[Enhancement] Add Cubic Spline Interpolator

[MothNik]

Edit If somebody read this before Monday Dec 30 I made a mistake in the matrix formulation of $\mathbf{A}$ and thus also $\mathbf{D}$ and $\mathbf{Q}$, but I fixed the explanations and the code. Now, the cubic spline really has second order derivatives of zero at the boundaries.

Motivation

Thank you very much for this amazing package! I'm still in the process of diving deeper into it, but I would like to apply it for problems that involve 1D-interpolation.

Unfortunately, the pylops.signalprocessing.Interp only offers interpolation methods which are either not accurate enough ("linear", "nearest") or too computationally expensive ("sinc").

Proposed Feature

I would like to propose to incorporate a cubic spline interpolation into Interp which would be a good middle ground in terms of accuracy and computational load.

While I managed to implement it, I don't know how easy this can be cast into the backend-agnostic nature of pylops. On top of that, I'm not 100% sure if this can easily be generalised to arbitrary dimensions, altough I think that a 1D-only implementation would already be great.

Theoretical Background

I derived the following matrix notation from the German Wikipedia article on spline interpolation. It was a bit of tedious work with index-bookkeeping and reformulation of the equations, but I got it to work.

The matrix formulation for the interpolation would be

where

• $\mathbf{X}$ is the design matrix that depends on the x-coordinates $x_{fit}$ of the data points to be fitted as well as the x-coordinates $x_{interp}$ of the data points to be interpolated
• $\mathbf{Q}$ is the matrix that obtains the coefficients for the cubic spline interpolation
• $\mathbf{y_{fit}}$ are the $n$ data points to be fitted
• $y_{interp}$ are the interpolated data points
• $\mathbf{O}$ is the interpolation operator

For natural boundary conditions (second derivative at the boundaries is zero), the matrix $\mathbf{Q}$ looks like

where

• $\mathbf{I}_{n}$ is the identity matrix of size $n\times n$
• $\mathbf{A}$ is a tridiagonal matrix (see below) of size $n \times n$
• $\mathbf{D}_{0+}$ is a second order finite difference matrix with zero-padding (see below) of size $n \times n$

The matrix $\mathbf{A}$ is a tridiagonal strictly diagonally dominant (but not symmetric) matrix with the following structure:

The matrix $\mathbf{D}_{0+}$ is a second order finite difference matrix with the following structure:

$\mathbf{X}$ has to be constructed to assign the correct x-coordinates to the data points to be fitted and interpolated. Each row in $\mathbf{X}$ assigns weights to the 2 nearest values in $y_{fit}$ as well as their 2 corresponding solutions of the linear system $\mathbf{A}^{-1}\cdot \mathbf{D}\cdot\mathbf{y_{fit}}$, so in total there is 4 nonzero entries per row.

Consequently, the transpose operation $\mathbf{O}^T$ looks like

In both the $\mathbf{O}$ and $\mathbf{O}^T$ case, the solution of $\mathbf{A}^{-1}$ and $\left(\mathbf{A}^{T}\right)^{-1}$ can be achieved very fast by using two separate banded LU decompositions.

Implementation

I would like to share my implementation that currently relies on scipy.sparse.linalg.LinearOperator because I didn't manage to inherit from Interp yet. Besides, it invokes scipy further for handing the sparse matrix X and solving the banded matrices with LAPACK. I also implemented a custom finite difference operator with Numba to be fast.

The code was written in a Jupyter notebook and all the %magic operations are commented out. The timing results can be found below.

Edit I adapted the code so that it reuses the factorizations fo $\mathbf{A}$ and $\mathbf{A}^{T}$ by invoking the LAPACK-wrappers for gttrf and gttrs. So, intead of a factorization + solve call for every matrix-vector product, the factorization is only called once and the results are used in one solve per matrix-vector product.

# === Imports ===

from dataclasses import dataclass
from typing import Callable, Tuple, Type

import numpy as np
from numba import jit
from matplotlib import pyplot as plt
from pylops.signalprocessing import Interp
from pylops.utils import dottest
from scipy.linalg import get_lapack_funcs
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import LinearOperator

# %matplotlib widget

# === Second Order Finite Differences ===


@jit(nopython=True)
def second_order_finite_differences_zero_padded(x: np.ndarray) -> np.ndarray:
    """
    Computes the second order finite differences of ``x`` and pads the result with a
    leading and trailing zero.

