Spline Padding [module]

Here are the functions of the module splinekit.spline_padding.

splinekit.spline_padding.pad_p(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Periodic padding.

Returns the value of the \(k\)-th data sample after the data have been extended by periodic padding. The padded data \(f\) satisfy that

\[\forall k\in{\mathbb{Z}}:f[k]=f[k+K],\]

where \(K\) is the length of the unpadded data.

Periodic padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	a	b	a	a	e
−19		a	b	c	b	b	f
−18		a	a	a	c	c	a
−17		a	b	b	d	d	b
−16		a	a	c	a	e	c
−15		a	b	a	b	a	d
−14		a	a	b	c	b	e
−13		a	b	c	d	c	f
−12		a	a	a	a	d	a
−11		a	b	b	b	e	b
−10		a	a	c	c	a	c
−9		a	b	a	d	b	d
−8		a	a	b	a	c	e
−7		a	b	c	b	d	f
−6		a	a	a	c	e	a
−5		a	b	b	d	a	b
−4		a	a	c	a	b	c
−3		a	b	a	b	c	d
−2		a	a	b	c	d	e
−1		a	b	c	d	e	f
0		A	A	A	A	A	A
1		a	B	B	B	B	B
2		a	a	C	C	C	C
3		a	b	a	D	D	D
4		a	a	b	a	E	E
5		a	b	c	b	a	F
6		a	a	a	c	b	a
7		a	b	b	d	c	b
8		a	a	c	a	d	c
9		a	b	a	b	e	d
10		a	a	b	c	a	e
11		a	b	c	d	b	f
12		a	a	a	a	c	a
13		a	b	b	b	d	b
14		a	a	c	c	e	c
15		a	b	a	d	a	d
16		a	a	b	a	b	e
17		a	b	c	b	c	f
18		a	a	a	c	d	a
19		a	b	b	d	e	b
20		a	a	c	a	a	c
21		a	b	a	b	b	d
22		a	a	b	c	c	e
23		a	b	c	d	d	f
24		a	a	a	a	e	a
25		a	b	b	b	a	b
26		a	a	c	c	b	c
27		a	b	a	d	c	d
28		a	a	b	a	d	e
29		a	b	c	b	e	f

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_p([1, 5, -3], at = -42)
1

---

splinekit.spline_padding.change_basis_p(data: ndarray[tuple[int], dtype[float64]], *, degree: int, source_basis: Bases, target_basis: Bases) → None

In-place conversion of data coefficients from one spline basis to another, under periodic padding.

At input time, data is a one-dimensional numpy.ndarray of coefficients that is expressing the periodic uniform spline \(f\) of nonnegative degree \(n\) as a weighted sum of integer-shifted bases of the type source_basis.

At output time, data is a one-dimensional numpy.ndarray of coefficients that is expressing the same spline \(f\) as a weighted sum of integer-shifted bases of the type target_basis.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from one basis to another.

• degree (int) – Nonnegative degree of the polynomial spline.

• source_basis (int) – Type of the basis at input time.

• target_basis (int) – Type of the basis at output time.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Arbitrary data samples with periodic padding.

>>> f = np.array([1, 5, -3], dtype = "float")

Cubic coefficients.

>>> c = f.copy()
>>> sk.change_basis_p(c, degree = 3, source_basis = sk.Bases.CARDINAL, target_basis = sk.Bases.BASIC)
>>> print(c)
[ 1.  9. -7.]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.samples_to_coeff_p(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under periodic padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The data samples are assumed to conform to a periodic padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Arbitrary data samples with periodic padding.

>>> f = np.array([1, 5, -3], dtype = "float")

Cubic coefficients.

>>> c = f.copy()
>>> sk.samples_to_coeff_p(c, degree = 3)
>>> print(c)
[ 1.  9. -7.]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.coeff_to_samples_p(data: ndarray[tuple[int], dtype[float64]], *, degree: int) → None

In-place conversion of spline coefficients into data samples under periodic padding.

Replaces a one-dimensional numpy.ndarray of spline coefficietnts \(c\) by data samples \(f\) such that

\[\forall k\in[0\ldots K-1]:\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q)=f[k],\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a periodic padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from coefficients to samples.

