ChebyshevBoundedField.cc

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// -*- LSST-C++ -*-
/*
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 * Copyright 2008-2014 LSST Corporation.
 *
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#include <memory>
 
#include "ndarray/eigen.h"
#include "lsst/afw/math/LeastSquares.h"
#include "lsst/afw/math/ChebyshevBoundedField.h"
#include "lsst/afw/math/detail/TrapezoidalPacker.h"
#include "lsst/afw/table/io/InputArchive.h"
#include "lsst/afw/table/io/OutputArchive.h"
#include "lsst/afw/table/io/CatalogVector.h"
#include "lsst/afw/table/aggregates.h"
#include "lsst/afw/table/io/Persistable.cc"
 
namespace lsst {
namespace afw {
 
template std::shared_ptr<math::ChebyshevBoundedField> table::io::PersistableFacade<
        math::ChebyshevBoundedField>::dynamicCast(std::shared_ptr<table::io::Persistable> const&);
 
namespace math {
 
int ChebyshevBoundedFieldControl::computeSize() const { return detail::TrapezoidalPacker(*this).size; }
 
// ------------------ Constructors and helpers ---------------------------------------------------------------
 
namespace {
 
// Compute an affine transform that maps an arbitrary box to [-1,1]x[-1,1]
lsst::geom::AffineTransform makeChebyshevRangeTransform(lsst::geom::Box2D const bbox) {
    return lsst::geom::AffineTransform(
            lsst::geom::LinearTransform::makeScaling(2.0 / bbox.getWidth(), 2.0 / bbox.getHeight()),
            lsst::geom::Extent2D(-(2.0 * bbox.getCenterX()) / bbox.getWidth(),
                                 -(2.0 * bbox.getCenterY()) / bbox.getHeight()));
}
 
}  // namespace
 
ChebyshevBoundedField::ChebyshevBoundedField(lsst::geom::Box2I const& bbox,
                                             ndarray::Array<double const, 2, 2> const& coefficients)
        : BoundedField(bbox),
          _toChebyshevRange(makeChebyshevRangeTransform(lsst::geom::Box2D(bbox))),
          _coefficients(coefficients) {}
 
ChebyshevBoundedField::ChebyshevBoundedField(lsst::geom::Box2I const& bbox)
        : BoundedField(bbox), _toChebyshevRange(makeChebyshevRangeTransform(lsst::geom::Box2D(bbox))) {}
 
ChebyshevBoundedField::ChebyshevBoundedField(ChebyshevBoundedField const&) = default;
ChebyshevBoundedField::ChebyshevBoundedField(ChebyshevBoundedField&&) = default;
ChebyshevBoundedField::~ChebyshevBoundedField() = default;
 
// ------------------ fit() and helpers ---------------------------------------------------------------------
 
namespace {
 
typedef ChebyshevBoundedField::Control Control;
typedef detail::TrapezoidalPacker Packer;
 
// fill an array with 1-d Chebyshev functions of the 1st kind T(x), evaluated at the given point x
void evaluateBasis1d(ndarray::Array<double, 1, 1> const& t, double x) {
    int const n = t.getSize<0>();
    if (n > 0) {
        t[0] = 1.0;
    }
    if (n > 1) {
        t[1] = x;
    }
    for (int i = 2; i < n; ++i) {
        t[i] = 2.0 * x * t[i - 1] - t[i - 2];
    }
}
 
