# Data

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

`Manopt.artificialIn_SAR_image`

— Method`artificialIn_SAR_image([pts=500])`

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

x `pts`

points.

`Manopt.artificial_S1_signal`

— Function`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

`Manopt.artificial_S1_signal`

— Method`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`

.

`Manopt.artificial_S1_slope_signal`

— Function`artificial_S1_slope_signal([pts=500, slope=4.])`

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

**Optional**

`pts`

– (`500`

) number of points to sample the function.`slope`

– (`4.0`

) initial slope that gets increased afterwards

`Manopt.artificial_S2_composite_bezier_curve`

— Method`artificial_S2_composite_bezier_curve()`

Create the artificial curve in the `Sphere(2)`

consisting of 3 segments between the four points

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

`Manopt.artificial_S2_lemniscate`

— Function`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.

`Manopt.artificial_S2_lemniscate`

— Function`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

`Manopt.artificial_S2_rotation_image`

— Function`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.

`Manopt.artificial_S2_whirl_image`

— Function`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.

`Manopt.artificial_S2_whirl_patch`

— Function`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.

`Manopt.artificial_SPD_image`

— Function`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`

.

`Manopt.artificial_SPD_image2`

— Function`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.