• degree (int) – Nonnegative degree of the polynomial B-spline.

Returns:

The returned data samples overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Arbitrary spline coefficients with periodic padding.

>>> c = np.array([1, 9, -7], dtype = "float")

Data samples.

>>> f = c.copy()
>>> sk.coeff_to_samples_p(f, degree = 3)
>>> print(f)
[ 1.  5. -3.]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.coeff_to_ortho_p(data: ndarray[tuple[int], dtype[float64]], *, degree: int) → None

In-place conversion of B-spline coefficients into orthonormal-spline coefficients under periodic padding.

Replaces a one-dimensional numpy.ndarray of spline coefficients \(c\) by orthonormal-spline coefficients \(g\) such that

\[\forall x\in{\mathbb{R}}:\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(x-q)=\sum_{q\in{\mathbb{Z}}}\,g[q]\,\phi^{n}(x-q),\]

where \(K\) is the length of the provided data, \(\beta^{n}\) is is a polynomial B-spline of nonnegative degree \(n,\) and \(\phi^{n}\) is an orthonormal polynomial spline. The spline coefficients are assumed to conform to a periodic padding of period \(K.\)

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from spline coefficients to orthonormal coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Arbitrary cubic dual-spline coefficients with periodic padding.

>>> a1 = np.array([1, 2.75714286, -0.75714286], dtype = "float")

Spline coefficients.

>>> c = a1.copy()
>>> sk.samples_to_coeff_p(c, degree = 2 * 3 + 1)

Orthonormal coefficients.

>>> g = c.copy()
>>> sk.coeff_to_ortho_p(g, degree = 3)

The re-application of coeff_to_ortho_p yields back the dual coefficients, up to numerical accuracy.

>>> a2 = g.copy()
>>> sk.coeff_to_ortho_p(a2, degree = 3)
>>> print(a1 - a2)
[-2.22044605e-16 -1.06581410e-14  1.18793864e-14]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.ortho_to_coeff_p(data: ndarray[tuple[int], dtype[float64]], *, degree: int) → None

In-place conversion of orthonormal-spline coefficients into B-spline coefficients under periodic padding.

Replaces a one-dimensional numpy.ndarray of orthonormal spline coefficients \(g\) by B-spline coefficients \(c\) such that

\[\forall x\in{\mathbb{R}}:\sum_{q\in{\mathbb{Z}}}\,g[q]\, \phi^{n}(x-q)=\sum_{q\in{\mathbb{Z}}}\,c[q]\,\beta^{n}(x-q),\]

where \(K\) is the length of the provided data, \(\phi^{n}\) is an orthonormal polynomial spline of nonnegative degree \(n,\) and \(\beta^{n}\) is a polynomial B-spline. The orthonormal spline coefficients are assumed to conform to a periodic padding of period \(K.\)

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from orthonormal coefficients to spline coefficients.

• degree (int) – Nonnegative degree of the polynomial orthonormal-spline.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Arbitrary cubic spline coefficients with periodic padding.

>>> c1 = np.array([1, 9, -7], dtype = "float")

Dual-spline coefficients.

>>> a = c1.copy()
>>> sk.coeff_to_samples_p(a, degree = 2 * 3 + 1)

Orthonormal coefficients.

>>> g1 = a.copy()
>>> sk.ortho_to_coeff_p(g1, degree = 3)

The re-application of ortho_to_coeff_p yields back the spline coefficients, up to numerical accuracy.

>>> g2 = g1.copy()
>>> sk.ortho_to_coeff_p(g2, degree = 3)
>>> print(c1 - g2)
[ 3.33066907e-16  3.55271368e-15 -2.66453526e-15]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.pad_n(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Narrow-mirror padding.

