src/forward_model.rs

/*!
Forward models from discrete measures to observations.
*/

use numeric_literals::replace_float_literals;
use nalgebra::base::{
    DMatrix,
    DVector
};
use std::iter::Zip;
use std::ops::RangeFrom;
use std::marker::PhantomData;

pub use alg_tools::linops::*;
use alg_tools::euclidean::Euclidean;
use alg_tools::norms::{
    L1, Linfinity, L2, Norm
};
use alg_tools::bisection_tree::*;
use alg_tools::mapping::RealMapping;
use alg_tools::lingrid::*;
use alg_tools::iter::{MapX, Mappable};
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::tabledump::write_csv;
use alg_tools::error::DynError;
use alg_tools::maputil::map2;

use crate::types::*;
use crate::measures::*;
use crate::seminorms::{
    ConvolutionOp,
    SimpleConvolutionKernel,
};
use crate::kernels::{
    Convolution,
    AutoConvolution,
    BoundedBy,
};
use crate::types::L2Squared;
use crate::transport::TransportLipschitz;

pub type RNDM<F, const N : usize> = DiscreteMeasure<Loc<F,N>, F>;

/// `ForwardeModel`s are bounded preadjointable linear operators  $A ∈ 𝕃(𝒵(Ω); E)$
/// where $𝒵(Ω) ⊂ ℳ(Ω)$ is the space of sums of delta measures, presented by
/// [`DiscreteMeasure`], and $E$ is a [`Euclidean`] space.
pub trait ForwardModel<Domain, F : Float + ToNalgebraRealField>
: BoundedLinear<DiscreteMeasure<Domain, F>, Codomain=Self::Observable, FloatType=F>
+ GEMV<F, DiscreteMeasure<Domain, F>, Self::Observable>
+ Linear<DeltaMeasure<Domain, F>, Codomain=Self::Observable>
+ Preadjointable<DiscreteMeasure<Domain, F>, Self::Observable> {
    /// The codomain or value space (of “observables”) for this operator.
    /// It is assumed to be a [`Euclidean`] space, and therefore also (identified with)
    /// the domain of the preadjoint.
    type Observable : Euclidean<F, Output=Self::Observable>
                      + AXPY<F>
                      + Clone;

    /// Return A_*A and A_* b
    fn findim_quadratic_model(
        &self,
        μ : &DiscreteMeasure<Domain, F>,
        b : &Self::Observable
    ) -> (DMatrix<F::MixedType>, DVector<F::MixedType>);

    /// Write an observable into a file.
    fn write_observable(&self, b : &Self::Observable, prefix : String) -> DynError;

    /// Returns a zero observable
    fn zero_observable(&self) -> Self::Observable;

    /// Returns an empty (uninitialised) observable.
    ///
    /// This is used as a placeholder for temporary [`std::mem::replace`] move operations.
    fn empty_observable(&self) -> Self::Observable;
}

pub type ShiftedSensor<F, S, P, const N : usize> = Shift<Convolution<S, P>, F, N>;

/// Trait for physical convolution models. Has blanket implementation for all cases.
pub trait Spread<F : Float, const N : usize>
: 'static + Clone + Support<F, N> + RealMapping<F, N> + Bounded<F> {}

impl<F, T, const N : usize> Spread<F, N> for T
where F : Float,
      T : 'static + Clone + Support<F, N> + Bounded<F> + RealMapping<F, N> {}

/// Trait for compactly supported sensors. Has blanket implementation for all cases.
pub trait Sensor<F : Float, const N : usize> : Spread<F, N> + Norm<F, L1> + Norm<F, Linfinity> {}

impl<F, T, const N : usize> Sensor<F, N> for T
where F : Float,
      T : Spread<F, N> + Norm<F, L1> + Norm<F, Linfinity> {}


pub trait SensorGridBT<F, S, P, const N : usize> :
Clone + BTImpl<F, N, Data=usize, Agg=Bounds<F>>
where F : Float,
      S : Sensor<F, N>,
      P : Spread<F, N> {}

impl<F, S, P, T, const N : usize>
SensorGridBT<F, S, P, N>
for T
where T : Clone + BTImpl<F, N, Data=usize, Agg=Bounds<F>>,
      F : Float,
      S : Sensor<F, N>,
      P : Spread<F, N> {}

