# Data

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

Manopt.artificialIn_SAR_imageMethod
artificialIn_SAR_image([pts=500])

generate an artificial InSAR image, i.e. phase valued data, of size pts x pts points.

This data set was introduced for the numerical examples in

Bergmann, R., Laus, F., Steidl, G., Weinmann, A.: Second Order Differences of Cyclic Data and Applications in Variational Denoising SIAM J. Imaging Sci., 7(4), 2916–2953, 2014. doi: 10.1137/140969993 arxiv: 1405.5349

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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

Bergmann, R., Laus, F., Steidl, G., Weinmann, A.: Second Order Differences of Cyclic Data and Applications in Variational Denoising SIAM J. Imaging Sci., 7(4), 2916–2953, 2014. doi: 10.1137/140969993 arxiv: 1405.5349

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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, R., Laus, F., Steidl, G., Weinmann, A.: Second Order Differences of Cyclic Data and Applications in Variational Denoising SIAM J. Imaging Sci., 7(4), 2916–2953, 2014. doi: 10.1137/140969993 arxiv: 1405.5349

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

This data set was introduced for the numerical examples in

Bergmann, R., Laus, F., Steidl, G., Weinmann, A.: Second Order Differences of Cyclic Data and Applications in Variational Denoising SIAM J. Imaging Sci., 7(4), 2916–2953, 2014. doi: 10.1137/140969993 arxiv: 1405.5349

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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}

This example was used within minimization of acceleration of the paper

Bergmann, R., Gousenbourger, P.-Y.: A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve, Front. Appl. Math. Stat. 12, 2018. doi: 10.3389/fams.2018.00059 arxiv: 1807.10090

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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.

This dataset was used in the numerical example of Section 5.1 of

Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images SIAM J. Sci. Comput. 38(1), A567–A597, 2016. doi: 10.1137/15M101988X arxiv: 1506.02409

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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

This dataset was used in the numerical example of Section 5.1 of

Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images SIAM J. Sci. Comput. 38(1), A567–A597, 2016. doi: 10.1137/15M101988X arxiv: 1506.02409

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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.

This dataset was used in the numerical example of Section 5.1 of

Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images SIAM J. Sci. Comput. 38(1), A567–A597, 2016. doi: 10.1137/15M101988X arxiv: 1506.02409

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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.

This example dataset was used in the numerical example in Section 5.5 of

Laus, F., Nikolova, M., Persch, J., Steidl, G.: A Nonlocal Denoising Algorithm for Manifold-Valued Images Using Second Order Statistics, SIAM J. Imaging Sci., 10(1), 416–448, 2017. doi: 10.1137/16M1087114 arxiv: 1607.08481

It is based on artificial_S2_rotation_image extended by small whirl patches.

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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.

This dataset was used in the numerical example of Section 5.2 of

Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images SIAM J. Sci. Comput. 38(1), A567–A597, 2016. doi: 10.1137/15M101988X arxiv: 1506.02409

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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.

This data set was introduced in the numerical examples of Section of

Bergmann, R., Persch, J., Steidl, G.: A Parallel Douglas Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds SIAM J. Imaging. Sci. 9(3), pp. 901-937, 2016. doi: 10.1137/15M1052858 arxiv: 1512.02814

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