Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Mathematical tools for analysis, modeling and simulation
of spatial networks on various length scales
Part II
Volker Schmidt
Ulm University, Institute of Stochastics
Blanton Museum of Art, UT Austin
May 21, 2015
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
1
Introduction
2
Spherical harmonics
3
Stochastic model for the inner structure of single cells
4
Model for networks of connected cells
5
Structural model validation
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Insert random graphs into cells (wired networks) and compute
the distribution of
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless
networks) and compute the distribution of
uncovered cell area (e.g., the area where the
signal-to-interference ratio is below a given threshold)
uncovered boundary length of cells (e.g., regions where
handover of mobile users might be problematic), etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Insert random graphs into cells (wired networks) and compute
the distribution of
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless
networks) and compute the distribution of
uncovered cell area (e.g., the area where the
signal-to-interference ratio is below a given threshold)
uncovered boundary length of cells (e.g., regions where
handover of mobile users might be problematic), etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Insert random graphs into cells (wired networks) and compute
the distribution of
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless
networks) and compute the distribution of
uncovered cell area (e.g., the area where the
signal-to-interference ratio is below a given threshold)
uncovered boundary length of cells (e.g., regions where
handover of mobile users might be problematic), etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Insert random graphs into cells (wired networks) and compute
the distribution of
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless
networks) and compute the distribution of
uncovered cell area (e.g., the area where the
signal-to-interference ratio is below a given threshold)
uncovered boundary length of cells (e.g., regions where
handover of mobile users might be problematic), etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Insert random graphs into cells (wired networks) and compute
the distribution of
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless
networks) and compute the distribution of
uncovered cell area (e.g., the area where the
signal-to-interference ratio is below a given threshold)
uncovered boundary length of cells (e.g., regions where
handover of mobile users might be problematic), etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Consider random tessellations with inner structure of cells
Insert random graphs into cells (wired networks) and compute
the distribution of
shortest-path lengths along the edge system
nmuber of hops to the root, etc.
Insert full-dimensional random sets into cells (wireless
networks) and compute the distribution of
uncovered cell area (e.g., the area where the
signal-to-interference ratio is below a given threshold)
uncovered boundary length of cells (e.g., regions where
handover of mobile users might be problematic), etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Develop a virtual network testing tool by
providing a formula library of analytical (simulation-based,
parametric) approximation formulas
which express the distributions of network performance
chararacteristics in terms of model parameters for
a wide spectrum of multiscale tessellation models, and
a wide spectrum of model parameters
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Develop a virtual network testing tool by
providing a formula library of analytical (simulation-based,
parametric) approximation formulas
which express the distributions of network performance
chararacteristics in terms of model parameters for
a wide spectrum of multiscale tessellation models, and
a wide spectrum of model parameters
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Develop a virtual network testing tool by
providing a formula library of analytical (simulation-based,
parametric) approximation formulas
which express the distributions of network performance
chararacteristics in terms of model parameters for
a wide spectrum of multiscale tessellation models, and
a wide spectrum of model parameters
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Develop a virtual network testing tool by
providing a formula library of analytical (simulation-based,
parametric) approximation formulas
which express the distributions of network performance
chararacteristics in terms of model parameters for
a wide spectrum of multiscale tessellation models, and
a wide spectrum of model parameters
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Multiscale Modeling and Simulation of Networks
Develop a virtual network testing tool by
providing a formula library of analytical (simulation-based,
parametric) approximation formulas
which express the distributions of network performance
chararacteristics in terms of model parameters for
a wide spectrum of multiscale tessellation models, and
a wide spectrum of model parameters
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Particulate materials
Many advanced materials are made up of networks of
connected particles — e.g., electrodes of lithium-ion batteries,
components of fuel cells, and photoactive layers of organic
solar cells.
The size and shape of the particles can play a significant role
in determining the functionality of the material.
In particular, regarding the transport of charge carriers
through the network of connected particles.
Often the particles are irregularly shaped and cannot be
modeled well by simple geometric objects such as spheres.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Particulate materials
Many advanced materials are made up of networks of
connected particles — e.g., electrodes of lithium-ion batteries,
components of fuel cells, and photoactive layers of organic
solar cells.
The size and shape of the particles can play a significant role
in determining the functionality of the material.
In particular, regarding the transport of charge carriers
through the network of connected particles.
Often the particles are irregularly shaped and cannot be
modeled well by simple geometric objects such as spheres.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Particulate materials
Many advanced materials are made up of networks of
connected particles — e.g., electrodes of lithium-ion batteries,
components of fuel cells, and photoactive layers of organic
solar cells.
The size and shape of the particles can play a significant role
in determining the functionality of the material.
In particular, regarding the transport of charge carriers
through the network of connected particles.
Often the particles are irregularly shaped and cannot be
modeled well by simple geometric objects such as spheres.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Particulate materials
Many advanced materials are made up of networks of
connected particles — e.g., electrodes of lithium-ion batteries,
components of fuel cells, and photoactive layers of organic
solar cells.
The size and shape of the particles can play a significant role
in determining the functionality of the material.
In particular, regarding the transport of charge carriers
through the network of connected particles.