// We need type alias bounds to access associated types
#[allow(type_alias_bounds)]
type SensorGridBTFN<F, S, P, BT : SensorGridBT<F, S, P, N>, const N : usize>
= BTFN<F, SensorGridSupportGenerator<F, S, P, N>, BT, N>;

/// Sensor grid forward model
#[derive(Clone)]
pub struct SensorGrid<F, S, P, BT, const N : usize>
where F : Float,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      BT : SensorGridBT<F, S, P, N>, {
    domain : Cube<F, N>,
    sensor_count : [usize; N],
    sensor : S,
    spread : P,
    base_sensor : Convolution<S, P>,
    bt : BT,
}

impl<F, S, P, BT, const N : usize> SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {

    pub fn new(
        domain : Cube<F, N>,
        sensor_count : [usize; N],
        sensor : S,
        spread : P,
        depth : BT::Depth
    ) -> Self {
        let base_sensor = Convolution(sensor.clone(), spread.clone());
        let bt = BT::new(domain, depth);
        let mut sensorgrid = SensorGrid {
            domain,
            sensor_count,
            sensor,
            spread,
            base_sensor,
            bt,
        };

        for (x, id) in sensorgrid.grid().into_iter().zip(0usize..) {
            let s = sensorgrid.shifted_sensor(x);
            sensorgrid.bt.insert(id, &s);
        }

        sensorgrid
    }

    pub fn grid(&self) -> LinGrid<F, N> {
        lingrid_centered(&self.domain, &self.sensor_count)
    }

    pub fn n_sensors(&self) -> usize {
        self.sensor_count.iter().product()
    }

    #[inline]
    fn shifted_sensor(&self, x : Loc<F, N>) -> ShiftedSensor<F, S, P, N> {
        self.base_sensor.clone().shift(x)
    }

    #[inline]
    fn _zero_observable(&self) -> DVector<F> {
        DVector::zeros(self.n_sensors())
    }
}

impl<F, S, P, BT, const N : usize> Apply<RNDM<F, N>> for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {

    type Output =  DVector<F>;

    #[inline]
    fn apply(&self, μ : RNDM<F, N>) -> DVector<F> {
        self.apply(&μ)
    }
}

impl<'a, F, S, P, BT, const N : usize> Apply<&'a RNDM<F, N>> for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {

    type Output =  DVector<F>;

    fn apply(&self, μ : &'a RNDM<F, N>) ->  DVector<F> {
        let mut res = self._zero_observable();
        self.apply_add(&mut res, μ);
        res
    }
}

impl<F, S, P, BT, const N : usize> Linear<RNDM<F, N>> for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {
    type Codomain = DVector<F>;
}


#[replace_float_literals(F::cast_from(literal))]
impl<F, S, P, BT, const N : usize> GEMV<F, RNDM<F, N>, DVector<F>> for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {

    fn gemv(&self, y : &mut DVector<F>, α : F, μ : &RNDM<F, N>, β : F) {
        let grid = self.grid();
        if β == 0.0 {
            y.fill(0.0)
        } else if β != 1.0 {
            *y *= β; // Need to multiply first, as we have to be able to add to y.
        }
        if α == 1.0 {
            self.apply_add(y, μ)
        } else {
            for δ in μ.iter_spikes() {
                for &d in self.bt.iter_at(&δ.x) {
                    let sensor = self.shifted_sensor(grid.entry_linear_unchecked(d));
                    y[d] += sensor.apply(&δ.x) * (α * δ.α);
                }
            }
        }
    }

    fn apply_add(&self, y : &mut DVector<F>, μ : &RNDM<F, N>) {
        let grid = self.grid();
        for δ in μ.iter_spikes() {
            for &d in self.bt.iter_at(&δ.x) {
                let sensor = self.shifted_sensor(grid.entry_linear_unchecked(d));
                y[d] += sensor.apply(&δ.x) * δ.α;
            }
        }
    }