Returns the value of the \(k\)-th data sample after the data have been extended by narrow-mirror padding. The padded data \(f\) satisfy that

\[\begin{split}\forall k\in{\mathbb{Z}}:f[k]=\left\{\begin{array}{rcl}f[k]&=&f[-k]\\ f[k+K-1]&=&f[K-1-k],\end{array}\right.\end{split}\]

where \(K\) is the length of the unpadded data. The conditions of narrow-mirror padding imply the periodicity

\[\forall k\in{\mathbb{Z}}:f[k]=f[k+2\,K-2].\]

Narrow-mirror padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	a	a	c	e	a
−19		a	b	b	b	d	b
−18		a	a	c	a	c	c
−17		a	b	b	b	b	d
−16		a	a	a	c	a	e
−15		a	b	b	d	b	f
−14		a	a	c	c	c	e
−13		a	b	b	b	d	d
−12		a	a	a	a	e	c
−11		a	b	b	b	d	b
−10		a	a	c	c	c	a
−9		a	b	b	d	b	b
−8		a	a	a	c	a	c
−7		a	b	b	b	b	d
−6		a	a	c	a	c	e
−5		a	b	b	b	d	f
−4		a	a	a	c	e	e
−3		a	b	b	d	d	d
−2		a	a	c	c	c	c
−1		a	b	b	b	b	b
0		A	A	A	A	A	A
1		a	B	B	B	B	B
2		a	a	C	C	C	C
3		a	b	b	D	D	D
4		a	a	a	c	E	E
5		a	b	b	b	d	F
6		a	a	c	a	c	e
7		a	b	b	b	b	d
8		a	a	a	c	a	c
9		a	b	b	d	b	b
10		a	a	c	c	c	a
11		a	b	b	b	d	b
12		a	a	a	a	e	c
13		a	b	b	b	d	d
14		a	a	c	c	c	e
15		a	b	b	d	b	f
16		a	a	a	c	a	e
17		a	b	b	b	b	d
18		a	a	c	a	c	c
19		a	b	b	b	d	b
20		a	a	a	c	e	a
21		a	b	b	d	d	b
22		a	a	c	c	c	c
23		a	b	b	b	b	d
24		a	a	a	a	a	e
25		a	b	b	b	b	f
26		a	a	c	c	c	e
27		a	b	b	d	d	d
28		a	a	a	c	e	c
29		a	b	b	b	d	b

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_n([1, 5, -3], at = -42)
-3

---

splinekit.spline_padding.samples_to_coeff_n(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under narrow-mirror padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a narrow-mirror padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Cubic coefficients of arbitrary data with narrow-mirror padding.

>>> f = np.array([1, 5, -3], dtype = "float")
>>> c = f.copy()
>>> sk.samples_to_coeff_n(c, degree = 3)
>>> print(c)
[ -4.  11. -10.]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.pad_w(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Wide-mirror padding.

Returns the value of the \(k\)-th data sample after the data have been extended by wide-mirror padding. The padded data \(f\) satisfy that

\[\begin{split}\forall k\in{\mathbb{Z}}:\left\{\begin{array}{rcl}f[x]&=&f[-1-x]\\ f[x+K]&=&f[K-1-x],\end{array}\right.\end{split}\]

where \(K\) is the length of the unpadded data. The conditions of wide-mirror padding imply the periodicity

\[\forall k\in{\mathbb{Z}}:f[k]=f[k+2\,K].\]

Wide-mirror padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	a	b	d	a	e
−19		a	b	a	c	b	f
−18		a	b	a	b	c	f
−17		a	a	b	a	d	e
−16		a	a	c	a	e	d
−15		a	b	c	b	e	c
−14		a	b	b	c	d	b
−13		a	a	a	d	c	a
−12		a	a	a	d	b	a
−11		a	b	b	c	a	b
−10		a	b	c	b	a	c
−9		a	a	c	a	b	d
−8		a	a	b	a	c	e
−7		a	b	a	b	d	f
−6		a	b	a	c	e	f
−5		a	a	b	d	e	e
−4		a	a	c	d	d	d
−3		a	b	c	c	c	c
−2		a	b	b	b	b	b
−1		a	a	a	a	a	a
0		A	A	A	A	A	A
1		a	B	B	B	B	B
2		a	b	C	C	C	C
3		a	a	c	D	D	D
4		a	a	b	d	E	E
5		a	b	a	c	e	F
6		a	b	a	b	d	f
7		a	a	b	a	c	e
8		a	a	c	a	b	d
9		a	b	c	b	a	c
10		a	b	b	c	a	b
11		a	a	a	d	b	a
12		a	a	a	d	c	a
13		a	b	b	c	d	b
14		a	b	c	b	e	c
15		a	a	c	a	e	d
16		a	a	b	a	d	e
17		a	b	a	b	c	f
18		a	b	a	c	b	f
19		a	a	b	d	a	e
20		a	a	c	d	a	d
21		a	b	c	c	b	c
22		a	b	b	b	c	b
23		a	a	a	a	d	a
24		a	a	a	a	e	a
25		a	b	b	b	e	b
26		a	b	c	c	d	c
27		a	a	c	d	c	d
28		a	a	b	d	b	e
29		a	b	a	c	a	f

