Higher Order Kernels and Locally Affine LDDMM Registration

To achieve sparse description that allows intuitive analysis, we aim to represent deformation with a basis containing interpretable elements, and we wish to use elements that have the description capacity to represent the deformation compactly. We accomplish this by introducing higher order kernels in the LDDMM registration framework. The kernels allow local description of affine transformations and subsequent compact description of non-translational movement and of the entire non-rigid deformation. This is obtained with a representation that contains directly interpretable information from both mathematical and modeling perspectives. We develop the mathematical construction behind the higher order kernels, we show the implications for sparse image registration and deformation description, and we provide examples of how the capacity of the kernels enables registration with a very low number of parameters. The capacity and interpretability of the kernels lead to natural modeling of articulated movement, and the kernels promise to be useful for quantifying ventricle expansion and progressing atrophy during Alzheimer's disease. 1. Introduction. Atrophy occurs in the human brain among patients suffering from Alzheimer's disease, and the progressing atrophy can be detected by the expansion of the ventricles [16, 13]. We wish to describe the deformation of the brain caused by the progressing disease using as few parameters as possible and with a representation which allows intuitive analysis: we search for sparse representations with basis elements that have the capacity to describe deformation with few parameters while being directly interpretable. Image registration algorithms often represent translational movement in a dense sampling of the image domain. Such approaches fail to satisfy the above goals: low dimensional deformations such as expansion of the ventricles will not be represented sparsely; the registration algorithm must optimize a large number of parameters; and the expansion cannot easily be interpreted from the registration result. In this paper, we introduce higher order kernels in the LDDMM registration framework to obtain a deformation representation promising sparsity, increased capacity , end interpretability. We show how higher order kernels allow local representation of affine transformations and that they increase the capacity of the representation at each point. We use the compact deformation description to register points and images using very few parameters, and we illustrate how the deformation coded by the kernels can be directly interpreted and that it represents information directly useful in applications: with low numbers of control points, we can detect the expanding ventricles of the patient shown in in …