}

impl<F, S, P, BT, const N : usize> Apply<DeltaMeasure<Loc<F, N>, F>>
for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {

    type Output =  DVector<F>;

    #[inline]
    fn apply(&self, δ : DeltaMeasure<Loc<F, N>, F>) -> DVector<F> {
        self.apply(&δ)
    }
}

impl<'a, F, S, P, BT, const N : usize> Apply<&'a DeltaMeasure<Loc<F, N>, F>>
for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {

    type Output =  DVector<F>;

    fn apply(&self, δ : &DeltaMeasure<Loc<F, N>, F>) -> DVector<F> {
        let mut res = DVector::zeros(self.n_sensors());
        let grid = self.grid();
        for &d in self.bt.iter_at(&δ.x) {
            let sensor = self.shifted_sensor(grid.entry_linear_unchecked(d));
            res[d] += sensor.apply(&δ.x) * δ.α;
        }
        res
    }
}

impl<F, S, P, BT, const N : usize> Linear<DeltaMeasure<Loc<F, N>, F>> for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {
    type Codomain = DVector<F>;
}

impl<F, S, P, BT, const N : usize> BoundedLinear<RNDM<F, N>> for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N, Agg=Bounds<F>>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N> {
    type FloatType = F;

    /// An estimate on the operator norm in $𝕃(ℳ(Ω); ℝ^n)$ with $ℳ(Ω)$ equipped
    /// with the Radon norm, and $ℝ^n$ with the Euclidean norm.
    fn opnorm_bound(&self) -> F {
        // With {x_i}_{i=1}^n the grid centres and φ the kernel, we have
        // |Aμ|_2 = sup_{|z|_2 ≤ 1} ⟨z,Αμ⟩ = sup_{|z|_2 ≤ 1} ⟨A^*z|μ⟩
        // ≤ sup_{|z|_2 ≤ 1} |A^*z|_∞ |μ|_ℳ
        // = sup_{|z|_2 ≤ 1} |∑ φ(· - x_i)z_i|_∞ |μ|_ℳ
        // ≤ sup_{|z|_2 ≤ 1} |φ|_∞ ∑ |z_i| |μ|_ℳ
        // ≤ sup_{|z|_2 ≤ 1} |φ|_∞ √n |z|_2 |μ|_ℳ
        // = |φ|_∞ √n |μ|_ℳ.
        // Hence
        let n = F::cast_from(self.n_sensors());
        self.base_sensor.bounds().uniform() * n.sqrt()
    }
}

type SensorGridPreadjoint<'a, A, F, const N : usize> = PreadjointHelper<'a, A, RNDM<F,N>>;


impl<F, S, P, BT, const N : usize>
Preadjointable<RNDM<F, N>, DVector<F>>
for SensorGrid<F, S, P, BT, N>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N>,
      Weighted<ShiftedSensor<F, S, P, N>, F> : LocalAnalysis<F, BT::Agg, N> {
    type PreadjointCodomain = BTFN<F, SensorGridSupportGenerator<F, S, P, N>, BT, N>;
    type Preadjoint<'a> = SensorGridPreadjoint<'a, Self, F, N> where Self : 'a;

    fn preadjoint(&self) -> Self::Preadjoint<'_> {
        PreadjointHelper::new(self)
    }
}

#[derive(Clone,Debug)]
pub struct SensorGridSupportGenerator<F, S, P, const N : usize>
where F : Float,
      S : Sensor<F, N>,
      P : Spread<F, N> {
    base_sensor : Convolution<S, P>,
    grid : LinGrid<F, N>,
    weights : DVector<F>
}

impl<F, S, P, const N : usize> SensorGridSupportGenerator<F, S, P, N>
where F : Float,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N> {

    #[inline]
    fn construct_sensor(&self, id : usize, w : F) -> Weighted<ShiftedSensor<F, S, P, N>, F> {
        let x = self.grid.entry_linear_unchecked(id);
        self.base_sensor.clone().shift(x).weigh(w)
    }