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_w([1, 5, -3], at = -42)
1

---

splinekit.spline_padding.samples_to_coeff_w(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under wide-mirror padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a wide-mirror padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Cubic coefficients of arbitrary data with wide-mirror padding.

>>> f = np.array([1, 5, -3], dtype = "float")
>>> c = f.copy()
>>> sk.samples_to_coeff_w(c, degree = 3)
>>> print(c)
[-0.6  9.  -5.4]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.pad_a(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Anti-mirror padding.

Returns the value of the \(k\)-th data sample after the data have been extended by anti-mirror padding. The padded data \(f\) satisfy that

\[\begin{split}\forall k\in{\mathbb{Z}}:\left\{\begin{array}{rcl} f[k]-f[0]&=&f[0]-f[-k]\\ f[k+K-1]-f[K-1]&=&f[K-1]-f[K-1-k]\end{array}\right.\end{split}\]

where \(K\) is the length of the unpadded data. The conditions of anti-mirror padding imply the pseudo periodicity

\[\forall k\in{\mathbb{Z}}:f[k]=f[k+2\,K-2]-2\,\left(f[K-1]-f[0]\right).\]

Anti-mirror padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	(21 a − 20 b)	(11 a − 10 c)	(8 a − c − 6 d)	(6 a − 5 e)	(5 a − 4 f)
−19		a	(20 a − 19 b)	(10 a + b − 10 c)	(8 a − b − 6 d)	(6 a − d − 4 e)	(4 a + b − 4 f)
−18		a	(19 a − 18 b)	(10 a − 9 c)	(7 a − 6 d)	(6 a − c − 4 e)	(4 a + c − 4 f)
−17		a	(18 a − 17 b)	(10 a − b − 8 c)	(6 a + b − 6 d)	(6 a − b − 4 e)	(4 a + d − 4 f)
−16		a	(17 a − 16 b)	(9 a − 8 c)	(6 a + c − 6 d)	(5 a − 4 e)	(4 a + e − 4 f)
−15		a	(16 a − 15 b)	(8 a + b − 8 c)	(6 a − 5 d)	(4 a + b − 4 e)	(4 a − 3 f)
−14		a	(15 a − 14 b)	(8 a − 7 c)	(6 a − c − 4 d)	(4 a + c − 4 e)	(4 a − e − 2 f)
−13		a	(14 a − 13 b)	(8 a − b − 6 c)	(6 a − b − 4 d)	(4 a + d − 4 e)	(4 a − d − 2 f)
−12		a	(13 a − 12 b)	(7 a − 6 c)	(5 a − 4 d)	(4 a − 3 e)	(4 a − c − 2 f)
−11		a	(12 a − 11 b)	(6 a + b − 6 c)	(4 a + b − 4 d)	(4 a − d − 2 e)	(4 a − b − 2 f)
−10		a	(11 a − 10 b)	(6 a − 5 c)	(4 a + c − 4 d)	(4 a − c − 2 e)	(3 a − 2 f)
−9		a	(10 a − 9 b)	(6 a − b − 4 c)	(4 a − 3 d)	(4 a − b − 2 e)	(2 a + b − 2 f)
−8		a	(9 a − 8 b)	(5 a − 4 c)	(4 a − c − 2 d)	(3 a − 2 e)	(2 a + c − 2 f)
−7		a	(8 a − 7 b)	(4 a + b − 4 c)	(4 a − b − 2 d)	(2 a + b − 2 e)	(2 a + d − 2 f)
−6		a	(7 a − 6 b)	(4 a − 3 c)	(3 a − 2 d)	(2 a + c − 2 e)	(2 a + e − 2 f)
−5		a	(6 a − 5 b)	(4 a − b − 2 c)	(2 a + b − 2 d)	(2 a + d − 2 e)	(2 a − f)
