = \int\displaylimits_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp{\left(-\frac{1}{2} v^2\right)} \text{d}v | ||
| 253 | /// \f} | ||
| 254 | /// which can be expressed with the error function | ||
| 255 | /// \anchor eq-normalDistCDF-erf \f{equation}{ \label{eq:eq-normalDistCDF-erf} | ||
| 256 | /// \Phi(x) = \frac{1}{2} \left[ 1 + \text{erf}{\left(\frac{x}{\sqrt{2}}\right)} \right] | ||
| 257 | /// \f} | ||
| 258 | /// Using the property | ||
| 259 | /// \anchor eq-erf-minus \f{equation}{ \label{eq:eq-erf-minus} | ||
| 260 | /// \text{erf}{\left( -x \right)} = -\text{erf}{\left( x \right)} | ||
| 261 | /// \f} | ||
| 262 | /// and the complementary error function | ||
| 263 | /// \anchor eq-erfc \f{equation}{ \label{eq:eq-erfc} | ||
| 264 | /// \text{erfc}{\left( x \right)} = 1 - \text{erf}{\left( x \right)} | ||
| 265 | /// \f} | ||
| 266 | /// we can simplify equation \ref eq-normalDistCDF-erf to | ||
| 267 | /// \anchor eq-normalDistCDF-erfc \f{equation}{ \label{eq:eq-normalDistCDF-erfc} | ||
| 268 | /// \begin{aligned} | ||
| 269 | /// \Phi(x) &= \frac{1}{2} \left[ 1 - \text{erf}{\left(-\frac{x}{\sqrt{2}}\right)} \right] \\ | ||
| 270 | /// &= \frac{1}{2} \text{erfc}{\left(-\frac{x}{\sqrt{2}}\right)} | ||
| 271 | /// \end{aligned} | ||
| 272 | /// \f} | ||
| 273 | /// | ||
| 274 | /// @param value Value to calculate the CDF for | ||
| 275 | double normalCDF(double value); | ||
| 276 | |||
| 277 | /// @brief Change the sign of x according to the value of y | ||
| 278 | /// @param[in] x input value | ||
| 279 | /// @param[in] y input value | ||
| 280 | /// @return -x or +x | ||
| 281 | template<typename T> | ||
| 282 | 2612 | T sign(const T& x, const T& y) | |
| 283 | { | ||
| 284 | // similar function 'sign' in fortran | ||
| 285 |
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2612 | if (y >= 0.0) |
| 286 | { | ||
| 287 | 498 | return fabs(x); | |
| 288 | } | ||
| 289 | 2114 | return -1.0 * fabs(x); | |
| 290 | } | ||
| 291 | |||
| 292 | /// @brief Linear interpolation between vectors | ||
| 293 | /// @param a Left value | ||
| 294 | /// @param b Right value | ||
| 295 | /// @param t Multiplier. [0, 1] for interpolation | ||
| 296 | /// @return a + t * (b - a) | ||
| 297 | template<typename Derived> | ||
| 298 | 101564 | typename Derived::PlainObject lerp(const Eigen::MatrixBase<Derived>& a, const Eigen::MatrixBase<Derived>& b, const typename Derived::Scalar& t) | |
| 299 | { | ||
| 300 |
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101564 | return a + t * (b - a); |
| 301 | } | ||
| 302 | |||
| 303 | /// Lerp Search Result | ||
| 304 | struct LerpSearchResult | ||
| 305 | { | ||
| 306 | size_t l; ///< Lower bound index | ||
| 307 | size_t u; ///< Upper bound index (l + 1) | ||
| 308 | double t; ///< [0, 1] for Interpolation, otherwise Extrapolation | ||
| 309 | }; | ||
| 310 | |||
| 311 | /// @brief Searches the value in the data container | ||
| 312 | /// @param[in] data Data container | ||
| 313 | /// @param[in] value Value to search | ||
| 314 | 3 | LerpSearchResult lerpSearch(const auto& data, const auto& value) | |
| 315 | { | ||
| 316 | 3 | auto i = static_cast<size_t>(std::distance(data.begin(), std::upper_bound(data.begin(), data.end(), value))); | |
| 317 |
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3 | if (i > 0) { i--; } |
| 318 |
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3 | if (i == data.size() - 1) { i--; } |
| 319 | 3 | const auto& lb = data.at(i); | |
| 320 | 3 | const auto& ub = data.at(i + 1); | |
| 321 | 3 | double t = (value - lb) / (ub - lb); | |
| 322 | |||
| 323 | 3 | return { .l = i, .u = i + 1, .t = t }; | |
| 324 | } | ||
| 325 | |||
| 326 | /// @brief Bilinear interpolation | ||
| 327 | /// @param[in] tx Distance in x component to interpolate [0, 1] | ||
| 328 | /// @param[in] ty Distance in y component to interpolate [0, 1] | ||
| 329 | /// @param[in] c00 Value for tx = ty = 0 | ||
| 330 | /// @param[in] c10 Value for tx = 1 and ty = 0 | ||
| 331 | /// @param[in] c01 Value for tx = 0 and ty = 1 | ||
| 332 | /// @param[in] c11 Value for tx = ty = 1 | ||
| 333 | /// | ||
| 334 | /// c01 ------ c11 | ||
| 335 | /// | | | ||
| 336 | /// | | | ||
| 337 | /// | | | ||
| 338 | /// c00 ------ c10 | ||
| 339 | /// | ||
| 340 | /// @note See https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/interpolation/bilinear-filtering.html | ||
| 341 | 9 | auto bilinearInterpolation(const double& tx, const double& ty, | |
| 342 | const auto& c00, const auto& c10, | ||
| 343 | const auto& c01, const auto& c11) | ||
| 344 | { | ||
| 345 | 9 | auto a = c00 * (1 - tx) + c10 * tx; | |
| 346 | 9 | auto b = c01 * (1 - tx) + c11 * tx; | |
| 347 | 9 | return a * (1 - ty) + b * ty; | |
| 348 | // Alternative implementation: | ||
| 349 | // return (1 - tx) * (1 - ty) * c00 + tx * (1 - ty) * c10 + (1 - tx) * ty * c01 + tx * ty * c11; | ||
| 350 | } | ||
| 351 | |||
| 352 | /// @brief Calculates the incomplete elliptical integral of the second kind | ||
| 353 | /// @param[in] phi Interval bound the integration uses from 0 to phi | ||
| 354 | /// @param[in] m Function parameter that is integrated 1-m*sin(t)^2 | ||
| 355 | /// @return Incomplete elliptical integral of the second kind | ||
| 356 | /// @note See http://www2.iap.fr/users/pichon/doc/html_xref/elliptic-es.html | ||
| 357 | double calcEllipticalIntegral(double phi, double m); | ||
| 358 | |||
| 359 | } // namespace NAV::math | ||
| 360 |