Often the particles are irregularly shaped and cannot be
modeled well by simple geometric objects such as spheres.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Example: Lithium-ion Battery Anode
2D cross-section of a network of LiC6 particles in the anode of a lithium-ion battery
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
The locations (and sizes) of the particles are determined using
a random Laguerre tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Finally, some morphological smoothing is performed on the
simulated particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
The locations (and sizes) of the particles are determined using
a random Laguerre tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Finally, some morphological smoothing is performed on the
simulated particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
The locations (and sizes) of the particles are determined using
a random Laguerre tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Finally, some morphological smoothing is performed on the
simulated particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
The locations (and sizes) of the particles are determined using
a random Laguerre tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Finally, some morphological smoothing is performed on the
simulated particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
2D cut-out of segmented image
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
2D cut-out of segmented image
1) Tessellation
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
1) Tessellation + 2) connectivity
graph + 3) connected particles
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
3) Connected particles (removing
tessellation + connectivity graph)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Modeling networks of connected particles
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
3) Connected particles (removing
tessellation + connectivity graph)
Volker Schmidt
4) Particle system after smoothing
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Analogies to modelling of cellular communication networks
Laguerre cells =⇒ serving zones
Particles embedded in Laguerre cells
=⇒ (sub-) areas with good service quality
(e.g. signal-to-interference ratio above a given threshold)
Pore space =⇒ area with bad service quality
(e.g. signal-to-interference ratio below a given threshold)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Analogies to modelling of cellular communication networks
Laguerre cells =⇒ serving zones
Particles embedded in Laguerre cells
=⇒ (sub-) areas with good service quality
(e.g. signal-to-interference ratio above a given threshold)
Pore space =⇒ area with bad service quality
(e.g. signal-to-interference ratio below a given threshold)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Analogies to modelling of cellular communication networks
Laguerre cells =⇒ serving zones
Particles embedded in Laguerre cells
=⇒ (sub-) areas with good service quality
(e.g. signal-to-interference ratio above a given threshold)
Pore space =⇒ area with bad service quality
(e.g. signal-to-interference ratio below a given threshold)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Analogies to modelling of cellular communication networks
Laguerre cells =⇒ serving zones
Particles embedded in Laguerre cells
=⇒ (sub-) areas with good service quality
(e.g. signal-to-interference ratio above a given threshold)
Pore space =⇒ area with bad service quality
(e.g. signal-to-interference ratio below a given threshold)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
In work at Ulm, we have:
Used spherical harmonics to parametrically represent
irregularly shaped particles extracted from image data.
Developed a parametric stochastic model to simulate particles
based on information about spherical harmonics coefficients
extracted from image data.
Used this stochastic model to get ‘virtual’ networks of
connected particles that are statistically very similar to those
in the ’real’ data sets.
Checked if structural characteristics of virtual particle
networks behave similar as those of particle networks
extracted from real data (model validation).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
In work at Ulm, we have:
Used spherical harmonics to parametrically represent
irregularly shaped particles extracted from image data.
Developed a parametric stochastic model to simulate particles
based on information about spherical harmonics coefficients
extracted from image data.
Used this stochastic model to get ‘virtual’ networks of
connected particles that are statistically very similar to those
in the ’real’ data sets.
Checked if structural characteristics of virtual particle
networks behave similar as those of particle networks
extracted from real data (model validation).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
In work at Ulm, we have:
Used spherical harmonics to parametrically represent
irregularly shaped particles extracted from image data.
Developed a parametric stochastic model to simulate particles
based on information about spherical harmonics coefficients
extracted from image data.
Used this stochastic model to get ‘virtual’ networks of
connected particles that are statistically very similar to those
in the ’real’ data sets.
Checked if structural characteristics of virtual particle
networks behave similar as those of particle networks
extracted from real data (model validation).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
In work at Ulm, we have:
Used spherical harmonics to parametrically represent
irregularly shaped particles extracted from image data.
Developed a parametric stochastic model to simulate particles
based on information about spherical harmonics coefficients
extracted from image data.
Used this stochastic model to get ‘virtual’ networks of
connected particles that are statistically very similar to those
in the ’real’ data sets.
Checked if structural characteristics of virtual particle
networks behave similar as those of particle networks
extracted from real data (model validation).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
In work at Ulm, we have:
Used spherical harmonics to parametrically represent
irregularly shaped particles extracted from image data.
Developed a parametric stochastic model to simulate particles
based on information about spherical harmonics coefficients
extracted from image data.
Used this stochastic model to get ‘virtual’ networks of
connected particles that are statistically very similar to those
in the ’real’ data sets.
Checked if structural characteristics of virtual particle
networks behave similar as those of particle networks
extracted from real data (model validation).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particulate Materials vs. Cellular Networks
Visual comparison of real and virtual particle networks
Left: 2D cut-out of real particle network; right: 2D cut-out of simulated particle network
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
We assume the particles are star-shaped with respect to their
barycenters.
We will discuss the 3D case, but in 2D everything goes
analogously.
For each particle, we can define a radius function,
r : [0, π] × [0, 2π) → [0, ∞).
These functions define the distance from the barycenters of
the particles to their boundaries, for each direction on the unit
sphere, S2 .
We can then obtain a representation of these particles using
spherical harmonics.
These are an analogue of Fourier series for functions defined
on S2 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
We assume the particles are star-shaped with respect to their
barycenters.
We will discuss the 3D case, but in 2D everything goes
analogously.
For each particle, we can define a radius function,
r : [0, π] × [0, 2π) → [0, ∞).
These functions define the distance from the barycenters of
the particles to their boundaries, for each direction on the unit
sphere, S2 .
We can then obtain a representation of these particles using
spherical harmonics.
These are an analogue of Fourier series for functions defined
on S2 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
We assume the particles are star-shaped with respect to their
barycenters.
We will discuss the 3D case, but in 2D everything goes
analogously.
For each particle, we can define a radius function,
r : [0, π] × [0, 2π) → [0, ∞).
These functions define the distance from the barycenters of
the particles to their boundaries, for each direction on the unit
sphere, S2 .
We can then obtain a representation of these particles using
spherical harmonics.
These are an analogue of Fourier series for functions defined
on S2 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
We assume the particles are star-shaped with respect to their
barycenters.
We will discuss the 3D case, but in 2D everything goes
analogously.
For each particle, we can define a radius function,
r : [0, π] × [0, 2π) → [0, ∞).
These functions define the distance from the barycenters of
the particles to their boundaries, for each direction on the unit
sphere, S2 .
We can then obtain a representation of these particles using
spherical harmonics.
These are an analogue of Fourier series for functions defined
on S2 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
We assume the particles are star-shaped with respect to their
barycenters.
We will discuss the 3D case, but in 2D everything goes
analogously.
For each particle, we can define a radius function,
r : [0, π] × [0, 2π) → [0, ∞).
