Data

# Data

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

artificialS1Signal([pts=500,pointType=S1Point])

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 default S1Point) the data is also wrapped to $[-\pi,\pi)$.

Optional

• pts – (500) number of points to sample the function
• pointType – (S1Point) provide the point type, i.e. manifold the data lives on. Possible values: S1Point, SnPoint, RnPoint, where for the second type, the data is embeypeded (point wise) in $\mathbb R^2$.
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artificialS1Signal(x)

evaluate the example signal $f(x), x\in [0,1]$, 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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artificialS1SlopeSignal([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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artificialInSARImage([pts=500, pointType=S1Point])

generate an artificial InSAR image, i.e. phase valued data, of size pts x pts points. The pointType – (S1Point) provide the point type, i.e. manifold the data lives on. Possible values: S1Point, SnPoint, RnPoint, where for the second type, the data is embeded (point wise) in $\mathbb R^2$.

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artificialS2Lemniscate(p [,pts=128,a=π/2,interval=[0,2π])

generate a Signal on the Sphere(2) $\mathbb S^2$ 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(2).

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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artificialS2Lemniscate(p,t; a=π/2)

generate a point from the signal on the Sphere(2) $\mathbb S^2$ by creating the Lemniscate of Bernoulli in the tangent space of p sampled at t and use èxpto obtain a point on the [Sphere](@ref)(2).

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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artificialS2WhirlPatch([pts=5])

create a whirl within the pts$\times$pts patch of Sphere(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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artificialS2WhirlImage([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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artificialS2Rot([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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artificialSPDImage([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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artificialSPDImage2([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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# Manifold Constructors

constructImageGraph(img,type)

construct the graph that can be used within the algorithms to model forward differences

Arguments

• img – the manifold-valued (MPoint-entries) image
• type – the type of graph, here firstOrderDifference and secondOrderDifference`.
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