    #[inline]
    fn construct_sensor_and_id<'a>(&'a self, (id, w) : (usize, &'a F))
    -> (usize, Weighted<ShiftedSensor<F, S, P, N>, F>) {
        (id.into(), self.construct_sensor(id, *w))
    }
}

impl<F, S, P, const N : usize> SupportGenerator<F, N>
for SensorGridSupportGenerator<F, S, P, N>
where F : Float,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N> {
    type Id = usize;
    type SupportType = Weighted<ShiftedSensor<F, S, P, N>, F>;
    type AllDataIter<'a> = MapX<'a, Zip<RangeFrom<usize>,
                                        std::slice::Iter<'a, F>>,
                                Self,
                                (Self::Id, Self::SupportType)>
                           where Self : 'a;

    #[inline]
    fn support_for(&self, d : Self::Id) -> Self::SupportType {
        self.construct_sensor(d, self.weights[d])
    }

    #[inline]
    fn support_count(&self) -> usize {
        self.weights.len()
    }

    #[inline]
    fn all_data(&self) -> Self::AllDataIter<'_> {
        (0..).zip(self.weights.as_slice().iter()).mapX(self, Self::construct_sensor_and_id)
    }
}

/// Helper structure for constructing preadjoints of `S` where `S : Linear<X>`.
/// [`Linear`] needs to be implemented for each instance, but [`Adjointable`]
/// and [`BoundedLinear`] have blanket implementations.
#[derive(Clone,Debug)]
pub struct PreadjointHelper<'a, S : 'a, X> {
    forward_op : &'a S,
    _domain : PhantomData<X>
}

impl<'a, S : 'a, X> PreadjointHelper<'a, S, X> {
    pub fn new(forward_op : &'a S) -> Self {
        PreadjointHelper { forward_op, _domain: PhantomData }
    }
}

impl<'a, X, Ypre, S> Adjointable<Ypre, X>
for PreadjointHelper<'a, S, X>
where Self : Linear<Ypre>,
      S : Clone + Linear<X> {
    type AdjointCodomain = S::Codomain;
    type Adjoint<'b> = S where Self : 'b;
    fn adjoint(&self) -> Self::Adjoint<'_> {
        self.forward_op.clone()
    }
}

impl<'a, X, Ypre, S> BoundedLinear<Ypre>
for PreadjointHelper<'a, S, X>
where Self : Linear<Ypre>,
      S : 'a + Clone + BoundedLinear<X> {
    type FloatType = S::FloatType;
    fn opnorm_bound(&self) -> Self::FloatType {
        self.forward_op.opnorm_bound()
    }
}


impl<'a, 'b, F, S, P, BT, const N : usize> Apply<&'b DVector<F>>
for PreadjointHelper<'a, SensorGrid<F, S, P, BT, N>, RNDM<F,N>>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N>,
      Weighted<ShiftedSensor<F, S, P, N>, F> : LocalAnalysis<F, BT::Agg, N> {

    type Output = SensorGridBTFN<F, S, P, BT, N>;

    fn apply(&self, x : &'b DVector<F>) -> Self::Output {
        self.apply(x.clone())
    }
}

impl<'a, F, S, P, BT, const N : usize> Apply<DVector<F>>
for PreadjointHelper<'a, SensorGrid<F, S, P, BT, N>, RNDM<F,N>>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N>,
      Weighted<ShiftedSensor<F, S, P, N>, F> : LocalAnalysis<F, BT::Agg, N> {

    type Output = SensorGridBTFN<F, S, P, BT, N>;

    fn apply(&self, x : DVector<F>) -> Self::Output {
        let fwd = &self.forward_op;
        let generator = SensorGridSupportGenerator{
            base_sensor : fwd.base_sensor.clone(),
            grid : fwd.grid(),
            weights : x
        };
        BTFN::new_refresh(&fwd.bt, generator)
    }
}