These functions define the distance from the barycenters of
the particles to their boundaries, for each direction on the unit
sphere, S2 .
We can then obtain a representation of these particles using
spherical harmonics.
These are an analogue of Fourier series for functions defined
on S2 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
We assume the particles are star-shaped with respect to their
barycenters.
We will discuss the 3D case, but in 2D everything goes
analogously.
For each particle, we can define a radius function,
r : [0, π] × [0, 2π) → [0, ∞).
These functions define the distance from the barycenters of
the particles to their boundaries, for each direction on the unit
sphere, S2 .
We can then obtain a representation of these particles using
spherical harmonics.
These are an analogue of Fourier series for functions defined
on S2 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
For each integer l ≥ 0 and for each m ∈ {−l, −(l − 1), . . . , l}, the
spherical harmonic function Yl,m : [0, π] × [0, 2π) → C is defined by
s
2l + 1 (l − m)!
Yl,m (θ, φ) =
Pl,m (cos(θ))eimφ ,
4π (l + m)!
where the functions (Pl,m (·))l≥0,m∈{−l,...,l} are the associated
Legendre functions.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Spherical Harmonics
For each integer l ≥ 0 and for each m ∈ {−l, −(l − 1), . . . , l}, the
spherical harmonic function Yl,m : [0, π] × [0, 2π) → C is defined by
s
2l + 1 (l − m)!
Yl,m (θ, φ) =
Pl,m (cos(θ))eimφ ,
4π (l + m)!
where the functions (Pl,m (·))l≥0,m∈{−l,...,l} are the associated
Legendre functions.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Representing Functions Using Spherical Harmonics
The spherical harmonic functions (Yl,m )l≥0,m∈{−l,...,l} form a
basis for square-integrable functions on S2 .
Thus, we can write any square-integrable real-valued function,
f , as a sum of the form
"
#
∞
l X
X
R
I
Yl,m
f (θ, φ) =
al,0 Yl,0 (θ, φ) + 2
+ Yl,m
,
m=1
l=0
where
R
Yl,m
= Re (al,m ) Re (Yl,m (θ, φ))
and
I
Yl,m
= Im (al,m ) Im (Yl,m (θ, φ)) .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Representing Functions Using Spherical Harmonics
The spherical harmonic functions (Yl,m )l≥0,m∈{−l,...,l} form a
basis for square-integrable functions on S2 .
Thus, we can write any square-integrable real-valued function,
f , as a sum of the form
"
#
∞
l X
X
R
I
Yl,m
f (θ, φ) =
al,0 Yl,0 (θ, φ) + 2
+ Yl,m
,
m=1
l=0
where
R
Yl,m
= Re (al,m ) Re (Yl,m (θ, φ))
and
I
Yl,m
= Im (al,m ) Im (Yl,m (θ, φ)) .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Representing Functions Using Spherical Harmonics
The spherical harmonic functions (Yl,m )l≥0,m∈{−l,...,l} form a
basis for square-integrable functions on S2 .
Thus, we can write any square-integrable real-valued function,
f , as a sum of the form
"
#
∞
l X
X
R
I
Yl,m
f (θ, φ) =
al,0 Yl,0 (θ, φ) + 2
+ Yl,m
,
m=1
l=0
where
R
Yl,m
= Re (al,m ) Re (Yl,m (θ, φ))
and
I
Yl,m
= Im (al,m ) Im (Yl,m (θ, φ)) .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Representing Functions Using Spherical Harmonics
In particular, we write the radius function as
"
#
∞
l X
X
R
I
r (θ, φ) =
al,0 Yl,0 (θ, φ) + 2
Yl,m
+ Yl,m
,
m=1
l=0
then approximate it by the finite sum
"
#
N
l X
X
R
I
r (θ, φ) ≈
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m ,
m=1
l=0
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Representing Functions Using Spherical Harmonics
In particular, we write the radius function as
"
#
∞
l X
X
R
I
r (θ, φ) =
al,0 Yl,0 (θ, φ) + 2
Yl,m
+ Yl,m
,
m=1
l=0
then approximate it by the finite sum
"
#
N
l X
X
R
I
r (θ, φ) ≈
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m ,
m=1
l=0
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Advantages of the Spherical Harmonics Representation
Representing particles in terms of spherical harmonics has a
number of significant advantages:
Instead of storing voxel data, we only need to store a vector of
coefficients for each particle.
We can control the ‘roughness’ of the particles (this increases
as the number of summands N increases).
A number of key particle shape and size characteristics —
including volume, surface area and Gaussian curvatures — can
be calculated using efficient numerical integration.
As we will see, this approach leads to a stochastic model for
star-shaped particles.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Advantages of the Spherical Harmonics Representation
Representing particles in terms of spherical harmonics has a
number of significant advantages:
Instead of storing voxel data, we only need to store a vector of
coefficients for each particle.
We can control the ‘roughness’ of the particles (this increases
as the number of summands N increases).
A number of key particle shape and size characteristics —
including volume, surface area and Gaussian curvatures — can
be calculated using efficient numerical integration.
As we will see, this approach leads to a stochastic model for
star-shaped particles.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Advantages of the Spherical Harmonics Representation
Representing particles in terms of spherical harmonics has a
number of significant advantages:
Instead of storing voxel data, we only need to store a vector of
coefficients for each particle.
We can control the ‘roughness’ of the particles (this increases
as the number of summands N increases).
A number of key particle shape and size characteristics —
including volume, surface area and Gaussian curvatures — can
be calculated using efficient numerical integration.
As we will see, this approach leads to a stochastic model for
star-shaped particles.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Advantages of the Spherical Harmonics Representation
Representing particles in terms of spherical harmonics has a
number of significant advantages:
Instead of storing voxel data, we only need to store a vector of
coefficients for each particle.
We can control the ‘roughness’ of the particles (this increases
as the number of summands N increases).
A number of key particle shape and size characteristics —
including volume, surface area and Gaussian curvatures — can
be calculated using efficient numerical integration.
