Pixel.h

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/*
 * LSST Data Management System
 * Copyright 2008-2016  AURA/LSST.
 *
 * This product includes software developed by the
 * LSST Project (http://www.lsst.org/).
 *
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the LSST License Statement and
 * the GNU General Public License along with this program.  If not,
 * see <http://www.lsstcorp.org/LegalNotices/>.
 */
 
#if !defined(LSST_AFW_IMAGE_PIXEL_H)
#define LSST_AFW_IMAGE_PIXEL_H
 
#include <cmath>
#include <functional>
#include <iostream>
#include <type_traits>
 
#include "lsst/utils/hashCombine.h"
 
namespace lsst {
namespace afw {
namespace image {
namespace pixel {
 
template <typename, typename, typename, typename, typename>
class BinaryExpr;
 
template <typename>
struct exprTraits;
 
template <typename>
struct bitwise_or;
template <typename>
struct variance_divides;
template <typename>
struct variance_multiplies;
template <typename>
struct variance_plus;
 
/*
 * Classes to provide utility functions for a "Pixel" to get at image/mask/variance operators
 *
 * These classes allow us to manipulate the tuples returned by MaskedImage iterators/locators as if they were
 * POD.  This provides convenient syntactic sugar, but it also permits us to write generic algorithms to
 * manipulate MaskedImages as well as Images
 *
 * We need SinglePixel as well as Pixel as the latter is just a reference to a pixel in an image, and we need
 * to be able to build temporary values too
 *
 * We use C++ template expressions to manipulate Pixel and SinglePixel; this permits us to avoid making
 * SinglePixel inherit from Pixel (or vice versa) and allows the compiler to do a much better job of
 * optimising mixed-mode expressions --- basically, it no longer needs to create SinglePixels as Pixels that
 * have their own storage but use the reference members of Pixel to get work done.  There may be better ways,
 * but this way works, and g++ 4.0.1 (and maybe other compilers/versions) failed to optimise the previous
 * solution very well.
 */
 
template <typename _ImagePixelT, typename _MaskPixelT, typename _VariancePixelT = double>
class SinglePixel : public detail::MaskedImagePixel_tag {
public:
    template <typename, typename, typename>
    friend class Pixel;
    template <typename T>
    friend class PixelTypeTraits;
 
    using ImagePixelT = _ImagePixelT;
    using MaskPixelT = _MaskPixelT;
    using VariancePixelT = _VariancePixelT;
 
    SinglePixel(ImagePixelT image, MaskPixelT mask = 0, VariancePixelT variance = 0)
            : _image(image), _mask(mask), _variance(variance) {}
 
    template <typename rhsExpr>
    SinglePixel(rhsExpr const& rhs,
                // ensure this ctor isn't invoked for simple numeric types, which should use
                // the overload above.
                typename std::enable_if<!std::is_fundamental<rhsExpr>::value, void*>::type dummy = nullptr)
            : _image(rhs.image()), _mask(rhs.mask()), _variance(rhs.variance()) {}
 
    ImagePixelT image() const { return _image; }
    MaskPixelT mask() const { return _mask; }
    VariancePixelT variance() const { return _variance; }
 
private:
    SinglePixel()
            : _image(std::numeric_limits<_ImagePixelT>::has_quiet_NaN
                             ? std::numeric_limits<_ImagePixelT>::quiet_NaN()
                             : 0),
              _mask(0),
              _variance(std::numeric_limits<_VariancePixelT>::has_quiet_NaN
                                ? std::numeric_limits<_VariancePixelT>::quiet_NaN()
                                : 0) {}
 
    ImagePixelT _image;
    MaskPixelT _mask;
    VariancePixelT _variance;
};
 
template <typename PixelT>
struct PixelTypeTraits {
    static inline const PixelT padValue() {
        return std::numeric_limits<PixelT>::has_quiet_NaN ? std::numeric_limits<PixelT>::quiet_NaN() : 0;
    }
};
 
template <typename _ImagePixelT, typename _MaskPixelT, typename _VariancePixelT>
struct PixelTypeTraits<SinglePixel<_ImagePixelT, _MaskPixelT, _VariancePixelT> > {
    using PixelT = SinglePixel<_ImagePixelT, _MaskPixelT, _VariancePixelT>;
 
    static inline const PixelT padValue() { return PixelT(); }
};
 
template <typename ImagePixelT, typename MaskPixelT, typename VariancePixelT>
SinglePixel<ImagePixelT, MaskPixelT, VariancePixelT> makeSinglePixel(ImagePixelT x, MaskPixelT m,
                                                                     VariancePixelT v) {
    return SinglePixel<ImagePixelT, MaskPixelT, VariancePixelT>(x, m, v);
}
 
template <typename _ImagePixelT, typename _MaskPixelT, typename _VariancePixelT = double>
class Pixel : public detail::MaskedImagePixel_tag {
public:
    using ImagePixelT = _ImagePixelT;
    using MaskPixelT = _MaskPixelT;
    using VariancePixelT = _VariancePixelT;
 
