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37 * Implements Gaussian function evaluations on lattices and related functionality
39 * \author Christian Blau <blau@kth.se>
41 * \ingroup module_math
45 #include "gausstransform.h"
52 #include "gromacs/math/functions.h"
53 #include "gromacs/math/multidimarray.h"
54 #include "gromacs/math/utilities.h"
59 /********************************************************************
60 * GaussianOn1DLattice::Impl
63 class GaussianOn1DLattice::Impl
66 Impl(int numGridPointsForSpreadingHalfWidth, real sigma);
68 Impl(const Impl &other) = default;
69 Impl &operator=(const Impl &other) = default;
71 /*! \brief evaluate Gaussian function at all lattice points
72 * \param[in] amplitude the amplitude of the Gaussian
73 * \param[in] dx distance from the center
75 void spread(double amplitude, real dx);
76 //! Largest distance in number of gridpoints from 0
77 int numGridPointsForSpreadingHalfWidth_;
78 /*! \brief Avoid overflow for E2^offset and underflow for E3(i).
80 * Occurs when sigma is much smaller than numGridPointsForSpreadingHalfWidth_.
82 * E2^offset smaller than maximum float requires
83 * \f$exp(dx / (2*square(sigma))^numGridPointsForSpreadingHalfWidth_ \leq max_float \f$
84 * The maximum expected distance of the Gaussian center to the next lattice point is dx = 0.5,
85 * thus the maximum spread distance here is \f$4 * sigma^2 * \log(\mathrm{maxfloat})\f$ .
87 * E3(i) larger than minmium float requires
88 * exp(i^2 / 2*(sigma)^2) > min_float
89 * Thus the maximum spread distance here is \f$\sigma \sqrt(-2\log(\mathrm{minfloat}))\f$
91 int maxEvaluatedSpreadDistance_;
92 //! Width of the Gaussian function
94 //! The result of the spreading calculation
95 std::vector<float> spreadingResult_;
96 //! Pre-calculated exp(-gridIndex^2/2 * (sigma^2)) named as in Greengard2004
97 std::vector<float> e3_;
98 /*! \brief Equal to std::floor(std::log(std::numeric_limits<float>::max())).
99 * Above expression is not constexpr and a const variable would implicitly delete default copy assignment.
100 * Therefore resorting to setting number manually.
102 static constexpr double c_logMaxFloat = 88.72284;
103 static constexpr double c_logMinFloat = -87.33654;
106 GaussianOn1DLattice::Impl::Impl(int numGridPointsForSpreadingHalfWidth, real sigma) :
107 numGridPointsForSpreadingHalfWidth_(numGridPointsForSpreadingHalfWidth),
109 spreadingResult_(2 * numGridPointsForSpreadingHalfWidth + 1)
111 maxEvaluatedSpreadDistance_ = std::min(numGridPointsForSpreadingHalfWidth_, static_cast<int>(std::floor(4 * square(sigma) * c_logMaxFloat)) - 1);
112 maxEvaluatedSpreadDistance_ = std::min(maxEvaluatedSpreadDistance_, static_cast<int>(std::floor(sigma * sqrt(-2.0 * c_logMinFloat))) - 1);
114 std::generate_n(std::back_inserter(e3_), maxEvaluatedSpreadDistance_ + 1,
115 [sigma, latticeIndex = 0]() mutable {
116 return std::exp(-0.5 * square(latticeIndex++ / sigma));
119 std::fill(std::begin(spreadingResult_), std::end(spreadingResult_), 0.);
122 void GaussianOn1DLattice::Impl::spread(double amplitude, real dx)
124 /* The spreading routine implements the fast gaussian gridding as in
126 * Leslie Greengard and June-Yub Lee,
127 * "Accelerating the Nonuniform Fast Fourier Transform"
128 * SIAM REV 2004 Vol. 46, No. 3, pp. 443-454 DOI. 10.1137/S003614450343200X
130 * Following the naming conventions for e1, e2 and e3, nu = 1, m = numGridPointsForSpreadingHalfWidth_.
132 * Speed up is achieved by factorization of the exponential that is evaluted
133 * at regular lattice points i, where the distance from the
134 * Gaussian center is \f$x-i\f$:
137 * a * \exp(-(x^2-2*i*x+ i^2)/(2*\sigma^2)) =
138 * a * \exp(-x^2/2*\sigma^2) * \exp(x/\sigma^2)^i * \exp(i/2*sigma^2) =
139 * e_1(x) * e_2(x)^i * e_3(i)
142 * Requiring only two exp evaluations per spreading operation.