As we will see, this approach leads to a stochastic model for
star-shaped particles.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Advantages of the Spherical Harmonics Representation
Representing particles in terms of spherical harmonics has a
number of significant advantages:
Instead of storing voxel data, we only need to store a vector of
coefficients for each particle.
We can control the ‘roughness’ of the particles (this increases
as the number of summands N increases).
A number of key particle shape and size characteristics —
including volume, surface area and Gaussian curvatures — can
be calculated using efficient numerical integration.
As we will see, this approach leads to a stochastic model for
star-shaped particles.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Spherical Harmonics Coefficients
We can determine the spherical harmonics coefficients,
(al,m )l≥0,m∈{−l,...,l} , using the relationships
Z π Z 2π
al,m =
r (θ, φ)Yl,m (θ, φ) sin(θ) dφ dθ
0
0
and
Yl,m (θ, φ) = (−1)m Yl,−m .
We evaluate this integral using quadrature with the points
i + 1/2
k
(θi , φi ) : π
, 2π
: 0 ≤ i, j, ≤ K ,
K
K
estimating r (θ, φ) at each point in the grid.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Spherical Harmonics Coefficients
We can determine the spherical harmonics coefficients,
(al,m )l≥0,m∈{−l,...,l} , using the relationships
Z π Z 2π
al,m =
r (θ, φ)Yl,m (θ, φ) sin(θ) dφ dθ
0
0
and
Yl,m (θ, φ) = (−1)m Yl,−m .
We evaluate this integral using quadrature with the points
i + 1/2
k
(θi , φi ) : π
, 2π
: 0 ≤ i, j, ≤ K ,
K
K
estimating r (θ, φ) at each point in the grid.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Example: Single Particle
Clockwise from top left: original particle; spherical harmonics approximations with N = 4, N = 10, N = 20
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Example: Particle System
Particles in voxelized form
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Example: Particle System
Spherical harmonics representation
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Stochastic Modeling of Particles
The idea of using spherical harmonics to represent particles
leads directly to a parametric stochastic model.
This model is in the form of a Gaussian random field on the
sphere.
It can easily be fitted to statistical information about spherical
harmonics coefficients extracted from empirical data.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Stochastic Modeling of Particles
The idea of using spherical harmonics to represent particles
leads directly to a parametric stochastic model.
This model is in the form of a Gaussian random field on the
sphere.
It can easily be fitted to statistical information about spherical
harmonics coefficients extracted from empirical data.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Stochastic Modeling of Particles
The idea of using spherical harmonics to represent particles
leads directly to a parametric stochastic model.
This model is in the form of a Gaussian random field on the
sphere.
It can easily be fitted to statistical information about spherical
harmonics coefficients extracted from empirical data.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
A real-valued isotropic Gaussian random field
ψ = {ψ(θ, φ), (θ, φ) ∈ S2 } on the unit sphere S2 admits a
spherical harmonics expansion of the form
"
#
∞
l X
X
R
I
ψ(θ, φ) =
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m ,
m=1
l=0
where
R
Yl,m
= Re (al,m ) Re (Yl,m (θ, φ))
and
I
Yl,m
= Im (al,m ) Im (Yl,m (θ, φ)) .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
A real-valued isotropic Gaussian random field
ψ = {ψ(θ, φ), (θ, φ) ∈ S2 } on the unit sphere S2 admits a
spherical harmonics expansion of the form
"
#
∞
l X
X
R
I
ψ(θ, φ) =
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m ,
m=1
l=0
where
R
Yl,m
= Re (al,m ) Re (Yl,m (θ, φ))
and
I
Yl,m
= Im (al,m ) Im (Yl,m (θ, φ)) .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
The coefficients a0,0 , . . . , al,0 , and both the real and imaginary
parts of the (al,m )l≥1,m∈{1,...,l} are independent and normal
distibuted random variables with
a0,0 ∼ N (µ, A0 ),
al,0 ∼ N (0, Al ) for l > 0,
Re (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
Im (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
where µ is the mean radius of the particles,
and the variance parameters (Al )l≥0 are called the angular
power spectrum of ψ.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
The coefficients a0,0 , . . . , al,0 , and both the real and imaginary
parts of the (al,m )l≥1,m∈{1,...,l} are independent and normal
distibuted random variables with
a0,0 ∼ N (µ, A0 ),
al,0 ∼ N (0, Al ) for l > 0,
Re (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
Im (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
where µ is the mean radius of the particles,
and the variance parameters (Al )l≥0 are called the angular
power spectrum of ψ.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
The coefficients a0,0 , . . . , al,0 , and both the real and imaginary
parts of the (al,m )l≥1,m∈{1,...,l} are independent and normal
distibuted random variables with
a0,0 ∼ N (µ, A0 ),
al,0 ∼ N (0, Al ) for l > 0,
Re (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
Im (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
where µ is the mean radius of the particles,
and the variance parameters (Al )l≥0 are called the angular
power spectrum of ψ.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
The coefficients a0,0 , . . . , al,0 , and both the real and imaginary
parts of the (al,m )l≥1,m∈{1,...,l} are independent and normal
distibuted random variables with
a0,0 ∼ N (µ, A0 ),
al,0 ∼ N (0, Al ) for l > 0,
Re (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
Im (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
where µ is the mean radius of the particles,
and the variance parameters (Al )l≥0 are called the angular
power spectrum of ψ.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
The coefficients a0,0 , . . . , al,0 , and both the real and imaginary
parts of the (al,m )l≥1,m∈{1,...,l} are independent and normal
distibuted random variables with
a0,0 ∼ N (µ, A0 ),
al,0 ∼ N (0, Al ) for l > 0,
Re (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
Im (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
where µ is the mean radius of the particles,
and the variance parameters (Al )l≥0 are called the angular
power spectrum of ψ.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
The coefficients a0,0 , . . . , al,0 , and both the real and imaginary
parts of the (al,m )l≥1,m∈{1,...,l} are independent and normal
distibuted random variables with
a0,0 ∼ N (µ, A0 ),
al,0 ∼ N (0, Al ) for l > 0,
Re (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
Im (al,m ) ∼ N (0, Al /2) for l > 0, m > 0,
where µ is the mean radius of the particles,
and the variance parameters (Al )l≥0 are called the angular
power spectrum of ψ.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
Approximating ψ by a finite sum, we have
"
#
N
l X
X
R
I
ψ(θ, φ) ≈
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m .