#if 0
    Pixel(ImagePixelT& image, MaskPixelT& mask, VariancePixelT& variance) :
        _image(image), _mask(mask), _variance(variance) {}
#else
    //
    // This constructor casts away const.  This should be fixed by making const Pixels.
    //
    Pixel(ImagePixelT const& image, MaskPixelT const& mask = 0x0, VariancePixelT const& variance = 0)
            : _image(const_cast<ImagePixelT&>(image)),
              _mask(const_cast<MaskPixelT&>(mask)),
              _variance(const_cast<VariancePixelT&>(variance)) {}
#endif
 
    Pixel(SinglePixel<ImagePixelT, MaskPixelT, VariancePixelT>& rhs)
            : _image(rhs._image), _mask(rhs._mask), _variance(rhs._variance) {}
 
    Pixel(Pixel const& rhs) = default;
    Pixel(Pixel&& rhs) = default;
    ~Pixel() = default;
 
    Pixel operator=(Pixel const& rhs) {  // the following template won't stop the compiler trying to generate
                                         // operator=
        _variance = rhs.variance();      // evaluate before we update image()
        _image = rhs.image();
        _mask = rhs.mask();
 
        return *this;
    }
    template <typename rhsExpr>
    Pixel operator=(rhsExpr const& rhs) {
        _variance = rhs.variance();  // evaluate before we update image()
        _image = rhs.image();
        _mask = rhs.mask();
 
        return *this;
    }
    // Delegate to copy-assignment for backwards compatibility
    Pixel operator=(Pixel&& rhs) { return *this = rhs; }
 
    Pixel operator=(double const& rhs_image) {
        _image = rhs_image;
        _mask = 0;
        _variance = 0;
 
        return *this;
    }
 
    Pixel operator=(int const& rhs_image) {
        _image = rhs_image;
        _mask = 0;
        _variance = 0;
 
        return *this;
    }
    ImagePixelT image() const { return _image; }
    MaskPixelT mask() const { return _mask; }
    VariancePixelT variance() const { return _variance; }
    //
    // Logical operators.  We don't need to construct BinaryExpr for them
    // as efficiency isn't a concern.
    //
    template <typename T1>
    friend bool operator==(Pixel const& lhs, T1 const& rhs) {
        return lhs.image() == rhs.image() && lhs.mask() == rhs.mask() && lhs.variance() == rhs.variance();
    }
 
    template <typename T1>
    friend bool operator!=(Pixel const& lhs, T1 const& rhs) {
        return !(lhs == rhs);
    }
 
    std::size_t hash_value() const noexcept {
        // Completely arbitrary seed
        // Convert to double to avoid inconsistently hashing equal numbers of different types
        return utils::hashCombine(17, static_cast<double>(image()), static_cast<double>(mask()),
                                  static_cast<double>(variance()));
    }
 
    //
    // Provide friend versions of the op= operators to permit argument promotion on their first arguments
    //
    template <typename ExprT>
    friend Pixel operator+=(Pixel const& e1, ExprT const& e2) {
        Pixel tmp(e1);  // n.b. shares storage with e1 but gets around "const" (which is required)
        tmp = BinaryExpr<Pixel, ExprT, std::plus<ImagePixelT>, bitwise_or<MaskPixelT>,
                         variance_plus<VariancePixelT> >(tmp, e2);
        return tmp;
    }
 
    template <typename ExprT>
    friend Pixel operator-=(Pixel const& e1, ExprT const& e2) {
        Pixel tmp(e1);  // n.b. shares storage with e1 but gets around "const" (which is required)
        tmp = BinaryExpr<Pixel, ExprT, std::minus<ImagePixelT>, bitwise_or<MaskPixelT>,
                         variance_plus<VariancePixelT> >(tmp, e2);
        return tmp;
    }
 