−4		a	(5 a − 4 b)	(3 a − 2 c)	(2 a + c − 2 d)	(2 a − e)	(2 a − e)
−3		a	(4 a − 3 b)	(2 a + b − 2 c)	(2 a − d)	(2 a − d)	(2 a − d)
−2		a	(3 a − 2 b)	(2 a − c)	(2 a − c)	(2 a − c)	(2 a − c)
−1		a	(2 a − b)	(2 a − b)	(2 a − b)	(2 a − b)	(2 a − b)
0		A	A	A	A	A	A
1		a	B	B	B	B	B
2		a	(−a + 2 b)	C	C	C	C
3		a	(−2 a + 3 b)	(−b + 2 c)	D	D	D
4		a	(−3 a + 4 b)	(−a + 2 c)	(−c + 2 d)	E	E
5		a	(−4 a + 5 b)	(−2 a + b + 2 c)	(−b + 2 d)	(−d + 2 e)	F
6		a	(−5 a + 6 b)	(−2 a + 3 c)	(−a + 2 d)	(−c + 2 e)	(−e + 2 f)
7		a	(−6 a + 7 b)	(−2 a − b + 4 c)	(−2 a + b + 2 d)	(−b + 2 e)	(−d + 2 f)
8		a	(−7 a + 8 b)	(−3 a + 4 c)	(−2 a + c + 2 d)	(−a + 2 e)	(−c + 2 f)
9		a	(−8 a + 9 b)	(−4 a + b + 4 c)	(−2 a + 3 d)	(−2 a + b + 2 e)	(−b + 2 f)
10		a	(−9 a + 10 b)	(−4 a + 5 c)	(−2 a − c + 4 d)	(−2 a + c + 2 e)	(−a + 2 f)
11		a	(−10 a + 11 b)	(−4 a − b + 6 c)	(−2 a − b + 4 d)	(−2 a + d + 2 e)	(−2 a + b + 2 f)
12		a	(−11 a + 12 b)	(−5 a + 6 c)	(−3 a + 4 d)	(−2 a + 3 e)	(−2 a + c + 2 f)
13		a	(−12 a + 13 b)	(−6 a + b + 6 c)	(−4 a + b + 4 d)	(−2 a − d + 4 e)	(−2 a + d + 2 f)
14		a	(−13 a + 14 b)	(−6 a + 7 c)	(−4 a + c + 4 d)	(−2 a − c + 4 e)	(−2 a + e + 2 f)
15		a	(−14 a + 15 b)	(−6 a − b + 8 c)	(−4 a + 5 d)	(−2 a − b + 4 e)	(−2 a + 3 f)
16		a	(−15 a + 16 b)	(−7 a + 8 c)	(−4 a − c + 6 d)	(−3 a + 4 e)	(−2 a − e + 4 f)
17		a	(−16 a + 17 b)	(−8 a + b + 8 c)	(−4 a − b + 6 d)	(−4 a + b + 4 e)	(−2 a − d + 4 f)
18		a	(−17 a + 18 b)	(−8 a + 9 c)	(−5 a + 6 d)	(−4 a + c + 4 e)	(−2 a − c + 4 f)
19		a	(−18 a + 19 b)	(−8 a − b + 10 c)	(−6 a + b + 6 d)	(−4 a + d + 4 e)	(−2 a − b + 4 f)
20		a	(−19 a + 20 b)	(−9 a + 10 c)	(−6 a + c + 6 d)	(−4 a + 5 e)	(−3 a + 4 f)
21		a	(−20 a + 21 b)	(−10 a + b + 10 c)	(−6 a + 7 d)	(−4 a − d + 6 e)	(−4 a + b + 4 f)
22		a	(−21 a + 22 b)	(−10 a + 11 c)	(−6 a − c + 8 d)	(−4 a − c + 6 e)	(−4 a + c + 4 f)
23		a	(−22 a + 23 b)	(−10 a − b + 12 c)	(−6 a − b + 8 d)	(−4 a − b + 6 e)	(−4 a + d + 4 f)
24		a	(−23 a + 24 b)	(−11 a + 12 c)	(−7 a + 8 d)	(−5 a + 6 e)	(−4 a + e + 4 f)
25		a	(−24 a + 25 b)	(−12 a + b + 12 c)	(−8 a + b + 8 d)	(−6 a + b + 6 e)	(−4 a + 5 f)
26		a	(−25 a + 26 b)	(−12 a + 13 c)	(−8 a + c + 8 d)	(−6 a + c + 6 e)	(−4 a − e + 6 f)
27		a	(−26 a + 27 b)	(−12 a − b + 14 c)	(−8 a + 9 d)	(−6 a + d + 6 e)	(−4 a − d + 6 f)
28		a	(−27 a + 28 b)	(−13 a + 14 c)	(−8 a − c + 10 d)	(−6 a + 7 e)	(−4 a − c + 6 f)
29		a	(−28 a + 29 b)	(−14 a + b + 14 c)	(−8 a − b + 10 d)	(−6 a − d + 8 e)	(−4 a − b + 6 f)