m=1
l=0
In particular, writing
a = (a0,0 , a1,0 , Re (a1,1 ) , Im (a1,1 ) , . . . , Im (aN,N ))|
y(θ, φ) = Y0,0 (θ, φ), Y1,0 (θ, φ), 2 · Re (Y1,1 (θ, φ)) ,
|
2 · Im (Y1,1 (θ, φ)) , . . . , 2 · Im (YN,N (θ, φ))
we have the vector representation
ψ(θ, φ) ≈ a| y(θ, φ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
Approximating ψ by a finite sum, we have
"
#
N
l X
X
R
I
ψ(θ, φ) ≈
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m .
m=1
l=0
In particular, writing
a = (a0,0 , a1,0 , Re (a1,1 ) , Im (a1,1 ) , . . . , Im (aN,N ))|
y(θ, φ) = Y0,0 (θ, φ), Y1,0 (θ, φ), 2 · Re (Y1,1 (θ, φ)) ,
|
2 · Im (Y1,1 (θ, φ)) , . . . , 2 · Im (YN,N (θ, φ))
we have the vector representation
ψ(θ, φ) ≈ a| y(θ, φ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
Approximating ψ by a finite sum, we have
"
#
N
l X
X
R
I
ψ(θ, φ) ≈
al,0 Yl,0 (θ, φ) + 2
Yl,m + Yl,m .
m=1
l=0
In particular, writing
a = (a0,0 , a1,0 , Re (a1,1 ) , Im (a1,1 ) , . . . , Im (aN,N ))|
y(θ, φ) = Y0,0 (θ, φ), Y1,0 (θ, φ), 2 · Re (Y1,1 (θ, φ)) ,
|
2 · Im (Y1,1 (θ, φ)) , . . . , 2 · Im (YN,N (θ, φ))
we have the vector representation
ψ(θ, φ) ≈ a| y(θ, φ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
Thus, we can approximately simulate a Gaussian random field
on the sphere
simply by simulating a multivariate Gaussian vector, a, of
length (N + 1)2 , where
a ∼ N (µ, Σ),
with
µ = (µ, 0, . . . , 0)|
Σ = diag A0 , A1 , A1 /2, A1 /2, A2 , A2 /2, A2 /2, A2 /2, A2 /2, . . . ,
AN , . . . , AN /2,
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
Thus, we can approximately simulate a Gaussian random field
on the sphere
simply by simulating a multivariate Gaussian vector, a, of
length (N + 1)2 , where
a ∼ N (µ, Σ),
with
µ = (µ, 0, . . . , 0)|
Σ = diag A0 , A1 , A1 /2, A1 /2, A2 , A2 /2, A2 /2, A2 /2, A2 /2, . . . ,
AN , . . . , AN /2,
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Gaussian Random Fields on the Sphere
Thus, we can approximately simulate a Gaussian random field
on the sphere
simply by simulating a multivariate Gaussian vector, a, of
length (N + 1)2 , where
a ∼ N (µ, Σ),
with
µ = (µ, 0, . . . , 0)|
Σ = diag A0 , A1 , A1 /2, A1 /2, A2 , A2 /2, A2 /2, A2 /2, A2 /2, . . . ,
AN , . . . , AN /2,
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Angular Power Spectrum
In order to fit the model, we need to determine
the angular power spectrum (Al )N
l=0
and the mean radius µ.
This is done by estimating spherical harmonics coefficients
from each particle in the data, to get
(1)
(M)
b
a0,0 , . . . , b
a0,0 ,
(1)
(M)
b
a1,0 , . . . , b
a1,0 ,
etc.
where M is the number of particles.
We then use the sample variances of these coefficients to
estimate the (Al )N
l=0 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Angular Power Spectrum
In order to fit the model, we need to determine
the angular power spectrum (Al )N
l=0
and the mean radius µ.
This is done by estimating spherical harmonics coefficients
from each particle in the data, to get
(1)
(M)
b
a0,0 , . . . , b
a0,0 ,
(1)
(M)
b
a1,0 , . . . , b
a1,0 ,
etc.
where M is the number of particles.
We then use the sample variances of these coefficients to
estimate the (Al )N
l=0 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Angular Power Spectrum
In order to fit the model, we need to determine
the angular power spectrum (Al )N
l=0
and the mean radius µ.
This is done by estimating spherical harmonics coefficients
from each particle in the data, to get
(1)
(M)
b
a0,0 , . . . , b
a0,0 ,
(1)
(M)
b
a1,0 , . . . , b
a1,0 ,
etc.
where M is the number of particles.
We then use the sample variances of these coefficients to
estimate the (Al )N
l=0 .
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Angular power spectrum, Al
Estimating the Angular Power Spectrum
30
20
10
0
1
2
3
4
5
l
6
7
8
9
10
Estimated angular power spectrum
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Mean Radius
(1)
(M)
In principle, we could use the sample mean of b
a0,0 , . . . , b
a0,0
to estimate µ.
However, we can control the expected volume EV of the
particles more precisely by solving the following equation for µ:
Z Z
N X
2
EV = µ
Var(aj )
Y1 (θ, φ)Yj (θ, φ) sin(θ) dθdφ ,
j=2
|
{z
=S
}
where the value of EV is chosen such that the volume fraction
of the solid phase is met (here: 73%),
and, for a given angular power spectrum, S is computed
numerically.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Mean Radius
(1)
(M)
In principle, we could use the sample mean of b
a0,0 , . . . , b
a0,0
to estimate µ.