impl<'a, F, S, P, BT, const N : usize> Linear<DVector<F>>
for PreadjointHelper<'a, SensorGrid<F, S, P, BT, N>, RNDM<F,N>>
where F : Float,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N>,
      Weighted<ShiftedSensor<F, S, P, N>, F> : LocalAnalysis<F, BT::Agg, N> {

    type Codomain = SensorGridBTFN<F, S, P, BT, N>;
}

impl<F, S, P, BT, const N : usize> ForwardModel<Loc<F, N>, F>
for SensorGrid<F, S, P, BT, N>
where F : Float + ToNalgebraRealField<MixedType=F> + nalgebra::RealField,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      ShiftedSensor<F, S, P, N> : LocalAnalysis<F, BT::Agg, N>,
      Weighted<ShiftedSensor<F, S, P, N>, F> : LocalAnalysis<F, BT::Agg, N> {
    type Observable = DVector<F>;

    fn findim_quadratic_model(
        &self,
        μ : &DiscreteMeasure<Loc<F, N>, F>,
        b : &Self::Observable
    ) -> (DMatrix<F::MixedType>, DVector<F::MixedType>) {
        assert_eq!(b.len(), self.n_sensors());
        let mut mA = DMatrix::zeros(self.n_sensors(), μ.len());
        let grid = self.grid();
        for (mut mAcol, δ) in mA.column_iter_mut().zip(μ.iter_spikes()) {
            for &d in self.bt.iter_at(&δ.x) {
                let sensor = self.shifted_sensor(grid.entry_linear_unchecked(d));
                mAcol[d] += sensor.apply(&δ.x);
            }
        }
        let mAt = mA.transpose();
        (&mAt * mA, &mAt * b)
    }

    fn write_observable(&self, b : &Self::Observable, prefix : String) -> DynError {
        let it = self.grid().into_iter().zip(b.iter()).map(|(x, &v)| (x, v));
        write_csv(it, prefix + ".txt")
    }

    #[inline]
    fn zero_observable(&self) -> Self::Observable {
        self._zero_observable()
    }

    #[inline]
    fn empty_observable(&self) -> Self::Observable {
        DVector::zeros(0)
    }

}

/// Implements the calculation a factor $L$ such that $A_*A ≤ L 𝒟$ for $A$ the forward model
/// and $𝒟$ a seminorm of suitable form.
///
/// **This assumes (but does not check) that the sensors are not overlapping.**
#[replace_float_literals(F::cast_from(literal))]
impl<'a, F, BT, S, P, K, const N : usize> Lipschitz<&'a ConvolutionOp<F, K, BT, N>>
for SensorGrid<F, S, P, BT, N>
where F : Float + nalgebra::RealField + ToNalgebraRealField,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N>,
      K : SimpleConvolutionKernel<F, N>,
      AutoConvolution<P> : BoundedBy<F, K> {

    type FloatType = F;

    fn lipschitz_factor(&self, seminorm : &'a ConvolutionOp<F, K, BT, N>) -> Option<F> {
        // Sensors should not take on negative values to allow
        // A_*A to be upper bounded by a simple convolution of `spread`.
        if self.sensor.bounds().lower() < 0.0 {
            return None
        }

        // Calculate the factor $L_1$ for betwee $ℱ[ψ * ψ] ≤ L_1 ℱ[ρ]$ for $ψ$ the base spread
        // and $ρ$ the kernel of the seminorm.
        let l1 = AutoConvolution(self.spread.clone()).bounding_factor(seminorm.kernel())?;

        // Calculate the factor for transitioning from $A_*A$ to `AutoConvolution<P>`, where A
        // consists of several `Convolution<S, P>` for the physical model `P` and the sensor `S`.
        let l0 = self.sensor.norm(Linfinity) * self.sensor.norm(L1);