    template <typename ExprT>
    friend Pixel operator*=(Pixel const& e1, ExprT const& e2) {
        Pixel tmp(e1);  // n.b. shares storage with e1 but gets around "const" (which is required)
        tmp = BinaryExpr<Pixel, ExprT, std::multiplies<ImagePixelT>, bitwise_or<MaskPixelT>,
                         variance_multiplies<VariancePixelT> >(tmp, e2);
        return tmp;
    }
 
    template <typename ExprT>
    friend Pixel operator/=(Pixel const& e1, ExprT const& e2) {
        Pixel tmp(e1);  // n.b. shares storage with e1 but gets around "const" (which is required)
        tmp = BinaryExpr<Pixel, ExprT, std::divides<ImagePixelT>, bitwise_or<MaskPixelT>,
                         variance_divides<VariancePixelT> >(tmp, e2);
        return tmp;
    }
 
private:
    ImagePixelT& _image;
    MaskPixelT& _mask;
    VariancePixelT& _variance;
};
 
template <typename ExprT>
struct exprTraits {
    using expr_type = ExprT;
    using ImagePixelT = typename ExprT::ImagePixelT;
    using MaskPixelT = typename ExprT::MaskPixelT;
    using VariancePixelT = typename ExprT::VariancePixelT;
};
 
template <>
struct exprTraits<double> {
    using ImagePixelT = double;
    using MaskPixelT = int;
    using VariancePixelT = double;
    using expr_type = SinglePixel<ImagePixelT, MaskPixelT>;
};
 
template <>
struct exprTraits<float> {
    using ImagePixelT = float;
    using MaskPixelT = exprTraits<double>::MaskPixelT;
    using VariancePixelT = exprTraits<double>::VariancePixelT;
    using expr_type = SinglePixel<ImagePixelT, MaskPixelT>;
};
 
template <>
struct exprTraits<int> {
    using ImagePixelT = int;
    using MaskPixelT = exprTraits<double>::MaskPixelT;
    using VariancePixelT = exprTraits<double>::VariancePixelT;
    using expr_type = SinglePixel<ImagePixelT, MaskPixelT>;
};
 
template <>
struct exprTraits<unsigned short> {
    using ImagePixelT = int;
    using MaskPixelT = exprTraits<double>::MaskPixelT;
    using VariancePixelT = exprTraits<double>::VariancePixelT;
    using expr_type = SinglePixel<ImagePixelT, MaskPixelT>;
};
 
template <typename T1>
struct noop : public std::unary_function<T1, T1> {
    T1 operator()(const T1& x) const { return x; }
};
 
template <typename T1>
struct bitwise_or : public std::binary_function<T1, T1, T1> {
    T1 operator()(const T1& x, const T1& y) const { return (x | y); }
    T1 operator()(const T1& x) const { return x; }
};
 
template <typename T1>
struct variance_divides {
    T1 operator()(T1 const& x, T1 const& y, T1 const& vx, T1 const& vy) const {
        T1 const x2 = x * x;
        T1 const y2 = y * y;
        T1 const iy2 = 1.0 / y2;
        return x2 * vy * iy2 * iy2 + vx * iy2;
    }
 
    T1 operator()(T1 const&, T1 const& y, T1 const& vx) const { return vx / (y * y); }
};
 
template <typename T1>
struct variance_multiplies {
    T1 operator()(T1 const& x, T1 const& y, T1 const& vx, T1 const& vy) const {
        T1 const x2 = x * x;
        T1 const y2 = y * y;
        return x2 * vy + y2 * vx;
    }
 
    T1 operator()(T1 const&, T1 const& y, T1 const& vx) const { return vx * y * y; }
};
 
template <typename T1>
struct variance_plus {
    T1 operator()(T1 const&, T1 const&, T1 const& vx, T1 const& vy) const { return vx + vy; }
 
    T1 operator()(T1 const&, T1 const&, T1 const& vx) const { return vx; }
};
 
template <typename T1>
struct variance_plus_covar {
    variance_plus_covar(double alpha = 0) : _alpha(alpha) {}
 
    T1 operator()(T1 const&, T1 const&, T1 const& vx, T1 const& vy) const {
        return vx + vy + 2 * _alpha * sqrt(vx * vy);
    }
    T1 operator()(T1 const&, T1 const&, T1 const& vx) const { return vx; }
 
private:
    double _alpha;
};
 
template <typename ExprT1, typename ImageBinOp, typename MaskBinOp, typename VarianceBinOp>
class UnaryExpr {
public:
    using ImagePixelT = typename exprTraits<ExprT1>::ImagePixelT;
    using MaskPixelT = typename exprTraits<ExprT1>::MaskPixelT;
    using VariancePixelT = typename exprTraits<ExprT1>::VariancePixelT;
    UnaryExpr(ExprT1 e1, ImageBinOp imageOp = ImageBinOp(), MaskBinOp maskOp = MaskBinOp(),
              VarianceBinOp varOp = VarianceBinOp())
            : _expr1(e1), _imageOp(imageOp), _maskOp(maskOp), _varOp(varOp) {}
 