However, we can control the expected volume EV of the
particles more precisely by solving the following equation for µ:
Z Z
N X
2
EV = µ
Var(aj )
Y1 (θ, φ)Yj (θ, φ) sin(θ) dθdφ ,
j=2
|
{z
=S
}
where the value of EV is chosen such that the volume fraction
of the solid phase is met (here: 73%),
and, for a given angular power spectrum, S is computed
numerically.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Mean Radius
(1)
(M)
In principle, we could use the sample mean of b
a0,0 , . . . , b
a0,0
to estimate µ.
However, we can control the expected volume EV of the
particles more precisely by solving the following equation for µ:
Z Z
N X
2
EV = µ
Var(aj )
Y1 (θ, φ)Yj (θ, φ) sin(θ) dθdφ ,
j=2
|
{z
=S
}
where the value of EV is chosen such that the volume fraction
of the solid phase is met (here: 73%),
and, for a given angular power spectrum, S is computed
numerically.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Estimating the Mean Radius
(1)
(M)
In principle, we could use the sample mean of b
a0,0 , . . . , b
a0,0
to estimate µ.
However, we can control the expected volume EV of the
particles more precisely by solving the following equation for µ:
Z Z
N X
2
EV = µ
Var(aj )
Y1 (θ, φ)Yj (θ, φ) sin(θ) dθdφ ,
j=2
|
{z
=S
}
where the value of EV is chosen such that the volume fraction
of the solid phase is met (here: 73%),
and, for a given angular power spectrum, S is computed
numerically.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Modeling Systems of Connected Particles
We model the system of connected particles in 4 steps:
The locations (and sizes) of the particles are determined using
a random tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Morphological smoothing is performed on the simulated
particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Modeling Systems of Connected Particles
We model the system of connected particles in 4 steps:
The locations (and sizes) of the particles are determined using
a random tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Morphological smoothing is performed on the simulated
particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Modeling Systems of Connected Particles
We model the system of connected particles in 4 steps:
The locations (and sizes) of the particles are determined using
a random tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Morphological smoothing is performed on the simulated
particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Modeling Systems of Connected Particles
We model the system of connected particles in 4 steps:
The locations (and sizes) of the particles are determined using
a random tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Morphological smoothing is performed on the simulated
particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Modeling Systems of Connected Particles
We model the system of connected particles in 4 steps:
The locations (and sizes) of the particles are determined using
a random tessellation.
A random graph is generated, conditional on the tessellation,
that describes the connectivity between particles.
The particles themselves are modeled as Gaussian random
fields on the sphere, whose parameters depend on the
tessellation and connection graph.
Morphological smoothing is performed on the simulated
particle system.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Overview
2D cut-out of segmented image
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Overview
2D cut-out of segmented image
1) Tessellation
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Overview
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Overview
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
1) Tessellation + 2) connectivity
graph + 3) connected particles
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Overview
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
3) Connected particles (removing
tessellation + connectivity graph)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Overview
2D cut-out of segmented image
1) Tessellation + 2) connectivity
graph
3) Connected particles (removing
tessellation + connectivity graph)
Volker Schmidt
4) Particle system after smoothing
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Locations
Step 1. Generating a random Laguerre tessellation
A random marked point process {(Si , Ri ), i ≥ 0} is generated, by a
random sequential absorption (RSA) process
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Locations
Step 1. Generating a random Laguerre tessellation
A Laguerre tessellation is generated from this, with cells
Ci = {x ∈ R3 : kx − Si k2 − Ri2 ≤ kx − Sj k2 − Rj2 ∀j 6= i}
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Locations
Step 1. Generating a random Laguerre tessellation
The tessellation is kept and the point process is discarded.
A particle will be placed in each tessellation cell.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
General idea: A random graph, with the seed points Si of the Laguerre tessellation as vertices, is simulated. The
edges indicate particles to be connected.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Given the Laguerre tessellation, with seed points Si and radii Ri ,
the random connectivity graph G = (V , E ) is constructed as
follows:
Consider the vertex set V = {Si } of all seed points Si .
Put edges eij = Si Sj between the pairs (Si , Sj ) of seed points
of all neighboring cells.
Weight the edges eij by 1/Aij , where Aij is the surface area of
the face separating the two corresponding cells.
Calculate the minimum spanning tree (V , Ems ) of this
weighted graph (which is fully connected by definition).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Given the Laguerre tessellation, with seed points Si and radii Ri ,
the random connectivity graph G = (V , E ) is constructed as
follows:
Consider the vertex set V = {Si } of all seed points Si .
Put edges eij = Si Sj between the pairs (Si , Sj ) of seed points
of all neighboring cells.
Weight the edges eij by 1/Aij , where Aij is the surface area of
the face separating the two corresponding cells.
Calculate the minimum spanning tree (V , Ems ) of this
weighted graph (which is fully connected by definition).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Given the Laguerre tessellation, with seed points Si and radii Ri ,
the random connectivity graph G = (V , E ) is constructed as
follows:
Consider the vertex set V = {Si } of all seed points Si .
Put edges eij = Si Sj between the pairs (Si , Sj ) of seed points
of all neighboring cells.
Weight the edges eij by 1/Aij , where Aij is the surface area of
the face separating the two corresponding cells.
Calculate the minimum spanning tree (V , Ems ) of this
weighted graph (which is fully connected by definition).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Given the Laguerre tessellation, with seed points Si and radii Ri ,
the random connectivity graph G = (V , E ) is constructed as
follows:
Consider the vertex set V = {Si } of all seed points Si .
Put edges eij = Si Sj between the pairs (Si , Sj ) of seed points
of all neighboring cells.
Weight the edges eij by 1/Aij , where Aij is the surface area of
the face separating the two corresponding cells.
Calculate the minimum spanning tree (V , Ems ) of this
weighted graph (which is fully connected by definition).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Given the Laguerre tessellation, with seed points Si and radii Ri ,
the random connectivity graph G = (V , E ) is constructed as
follows:
Consider the vertex set V = {Si } of all seed points Si .
