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Section: Application Domains

Inverse problems in Neuroimaging

Many problems in neuroimaging can be framed as forward and inverse problems. For instance, the neuroimaging inverse problem consists in predicting individual information (behavior, phenotype) from neuroimaging data, while an important the forward problem consists in fitting neuroimaging data with high-dimensional (e.g. genetic) variables. Solving these problems entails the definition of two terms: a loss that quantifies the goodness of fit of the solution (does the model explain the data reasonably well ?), and a regularization schemes that represents a prior on the expected solution of the problem. In particular some priors enforce some properties of the solutions, such as sparsity, smoothness or being piecewise constant.

Let us detail the model used in the inverse problem: Let 𝐗 be a neuroimaging dataset as an (nsubj,nvoxels) matrix, where nsubj and nvoxels are the number of subjects under study, and the image size respectively, 𝐘 an array of values that represent characteristics of interest in the observed population, written as (nsubj,nf) matrix, where nf is the number of characteristics that are tested, and β an array of shape (nvoxels,nf) that represents a set of pattern-specific maps. In the first place, we may consider the columns 𝐘1,..,𝐘nf of Y independently, yielding nf problems to be solved in parallel:

𝐘 i = 𝐗 β i + ϵ i , i { 1 , . . , n f } ,

where the vector contains βi is the ith row of β. As the problem is clearly ill-posed, it is naturally handled in a regularized regression framework:

where Ψ is an adequate penalization used to regularize the solution:

with λ1,λ2,η1,η20. In general, only one or two of these constraints is considered (hence is enforced with a non-zero coefficient):

Figure 1. Example of the regularization of a brain map with total variation in an inverse problem. The problems here consists in predicting the spatial scale of an object presented as a stimulus, given functional neuroimaging data acquired during the observation of an image. Learning and test are performed across individuals. Unlike other approaches, Total Variation regularization yields a sparse and well-localized solution that enjoys particularly high accuracy.

The performance of the predictive model can simply be evaluated as the amount of variance in 𝐘i fitted by the model, for each i{1,..,nf}. This can be computed through cross-validation, by learning β^i on some part of the dataset, and then estimating (Yi-Xβ^i) using the remainder of the dataset.

This framework is easily extended by considering