    ImagePixelT image() const { return _imageOp(_expr1.image()); }
 
    MaskPixelT mask() const { return _maskOp(_expr1.mask()); }
 
    VariancePixelT variance() const { return _varOp(_expr1.variance()); }
 
private:
    typename exprTraits<ExprT1>::expr_type _expr1;
    ImageBinOp _imageOp;
    MaskBinOp _maskOp;
    VarianceBinOp _varOp;
};
 
template <typename ExprT1, typename ExprT2, typename ImageBinOp, typename MaskBinOp, typename VarianceBinOp>
class BinaryExpr final {
public:
    using ImagePixelT = typename exprTraits<ExprT1>::ImagePixelT;
    using MaskPixelT = typename exprTraits<ExprT1>::MaskPixelT;
    using VariancePixelT = typename exprTraits<ExprT1>::VariancePixelT;
    BinaryExpr(ExprT1 e1, ExprT2 e2, ImageBinOp imageOp = ImageBinOp(), MaskBinOp maskOp = MaskBinOp(),
               VarianceBinOp varOp = VarianceBinOp())
            : _expr1(e1), _expr2(e2), _imageOp(imageOp), _maskOp(maskOp), _varOp(varOp) {}
 
    BinaryExpr(ExprT1 e1, ExprT2 e2, double const alpha, ImageBinOp imageOp = ImageBinOp(),
               MaskBinOp maskOp = MaskBinOp(), VarianceBinOp = VarianceBinOp())
            : _expr1(e1), _expr2(e2), _imageOp(imageOp), _maskOp(maskOp), _varOp(VarianceBinOp(alpha)) {}
    ImagePixelT image() const { return _imageOp(_expr1.image(), _expr2.image()); }
 
    MaskPixelT mask() const { return _maskOp(_expr1.mask(), _expr2.mask()); }
 
    VariancePixelT variance() const {
        return _varOp(_expr1.image(), _expr2.image(), _expr1.variance(), _expr2.variance());
    }
 
private:
    typename exprTraits<ExprT1>::expr_type _expr1;
    typename exprTraits<ExprT2>::expr_type _expr2;
    ImageBinOp _imageOp;
    MaskBinOp _maskOp;
    VarianceBinOp _varOp;
};
 
template <typename ExprT1, typename ImageBinOp, typename MaskBinOp, typename VarianceBinOp>
class BinaryExpr<ExprT1, double, ImageBinOp, MaskBinOp, VarianceBinOp> final {
public:
    using ImagePixelT = typename exprTraits<ExprT1>::ImagePixelT;
    using MaskPixelT = typename exprTraits<ExprT1>::MaskPixelT;
    using VariancePixelT = typename exprTraits<ExprT1>::VariancePixelT;
    BinaryExpr(ExprT1 e1, double e2, ImageBinOp imageOp = ImageBinOp(), MaskBinOp maskOp = MaskBinOp(),
               VarianceBinOp varOp = VarianceBinOp())
            : _expr1(e1), _expr2(e2), _imageOp(imageOp), _maskOp(maskOp), _varOp(varOp) {}
 
    BinaryExpr(ExprT1 e1, double e2, double const alpha, ImageBinOp imageOp = ImageBinOp(),
               MaskBinOp maskOp = MaskBinOp(), VarianceBinOp = VarianceBinOp())
            : _expr1(e1), _expr2(e2), _imageOp(imageOp), _maskOp(maskOp), _varOp(VarianceBinOp(alpha)) {}
    ImagePixelT image() const { return _imageOp(_expr1.image(), _expr2); }
 