    Parameters
    ----------
    x : :class:`numpy.ndarray` of shape (n,)
        The input array.

    Returns
    -------
    x_diffs : :class:`numpy.ndarray` of shape (n,)
        The second order finite differences of ``x`` padded with a leading and trailing
        zero.

    """

    assert x.ndim == 1
    assert x.size >= 3
    x_diffs = np.empty(shape=(x.size,), dtype=x.dtype)

    x_diffs[0] = 0.0

    for index in range(1, x.size - 1):
        x_diffs[index] = x[index + 1] - 2.0 * x[index] + x[index - 1]

    x_diffs[x.size - 1] = 0.0

    return x_diffs


@jit(nopython=True)
def second_order_finite_differences_zero_padded_transposed(x: np.ndarray) -> np.ndarray:
    """
    Computes the transposed operation of the second order finite differences operator
    with subsequent zero padding, i.e.,

    - ``x[0]`` and ``x[-1]`` are not considered,
    - ``x[1:-1]`` is padded with 2 leading and 2 trailing zero,
    - the second order finite differences of the padded array are computed

    Parameters
    ----------
    x : :class:`numpy.ndarray` of shape (n,)
        The input array.

    Returns
    -------
    x_diffs : :class:`numpy.ndarray` of shape (n,)
        The result of the transposed second order finite differences operator with
        subsequent zero padding.

    """

    assert x.ndim == 1
    assert x.size >= 3

    if x.size == 3:
        return np.array((x[1], -2.0 * x[1], x[1]))
    if x.size == 4:
        return np.array((x[1], -2.0 * x[1] + x[2], x[1] - 2.0 * x[2], x[2]))

    x_diffs = np.empty(shape=(x.size,), dtype=x.dtype)

    x_diffs[0] = x[1]
    x_diffs[1] = -2.0 * x[1] + x[2]

    for index in range(2, x.size - 2):
        x_diffs[index] = x[index - 1] - 2.0 * x[index] + x[index + 1]

    x_diffs[x.size - 2] = x[x.size - 3] - 2.0 * x[x.size - 2]
    x_diffs[x.size - 1] = x[x.size - 2]

    return x_diffs


# === Tridiagonal Matrices ===


@dataclass
class TridiagonalMatrix:
    """
    Represents a tridiagonal matrix with

    - the main diagonal ``main_diagonal``,
    - the super-diagonal ``super_diagonal``,
    - the sub-diagonal ``sub_diagonal``.

    """

    main_diagonal: np.ndarray
    super_diagonal: np.ndarray
    sub_diagonal: np.ndarray

    def __post_init__(self) -> None:
        """
        Validates the input.

        """

        assert self.main_diagonal.ndim == 1
        assert self.super_diagonal.ndim == 1
        assert self.sub_diagonal.ndim == 1

        assert self.main_diagonal.size == self.super_diagonal.size + 1
        assert self.main_diagonal.size == self.sub_diagonal.size + 1

    def __iter__(self):
        """
        Returns an iterator over the sub-diagonal, main diagonal and super-diagonal
        (in that order) as required for the LAPACK routines ``?gttrf``.

        """
        return iter((self.sub_diagonal, self.main_diagonal, self.super_diagonal))

    @property
    def T(self) -> "TridiagonalMatrix":
        """
        Returns the transpose of the tridiagonal matrix.

        """

        return TridiagonalMatrix(
            main_diagonal=self.main_diagonal,
            super_diagonal=self.sub_diagonal,
            sub_diagonal=self.super_diagonal,
        )


@dataclass
class TridiagonalLUDecomposition:
    """
    Represents the LU decomposition of a tridiagonal matrix as performed by the LAPACK
    routines ``?gttrf``.

    """

    sub_diagonal_lu: np.ndarray
    main_diagonal_lu: np.ndarray
    super_diagonal_lu: np.ndarray
    super_two_diagonal_lu: np.ndarray
    pivot_indices: np.ndarray

    def __iter__(self):
        """
        Returns an iterator over the sub-diagonal, main diagonal, super-diagonal,
        the second super-diagonal (filled by pivoting) and the pivot indices
        (in that order) as required for the LAPACK routines ``?gttrs``.