145 const double e1 = amplitude * exp(-0.5 * dx * dx / square(sigma_)) / (sqrt(2 * M_PI) * sigma_);
146 spreadingResult_[numGridPointsForSpreadingHalfWidth_] = e1;
148 const double e2 = exp(dx / square(sigma_));
150 double e2pow = e2; //< powers of e2, e2^offset
152 // Move outwards from mid-point, using e2pow value for both points simultaneously
153 // o o o<----O---->o o o
154 for (int offset = 1; offset < maxEvaluatedSpreadDistance_; offset++)
156 const double e1_3 = e1 * e3_[offset];
157 spreadingResult_[numGridPointsForSpreadingHalfWidth_ + offset] = e1_3 * e2pow;
158 spreadingResult_[numGridPointsForSpreadingHalfWidth_ - offset] = e1_3 / e2pow;
161 // separate statement for gridpoints at the end of the range avoids
162 // overflow for large sigma and saves one e2 multiplication operation
163 spreadingResult_[numGridPointsForSpreadingHalfWidth_ - maxEvaluatedSpreadDistance_] = (e1 / e2pow) * e3_[maxEvaluatedSpreadDistance_];
164 spreadingResult_[numGridPointsForSpreadingHalfWidth_ + maxEvaluatedSpreadDistance_] = (e1 * e2pow) * e3_[maxEvaluatedSpreadDistance_];
167 /********************************************************************
168 * GaussianOn1DLattice
171 GaussianOn1DLattice::GaussianOn1DLattice(int numGridPointsForSpreadingHalfWidth_, real sigma) : impl_(new Impl(numGridPointsForSpreadingHalfWidth_, sigma))
175 GaussianOn1DLattice::~GaussianOn1DLattice () {}
177 void GaussianOn1DLattice::spread(double amplitude, real dx)
179 impl_->spread(amplitude, dx);
182 ArrayRef<const float> GaussianOn1DLattice::view()
184 return impl_->spreadingResult_;
187 GaussianOn1DLattice::GaussianOn1DLattice(const GaussianOn1DLattice &other)
188 : impl_(new Impl(*other.impl_))
192 GaussianOn1DLattice &GaussianOn1DLattice::operator=(const GaussianOn1DLattice &other)
194 *impl_ = *other.impl_;
198 GaussianOn1DLattice::GaussianOn1DLattice(GaussianOn1DLattice &&) noexcept = default;
200 GaussianOn1DLattice &GaussianOn1DLattice::operator=(GaussianOn1DLattice &&) noexcept = default;
205 //! rounds real-valued coordinate to the closest integer values
206 IVec closestIntegerPoint(const RVec &coordinate)
209 roundToInt(coordinate[XX]),
210 roundToInt(coordinate[YY]),
211 roundToInt(coordinate[ZZ])
215 /*! \brief Substracts a range from a three-dimensional integer coordinate and ensures
216 * the resulting coordinate is within a lattice.
217 * \param[in] index point in lattice
218 * \param[in] range to be shifted
219 * \returns Shifted index or zero if shifted index is smaller than zero.
221 IVec rangeBeginWithinLattice(const IVec &index, const IVec &range)
223 return elementWiseMax({0, 0, 0}, index - range);
226 /*! \brief Adds a range from a three-dimensional integer coordinate and ensures
227 * the resulting coordinate is within a lattice.
228 * \param[in] index point in lattice
229 * \param[in] extents extent of the lattice
230 * \param[in] range to be shifted
231 * \returns Shifted index or the lattice extent if shifted index is larger than the extent
233 IVec rangeEndWithinLattice(const IVec &index, const dynamicExtents3D &extents, const IVec &range)
235 IVec extentAsIvec(static_cast<int>(extents.extent(ZZ)), static_cast<int>(extents.extent(YY)), static_cast<int>(extents.extent(XX)));
236 return elementWiseMin(extentAsIvec, index + range);
242 /********************************************************************
243 * OuterProductEvaluator
246 mdspan<const float, dynamic_extent, dynamic_extent>
247 OuterProductEvaluator::operator()(ArrayRef<const float> x, ArrayRef<const float> y)
249 data_.resize(ssize(x), ssize(y));
250 for (int xIndex = 0; xIndex < ssize(x); ++xIndex)
252 const auto xValue = x[xIndex];
253 std::transform(std::begin(y), std::end(y), begin(data_.asView()[xIndex]),
254 [xValue](float yValue) { return xValue * yValue; });
256 return data_.asConstView();
259 /********************************************************************
263 IntegerBox::IntegerBox(const IVec &begin, const IVec &end) : begin_ {begin}, end_ {
268 const IVec &IntegerBox::begin() const{return begin_; }
269 const IVec &IntegerBox::end() const { return end_; }
271 bool IntegerBox::empty() const { return !((begin_[XX] < end_[XX] ) && (begin_[YY] < end_[YY]) && (begin_[ZZ] < end_[ZZ])); }
273 IntegerBox spreadRangeWithinLattice(const IVec ¢er, dynamicExtents3D extent, IVec range)
275 const IVec begin = rangeBeginWithinLattice(center, range);
276 const IVec end = rangeEndWithinLattice(center, extent, range);
279 /********************************************************************
280 * GaussianSpreadKernel
283 IVec GaussianSpreadKernelParameters::Shape::latticeSpreadRange() const
285 DVec range(std::ceil(sigma_[XX] * spreadWidthMultiplesOfSigma_), std::ceil(sigma_[YY] * spreadWidthMultiplesOfSigma_), std::ceil(sigma_[ZZ] * spreadWidthMultiplesOfSigma_));
286 return range.toIVec();
289 /********************************************************************
290 * GaussTransform3D::Impl
294 * Private implementation class for GaussTransform3D.