// Create a matrix of 2-d Chebyshev functions evaluated at a set of positions, with
// Chebyshev order along columns and evaluation positions along rows.  We pack the
// 2-d functions using the TrapezoidalPacker class, because we don't want any columns
// that correspond to coefficients that should be set to zero.
ndarray::Array<double, 2, 2> makeMatrix(ndarray::Array<double const, 1> const& x,
                                        ndarray::Array<double const, 1> const& y,
                                        lsst::geom::AffineTransform const& toChebyshevRange,
                                        Packer const& packer, Control const& ctrl) {
    int const nPoints = x.getSize<0>();
    ndarray::Array<double, 1, 1> tx = ndarray::allocate(packer.nx);
    ndarray::Array<double, 1, 1> ty = ndarray::allocate(packer.ny);
    ndarray::Array<double, 2, 2> out = ndarray::allocate(nPoints, packer.size);
    // Loop over x and y together, computing T_i(x) and T_j(y) arrays for each point,
    // then packing them together.
    for (int p = 0; p < nPoints; ++p) {
        lsst::geom::Point2D sxy = toChebyshevRange(lsst::geom::Point2D(x[p], y[p]));
        evaluateBasis1d(tx, sxy.getX());
        evaluateBasis1d(ty, sxy.getY());
        packer.pack(out[p], tx, ty);  // this sets a row of out to the packed outer product of tx and ty
    }
    return out;
}
 
// Create a matrix of 2-d Chebyshev functions evaluated on a grid of positions, with
// Chebyshev order along columns and evaluation positions along rows.  We pack the
// 2-d functions using the TrapezoidalPacker class, because we don't want any columns
// that correspond to coefficients that should be set to zero.
ndarray::Array<double, 2, 2> makeMatrix(lsst::geom::Box2I const& bbox,
                                        lsst::geom::AffineTransform const& toChebyshevRange,
                                        Packer const& packer, Control const& ctrl) {
    // Create a 2-d array that contains T_j(x) for each x value, with x values in rows and j in columns
    ndarray::Array<double, 2, 2> tx = ndarray::allocate(bbox.getWidth(), packer.nx);
    for (int x = bbox.getBeginX(), p = 0; p < bbox.getWidth(); ++p, ++x) {
        evaluateBasis1d(tx[p], toChebyshevRange[lsst::geom::AffineTransform::XX] * x +
                                       toChebyshevRange[lsst::geom::AffineTransform::X]);
    }
 
    // Loop over y values, and at each point, compute T_i(y), then loop over x and multiply by the T_j(x)
    // we already computed and stored above.
    ndarray::Array<double, 2, 2> out = ndarray::allocate(bbox.getArea(), packer.size);
    ndarray::Array<double, 2, 2>::Iterator outIter = out.begin();
    ndarray::Array<double, 1, 1> ty = ndarray::allocate(ctrl.orderY + 1);
    for (int y = bbox.getBeginY(), i = 0; i < bbox.getHeight(); ++i, ++y) {
        evaluateBasis1d(ty, toChebyshevRange[lsst::geom::AffineTransform::YY] * y +
                                    toChebyshevRange[lsst::geom::AffineTransform::Y]);
        for (int j = 0; j < bbox.getWidth(); ++j, ++outIter) {
            // this sets a row of out to the packed outer product of tx and ty
            packer.pack(*outIter, tx[j], ty);
        }
    }
    return out;
}
 
}  // namespace
 
std::shared_ptr<ChebyshevBoundedField> ChebyshevBoundedField::fit(lsst::geom::Box2I const& bbox,
                                                                  ndarray::Array<double const, 1> const& x,
                                                                  ndarray::Array<double const, 1> const& y,
                                                                  ndarray::Array<double const, 1> const& z,
                                                                  Control const& ctrl) {
    // Initialize the result object, so we can make use of the AffineTransform it builds
    std::shared_ptr<ChebyshevBoundedField> result(new ChebyshevBoundedField(bbox));
    // This packer object knows how to map the 2-d Chebyshev functions onto a 1-d array,
    // using only those that the control says should have nonzero coefficients.
    Packer const packer(ctrl);
    // Create a "design matrix" for the linear least squares problem (A in min||Ax-b||)
    ndarray::Array<double, 2, 2> matrix = makeMatrix(x, y, result->_toChebyshevRange, packer, ctrl);
    // Solve the linear least squares problem.
    LeastSquares lstsq = LeastSquares::fromDesignMatrix(matrix, z, LeastSquares::NORMAL_EIGENSYSTEM);
    // Unpack the solution into a 2-d matrix, with zeros for values we didn't fit.
    result->_coefficients = packer.unpack(lstsq.getSolution());
    return result;
}
 
