# Data

For some manifolds there are artificial or real application data available that can be loaded using the following data functions

Manopt.artificial_S1_signalFunction
artificial_S1_signal([pts=500])

generate a real-valued signal having piecewise constant, linear and quadratic intervals with jumps in between. If the resulting manifold the data lives on, is the Circle the data is also wrapped to $[-\pi,\pi)$.

Optional

• pts – (500) number of points to sample the function
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Manopt.artificial_S1_signalMethod
artificial_S1_signal(x)

evaluate the example signal $f(x), x ∈ [0,1]$, of phase-valued data introduces in Sec. 5.1 of

Bergmann, Laus, Steidl, Weinmann, Second Order Differences of Cyclic Data and Applications in Variational Denoising, SIAM J. Imaging Sci., 7(4), 2916–2953, 2014. doi: 10.1137/140969993

for values outside that intervall, this Signal is missing.

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Manopt.artificial_S1_slope_signalFunction
artificial_S1_slope_signal([pts=500, slope=4.])

Creates a Signal of (phase-valued) data represented on the CircleManifold with increasing slope.

Optional

• pts – (500) number of points to sample the function.
• slope – (4.0) initial slope that gets increased afterwards
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Manopt.artificial_S2_composite_bezier_curveMethod
artificial_S2_composite_bezier_curve()

Create the artificial curve in the Sphere(2) consisting of 3 segments between the four points

$p_0 = \begin{bmatrix}0&0&1\end{bmatrix}^{\mathrm{T}}, p_1 = \begin{bmatrix}0&-1&0\end{bmatrix}^{\mathrm{T}}, p_2 = \begin{bmatrix}-1&0&0\end{bmatrix}^{\mathrm{T}}, p_3 = \begin{bmatrix}0&0&-1\end{bmatrix}^{\mathrm{T}},$

where each segment is a cubic Bezér curve, i.e. each point, except $p_3$ has a first point within the following segment $b_i^+$, $i=0,1,2$ and a last point within the previous segment, except for $p_0$, which are denoted by $b_i^-$, $i=1,2,3$. This curve is differentiable by the conditions $b_i^- = \gamma_{b_i^+,p_i}(2)$, $i=1,2$, where $\gamma_{a,b}$ is the shortest_geodesic connecting $a$ and $b$. The remaining points are defined as

\begin{aligned} b_0^+ &= \exp_{p_0}\frac{\pi}{8\sqrt{2}}\begin{pmatrix}1&-1&0\end{pmatrix}^{\mathrm{T}},& b_1^+ &= \exp_{p_1}-\frac{\pi}{4\sqrt{2}}\begin{pmatrix}-1&0&1\end{pmatrix}^{\mathrm{T}},\\ b_2^+ &= \exp_{p_2}\frac{\pi}{4\sqrt{2}}\begin{pmatrix}0&1&-1\end{pmatrix}^{\mathrm{T}},& b_3^- &= \exp_{p_3}-\frac{\pi}{8\sqrt{2}}\begin{pmatrix}-1&1&0\end{pmatrix}^{\mathrm{T}}. \end{aligned}
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Manopt.artificial_S2_lemniscateFunction
artificial_S2_lemniscate(p,t; a=π/2)

generate a point from the signal on the Sphere $\mathbb S^2$ by creating the Lemniscate of Bernoulli in the tangent space of p sampled at t and use èxp to obtain a point on the Sphere.

Input

• p – the tangent space the Lemniscate is created in
• t – value to sample the Lemniscate at

Optional Values

• a – (π/2) defines a half axis of the Lemniscate to cover a half sphere.
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Manopt.artificial_S2_lemniscateFunction
artificial_S2_lemniscate(p [,pts=128,a=π/2,interval=[0,2π])

generate a Signal on the Sphere $\mathbb S^2$ by creating the Lemniscate of Bernoulli in the tangent space of p sampled at pts points and use exp to get a signal on the Sphere.

Input

• p – the tangent space the Lemniscate is created in
• pts – (128) number of points to sample the Lemniscate
• a – (π/2) defines a half axis of the Lemniscate to cover a half sphere.
• interval – ([0,2*π]) range to sample the lemniscate at, the default value refers to one closed curve
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Manopt.artificial_S2_rotation_imageFunction
artificial_S2_rotation_image([pts=64, rotations=(.5,.5)])

creates an image with a rotation on each axis as a parametrization.

Optional Parameters

• pts – (64) number of pixels along one dimension
• rotations – ((.5,.5)) number of total rotations performed on the axes.
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Manopt.artificial_S2_whirl_imageFunction
artificial_S2_whirl_image([pts=64])

generate an artificial image of data on the 2 sphere,

Arguments

• pts – (64) size of the image in pts$\times$pts pixel.
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Manopt.artificial_S2_whirl_patchFunction
artificial_S2_whirl_patch([pts=5])

create a whirl within the pts$\times$pts patch of Sphere(@ref)(2)-valued image data.

Optional Parameters

• pts – (5) size of the patch. If the number is odd, the center is the north pole.
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Manopt.artificial_SPD_imageFunction
artificial_SPD_image([pts=64, stepsize=1.5])

create an artificial image of symmetric positive definite matrices of size pts$\times$pts pixel with a jump of size stepsize.

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Manopt.artificial_SPD_image2Function
artificial_SPD_image2([pts=64, fraction=.66])

create an artificial image of symmetric positive definite matrices of size pts$\times$pts pixel with right hand side fraction` is moved upwards.

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