        """
        return iter(
            (
                self.sub_diagonal_lu,
                self.main_diagonal_lu,
                self.super_diagonal_lu,
                self.super_two_diagonal_lu,
                self.pivot_indices,
            )
        )

    @staticmethod
    def from_tridiagonal_matrix(
        matrix: TridiagonalMatrix,
        lapack_factorizer: Callable,
    ) -> "TridiagonalLUDecomposition":
        """
        Computes the LU decomposition of a tridiagonal matrix using the LAPACK routine
        ``?gttrf``.

        Parameters
        ----------
        matrix : :class:`TridiagonalMatrix`
            The tridiagonal matrix to decompose.
        lapack_factorizer : callable
            The LAPACK routine ``?gttrf`` to use for the decomposition.

        Returns
        -------
        lu_decomposition : :class:`TridiagonalLUDecomposition`
            The LU decomposition of the tridiagonal matrix.

        """

        (
            sub_diagonal_lu,
            main_diagonal_lu,
            super_diagonal_lu,
            super_two_diagonal_lu,
            pivot_indices,
            info,
        ) = lapack_factorizer(*matrix)

        assert info == 0

        return TridiagonalLUDecomposition(
            sub_diagonal_lu=sub_diagonal_lu,
            main_diagonal_lu=main_diagonal_lu,
            super_diagonal_lu=super_diagonal_lu,
            super_two_diagonal_lu=super_two_diagonal_lu,
            pivot_indices=pivot_indices,
        )

    def solve(self, rhs: np.ndarray, lapack_solver: Callable) -> np.ndarray:
        """
        Solves the linear system of equations ``A @ x = rhs`` where ``A`` is the
        tridiagonal matrix represented by the LU decomposition. For this, the LAPACK
        routine ``?gttrs`` is used.

        Parameters
        ----------
        rhs : :class:`numpy.ndarray` of shape (n,)
            The right-hand side of the linear system of equations.
        lapack_solver : callable
            The LAPACK routine ``?gttrs`` to use for solving the system.

        Returns
        -------
        x : :class:`numpy.ndarray` of shape (n,)
            The solution of the linear system of equations.

        """

        x, info = lapack_solver(*self, rhs)

        assert info == 0

        return x


def _make_cubic_spline_left_hand_side(
    dims: int,
) -> TridiagonalMatrix:
    """
    Constructs the banded matrix ``A`` for the linear system of equations ``A @ m = b``
    where

    - ``A`` is a diagonally dominant tridiagonal matrix with the main diagonal
        being ``[1, 2/3, 2/3, ..., 2/3, 2/3, 1]``, the super-diagonal being
        ``[0, 1/6, 1/6, ..., 1/6, 1/6]`` and the sub-diagonal being
        ``[1/6, 1/6, ..., 1/6, 1/6, 0]``,
    - ``m`` is the unknown vector of spline coefficients,
    - ``b`` is the second order finite differences of the ``y`` values for which the
        spline should interpolate.

    Parameters
    ----------
    dims : :class:`int`
        The number of points the spline should interpolate.

    Returns
    -------
    matrix_a : :class:`TridiagonalMatrix`
        The tridiagonal matrix ``A``.

    """

    main_diag = np.empty(shape=(dims,), dtype=np.float64)
    super_diag = np.empty(shape=(dims - 1,), dtype=np.float64)

    main_diag[0] = 1.0
    main_diag[1 : dims - 1] = 0.6666666666666666
    main_diag[dims - 1] = 1.0

    super_diag[0] = 0.0
    super_diag[1 : dims - 1] = 0.16666666666666666

    return TridiagonalMatrix(
        main_diagonal=main_diag,
        super_diagonal=super_diag,
        sub_diagonal=np.flip(super_diag).copy(),  # to avoid a view
    )

# === Custom Cubic Spline Interpolator ===

@jit(nopython=True)
def _make_cubic_spline_x_csr_specs(
    dims: int,
    iava: np.ndarray,
    base_indices: np.ndarray,
    iava_remainders: np.ndarray,
) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    Constructs the specifications ``data``, ``indices``, and ``indptr`` for a
    :class:`scipy.sparse.csr_matrix` ``X`` that can be used to interpolate the
    equally spaced ``y`` values with a cubic spline like ``X @ Q @ y`` where

    - ``X`` is the interpolation matrix to be constructed,
    - ``Q`` is the linear operator that obtains the spline coefficients ``y``
        (a vertical concatenation of the ``y`` values and the spline coefficients
        ``m``),
    - ``y`` are the values to be interpolated.