std::shared_ptr<ChebyshevBoundedField> ChebyshevBoundedField::fit(lsst::geom::Box2I const& bbox,
                                                                  ndarray::Array<double const, 1> const& x,
                                                                  ndarray::Array<double const, 1> const& y,
                                                                  ndarray::Array<double const, 1> const& z,
                                                                  ndarray::Array<double const, 1> const& w,
                                                                  Control const& ctrl) {
    // Initialize the result object, so we can make use of the AffineTransform it builds
    std::shared_ptr<ChebyshevBoundedField> result(new ChebyshevBoundedField(bbox));
    // This packer object knows how to map the 2-d Chebyshev functions onto a 1-d array,
    // using only those that the control says should have nonzero coefficients.
    Packer const packer(ctrl);
    // Create a "design matrix" for the linear least squares problem ('A' in min||Ax-b||)
    ndarray::Array<double, 2, 2> matrix = makeMatrix(x, y, result->_toChebyshevRange, packer, ctrl);
    // We want to do weighted least squares, so we multiply both the data vector 'b' and the
    // matrix 'A' by the weights.
    ndarray::asEigenArray(matrix).colwise() *= ndarray::asEigenArray(w);
    ndarray::Array<double, 1, 1> wz = ndarray::copy(z);
    ndarray::asEigenArray(wz) *= ndarray::asEigenArray(w);
    // Solve the linear least squares problem.
    LeastSquares lstsq = LeastSquares::fromDesignMatrix(matrix, wz, LeastSquares::NORMAL_EIGENSYSTEM);
    // Unpack the solution into a 2-d matrix, with zeros for values we didn't fit.
    result->_coefficients = packer.unpack(lstsq.getSolution());
    return result;
}
 
template <typename T>
std::shared_ptr<ChebyshevBoundedField> ChebyshevBoundedField::fit(image::Image<T> const& img,
                                                                  Control const& ctrl) {
    // Initialize the result object, so we can make use of the AffineTransform it builds
    lsst::geom::Box2I bbox = img.getBBox(image::PARENT);
    std::shared_ptr<ChebyshevBoundedField> result(new ChebyshevBoundedField(bbox));
    // This packer object knows how to map the 2-d Chebyshev functions onto a 1-d array,
    // using only those that the control says should have nonzero coefficients.
    Packer const packer(ctrl);
    ndarray::Array<double, 2, 2> matrix = makeMatrix(bbox, result->_toChebyshevRange, packer, ctrl);
    // Flatten the data image into a 1-d vector.
    ndarray::Array<double, 2, 2> imgCopy = ndarray::allocate(img.getArray().getShape());
    imgCopy.deep() = img.getArray();
    ndarray::Array<double const, 1, 1> z = ndarray::flatten<1>(imgCopy);
    // Solve the linear least squares problem.
    LeastSquares lstsq = LeastSquares::fromDesignMatrix(matrix, z, LeastSquares::NORMAL_EIGENSYSTEM);
    // Unpack the solution into a 2-d matrix, with zeros for values we didn't fit.
    result->_coefficients = packer.unpack(lstsq.getSolution());
    return result;
}
 
// ------------------ modifier factories ---------------------------------------------------------------
 
std::shared_ptr<ChebyshevBoundedField> ChebyshevBoundedField::truncate(Control const& ctrl) const {
    if (static_cast<std::size_t>(ctrl.orderX) >= _coefficients.getSize<1>()) {
        throw LSST_EXCEPT(pex::exceptions::LengthError,
                          (boost::format("New x order (%d) exceeds old x order (%d)") % ctrl.orderX %
                           (_coefficients.getSize<1>() - 1))
                                  .str());
    }
    if (static_cast<std::size_t>(ctrl.orderY) >= _coefficients.getSize<0>()) {
        throw LSST_EXCEPT(pex::exceptions::LengthError,
                          (boost::format("New y order (%d) exceeds old y order (%d)") % ctrl.orderY %
                           (_coefficients.getSize<0>() - 1))
                                  .str());
    }
    ndarray::Array<double, 2, 2> coefficients = ndarray::allocate(ctrl.orderY + 1, ctrl.orderX + 1);
    coefficients.deep() = _coefficients[ndarray::view(0, ctrl.orderY + 1)(0, ctrl.orderX + 1)];
    if (ctrl.triangular) {
        Packer packer(ctrl);
        ndarray::Array<double, 1, 1> packed = ndarray::allocate(packer.size);
        packer.pack(packed, coefficients);
        packer.unpack(coefficients, packed);
    }
    return std::make_shared<ChebyshevBoundedField>(getBBox(), coefficients);
}
 
