View Source Scholar.Interpolation.BezierSpline (Scholar v0.4.0)

Cubic Bezier Spline interpolation.

This kind of interpolation is calculated by fitting a set of continuous cubic polynomials of which the first and second derivatives are also continuous (which makes them curves of class $C^2$ ). In contrast to the Scholar.Interpolation.CubicSpline algorithm, the Bezier curves aren't necessarily of class $C^2$ , but this interpolation forces this so it yields a smooth function as the result.

The interpolated curve also has local control, meaning that a point that would be problematic for other interpolations, such as the Scholar.Interpolation.CubicSpline or Scholar.Interpolation.Linear algorithms, will only affect the segments right next to it, instead of affecting the curve as a whole.

Computing Bezier curve is $O(N^2)$ where $N$ is the number of points.

Reference:

[1] - Bezier theory
[2] - Spline equation derivation

Summary

Functions

fit(x, y)

Fits a cubic Bezier spline interpolation of the given (x, y) points.

predict(model, target_x, opts \\ [])

Returns the value fit by fit/2 corresponding to the target_x input.

Functions

fit(x, y)

Fits a cubic Bezier spline interpolation of the given (x, y) points.

Inputs are expected to be rank-1 tensors with the same shape and at least 4 entries.

Examples

iex> x = Nx.iota({4})
iex> y = Nx.tensor([2.0, 0.0, 1.0, 0.5])
iex> Scholar.Interpolation.BezierSpline.fit(x, y)
%Scholar.Interpolation.BezierSpline{
  coefficients: Nx.tensor(
    [
      [
        [0.0, 2.0],
        [0.3333331048488617, 1.0333333015441895],
        [0.6666665077209473, 0.06666667759418488],
        [1.0, 0.0]
      ],
      [
        [1.0, 0.0],
        [1.3333334922790527, -0.06666667759418488],
        [1.6666665077209473, 0.7666666507720947],
        [2.0, 1.0]
      ],
      [
        [2.0, 1.0],
        [2.3333334922790527, 1.2333333492279053],
        [2.6666667461395264, 0.8666666746139526],
        [3.0, 0.5]
      ]
    ]
  ),
  k: Nx.tensor(
    [
      [0.0, 2.0],
      [1.0, 0.0],
      [2.0, 1.0],
      [3.0, 0.5]
    ]
  )
}

predict(model, target_x, opts \\ [])

Returns the value fit by fit/2 corresponding to the target_x input.

Options

:max_iter (pos_integer/0) - determines the maximum iterations for converging the t parameter for each spline polynomial The default value is 15.
:eps (float/0) - determines the tolerance to be used for converging the t parameter for each spline polynomial The default value is 1.0e-6.

Examples

iex> x = Nx.iota({4})
iex> y = Nx.tensor([2.0, 0.0, 1.0, 0.5])
iex> model = Scholar.Interpolation.BezierSpline.fit(x, y)
iex> Scholar.Interpolation.BezierSpline.predict(model, Nx.tensor([3.0, 4.0, 2.0, 7.0]))
Nx.tensor(
  [0.5000335574150085, -4.2724612285383046e-5, 0.9999786615371704, 34.5]
)