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_a([1, 5, -3], at = -42)
85

---

splinekit.spline_padding.samples_to_coeff_a(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under anti-mirror padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a anti-mirror padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Cubic coefficients of arbitrary data with anti-mirror padding.

>>> f = np.array([1, 5, -3], dtype = "float")
>>> c = f.copy()
>>> sk.samples_to_coeff_a(c, degree = 3)
>>> print(c)
[ 1.  8. -3.]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.pad_np(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Nega-periodic padding.

Returns the value of the \(k\)-th data sample after the data have been extended by nega-periodic padding. The padded data \(f\) satisfy that

\[\forall k\in{\mathbb{Z}}:f[k+K]=-f[k],\]

where \(K\) is the length of the unpadded data. The conditions of nega-periodic padding imply the periodicity

\[\forall k\in{\mathbb{Z}}:f[k]=f[k+2\,K].\]

Nega-periodic padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	a	−b	−a	a	e
−19		−a	b	−c	−b	b	f
−18		a	−a	a	−c	c	−a
−17		−a	−b	b	−d	d	−b
−16		a	a	c	a	e	−c
−15		−a	b	−a	b	−a	−d
−14		a	−a	−b	c	−b	−e
−13		−a	−b	−c	d	−c	−f
−12		a	a	a	−a	−d	a
−11		−a	b	b	−b	−e	b
−10		a	−a	c	−c	a	c
−9		−a	−b	−a	−d	b	d
−8		a	a	−b	a	c	e
−7		−a	b	−c	b	d	f
−6		a	−a	a	c	e	−a
−5		−a	−b	b	d	−a	−b
−4		a	a	c	−a	−b	−c
−3		−a	b	−a	−b	−c	−d
−2		a	−a	−b	−c	−d	−e
−1		−a	−b	−c	−d	−e	−f
0		A	A	A	A	A	A
1		−a	B	B	B	B	B
2		a	−a	C	C	C	C
3		−a	−b	−a	D	D	D
4		a	a	−b	−a	E	E
5		−a	b	−c	−b	−a	F
6		a	−a	a	−c	−b	−a
7		−a	−b	b	−d	−c	−b
8		a	a	c	a	−d	−c
9		−a	b	−a	b	−e	−d
10		a	−a	−b	c	a	−e
11		−a	−b	−c	d	b	−f
12		a	a	a	−a	c	a
13		−a	b	b	−b	d	b
14		a	−a	c	−c	e	c
15		−a	−b	−a	−d	−a	d
16		a	a	−b	a	−b	e
17		−a	b	−c	b	−c	f
18		a	−a	a	c	−d	−a
19		−a	−b	b	d	−e	−b
20		a	a	c	−a	a	−c
21		−a	b	−a	−b	b	−d
22		a	−a	−b	−c	c	−e
23		−a	−b	−c	−d	d	−f
24		a	a	a	a	e	a
25		−a	b	b	b	−a	b
26		a	−a	c	c	−b	c
27		−a	−b	−a	d	−c	d
28		a	a	−b	−a	−d	e
29		−a	b	−c	−b	−e	f

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_np([1, 5, -3], at = -42)
1

---

splinekit.spline_padding.samples_to_coeff_np(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under nega-periodic padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a nega-periodic padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Cubic coefficients of arbitrary data with nega-periodic padding.