std::shared_ptr<ChebyshevBoundedField> ChebyshevBoundedField::relocate(lsst::geom::Box2I const& bbox) const {
    return std::make_shared<ChebyshevBoundedField>(bbox, _coefficients);
}
 
// ------------------ evaluate() and helpers ---------------------------------------------------------------
 
namespace {
 
// To evaluate a 1-d Chebyshev function without needing to have workspace, we use the
// Clenshaw algorith, which is like going through the recurrence relation in reverse.
// The CoeffGetter argument g is something that behaves like an array, providing access
// to the coefficients.
template <typename CoeffGetter>
double evaluateFunction1d(CoeffGetter g, double x, int size) {
    double b_kp2 = 0.0, b_kp1 = 0.0;
    for (int k = (size - 1); k > 0; --k) {
        double b_k = g[k] + 2 * x * b_kp1 - b_kp2;
        b_kp2 = b_kp1;
        b_kp1 = b_k;
    }
    return g[0] + x * b_kp1 - b_kp2;
}
 
// This class imitates a 1-d array, by running evaluateFunction1d on a nested dimension;
// this lets us reuse the logic in evaluateFunction1d for both dimensions.  Essentially,
// we run evaluateFunction1d on a column of coefficients to evaluate T_i(x), then pass
// the result of that to evaluateFunction1d with the results as the "coefficients" associated
// with the T_j(y) functions.
struct RecursionArrayImitator {
    double operator[](int i) const {
        return evaluateFunction1d(coefficients[i], x, coefficients.getSize<1>());
    }
 
    RecursionArrayImitator(ndarray::Array<double const, 2, 2> const& coefficients_, double x_)
            : coefficients(coefficients_), x(x_) {}
 
    ndarray::Array<double const, 2, 2> coefficients;
    double x;
};
 
}  // namespace
 
double ChebyshevBoundedField::evaluate(lsst::geom::Point2D const& position) const {
    lsst::geom::Point2D p = _toChebyshevRange(position);
    return evaluateFunction1d(RecursionArrayImitator(_coefficients, p.getX()), p.getY(),
                              _coefficients.getSize<0>());
}
 
// The integral of T_n(x) over [-1,1]:
// https://en.wikipedia.org/wiki/Chebyshev_polynomials#Differentiation_and_integration
double integrateTn(int n) {
    if (n % 2 == 1)
        return 0;
    else
        return 2.0 / (1.0 - double(n * n));
}
 
double ChebyshevBoundedField::integrate() const {
    double result = 0;
    double determinant = getBBox().getArea() / 4.0;
    for (ndarray::Size j = 0; j < _coefficients.getSize<0>(); j++) {
        for (ndarray::Size i = 0; i < _coefficients.getSize<1>(); i++) {
            result += _coefficients[j][i] * integrateTn(i) * integrateTn(j);
        }
    }
    return result * determinant;
}
 
double ChebyshevBoundedField::mean() const { return integrate() / getBBox().getArea(); }
 
// ------------------ persistence ---------------------------------------------------------------------------
 
namespace {
 
struct PersistenceHelper {
    table::Schema schema;
    table::Key<int> orderX;
    table::Box2IKey bbox;
    table::Key<table::Array<double> > coefficients;
 
    PersistenceHelper(int nx, int ny)
            : schema(),
              orderX(schema.addField<int>("order_x", "maximum Chebyshev function order in x")),
              bbox(table::Box2IKey::addFields(schema, "bbox", "bounding box", "pixel")),
              coefficients(schema.addField<table::Array<double> >(
                      "coefficients", "Chebyshev function coefficients, ordered by y then x", nx * ny)) {}
 
