Gaussian processes, sampling and marginalization
In this tutorial we will experiment with sampling and marginalization of Gaussian processes. Furthermore, we will learn how to compare the likelihood of instances of our model.
Related resources
The following resources from our online course may provide some helpful context for this tutorial:
Preparation
As in the previous tutorials, we start by importing some commonly used objects and initializing the system.
Discrete and Continuous Gaussian processes
We have seen in the last tutorial that a Point Distribution Model (PDM) is represented in Scalismo as a (discrete) Gaussian process over deformation fields, defined on a reference mesh.
To continue our exploration of Gaussian processes, we therefore start by loading (and visualizing) an existing PDM and retrieve its underlying Gaussian process
We can retrieve random samples from the Gaussian process by calling sample
on the gp object:
Note that the sampled vector field is discrete; I.e. is defined over a discrete set of points. This is due to the fact that our Gaussian Process is stored in a file and was therefore discretized over the points of the reference mesh.
As seen in the previous tutorial, we could interpolate the
sample sampleDf to obtain a continuous version of the deformation field.
A more convenient approach is, however, to interpolate the
Gaussian process directly:
When we sample now from the continuous GP, we obtain a vector-valued function, which is defined on the entire 3D Space:
Attention: While the interpolated Gaussian process is now defined on the entire 3D Space, the interpolation really only makes sense close to the mesh points.
From continuous to discrete: marginalization and discretization
In practice, we will never work with a continuous Gaussian process directly. We are always interested in the distribution on a finite set of points. The real advantage of having a continuous Gaussian process is, that we can get samples at any finite set of points and thereby choosing the discretization according to the needs of our application.
To illustrate this, we could, for example obtain a sample, which is defined on all the points of the original reference mesh.
The marginalization property of a Gaussian process makes it possible not only
to obtain samples at an arbitrary set of points, but also the
distribution at these points. We can
obtain this distribution, by calling the method marginal
on the Gaussian process instance:
The result of marginalization is again a discrete Gaussian process. As we have specified individual points, on which
to evaluate the Gaussian process, but not how these points are connected, the resulting
discrete Gaussian process is defined over an UnstructuredPointsDomain.
To obtain a discrete Gaussian Process with a richer structure, we can use the discretize method,
which takes as an argument a domain and result in a discrete Gaussian Process defined on that domain.
To obtain the Gaussian Process that we started with again, we can call the discretize method
with the reference mesh:
This mechanism of interpolation followed by discretization gives us the ability to freely change the resolution of the domain on which the Gaussian process is defined.
Changing the reference of a point distribution model
Given that a point distribution model is really just a wrapper around a Gaussian process, it
is not surprising that we can apply the same ideas to these models. In particular, we often
would like to change the domain (I.e., the reference) of a point distribution model.
This can be done using the method newReference of the point distribution model. Under the hood, the method newReference
interpolates the gaussian process and discretizes it with the new reference.
In the following example we use this method to obtain a model which is defined on a low-resolution mesh:
Other common applications of this method include restricting the model to only a part of the domain.
Probability of shapes and deformations:
It is often interesting to assess how probable a model instance is.
This can be done in Scalismo by means of the method pdf
(which stands for probability density function) of the class GaussianProcess
and StatisticalMeshModel respectively.
The value of the pdf is often not interesting as such. But it allows us to compare the likelihood of different instances, by comparing their density value. For numerical reasons, we usually work with the log probability: