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Analysis of Optical Imaging Data

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John Porrill, James V Stone, Jason Berwick, John Mayhew and Peter Coffey.
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Summary

This chapter will present techniques
for the recovery of component spatial and temporal modes
from spatio-temporal data sets,
in particular from medical imaging data
such as that obtained by functional magnetic resonance imaging
\index{functional magnetic resonance imaging} (fMRI)
and optical imaging
\index{optical imaging} (OI) of brain activity.
These techniques were developed in order to help address some current issues
involving the nature of the haemodynamic response to neural activity
(for more details of this research and further references see \cite{mayhew}).
Current analysis techniques can be divided, roughly speaking,
into two classes: data-driven and model-driven.
An example of a data driven technique is principal component analysis (PCA).
This requires no prior knowledge about the form of the components
but physical interpretation of the decomposition is difficult
and there is no obvious way to test for statistical significance.
In contrast model-driven techniques such as general linear model analysis
require that rather constrained models of the expected behaviour
be prescribed in advance
but have the very significant advantage that they allow the use
of standard statistical tests.

In our application it is important that the components we recover correspond
to underlying physical processes.
Since current biophysical knowledge of these processes is limited,
the decomposition must be based, as far as is possible, on generic principles
rather than on highly constrained models.
We therefore define an intermediate class of \emph{weak temporal models},
designed to utilise incomplete prior information about temporal responses
(available, for example, from experimental protocols)
without having to specify the expected behaviour in detail.

We will show how to define a
\emph{weak causal model} for the response to a single stimulus and
a \emph{weak periodic model} for the response
to a periodic (e.g. box-car) stimulus,
and discuss how the statistical significance of temporal components
recovered using such models might be assessed.

The weak models we define here prescribe an optimal temporal response
but leave the problem of recovering
the associated spatial activity map under-constrained.
We show that this ill-posed problem can be regularised effectively
using the entropy measure underlying spatial independent component analysis
\index{independent component analysis}.
We will argue that this regulariser can be regarded as implementing a
spatial prior based on knowledge of the spatial statistics of various
classes of activity maps.

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