    MaskPixelT mask() const { return _maskOp(_expr1.mask()); }
 
    VariancePixelT variance() const { return _varOp(_expr1.image(), _expr2, _expr1.variance()); }
 
private:
    typename exprTraits<ExprT1>::expr_type _expr1;
    double _expr2;
    ImageBinOp _imageOp;
    MaskBinOp _maskOp;
    VarianceBinOp _varOp;
};
 
template <typename ExprT1>
UnaryExpr<ExprT1, std::negate<typename exprTraits<ExprT1>::ImagePixelT>,
          noop<typename exprTraits<ExprT1>::MaskPixelT>, noop<typename exprTraits<ExprT1>::VariancePixelT> >
operator-(ExprT1 e1) {
    return UnaryExpr<ExprT1, std::negate<typename exprTraits<ExprT1>::ImagePixelT>,
                     noop<typename exprTraits<ExprT1>::MaskPixelT>,
                     noop<typename exprTraits<ExprT1>::VariancePixelT> >(e1);
}
 
//------------------------------------------
template <class T, template <class...> class Template>
struct is_specialization : std::false_type {};
 
template <template <class...> class Template, class... Args>
struct is_specialization<Template<Args...>, Template> : std::true_type {};
// avoid invocation with pybind11::detail::descr::operator+ for pybind11 >= 2.3.0
template <typename ExprT1, typename ExprT2,
   typename = std::enable_if_t <
          ( std::is_integral<ExprT1>::value  ||
            is_specialization<ExprT1,BinaryExpr>{} ||
            is_specialization<ExprT1,SinglePixel>{} ||
            is_specialization<ExprT1,Pixel>{})
            &&
          ( std::is_integral<ExprT2>::value  ||
            is_specialization<ExprT2,BinaryExpr>{} ||
            is_specialization<ExprT2,SinglePixel>{} ||
            is_specialization<ExprT2,Pixel>{} )
         >
>
BinaryExpr<ExprT1, ExprT2, std::plus<typename exprTraits<ExprT1>::ImagePixelT>,
           bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
           variance_plus<typename exprTraits<ExprT1>::VariancePixelT> >
operator+(ExprT1 e1, ExprT2 e2) {
    return BinaryExpr<ExprT1, ExprT2, std::plus<typename exprTraits<ExprT1>::ImagePixelT>,
                      bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                      variance_plus<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
}
 
template <typename ExprT1, typename ExprT2>
ExprT1 operator+=(ExprT1& e1, ExprT2 e2) {
    e1 = BinaryExpr<ExprT1, ExprT2, std::plus<typename exprTraits<ExprT1>::ImagePixelT>,
                    bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                    variance_plus<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
    return e1;
}
 
//
// Implementations of add that work for arithmetic or MaskedImage pixels
//
// The choice is made on the basis of std::is_arithmetic
namespace {
template <typename ExprT1, typename ExprT2>
ExprT1 doPlus(ExprT1 e1, ExprT2 e2, double const, boost::mpl::true_) {
    return e1 + e2;
}
 
template <typename ExprT1, typename ExprT2>
BinaryExpr<ExprT1, ExprT2, std::plus<typename exprTraits<ExprT1>::ImagePixelT>,
           bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
           variance_plus_covar<typename exprTraits<ExprT1>::VariancePixelT> >
doPlus(ExprT1 e1, ExprT2 e2, double const alpha, boost::mpl::false_) {
    return BinaryExpr<ExprT1, ExprT2, std::plus<typename exprTraits<ExprT1>::ImagePixelT>,
                      bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                      variance_plus_covar<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2, alpha);
}
}  // namespace
 
template <typename ExprT1, typename ExprT2>
inline ExprT1 plus(
        ExprT1& lhs,        
        ExprT2 const& rhs,  
        float covariance    
) {
    return doPlus(lhs, rhs, covariance, typename std::is_arithmetic<ExprT1>::type());
}
 
//------------------------------------------
template <typename ExprT1, typename ExprT2>
BinaryExpr<ExprT1, ExprT2, std::minus<typename exprTraits<ExprT1>::ImagePixelT>,
           bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
           variance_plus<typename exprTraits<ExprT1>::VariancePixelT> >
operator-(ExprT1 e1, ExprT2 e2) {
    return BinaryExpr<ExprT1, ExprT2, std::minus<typename exprTraits<ExprT1>::ImagePixelT>,
                      bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                      variance_plus<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
}
 
template <typename ExprT1, typename ExprT2>
ExprT1 operator-=(ExprT1& e1, ExprT2 e2) {
    e1 = BinaryExpr<ExprT1, ExprT2, std::minus<typename exprTraits<ExprT1>::ImagePixelT>,
                    bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                    variance_plus<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
    return e1;
}
 