    Parameters
    ----------
    dims : :class:`int`
        The number of points the spline should interpolate.
    iava : :class:`numpy.ndarray` of shape (n,)
        Floating indices of the locations to which the spline should interpolate.
    base_indices : :class:`numpy.ndarray` of shape (n,)
        The indices of the respective first data point in ``y`` for the intervals
        in which the corresponding ``iava`` values lie.
    iava_remainders : :class:`numpy.ndarray` of shape (n,)
        The remainders of the ``iava`` values after subtracting the respective
        ``base_indices``.

    Returns
    -------
    data : :class:`numpy.ndarray` of shape (4 * ``iava.size``,)
        The data array of the CSR matrix ``X``
    indices : :class:`numpy.ndarray` of shape (4 * ``iava.size``,)
        The column indices of the CSR matrix ``X``
    indptr : :class:`numpy.ndarray` of shape (``iava.size`` + 1,)
        The row pointers of the CSR matrix ``X``

    """

    # some auxiliary variables are required
    iava_remainders_cubic = (  # (x - x[i])**3
        iava_remainders * iava_remainders * iava_remainders
    )
    one_minus_iava_remainders = 1.0 - iava_remainders  # (x[i + 1] - x)
    one_minus_iava_remainders_cubic = (  # (x[i + 1] - x)**33
        one_minus_iava_remainders
        * one_minus_iava_remainders
        * one_minus_iava_remainders
    )

    # for each data point, except for the first and the last one, we need 4 entries
    # to multiply with ``y[i]``, ``y[i + 1]``, ``m[i]``, and ``m[i + 1]``;
    data = np.empty(shape=(4 * iava.size,), dtype=np.float64)
    indices = np.empty(shape=(4 * iava.size,), dtype=np.intp)
    indptr = np.arange(0, 4 * (iava.size + 1), 4, dtype=np.int64)

    # the rows are specified by means of a loop
    for row_index in range(0, iava.size):
        index_offset = 4 * row_index
        data[index_offset] = one_minus_iava_remainders[row_index]
        data[index_offset + 1] = iava_remainders[row_index]
        data[index_offset + 2] = 0.16666666666666666 * (
            one_minus_iava_remainders_cubic[row_index]
            - one_minus_iava_remainders[row_index]
        )
        data[index_offset + 3] = 0.16666666666666666 * (
            iava_remainders_cubic[row_index] - iava_remainders[row_index]
        )

        indices[index_offset] = base_indices[row_index]
        indices[index_offset + 1] = base_indices[row_index] + 1
        indices[index_offset + 2] = base_indices[row_index] + dims
        indices[index_offset + 3] = base_indices[row_index] + dims + 1

    return data, indices, indptr


class CubicSplineInterp(LinearOperator):
    """
    Custom cubic spline interpolator.

    Parameters
    ----------
    dims : :class:`int`
        The number of points the spline should interpolate.
    iava : :class:`numpy.ndarray` of shape (n,)
        Floating indices of the locations to which the spline should interpolate.
    dtype : type
        The data type of the input and output arrays.

    """

    def __init__(
        self,
        dims: int,
        iava: np.ndarray,
        dtype: Type = np.float64,
    ) -> None:

        self.dims: int = dims
        self.iava: np.ndarray = iava
        self.shape: Tuple[int, int] = (iava.size, dims)
        self.dtype: Type = dtype

        self.base_indices = np.clip(
            iava.astype(np.int64),
            a_min=0,
            a_max=dims - 2,
        )
        self.iava_remainders = iava - self.base_indices

        self._tridiag_factorize = get_lapack_funcs(("gttrf",), dtype=np.float64)[0]
        self._tridiag_lu_solve = get_lapack_funcs(("gttrs",), dtype=self.dtype)[0]

