Data
For some manifolds there are artificial or real application data available that can be loaded using the following data functions. Note that these need additionally Manifolds.jl
to be loaded.
Manopt.artificialIn_SAR_image
— MethodartificialIn_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 et. al., SIAM J Imag Sci, 2014.
Manopt.artificial_S1_signal
— Functionartificial_S1_signal([pts=500])
generate a realvalued 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 $[\pi,\pi)$. This is data for an example from Bergmann et. al., SIAM J Imag Sci, 2014.
Optional
pts
– (500
) number of points to sample the function
Manopt.artificial_S1_signal
— Methodartificial_S1_signal(x)
evaluate the example signal $f(x), x ∈ [0,1]$, of phasevalued data introduces in Sec. 5.1 of Bergmann et. al., SIAM J Imag Sci, 2014 for values outside that intervall, this Signal is missing
.
Manopt.artificial_S1_slope_signal
— Functionartificial_S1_slope_signal([pts=500, slope=4.])
Creates a Signal of (phasevalued) data represented on the Circle
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 et. al., SIAM J Imag Sci, 2014
Manopt.artificial_S2_composite_bezier_curve
— Methodartificial_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, Gousenbourger, Front. Appl. Math. Stat., 2018.
Manopt.artificial_S2_lemniscate
— Functionartificial_S2_lemniscate(p, t::Float64; a::Float64=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 èxpto obtain a point on the [
Sphere`](https://juliamanifolds.github.io/Manifolds.jl/stable/manifolds/sphere.html).
Input
p
– the tangent space the Lemniscate is created int
– 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 et al., SIAM J Sci Comput, 2016.
Manopt.artificial_S2_lemniscate
— Functionartificial_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 inpts
– (128
) number of points to sample the Lemniscatea
– (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 et al., SIAM J Sci Comput, 2016.
Manopt.artificial_S2_rotation_image
— Methodartificial_S2_rotation_image([pts=64, rotations=(.5,.5)])
Create an image with a rotation on each axis as a parametrization.
Optional Parameters
pts
– (64
) number of pixels along one dimensionrotations
– ((.5,.5)
) number of total rotations performed on the axes.
This dataset was used in the numerical example of Section 5.1 of Bačák et al., SIAM J Sci Comput, 2016.
Manopt.artificial_S2_whirl_image
— Methodartificial_S2_whirl_image([pts::Int=64])
Generate an artificial image of data on the 2 sphere,
Arguments
pts
– (64
) size of the image inpts
$\times$pts
pixel.
This example dataset was used in the numerical example in Section 5.5 of Laus et al., SIAM J Imag Sci., 2017
It is based on artificial_S2_rotation_image
extended by small whirl patches.
Manopt.artificial_S2_whirl_patch
— Functionartificial_S2_whirl_patch([pts=5])
create a whirl within the pts
$\times$pts
patch of Sphere(@ref)(2)
valued image data.
These patches are used within artificial_S2_whirl_image
.
Optional Parameters
pts
– (5
) size of the patch. If the number is odd, the center is the north pole.
Manopt.artificial_SPD_image
— Functionartificial_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 et al., SIAM J Sci Comput, 2016.
Manopt.artificial_SPD_image2
— Methodartificial_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, Presch, Steidl, SIAM J Imag Sci, 2016
Literature
 [BBSW16]

M. Bačák, R. Bergmann, G. Steidl and A. Weinmann. A second order nonsmooth variational model for restoring manifoldvalued images. SIAM Journal on Scientific Computing 38, A567–A597 (2016), arxiv: [1506.02409](https://arxiv.org/abs/1506.02409).
 [BG18]

R. Bergmann and P.Y. Gousenbourger. A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve. Frontiers in Applied Mathematics and Statistics 4 (2018), arXiv: [1807.10090](https://arxiv.org/abs/1807.10090).
 [BLSW14]

R. Bergmann, F. Laus, G. Steidl and A. Weinmann. Second order differences of cyclic data and applications in variational denoising. SIAM Journal on Imaging Sciences 7, 2916–2953 (2014), arxiv: [1405.5349](https://arxiv.org/abs/1405.5349).
 [BPS16]

R. Bergmann, J. Persch and G. Steidl. A parallel Douglas Rachford algorithm for minimizing ROFlike functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences 9, 901–937 (2016), arxiv: [1512.02814](https://arxiv.org/abs/1512.02814).
 [LNPS17]

F. Laus, M. Nikolova, J. Persch and G. Steidl. A nonlocal denoising algorithm for manifoldvalued images using second order statistics. SIAM Journal on Imaging Sciences 10, 416–448 (2017).