pylops.signalprocessing.Interp

pylops.signalprocessing.Interp(dims, iava, axis=-1, kind='linear', dtype='float64', name='I')[source]

Interpolation operator.

Apply interpolation along axis from regularly sampled input vector into fractionary positions iava using one of the following algorithms:

  • Nearest neighbour interpolation is a thin wrapper around pylops.Restriction at np.round(iava) locations.

  • Linear interpolation extracts values from input vector at locations np.floor(iava) and np.floor(iava)+1 and linearly combines them in forward mode, places weighted versions of the interpolated values at locations np.floor(iava) and np.floor(iava)+1 in an otherwise zero vector in adjoint mode.

  • Sinc interpolation performs sinc interpolation at locations iava. Note that this is the most accurate method but it has higher computational cost as it involves multiplying the input data by a matrix of size \(N \times M\).

  • Cubic Spline interpolation relies on a cubic spline, i.e., a 2-times continuously differentiable piecewise third order polynomial with equally spaced knots. It is interpolated at the locations iava by evaluating the respective polynomial fitted between np.floor(iava) and np.floor(iava) + 1. It offers an excellent tradeoff between accuracy and computational complexity and its results oscillate less than those obtained from sinc interpolation. It can also be accessed directly via pylops.signalprocessing.InterpCubicSpline.

Note

The vector iava should contain unique values. If the same index is repeated twice an error will be raised. This also applies when values beyond the last element of the input array for linear interpolation and cubic spline interpolation as those values are forced to be just before this element.

Parameters:
dimslist or int

Number of samples for each dimension

iavalist or numpy.ndarray

Floating indices of locations of available samples for interpolation.

axisint, optional

Added in version 2.0.0.

Axis along which interpolation is applied.

kindstr, optional

Kind of interpolation. Currently, "nearest", "linear", "sinc", and "cubic_spline" are available.

Added in version 2.7.0.

The "cubic_spline"-interpolation was added.

dtypestr, optional

Type of elements in input array.

namestr, optional

Added in version 2.0.0.

Name of operator (to be used by pylops.utils.describe.describe)

Returns:
oppylops.LinearOperator

Linear interpolation operator

iavalist or numpy.ndarray

Corrected indices of locations of available samples (samples at M-1 or beyond are forced to be at M-1-eps)

Raises:
ValueError

If the vector iava contains repeated values.

NotImplementedError

If kind is not nearest, linear, sinc, or cubic_spline

See also

pylops.Restriction

Restriction operator

Notes

Linear interpolation of a subset of \(N\) values at locations iava from an input (or model) vector \(\mathbf{x}\) of size \(M\) can be expressed as:

\[y_i = (1-w_i) x_{l^{l}_i} + w_i x_{l^{r}_i} \quad \forall i=1,2,\ldots,N\]

where \(\mathbf{l^l}=[\lfloor l_1 \rfloor, \lfloor l_2 \rfloor,\ldots, \lfloor l_N \rfloor]\) and \(\mathbf{l^r}=[\lfloor l_1 \rfloor +1, \lfloor l_2 \rfloor +1,\ldots, \lfloor l_N \rfloor +1]\) are vectors containing the indeces of the original array at which samples are taken, and \(\mathbf{w}=[l_1 - \lfloor l_1 \rfloor, l_2 - \lfloor l_2 \rfloor, ..., l_N - \lfloor l_N \rfloor]\) are the linear interpolation weights. This operator can be implemented by simply summing two pylops.Restriction operators which are weighted using pylops.basicoperators.Diagonal operators.

Sinc interpolation of a subset of \(N\) values at locations iava from an input (or model) vector \(\mathbf{x}\) of size \(M\) can be expressed as:

\[\DeclareMathOperator{\sinc}{sinc} y_i = \sum_{j=0}^{M} x_j \sinc(i-j) \quad \forall i=1,2,\ldots,N\]

This operator can be implemented using the pylops.MatrixMult operator with a matrix containing the values of the sinc function at all \(i,j\) possible combinations.

Examples using pylops.signalprocessing.Interp

Restriction and Interpolation

Restriction and Interpolation

22. Time-shift estimation

22. Time-shift estimation