    PersistenceHelper(table::Schema const& s)
            : schema(s), orderX(s["order_x"]), bbox(s["bbox"]), coefficients(s["coefficients"]) {}
};
 
class ChebyshevBoundedFieldFactory : public table::io::PersistableFactory {
public:
    explicit ChebyshevBoundedFieldFactory(std::string const& name)
            : afw::table::io::PersistableFactory(name) {}
 
    std::shared_ptr<table::io::Persistable> read(InputArchive const& archive,
                                                 CatalogVector const& catalogs) const override {
        LSST_ARCHIVE_ASSERT(catalogs.size() == 1u);
        LSST_ARCHIVE_ASSERT(catalogs.front().size() == 1u);
        table::BaseRecord const& record = catalogs.front().front();
        PersistenceHelper const keys(record.getSchema());
        lsst::geom::Box2I bbox(record.get(keys.bbox));
        int nx = record.get(keys.orderX) + 1;
        int ny = keys.coefficients.getSize() / nx;
        LSST_ARCHIVE_ASSERT(nx * ny == keys.coefficients.getSize());
        ndarray::Array<double, 2, 2> coefficients = ndarray::allocate(ny, nx);
        ndarray::flatten<1>(coefficients) = record.get(keys.coefficients);
        return std::make_shared<ChebyshevBoundedField>(bbox, coefficients);
    }
};
 
std::string getChebyshevBoundedFieldPersistenceName() { return "ChebyshevBoundedField"; }
 
ChebyshevBoundedFieldFactory registration(getChebyshevBoundedFieldPersistenceName());
 
}  // namespace
 
std::string ChebyshevBoundedField::getPersistenceName() const {
    return getChebyshevBoundedFieldPersistenceName();
}
 
std::string ChebyshevBoundedField::getPythonModule() const { return "lsst.afw.math"; }
 
void ChebyshevBoundedField::write(OutputArchiveHandle& handle) const {
    PersistenceHelper const keys(_coefficients.getSize<1>(), _coefficients.getSize<0>());
    table::BaseCatalog catalog = handle.makeCatalog(keys.schema);
    std::shared_ptr<table::BaseRecord> record = catalog.addNew();
    record->set(keys.orderX, _coefficients.getSize<1>() - 1);
    record->set(keys.bbox, getBBox());
    (*record)[keys.coefficients].deep() = ndarray::flatten<1>(_coefficients);
    handle.saveCatalog(catalog);
}
 
// ------------------ operators -----------------------------------------------------------------------------
 
std::shared_ptr<BoundedField> ChebyshevBoundedField::operator*(double const scale) const {
    return std::make_shared<ChebyshevBoundedField>(getBBox(), ndarray::copy(getCoefficients() * scale));
}
 
bool ChebyshevBoundedField::operator==(BoundedField const& rhs) const {
    auto rhsCasted = dynamic_cast<ChebyshevBoundedField const*>(&rhs);
    if (!rhsCasted) return false;
 
    return (getBBox() == rhsCasted->getBBox()) &&
           (_coefficients.getShape() == rhsCasted->_coefficients.getShape()) &&
           all(equal(_coefficients, rhsCasted->_coefficients));
}
 
std::string ChebyshevBoundedField::toString() const {
    std::ostringstream os;
    os << "ChebyshevBoundedField (" << _coefficients.getShape() << " coefficients in y,x)";
    return os.str();
}
 
// ------------------ explicit instantiation ----------------------------------------------------------------
 
#ifndef DOXYGEN
 
#define INSTANTIATE(T)                                                                                       \
    template std::shared_ptr<ChebyshevBoundedField> ChebyshevBoundedField::fit(image::Image<T> const& image, \
                                                                               Control const& ctrl)
 
INSTANTIATE(float);
INSTANTIATE(double);
 
#endif
}  // namespace math
}  // namespace afw
}  // namespace lsst