        self.lhs_matrix: TridiagonalMatrix = (
            _make_cubic_spline_left_hand_side(dims=dims)
        )
        self.lhs_matrix_lu = TridiagonalLUDecomposition.from_tridiagonal_matrix(
            matrix=self.lhs_matrix,
            lapack_factorizer=self._tridiag_factorize,
        )
        self.lhs_matrix_transposed_lu = (
            TridiagonalLUDecomposition.from_tridiagonal_matrix(
                matrix=self.lhs_matrix.T,
                lapack_factorizer=self._tridiag_factorize,
            )
        )

        self.X_matrix: csr_matrix = csr_matrix(
            _make_cubic_spline_x_csr_specs(
                dims=dims,
                iava=iava,
                base_indices=self.base_indices,
                iava_remainders=self.iava_remainders,
            ),
            shape=(iava.size, 2 * dims),
        )
        self.X_matrix_transposed: csr_matrix = self.X_matrix.transpose().tocsr()  # type: ignore

    def _matvec(self, x: np.ndarray) -> np.ndarray:
        m_coeffs = self.lhs_matrix_lu.solve(
            rhs=second_order_finite_differences_zero_padded(x),
            lapack_solver=self._tridiag_lu_solve,
        )

        return self.X_matrix @ np.concatenate((x, m_coeffs))

    def _rmatvec(self, x: np.ndarray) -> np.ndarray:
        x_mod = self.X_matrix_transposed @ x
        return (
            x_mod[0:self.dims]
            + second_order_finite_differences_zero_padded_transposed(
                self.lhs_matrix_transposed_lu.solve(
                    rhs=x_mod[self.dims: x_mod.size],
                    lapack_solver=self._tridiag_lu_solve,
                )
            )
        )


# === Interpolators Test ===

# A complex signal is created to test the interpolators

x = np.arange(50)
x_new = np.linspace(0, x.max() - np.spacing(x.max()), 50_001)
y = np.sin(6.0 * np.pi * (x / x.size)) + 1.0j * np.cos(4.0 * np.pi * (x / x.size))
y = y.astype(np.complex128)

linop_linear = Interp(
    dims=x.size,
    iava=x_new,
    kind="linear",
)[0]
y_new_linear = linop_linear @ y

print("Timing PyLops Linear Interpolator")
print("Creation")
# %timeit linop_linear = Interp(dims=x.size, iava=x_new, kind="linear", dtype="complex128",)
print("Matrix-Vector Multiplication")
# %timeit y_new_linear = linop_linear @ y
print("Transpose Operation")
# %timeit linop_linear.T @ y_new_linear

linop_sinc = Interp(
    dims=x.size,
    iava=x_new,
    kind="sinc",
)[0]
y_new_sinc = linop_sinc @ y

print("\nTiming PyLops Sinc Interpolator")
print("Creation")
# %timeit linop_sinc = Interp(dims=x.size, iava=x_new, kind="sinc", dtype="complex128",)
print("Matrix-Vector Multiplication")
# %timeit y_new_sinc = linop_sinc @ y
print("Transpose Operation")
# %timeit linop_sinc.T @ y_new_sinc

linop_spline = CubicSplineInterp(dims=x.size, iava=x_new, dtype=np.complex128,)
print("\nTesting custom Spline Interpolator")
assert dottest(linop_spline, nr=x_new.size, nc=x.size, complexflag=3)
print("Passed!")
y_new_spline = linop_spline @ y

print("\nTiming custom Spline Interpolator")
print("Creation")
# %timeit linop_spline = CubicSplineInterp(dims=x.size, iava=x_new, dtype=np.complex128,)
print("Matrix-Vector Multiplication")
# %timeit y_new_sline = linop_spline @ y
print("Transpose Operation")
# %timeit linop_spline.T @ y_new_spline