        // The final transition factor is:
        Some(l0 * l1)
    }
}

#[replace_float_literals(F::cast_from(literal))]
impl<F, BT, S, P, const N : usize> TransportLipschitz<L2Squared>
for SensorGrid<F, S, P, BT, N>
where F : Float + ToNalgebraRealField,
      BT : SensorGridBT<F, S, P, N>,
      S : Sensor<F, N>,
      P : Spread<F, N>,
      Convolution<S, P> : Spread<F, N> + Lipschitz<L2, FloatType = F> {
    type FloatType = F;

    fn transport_lipschitz_factor(&self, L2Squared : L2Squared) -> Self::FloatType {
        // We estimate the factor by N_ψL^2, where L is the 2-norm Lipschitz factor of
        // the base sensor (sensor * base_spread), and N_ψ the maximum overlap.
        let l = self.base_sensor.lipschitz_factor(L2).unwrap();
        let w = self.base_sensor.support_hint().width();
        let d = map2(self.domain.width(), &self.sensor_count, |wi, &i| wi/F::cast_from(i));
        let n = w.iter()
                 .zip(d.iter())
                 .map(|(&wi, &di)| (wi/di).ceil())
                 .reduce(F::mul)
                 .unwrap();
        2.0 * n * l.powi(2)
    }
}


macro_rules! make_sensorgridsupportgenerator_scalarop_rhs {
    ($trait:ident, $fn:ident, $trait_assign:ident, $fn_assign:ident) => {
        impl<F, S, P, const N : usize>
        std::ops::$trait_assign<F>
        for SensorGridSupportGenerator<F, S, P, N>
        where F : Float,
              S : Sensor<F, N>,
              P : Spread<F, N>,
              Convolution<S, P> : Spread<F, N> {
            fn $fn_assign(&mut self, t : F) {
                self.weights.$fn_assign(t);
            }
        }

        impl<F, S, P, const N : usize>
        std::ops::$trait<F>
        for SensorGridSupportGenerator<F, S, P, N>
        where F : Float,
              S : Sensor<F, N>,
              P : Spread<F, N>,
              Convolution<S, P> : Spread<F, N> {
            type Output = SensorGridSupportGenerator<F, S, P, N>;
            fn $fn(mut self, t : F) -> Self::Output {
                std::ops::$trait_assign::$fn_assign(&mut self.weights, t);
                self
            }
        }

        impl<'a, F, S, P, const N : usize>
        std::ops::$trait<F>
        for &'a SensorGridSupportGenerator<F, S, P, N>
        where F : Float,
              S : Sensor<F, N>,
              P : Spread<F, N>,
              Convolution<S, P> : Spread<F, N> {
            type Output = SensorGridSupportGenerator<F, S, P, N>;
            fn $fn(self, t : F) -> Self::Output {
                SensorGridSupportGenerator{
                    base_sensor : self.base_sensor.clone(),
                    grid : self.grid,
                    weights : (&self.weights).$fn(t)
                }
            }
        }
    }
}

make_sensorgridsupportgenerator_scalarop_rhs!(Mul, mul, MulAssign, mul_assign);
make_sensorgridsupportgenerator_scalarop_rhs!(Div, div, DivAssign, div_assign);

macro_rules! make_sensorgridsupportgenerator_unaryop {
    ($trait:ident, $fn:ident) => {
        impl<F, S, P, const N : usize>
        std::ops::$trait
        for SensorGridSupportGenerator<F, S, P, N>
        where F : Float,
              S : Sensor<F, N>,
              P : Spread<F, N>,
              Convolution<S, P> : Spread<F, N> {
            type Output = SensorGridSupportGenerator<F, S, P, N>;
            fn $fn(mut self) -> Self::Output {
                self.weights = self.weights.$fn();
                self
            }
        }

        impl<'a, F, S, P, const N : usize>
        std::ops::$trait
        for &'a SensorGridSupportGenerator<F, S, P, N>
        where F : Float,
              S : Sensor<F, N>,
              P : Spread<F, N>,
              Convolution<S, P> : Spread<F, N> {
            type Output = SensorGridSupportGenerator<F, S, P, N>;
            fn $fn(self) -> Self::Output {
                SensorGridSupportGenerator{
                    base_sensor : self.base_sensor.clone(),
                    grid : self.grid,
                    weights : (&self.weights).$fn()
                }
            }
        }
    }
}

make_sensorgridsupportgenerator_unaryop!(Neg, neg);