>>> f = np.array([1, 5, -3], dtype = "float")
>>> c = f.copy()
>>> sk.samples_to_coeff_np(c, degree = 3)
>>> print(c)
[-3.  10.2 -7.8]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.pad_nn(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Nega-narrow-mirror padding.

Returns the value of the \(k\)-th data sample after the data have been extended by nega-narrow-mirror padding. The padded data \(f\) satisfy that

\[\begin{split}\forall k\in{\mathbb{Z}}:\left\{\begin{array}{rcl}f[k-1]&=&-f[-1-k]\\ f[k+K+1]&=&-f[K-1-k],\end{array}\right.\end{split}\]

where \(K\) is the length of the unpadded data. The conditions of nega-narrow-mirror padding imply the periodicity

\[\begin{split}\forall k\in{\mathbb{Z}}:\left\{\begin{array}{rcl}f[k]&=&f[k+2\,K+2]\\ f[k\,\left(K+1\right)-1]&=&0.\end{array}\right.\end{split}\]

Nega-narrow-mirror padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	−a	−c	a	e	−e
−19		0	0	−b	b	0	−d
−18		−a	a	−a	c	−e	−c
−17		0	b	0	d	−d	−b
−16		a	0	a	0	−c	−a
−15		0	−b	b	−d	−b	0
−14		−a	−a	c	−c	−a	a
−13		0	0	0	−b	0	b
−12		a	a	−c	−a	a	c
−11		0	b	−b	0	b	d
−10		−a	0	−a	a	c	e
−9		0	−b	0	b	d	f
−8		a	−a	a	c	e	0
−7		0	0	b	d	0	−f
−6		−a	a	c	0	−e	−e
−5		0	b	0	−d	−d	−d
−4		a	0	−c	−c	−c	−c
−3		0	−b	−b	−b	−b	−b
−2		−a	−a	−a	−a	−a	−a
−1		0	0	0	0	0	0
0		A	A	A	A	A	A
1		0	B	B	B	B	B
2		−a	0	C	C	C	C
3		0	−b	0	D	D	D
4		a	−a	−c	0	E	E
5		0	0	−b	−d	0	F
6		−a	a	−a	−c	−e	0
7		0	b	0	−b	−d	−f
8		a	0	a	−a	−c	−e
9		0	−b	b	0	−b	−d
10		−a	−a	c	a	−a	−c
11		0	0	0	b	0	−b
12		a	a	−c	c	a	−a
13		0	b	−b	d	b	0
14		−a	0	−a	0	c	a
15		0	−b	0	−d	d	b
16		a	−a	a	−c	e	c
17		0	0	b	−b	0	d
18		−a	a	c	−a	−e	e
19		0	b	0	0	−d	f
20		a	0	−c	a	−c	0
21		0	−b	−b	b	−b	−f
22		−a	−a	−a	c	−a	−e
23		0	0	0	d	0	−d
24		a	a	a	0	a	−c
25		0	b	b	−d	b	−b
26		−a	0	c	−c	c	−a
27		0	−b	0	−b	d	0
28		a	−a	−c	−a	e	a
29		0	0	−b	0	0	b

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_nn([1, 5, -3], at = -42)
-1

---

splinekit.spline_padding.samples_to_coeff_nn(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under nega-narrow-mirror padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a nega-narrow-mirror padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Cubic coefficients of arbitrary data with nega-narrow-mirror padding.

>>> f = np.array([1, 5, -3], dtype = "float")
>>> c = f.copy()
>>> sk.samples_to_coeff_nn(c, degree = 3)
>>> print(c)
[-0.85714286  9.42857143 -6.85714286]

Notes

The computations are performed in-place.

---

splinekit.spline_padding.pad_nw(data: ndarray[tuple[int], dtype[float64]], *, at: int) → float

Nega-wide-mirror padding.