//------------------------------------------
template <typename ExprT1, typename ExprT2>
BinaryExpr<ExprT1, ExprT2, std::multiplies<typename exprTraits<ExprT1>::ImagePixelT>,
           bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
           variance_multiplies<typename exprTraits<ExprT1>::VariancePixelT> >
operator*(ExprT1 e1, ExprT2 e2) {
    return BinaryExpr<ExprT1, ExprT2, std::multiplies<typename exprTraits<ExprT1>::ImagePixelT>,
                      bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                      variance_multiplies<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
}
 
template <typename ExprT1, typename ExprT2>
ExprT1 operator*=(ExprT1& e1, ExprT2 e2) {
    e1 = BinaryExpr<ExprT1, ExprT2, std::multiplies<typename exprTraits<ExprT1>::ImagePixelT>,
                    bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                    variance_multiplies<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
    return e1;
}
 
//------------------------------------------
template <typename ExprT1, typename ExprT2>
BinaryExpr<ExprT1, ExprT2, std::divides<typename exprTraits<ExprT1>::ImagePixelT>,
           bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
           variance_divides<typename exprTraits<ExprT1>::VariancePixelT> >
operator/(ExprT1 e1, ExprT2 e2) {
    return BinaryExpr<ExprT1, ExprT2, std::divides<typename exprTraits<ExprT1>::ImagePixelT>,
                      bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                      variance_divides<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
}
 
template <typename ExprT1, typename ExprT2>
ExprT1 operator/=(ExprT1& e1, ExprT2 e2) {
    e1 = BinaryExpr<ExprT1, ExprT2, std::divides<typename exprTraits<ExprT1>::ImagePixelT>,
                    bitwise_or<typename exprTraits<ExprT1>::MaskPixelT>,
                    variance_divides<typename exprTraits<ExprT1>::VariancePixelT> >(e1, e2);
    return e1;
}
 
template <typename ImagePixelT, typename MaskPixelT, typename VariancePixelT>
std::ostream& operator<<(std::ostream& os, SinglePixel<ImagePixelT, MaskPixelT, VariancePixelT> const& v) {
    return os << "(" << v.image() << ", " << v.mask() << ", " << v.variance() << ")";
}
 
template <typename ImagePixelT, typename MaskPixelT, typename VariancePixelT>
std::ostream& operator<<(std::ostream& os, Pixel<ImagePixelT, MaskPixelT, VariancePixelT> const& v) {
    return os << "(" << v.image() << ", " << v.mask() << ", " << v.variance() << ")";
}
 
template <typename ExprT1, typename ExprT2, typename BinOp, typename MaskBinOp, typename VarBinOp>
std::ostream& operator<<(std::ostream& os, BinaryExpr<ExprT1, ExprT2, BinOp, MaskBinOp, VarBinOp> const& v) {
    return os << "(" << v.image() << ", " << v.mask() << ", " << v.variance() << ")";
}
}  // namespace pixel
}  // namespace image
}  // namespace afw
}  // namespace lsst
 
namespace std {
template <typename ImagePixelT, typename MaskPixelT, typename VariancePixelT>
struct hash<lsst::afw::image::pixel::Pixel<ImagePixelT, MaskPixelT, VariancePixelT>> {
    using argument_type = lsst::afw::image::pixel::Pixel<ImagePixelT, MaskPixelT, VariancePixelT>;
    using result_type = size_t;
    size_t operator()(argument_type const& obj) const noexcept { return obj.hash_value(); }
};
 
/*
 * SinglePixel is an awkward case because Pixel == SinglePixel is a valid comparison
 * but SinglePixel == SinglePixel is not.
 * Give SinglePixel a hash consistent with Pixel's, just in case somebody tries something funny.
 */
template <typename ImagePixelT, typename MaskPixelT, typename VariancePixelT>
struct hash<lsst::afw::image::pixel::SinglePixel<ImagePixelT, MaskPixelT, VariancePixelT>> {
    using argument_type = lsst::afw::image::pixel::SinglePixel<ImagePixelT, MaskPixelT, VariancePixelT>;
    using result_type = size_t;
    size_t operator()(argument_type const& obj) const noexcept {
        // Don't want to define SinglePixel::hash_value when SinglePixel::operator== doesn't exist
        using EquivalentPixel = lsst::afw::image::pixel::Pixel<ImagePixelT, MaskPixelT, VariancePixelT>;
        // Work around Pixel needing references to external data
        ImagePixelT image = obj.image();
        MaskPixelT mask = obj.mask();
        VariancePixelT variance = obj.variance();
        return EquivalentPixel(image, mask, variance).hash_value();
    }
};
}  // namespace std
#endif