# === Plotting ===

fig, ax = plt.subplots(nrows=2, sharex=True, figsize=(10, 6))

ax[0].plot(x, y.real, "o", label="Original data")
ax[0].plot(x_new, np.sin(6.0 * np.pi * (x_new / x.size)), label="Ground truth")
ax[0].plot(x_new, y_new_spline.real, label="Cubic spline interpolation")
ax[0].plot(x_new, y_new_linear.real, label="Linear interpolation", linestyle=":")
ax[0].plot(x_new, y_new_sinc.real, label="Sinc interpolation", linestyle="--")

ax[0].set_ylabel("Real part")

ax[0].legend()

ax[1].plot(x, y.imag, "o", label="Original data")
ax[1].plot(x_new, np.cos(4.0 * np.pi * (x_new / x.size)), label="Ground truth")
ax[1].plot(x_new, y_new_spline.imag, label="Cubic spline interpolation")
ax[1].plot(x_new, y_new_linear.imag, label="Linear interpolation", linestyle=":")
ax[1].plot(x_new, y_new_sinc.imag, label="Sinc interpolation", linestyle="--")

ax[1].set_xlabel("Index")
ax[1].set_ylabel("Imaginary part")

ax[1].legend()

plt.show()

Results

Correctness and Timing

The timings for this test with a relatively large complex Array show that the cubic spline interpolation in this basic version is competitive. It passes dottest and its runtime is comparable to the linear interpolation.

Timing PyLops Linear Interpolator
Creation
2.04 ms ± 106 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
Matrix-Vector Multiplication
314 µs ± 4.53 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
Transpose Operation
218 µs ± 11.4 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)

Timing PyLops Sinc Interpolator
Creation
12.7 ms ± 228 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
Matrix-Vector Multiplication
3.67 ms ± 146 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
Transpose Operation
5.43 ms ± 389 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

Testing custom Spline Interpolator
Passed!

Timing custom Spline Interpolator
Creation
750 µs ± 14.7 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
Matrix-Vector Multiplication
260 µs ± 12.1 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
Transpose Operation
290 µs ± 4.53 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)

Visualisation

In one go, the real and imaginary part of the complex test data are interpolated separately as intended. For this example, the cubic spline gives higher accuracy than the linear method. Besides, it also does not suffer from the ringing introduces by the sinc-interpolation of the imaginary part.

[MothNik]

If somebody read the initial issue before Monday Dec 30 I made a mistake in the matrix formulation of $\mathbf{A}$ and thus also $\mathbf{D}$ and $\mathbf{Q}$, but I fixed the explanations and the code. Now, the cubic spline really has second order derivatives of zero at the boundaries.

[mrava87]

@MothNik thanks for this contribution, it looks great at first sight!

Indeed, we do have only pretty basic interpolation method so far... the main reason being that anything more complex requires more thinking when framed in a forward/adjoint setting like the one we require in PyLops (as you have probably realized writing your code); for example, spicy has nice interpolators but they only provide what we call the forward pass so they can't be incorporated easily (unless one writes the equivalent adjoint).

Let me take a more detailed look in coming weeks and help you with incorporating this into the plops Interp... ideally as you say we would always like to have something that broadcast, so 1D interpolation but applied to multiple traces at the same time. And yes, even if it is only 1D, it would be a great addition 😄

[MothNik]

@mrava87 Thanks for the reply!

Please let me know if you have any questions regarding the approach or implementation!

I also noticed that the adjoints can be quite painful because they often have no real physical meaning. The cubic spline is a good example because it sums up values, solves a linear system, and then computes a derivative, all of which does not really make sense. I'm so glad I found your documentation of the dottest. That really turned out into a lifesaver because in the end I was always hoping for everything to be correct when the results looked reasonable.

Regarding the dimensionality: In the end, the piecewise third order polynomials in all dimensions always need to join on the knot points in a way that the value and the first two directional derivatives are continuous. That can probably be generalized and requires solving a sparse matrix with a less convenient sparsity pattern than being tridiagonal. I will let it simmer a bit more and hopefully come back with a solution.

[mrava87]

Hi @MothNik, I have finally got some time to go through you interpolator :) I must say that I haven't looked in great details at all the internal methods but thanks to the German wiki page (and the Google Chrome translator) I can follow how your equations relate to that explanation and what you are doing in the code.

A few comments/questions:

• I think the easiest and cleanest way for you to incorporate your interpolator as one of the options for Interp is to simply create the class as you did but subclassing pylons's LinearOperator like this:

...
from pylops import LinearOperator

class CubicSplineInterp(LinearOperator):
    """
    Custom cubic spline interpolator.