Returns the value of the \(k\)-th data sample after the data have been extended by nega-wide-mirror padding. The padded data \(f\) satisfy that

\[\begin{split}\forall k\in{\mathbb{Z}}:\left\{\begin{array}{rcl}f[k]&=&-f[-1-k]\\ f[k+K]&=&-f[K-1-k],\end{array}\right.\end{split}\]

where \(K\) is the length of the unpadded data. The conditions of nega-wide-mirror padding imply the periodicity

\[\forall k\in{\mathbb{Z}}:f[k]=f[k+2\,K].\]

Nega-wide-mirror padding of data samples

k		ƒ1[k]	ƒ2[k]	ƒ3[k]	ƒ4[k]	ƒ5[k]	ƒ6[k]
−20		a	a	−b	−d	a	e
−19		−a	b	−a	−c	b	f
−18		a	−b	a	−b	c	−f
−17		−a	−a	b	−a	d	−e
−16		a	a	c	a	e	−d
−15		−a	b	−c	b	−e	−c
−14		a	−b	−b	c	−d	−b
−13		−a	−a	−a	d	−c	−a
−12		a	a	a	−d	−b	a
−11		−a	b	b	−c	−a	b
−10		a	−b	c	−b	a	c
−9		−a	−a	−c	−a	b	d
−8		a	a	−b	a	c	e
−7		−a	b	−a	b	d	f
−6		a	−b	a	c	e	−f
−5		−a	−a	b	d	−e	−e
−4		a	a	c	−d	−d	−d
−3		−a	b	−c	−c	−c	−c
−2		a	−b	−b	−b	−b	−b
−1		−a	−a	−a	−a	−a	−a
0		A	A	A	A	A	A
1		−a	B	B	B	B	B
2		a	−b	C	C	C	C
3		−a	−a	−c	D	D	D
4		a	a	−b	−d	E	E
5		−a	b	−a	−c	−e	F
6		a	−b	a	−b	−d	−f
7		−a	−a	b	−a	−c	−e
8		a	a	c	a	−b	−d
9		−a	b	−c	b	−a	−c
10		a	−b	−b	c	a	−b
11		−a	−a	−a	d	b	−a
12		a	a	a	−d	c	a
13		−a	b	b	−c	d	b
14		a	−b	c	−b	e	c
15		−a	−a	−c	−a	−e	d
16		a	a	−b	a	−d	e
17		−a	b	−a	b	−c	f
18		a	−b	a	c	−b	−f
19		−a	−a	b	d	−a	−e
20		a	a	c	−d	a	−d
21		−a	b	−c	−c	b	−c
22		a	−b	−b	−b	c	−b
23		−a	−a	−a	−a	d	−a
24		a	a	a	a	e	a
25		−a	b	b	b	−e	b
26		a	−b	c	c	−d	c
27		−a	−a	−c	d	−c	d
28		a	a	−b	−d	−b	e
29		−a	b	−a	−c	−a	f

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to extend.

• at (int) – Arbitrary index.

Returns:

The value of the padded data at the requested index.

Return type:

float

Examples

Load the library.

>>> import splinekit as sk

Short data at large index -42.

>>> sk.pad_nw([1, 5, -3], at = -42)
1

---

splinekit.spline_padding.samples_to_coeff_nw(data: ndarray[tuple[int], dtype[float64]], *, degree: int, pure_python: bool = False) → None

In-place conversion of data samples into spline coefficients under nega-wide-mirror padding.

Replaces a one-dimensional numpy.ndarray of data samples \(f\) by spline coefficients \(c\) such that

\[\forall k\in[0\ldots K-1]:f[k]=\sum_{q\in{\mathbb{Z}}}\,c[q]\, \beta^{n}(k-q),\]

where \(K\) is the length of the provided data and \(\beta^{n}\) is a polynomial B-spline of nonnegative degree \(n.\) The spline coefficients are assumed to conform to a nega-wide-mirror padding.

Parameters:

• data (np.ndarray[tuple[int], np.dtype[np.float64]],) – Data to convert from samples to coefficients.

• degree (int) – Nonnegative degree of the polynomial B-spline.

• pure_python (bool) – The directive that forces the computations to be performed in Python.

Returns:

The returned coefficients overwrite data.

Return type:

None

Examples

Load the libraries.

>>> import numpy as np
>>> import splinekit as sk

Cubic coefficients of arbitrary data with nega-wide-mirror padding.

>>> f = np.array([1, 5, -3], dtype = "float")
>>> c = f.copy()
>>> sk.samples_to_coeff_nw(c, degree = 3)
>>> print(c)
[-1.4 10.2 -9.4]

Notes

The computations are performed in-place.