Grouping Total Variation and Sparsity: Statistical Learning with Segmenting Penalties
Prediction from medical images is a valuable aid to diagnosis. For instance, anatomical MR images can reveal certain disease conditions, while their functional counterparts can predict neuropsychiatric phenotypes. However, a physician will not rely on pre
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Abstract. Prediction from medical images is a valuable aid to diagnosis. For instance, anatomical MR images can reveal certain disease conditions, while their functional counterparts can predict neuropsychiatric phenotypes. However, a physician will not rely on predictions by black-box models: understanding the anatomical or functional features that underpin decision is critical. Generally, the weight vectors of classifiers are not easily amenable to such an examination: Often there is no apparent structure. Indeed, this is not only a prediction task, but also an inverse problem that calls for adequate regularization. We address this challenge by introducing a convex region-selecting penalty. Our penalty combines total-variation regularization, enforcing spatial contiguity, and 1 regularization, enforcing sparsity, into one group: Voxels are either active with non-zero spatial derivative or zero with inactive spatial derivative. This leads to segmenting contiguous spatial regions (inside which the signal can vary freely) against a background of zeros. Such segmentation of medical images in a target-informed manner is an important analysis tool. On several prediction problems from brain MRI, the penalty shows good segmentation. Given the size of medical images, computational efficiency is key. Keeping this in mind, we contribute an efficient optimization scheme that brings significant computational gains.
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Introduction
For certain pathologies, medical images carry weak indicators of external phenotype. For instance, in Magnetic Resonance images, a pattern of brain atrophy centered on the thalamus predicts the evolution in Alzheimer’s disease [19]. Functional Magnetic Resonance Imaging (fMRI) can be used to infer subjects’ behavioral state from their brain activity [11]. Machine learning methods can identify these biomarkers. With linear predictors, the weight vectors form spatial maps in the image domain. However, minimizing a prediction error gives little control on the corresponding maps. Indeed, the prediction problem is often an ill-posed inverse problem in the sense that there are less samples than features available: many different weight maps can generate exactly the same predictions. A choice among these candidates is implicitly taken by the estimator employed. In the empirical risk minimization framework, this choice is imposed via a penalty which favors maps according to certain criteria, interpretable as a “prior”. Sparsity for c Springer International Publishing Switzerland 2015 N. Navab et al. (Eds.): MICCAI 2015, Part I, LNCS 9349, pp. 685–693, 2015. DOI: 10.1007/978-3-319-24553-9_84
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instance, imposable in convex optimization via the 1 norm, is very useful as it selects a small number of voxels for the prediction. It has been widely used in medical imaging, from fMRI [21] to regularizing diffeomorphic registration [8]. However, imposing sparsity can often lead to less stable weight maps. Indeed, for images with high spatial correlations, adjacent voxels contain similar infor
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