    Parameters
    ----------
    dims : :class:`int`
        The number of points the spline should interpolate.
    iava : :class:`numpy.ndarray` of shape (n,)
        Floating indices of the locations to which the spline should interpolate.
    dtype : type
        The data type of the input and output arrays.

    """

    def __init__(
        self,
        dims: int,
        iava: np.ndarray,
        dtype: Type = np.float64,
        name: str = "S",
    ) -> None:

        self.iava: np.ndarray = iava
        super().__init__(dtype=np.dtype(dtype), shape=(iava.size, dims), name=name)

(everything else is unchanged...), and then add it as one of the options - i.e. add elif kind == "cubicspline": at https://github.com/PyLops/pylops/blob/6c92032519eb778e5f801abc88e8b0cbdafff8aa/pylops/signalprocessing/interp.py#L229 line by just creating the operator as you do in the code above. You can put all your methods and class in a file called _cubicsplineinterp.py and the just import the class in the interp file.

• I see why you created the operator the way you did - i.e., the forward does the interpolation by taking a signal at coarse locations and building a signal along a denser grid. However, Pylops is a library for inverse problems, so we build all our operators with that in mind. In other words, like we do for linear/sinc/nearest, the input should be dense and the output coarse so that if we put the interpolation operator as part of a chain of operators we want to invert (e.g., Op = Smoothing @ Interp - the observed data is a smooth version of the coarse input signal), we can setup and inverse problem to undo its effect (e.g., reconstruct the dense signal from a coarse smooth version of it). I am not sure if it makes sense and if your code actually needs any change to align with this - basically, if instead of going to coarse to dense, we go from dense to coarse (at locations that are not available in the original dense signal), would your interpolator still make sense.

• I see in the Wiki that the matrix A should have hs (I guess assuming that the input points may not be equally spaced), but you don't.... are you always assuming they are? I would say we shouldn't to make sure CubicSplineInterp is consistent with the other options exposed in Interp.

• Right now, I am not that concerned with extending what you have to multiple dimensions (this would be nice but in a second moment). What I would like to see instead (again to be consistent with the other kind of interpolators in Interp) is that if the method is able to interpolate multiple traces at the same time, all with the same sampling pattern (basically allowing batching over the remaining dimensions (where axis is the dimension over which interpolation happens). I think it should be rather straightforward as the X part would remain unchanged, but instead of having self.X_matrix @ np.concatenate((x, m_coeffs)) you will have something like many x concatenated over the second axis, and then the output will be reshaped... look at for example https://github.com/PyLops/pylops/blob/dev/pylops/basicoperators/flip.py to see how we do this by simply using the decorator reshaped

• Your codes look already quite good... a few small changes that I suggest to align with Pylops style are:

1. No assert --> if cond: raise WhateverError(message)
2. Use our from pylops.utils.typing import DTypeLike, InputDimsLike, IntNDArray for type annotations of numpy array, not raw np.ndarray
3. When making the codes ready for PyLops, I highly recommend to write the Notes section explaining how the operator works with text and equations. If you put what you have written here (including the equations), this is more than enough for someone to follow when later looking at the code :smile:

Feel free to have a go at it and make a PR, then we can help you finalize the code :)

[mrava87]

@MothNik just checking you saw my message 😄

[MothNik]

@mrava87 Sorry, I got so caught up in work issues, that the message slipped through. I will try to answer you this weekend because I have a few questions regarding your review and feedback (thanks a lot for that btw!).

Have a great rest of the week!

[MothNik]

@mrava87 So, I finally found some time to ask my questions regarding your review and feedback (again, thanks a lot for that).

I think the easiest and cleanest way for you to incorporate your interpolator as one of the options for Interp is to simply create the class as you did but subclassing pylons's LinearOperator like this:

In my code, there is some pre-computations like the LU-decomposition of the banded matrix $\mathbf{A}$ and the interpolation matrix $\mathbf{X}$. I initialised them in the constructor in my original code, but this is not the way to go for proper code. What is the convention with such pre-computations in pylops. Should I initialise everything lazily, e.g., via a _setup(self)-method that is only called in case the pre-computations were not carried out yet or some attributes changed?

I see why you created the operator the way you did - i.e., the forward does the interpolation by taking a signal at coarse locations and building a signal along a denser grid. However, Pylops is a library for inverse problems, ...