Put edges eij = Si Sj between the pairs (Si , Sj ) of seed points
of all neighboring cells.
Weight the edges eij by 1/Aij , where Aij is the surface area of
the face separating the two corresponding cells.
Calculate the minimum spanning tree (V , Ems ) of this
weighted graph (which is fully connected by definition).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Add an additional set Eadd of edges to Ems , so that the
connectivity structure resembles that of the material,
where the seed points Si and Sj of two neighboring cells are
connected independently from each other, and
with the (surface-area-dependent) connection probability
`(A) = P(Si and Sj are connected | Aij = A) .
The random connectivity graph is then given by
G = (V , Ems ∪ Eadd ).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Add an additional set Eadd of edges to Ems , so that the
connectivity structure resembles that of the material,
where the seed points Si and Sj of two neighboring cells are
connected independently from each other, and
with the (surface-area-dependent) connection probability
`(A) = P(Si and Sj are connected | Aij = A) .
The random connectivity graph is then given by
G = (V , Ems ∪ Eadd ).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Add an additional set Eadd of edges to Ems , so that the
connectivity structure resembles that of the material,
where the seed points Si and Sj of two neighboring cells are
connected independently from each other, and
with the (surface-area-dependent) connection probability
`(A) = P(Si and Sj are connected | Aij = A) .
The random connectivity graph is then given by
G = (V , Ems ∪ Eadd ).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Connectivity of Particles
Step 2. Generating a random connectivity graph
Add an additional set Eadd of edges to Ems , so that the
connectivity structure resembles that of the material,
where the seed points Si and Sj of two neighboring cells are
connected independently from each other, and
with the (surface-area-dependent) connection probability
`(A) = P(Si and Sj are connected | Aij = A) .
The random connectivity graph is then given by
G = (V , Ems ∪ Eadd ).
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
Particles are simulated in each cell, with sizes conditional on the size of their corresponding cells.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
The spherical harmonics coefficients of the particles are generated so that they connect according to the graph.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
Choose the centroids of Laguerre cells as locations of particles.
Force the particles to connect according to the connectivity
graph,
by forcing them to make contact with the corresponding faces
of their cells in the Laquerre tessellation.
Generate the spherical harmonics coefficients of the particles
e .
by simulating a (conditional) Gaussian vector a ∼ N (e
µ, Σ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
Choose the centroids of Laguerre cells as locations of particles.
Force the particles to connect according to the connectivity
graph,
by forcing them to make contact with the corresponding faces
of their cells in the Laquerre tessellation.
Generate the spherical harmonics coefficients of the particles
e .
by simulating a (conditional) Gaussian vector a ∼ N (e
µ, Σ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
Choose the centroids of Laguerre cells as locations of particles.
Force the particles to connect according to the connectivity
graph,
by forcing them to make contact with the corresponding faces
of their cells in the Laquerre tessellation.
Generate the spherical harmonics coefficients of the particles
e .
by simulating a (conditional) Gaussian vector a ∼ N (e
µ, Σ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
Choose the centroids of Laguerre cells as locations of particles.
Force the particles to connect according to the connectivity
graph,
by forcing them to make contact with the corresponding faces
of their cells in the Laquerre tessellation.
Generate the spherical harmonics coefficients of the particles
e .
by simulating a (conditional) Gaussian vector a ∼ N (e
µ, Σ)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
The (blue) points of a tessellation face (corresponding to an edge) that a particle must touch
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
This is done by imposing the linear

 
y(θ1 , φ1 )| a
r1

  ..
..

= .
.
y(θK , φK
)| a,
constraints


,
i.e.,
Ya = r
rK
where K is the number of points of the faces, which the
particle needs to touch.
Thus, instead of drawing a from the (unconditional) normal
distribution N (µ, Σ),
e conditional on these constraints, using
we draw a ∼ N (e
µ, Σ)
a singular value decomposition of Y.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
This is done by imposing the linear

 
y(θ1 , φ1 )| a
r1

  ..
..

= .
.
y(θK , φK
)| a,
constraints


,
i.e.,
Ya = r
rK
where K is the number of points of the faces, which the
particle needs to touch.
Thus, instead of drawing a from the (unconditional) normal
distribution N (µ, Σ),
e conditional on these constraints, using
we draw a ∼ N (e
µ, Σ)
a singular value decomposition of Y.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Particle Sizes and Shapes
Step 3. Generating (connected) particles
This is done by imposing the linear

 
y(θ1 , φ1 )| a
r1

  ..
..

= .
.
y(θK , φK
)| a,
constraints


,
i.e.,
Ya = r
rK
where K is the number of points of the faces, which the
particle needs to touch.
Thus, instead of drawing a from the (unconditional) normal
distribution N (µ, Σ),
e conditional on these constraints, using
we draw a ∼ N (e
µ, Σ)
a singular value decomposition of Y.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Binder
Step 4. Morphological smoothing of particles
The particles in the material are actually encased in a thin
layer of binder.
However, the volume fraction of this binder is too small to be
modeled directly.
Instead, we carry out a morphological smoothing, which
mimics this effect.
