noobase 0.0.5

Foundational pure-function utilities for astronomy analysis
Documentation 
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//! Shared separable bicubic Catmull-Rom interpolation kernel.
//!
//! This is the single source of the interpolation weights used by both
//! the forward operator ([`super::render`]) and its adjoint
//! ([`super::accumulate`]). Sharing one weight function is the structural
//! guarantee that `accumulate` is the *exact* transpose of `render`:
//! `render` gathers `value += epsf[ru, rv] * w_u * w_v` while
//! `accumulate` scatters `epsf_acc[ru, rv] += coeff * w_u * w_v` over the
//! very same per-tap weights, the same `floor`-derived tap base, and the
//! same zero-padding rule. Were the two to compute weights independently,
//! any drift between them would silently break the inner-product
//! transpose identity that the solver relies on.
//!
//! Keys cubic convolution with `a = -1/2` (the Catmull-Rom member),
//! four taps per axis. It is a pure-Rust, psf-module-internal helper that
//! never crosses the FFI boundary and is never visible to the caller.

use ndarray::ArrayView2;

/// Keys cubic-convolution (Catmull-Rom, `a = -1/2`) weights for the four
/// taps surrounding a fractional position `frac` in `[0, 1)`. The
/// returned array is ordered `[w(-1), w(0), w(+1), w(+2)]` relative to
/// `floor(coordinate)`.
///
/// The weights sum to one (partition of unity) and at `frac = 0` collapse
/// to `[0, 1, 0, 0]`, which is what makes an integer-aligned sample an
/// exact, interpolation-free grid pick (and, on the adjoint side, an
/// exact grid scatter).
#[inline]
pub(super) fn catmull_rom_weights(frac: f64) -> [f64; 4] {
    // Catmull-Rom basis on the unit interval, parameter t = frac:
    //   w(-1) = -0.5 t^3 +      t^2 - 0.5 t
    //   w( 0) =  1.5 t^3 -  2.5 t^2       + 1
    //   w(+1) = -1.5 t^3 +  2.0 t^2 + 0.5 t
    //   w(+2) =  0.5 t^3 -  0.5 t^2
    let t = frac;
    [
        ((-0.5 * t + 1.0) * t - 0.5) * t,
        ((1.5 * t - 2.5) * t) * t + 1.0,
        ((-1.5 * t + 2.0) * t + 0.5) * t,
        ((0.5 * t - 0.5) * t) * t,
    ]
}

/// Derivatives of the four Keys cubic-convolution (Catmull-Rom,
/// `a = -1/2`) tap weights with respect to the fractional position
/// `frac`. The returned array is ordered `[w'(-1), w'(0), w'(+1),
/// w'(+2)]`, tap-for-tap aligned with [`catmull_rom_weights`].
///
/// These are the exact term-by-term derivatives of the polynomials in
/// [`catmull_rom_weights`]. Differentiating the partition-of-unity
/// identity `sum w = 1` gives `sum w' = 0`, so the four returned values
/// always sum to zero. They are the structural source of the analytic
/// `dS/ddelta` Jacobian used by [`super::nuisance`]: with the render
/// sampling coordinate `k = K_c + os * ((i - c_det) - delta)` we have
/// `dk/ddelta = -os` and `dfrac/dk = 1`, so the centroid Jacobian is
/// these weight derivatives scaled by `-os` (pinned against a finite
/// difference of `render` in the nuisance tests). Sharing this single
/// derivative source with the forward kernel is the same structural
/// guarantee that keeps `accumulate` an exact transpose of `render`.
#[inline]
pub(super) fn catmull_rom_weight_derivatives(frac: f64) -> [f64; 4] {
    // d/dt of the basis written out in `catmull_rom_weights`:
    //   w(-1)' = -1.5 t^2 + 2.0 t - 0.5
    //   w( 0)' =  4.5 t^2 - 5.0 t
    //   w(+1)' = -4.5 t^2 + 4.0 t + 0.5
    //   w(+2)' =  1.5 t^2 - 1.0 t
    let t = frac;
    [
        (-1.5 * t + 2.0) * t - 0.5,
        (4.5 * t - 5.0) * t,
        (-4.5 * t + 4.0) * t + 0.5,
        (1.5 * t - 1.0) * t,
    ]
}

/// Separable bicubic Catmull-Rom point sample of `epsf` at the continuous
/// model coordinate `(k_u, k_v)`. Taps that fall outside the model grid
/// are zero (zero padding).
///
/// The adjoint scatters the identical per-tap weights returned by
/// [`catmull_rom_weights`] at the identical tap base derived here, so this
/// function and the scatter stay exact transposes by construction.
pub(super) fn catmull_rom_sample(epsf: &ArrayView2<f64>, k_u: f64, k_v: f64) -> f64 {
    let side = epsf.shape()[0] as i64; // square: shape()[0] == shape()[1]

    let u_floor = k_u.floor();
    let v_floor = k_v.floor();
    let weights_u = catmull_rom_weights(k_u - u_floor);
    let weights_v = catmull_rom_weights(k_v - v_floor);

    // The four taps of a Catmull-Rom segment sit at integer offsets
    // -1, 0, +1, +2 from floor(k); `weights_*[t]` is the weight of tap
    // `base + t`. The adjoint must scatter these very same weights.
    let base_u = u_floor as i64 - 1;
    let base_v = v_floor as i64 - 1;

    let mut value = 0.0_f64;
    for (tap_u, &weight_u) in weights_u.iter().enumerate() {
        let row = base_u + tap_u as i64;
        if row < 0 || row >= side {
            continue; // zero-pad outside the model grid
        }
        for (tap_v, &weight_v) in weights_v.iter().enumerate() {
            let column = base_v + tap_v as i64;
            if column < 0 || column >= side {
                continue; // zero-pad outside the model grid
            }
            value += epsf[(row as usize, column as usize)] * weight_u * weight_v;
        }
    }
    value
}

#[cfg(test)]
mod tests {
    use super::{catmull_rom_weight_derivatives, catmull_rom_weights};

    #[test]
    fn weight_derivatives_match_central_difference() {
        // Pin the analytic 4-tap weight derivatives against a central
        // finite difference of `catmull_rom_weights`. A sign or
        // coefficient slip in the polynomial derivative (which would
        // flip the GN-delta step) is gross relative to this bound.
        let step = 1e-6;
        for &frac in &[0.05, 0.2, 0.37, 0.5, 0.63, 0.8, 0.95] {
            let analytic = catmull_rom_weight_derivatives(frac);
            let forward = catmull_rom_weights(frac + step);
            let backward = catmull_rom_weights(frac - step);
            for tap in 0..4 {
                let finite_difference = (forward[tap] - backward[tap]) / (2.0 * step);
                assert!(
                    (analytic[tap] - finite_difference).abs() < 1e-6,
                    "tap {tap} at frac {frac}: analytic {} vs finite difference {}",
                    analytic[tap],
                    finite_difference
                );
            }
        }
    }

    #[test]
    fn weight_derivatives_sum_to_zero() {
        // Differentiating the partition-of-unity identity (sum w = 1)
        // gives sum w' = 0 for every fractional position.
        for &frac in &[0.0, 0.1, 0.25, 0.5, 0.75, 0.999] {
            let derivatives = catmull_rom_weight_derivatives(frac);
            let sum: f64 = derivatives.iter().sum();
            assert!(sum.abs() < 1e-12, "sum {sum} at frac {frac}");
        }
    }
}