The class I implemented can handle both

• decimation: dense → coarse
• interpolation: coarse → dense

because all the relevant mapping is defined in the $\mathbf{X}$-matrix while the matrix $\mathbf{Q}$ only computes the respective coefficients. Basically, $\mathbf{Q}$ just provides coefficients (and thus the spline) and leaves it to $\mathbf{X}$ on where to evaluate the spline, which can be at more or fewer points than those used for the spline fit. So, from this point of view, the code should handle this properly.

I see in the Wiki that the matrix A should have hs (I guess assuming that the input points may not be equally spaced), but you don't.... are you always assuming they are?

If I saw this correctly, the evenly spaced points in dims are interpolated to the more or less arbitrarily spaced points iava. At least when I run

import numpy as np
from pylops.signalprocessing import Interp

linop_linear = Interp(
    dims=51,
    iava=np.sort(50.0 * np.random.rand(50_000)),
    kind="linear",
)[0]

print(linop_linear.shape)  # <- gives (50000, 51)

So, dims determines the grid on which the spline is fitted and this seems to be assumed to be evenly spaced. Since the coefficients are evaluated for the points at dims and hs only appears in this fit, I assumed hs to be equal for all point and did not incorporate it in the matrix $\mathbf{Q}$ that handles this fit. On the other hand, iava can then be arbitrarily spaced (relative to the spacing hs implicit in dims) and this needs to be encoded in the matrix $\mathbf{X}$ alone.

I was a bit confused an thus wrote the documentation in my code like

"""
    dims : :class:`int`
        The number of points the spline should interpolate.
    iava : :class:`numpy.ndarray` of shape (n,)
        Floating indices of the locations to which the spline should interpolate.
""" 

to be a bit more specific.

Please correct me if I got this point wrong, but I derived it from the shape of the operator itself.

Regarding the multidimensional fit, I did not get a good running version so far. If I understood this correctly, you would for simplicity just interpolate over the first axis, use the results and interpolate them over the second axis, ..., and so on?

Thanks for the heads up regarding the pylops code style conventions. I will adhere to them and I'm already looking forward to feedback!

[mrava87]

@MothNik thanks for your detailed reply!

Alright I think we are getting closes to something that would nicely align with the other interpolators:

• pre-computations: this is not always the case in PyLops operators but we have a few that follow a similar pattern. One example is https://github.com/PyLops/pylops/blob/dev/pylops/waveeqprocessing/twoway.py, where the methods _create_model and _create_geometry are responsible to create some variables to be used later in the _matvec/_rmatvec methods. We call these methods directly in the constructor, to me this is totally fine, but hiding away lengthy code so to keep the constructor as concise as possible. You could do something similar, wrapping all your precomputations in a _setup method as you suggest and calling this in the __init__. Note however that we DO not favor users to change attributes, our operator as somehow immutable in that once they are created they are meant to be use many times (within an iterative solver) but not change (as in linear solvers the operator cannot change along iterations)... if a user wants to change some of the parameters of the operator to say run another inversion, at that point we would recommend someone to just remake a new operator. So the pre-computation will only be done once upfront 😄
• decimation/interpolation: this is great and it is great that it is setup to go from regular grid to anything else (could be regular or irrugular based on how one chooses iava. Then it fits perfectly with all the other 1D interpolators we have.
• I get your point about having by definition evenly spaced input and so not needing the step-size... indeed, if we follow the convention of Interp (which I think is what we should do), this is totally fine
• No worries about the multidimensional case just yet.. anyways this will have to be its own class, like we have a linear interpolator for 1d signals in Interp and a bilinear interpolator for 2d signals in Bilinear. And yes, I guess a simple way could be do to multiple 1D interpolations over various axes. But my point about ND-arrays was simpler: a common scenario that we see in many problems is that one has an ensemble of traces that are all sampled in the same manner and you want to interpolate all of them in the same manner... so your code should be able to broadcast such that when you apply the various matrices you don't apply them to a vector but to a matrix (where each column is a signal of the ensemble) such that we can interpolate in one pass multiple traces instead of having to do a (slow) for loop over them.

Hopefully you have all the pointers to start drafting the CubicSplineInterp operator, and we can take it from there 😄