In particular, it smooths the sharp edges around particle
connections, where
a so-called morphological closing is performed,
using a ball with radius 1 as the structuring element,
i.e., the union set of particles is first dilated, then eroded.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary
Algorithm
Generate a random Laguerre tessellation
Generate a random connectivity graph
For each Laguerre cell,
choose its centroid as location of a particle,
impose linear constraints to force the particles to connect
according to the connectivity graph,
draw the vector of spherical harmonics coefficients from a
(conditional) normal distribution,
Smooth the obtained structure using morphological closing.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary
Algorithm
Generate a random Laguerre tessellation
Generate a random connectivity graph
For each Laguerre cell,
choose its centroid as location of a particle,
impose linear constraints to force the particles to connect
according to the connectivity graph,
draw the vector of spherical harmonics coefficients from a
(conditional) normal distribution,
Smooth the obtained structure using morphological closing.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary
Algorithm
Generate a random Laguerre tessellation
Generate a random connectivity graph
For each Laguerre cell,
choose its centroid as location of a particle,
impose linear constraints to force the particles to connect
according to the connectivity graph,
draw the vector of spherical harmonics coefficients from a
(conditional) normal distribution,
Smooth the obtained structure using morphological closing.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary
Algorithm
Generate a random Laguerre tessellation
Generate a random connectivity graph
For each Laguerre cell,
choose its centroid as location of a particle,
impose linear constraints to force the particles to connect
according to the connectivity graph,
draw the vector of spherical harmonics coefficients from a
(conditional) normal distribution,
Smooth the obtained structure using morphological closing.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Visual Comparison of Microstructures
Left: 2D cut-out of the experimental structure; right: 2D cut-out of the simulated structure
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Comparison of Basic Structural Characteristics
Comparison of basic mean values, computed from experimental
and simulated data, shows a very good agreement:
Volume Fraction
of Solid Phase
Specific Surface Area
Mean Tortuosity
of Pore Phase
Experimental
0.7331366
Simulated
0.7373997
Relative Error
0.6 %
308858.3
1.568
313988.4
1.559
1.7 %
0.574 %
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Solid Phase
Spherical contact distances
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Solid Phase
Spherical contact distribution function:
Red: experimental; blue: simulated
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Solid Phase
Chord lengths
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Solid Phase
Chord length distribution function
Red: experimental; blue: simulated
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Solid Phase
Connectivity graph (computed from experimental data)
Points indicate the centroids of particles and edges indicate connections between particles
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Solid Phase
Distribution function of coordination numbers
(of the connectivity graph)
Red: experimental; blue: simulated
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Probability densities of local tortuosities
Probability density
5
4
3
2
1
0
1.2
1.4
1.6
1.8
2.0
Local tortuosity (in voxels)
Mean values are indicated by the dark lines and the ranges between the 0.05 and 0.95 quantiles are shown in lighter
colors. Red: experimental; blue: simulated
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Pore size distribution
Red: experimental; blue: simulated
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Shortest paths from phase boundary to separator (indicate the
efficiency of ion transport)
Paths are calculated for all surface voxels
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Probability density of shortest-paths lengths
Red: experimental; blue: simulated
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Constrictivity of pore phase
β=
rmin
rmax
2
Definition rmax
Definition rmin
rmax is the maximal radius r such that 50% of the pore
phase can be covered by spheres of radius r which are
completely contained in the pore phase.
rmin is the maximal radius r such that 50% of the pore
phase can be filled by an intrusion of spheres with
radius r from the bottom.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Constrictivity of pore phase
β=
rmin
rmax
2
Definition rmax
Definition rmin
rmax is the maximal radius r such that 50% of the pore
phase can be covered by spheres of radius r which are
completely contained in the pore phase.
rmin is the maximal radius r such that 50% of the pore
phase can be filled by an intrusion of spheres with
radius r from the bottom.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Structural Characteristics of Pore Phase
Constrictivity of pore phase (for 50 samples)
Red: experimental; blue: simulated. Relative error of mean values: 8.7%
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Summary & Outlook
We have...
...developed a stochastic 3D microstructure model
...fitted our model to experimental image data
...validated our model comparing structural characteristics
...validated our model using electrochemical simulations
Next steps:
develop 3D models for other battery electrode materials
capture structural degradation
perform virtual materials testing, i.e., generate and analyze a
wide spectrum of virtual microstructures, varying volume
fraction, specific surface area, tortuosity, constrictivity, etc.
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Literature
J. Feinauer, A. Spettl, I. Manke, S. Strege, A. Kwade, A. Pott
and V. Schmidt (2015). Structural characterization of particle
systems using spherical harmonics.Materials Characterization
(in print).
J. Feinauer, T. Brereton, A. Spettl, M. Weber, I. Manke, and
V. Schmidt (2015). Stochastic 3D modeling of the
microstructure of lithium-ion battery annodes via Gaussian
random fields on the sphere. Computational Materials Science
(submitted).
S. Hein, J. Feinauer, D. Westhoff, I. Manke, V. Schmidt and
A. Latz (2015). Stochastic microstructure modelling and
electrochemical simulation of lithium-ion cell anodes in 3D.
Working paper (under preparation)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Literature
J. Feinauer, A. Spettl, I. Manke, S. Strege, A. Kwade, A. Pott
and V. Schmidt (2015). Structural characterization of particle
systems using spherical harmonics.Materials Characterization
(in print).
J. Feinauer, T. Brereton, A. Spettl, M. Weber, I. Manke, and
V. Schmidt (2015). Stochastic 3D modeling of the
microstructure of lithium-ion battery annodes via Gaussian
random fields on the sphere. Computational Materials Science
(submitted).
S. Hein, J. Feinauer, D. Westhoff, I. Manke, V. Schmidt and
A. Latz (2015). Stochastic microstructure modelling and
electrochemical simulation of lithium-ion cell anodes in 3D.
Working paper (under preparation)
Volker Schmidt
Mathematical tools for spatial networks on various length scales
Introduction
Spherical harmonics
Stochastic model for the inner structure of single cells
Model for networks of connected cells
Structural model validation
Literature
J. Feinauer, A. Spettl, I. Manke, S. Strege, A. Kwade, A. Pott
and V. Schmidt (2015). Structural characterization of particle
systems using spherical harmonics.Materials Characterization
(in print).
J. Feinauer, T. Brereton, A. Spettl, M. Weber, I. Manke, and
V. Schmidt (2015). Stochastic 3D modeling of the
microstructure of lithium-ion battery annodes via Gaussian
random fields on the sphere. Computational Materials Science
(submitted).
S. Hein, J. Feinauer, D. Westhoff, I. Manke, V. Schmidt and
A. Latz (2015). Stochastic microstructure modelling and
electrochemical simulation of lithium-ion cell anodes in 3D.
Working paper (under preparation)
Volker Schmidt
Mathematical tools for spatial networks on various length scales