296 class GaussTransform3D::Impl
299 //! Construct from extent and spreading width and range
300 Impl(const dynamicExtents3D &extent,
301 const GaussianSpreadKernelParameters::Shape &kernelShapeParameters);
304 Impl(const Impl &other) = default;
306 Impl &operator=(const Impl &other) = default;
307 //! Add another gaussian
308 void add(const GaussianSpreadKernelParameters::PositionAndAmplitude &localParamters );
309 //! The width of the Gaussian in lattice spacing units
310 BasicVector<double> sigma_;
311 //! The spread range in lattice points
313 //! The result of the Gauss transform
314 MultiDimArray<std::vector<float>, dynamicExtents3D> data_;
315 //! The outer product of a Gaussian along the z and y dimension
316 OuterProductEvaluator outerProductZY_;
317 //! The three one-dimensional Gaussians, whose outer product is added to the Gauss transform
318 std::array<GaussianOn1DLattice, DIM> gauss1d_;
321 GaussTransform3D::Impl::Impl(const dynamicExtents3D &extent,
322 const GaussianSpreadKernelParameters::Shape &kernelShapeParameters)
323 : sigma_ {kernelShapeParameters.sigma_ },
325 kernelShapeParameters.latticeSpreadRange()
330 gauss1d_( {GaussianOn1DLattice(spreadRange_[XX], sigma_[XX]),
331 GaussianOn1DLattice(spreadRange_[YY], sigma_[YY]),
332 GaussianOn1DLattice(spreadRange_[ZZ], sigma_[ZZ]) } )
336 void GaussTransform3D::Impl::add(const GaussianSpreadKernelParameters::PositionAndAmplitude &localParameters)
338 const IVec closestLatticePoint = closestIntegerPoint(localParameters.coordinate_);
339 const auto spreadRange = spreadRangeWithinLattice(closestLatticePoint, data_.asView().extents(), spreadRange_);
341 // do nothing if the added Gaussian will never reach the lattice
342 if (spreadRange.empty())
347 for (int dimension = XX; dimension <= ZZ; ++dimension)
349 // multiply with amplitude so that Gauss3D = (amplitude * Gauss_x) * Gauss_y * Gauss_z
350 const float gauss1DAmplitude = dimension > XX ? 1.0 : localParameters.amplitude_;
351 gauss1d_[dimension].spread(gauss1DAmplitude, localParameters.coordinate_[dimension] - closestLatticePoint[dimension]);
354 const auto spreadZY = outerProductZY_(gauss1d_[ZZ].view(), gauss1d_[YY].view());
355 const auto spreadX = gauss1d_[XX].view();
356 const IVec spreadGridOffset = spreadRange_ - closestLatticePoint;
358 // \todo optimize these loops if performance critical
359 // The looping strategy uses that the last, x-dimension is contiguous in the memory layout
360 for (int zLatticeIndex = spreadRange.begin()[ZZ]; zLatticeIndex < spreadRange.end()[ZZ]; ++zLatticeIndex)
362 const auto zSlice = data_.asView()[zLatticeIndex];
364 for (int yLatticeIndex = spreadRange.begin()[YY]; yLatticeIndex < spreadRange.end()[YY]; ++yLatticeIndex)
366 const auto ySlice = zSlice[yLatticeIndex];
367 const float zyPrefactor = spreadZY(zLatticeIndex + spreadGridOffset[ZZ], yLatticeIndex + spreadGridOffset[YY]);
369 for (int xLatticeIndex = spreadRange.begin()[XX]; xLatticeIndex < spreadRange.end()[XX]; ++xLatticeIndex)
371 const float xPrefactor = spreadX[xLatticeIndex + spreadGridOffset[XX]];
372 ySlice[xLatticeIndex] += zyPrefactor * xPrefactor;
378 /********************************************************************
382 GaussTransform3D::GaussTransform3D(const dynamicExtents3D &extent,
383 const GaussianSpreadKernelParameters::Shape &kernelShapeParameters) : impl_(new Impl(extent, kernelShapeParameters))
387 void GaussTransform3D::add(const GaussianSpreadKernelParameters::PositionAndAmplitude &localParameters)
389 impl_->add(localParameters);
392 void GaussTransform3D::setZero()
394 std::fill(begin(impl_->data_), end(impl_->data_), 0.);
397 const basic_mdspan<const float, dynamicExtents3D> GaussTransform3D::view()
399 return impl_->data_.asConstView();
402 GaussTransform3D::~GaussTransform3D()
405 GaussTransform3D::GaussTransform3D(const GaussTransform3D &other)
406 : impl_(new Impl(*other.impl_))
410 GaussTransform3D &GaussTransform3D::operator=(const GaussTransform3D &other)
412 *impl_ = *other.impl_;
416 GaussTransform3D::GaussTransform3D(GaussTransform3D &&) noexcept = default;
418 GaussTransform3D &GaussTransform3D::operator=(GaussTransform3D &&) noexcept = default;