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Diffeomorphic density registration. (English) Zbl 07274053
Pennec, Xavier (ed.) et al., Riemannian geometric statistics in medical image analysis. Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-814725-2/pbk; 978-0-12-814726-9/ebook). The Elsevier and Miccai Society Book Series, 577-603 (2020).
Summary: Over the last decade image registration has received intense interest, both with respect to medical imaging applications and to the mathematical foundations of the general problem of estimating a transformation that brings two or more given medical images into a common coordinate system. In this chapter we focus on a subclass of registration problems referred to as density registration. The primary difference between density registration and general image registration is in how the registration transformation acts on the image being transformed. In density registration the transformation not only deforms the underlying coordinate system, but also scales the image intensity by the local change in volume. In numerous medical imaging applications this is of critical importance and is a fundamental property of the registration problem. The primary motivating clinical application is that of estimating the complex changes in anatomy due to breathing as imaged via 4D respiratory correlated computed tomography (4DRCCT). Given the physical quantitative nature of CT imaging, the natural action of a transformation on a CT image is that of density action: Any local compression induces a corresponding change in local density, resulting in changes in the local attenuation coefficient. We will also see that this difference in action of the transformation on the image being registered has wide ranging implications to the structure of the estimation problem. In this chapter we will study the fundamental geometrical structure of the problem and exemplify its application. The basic outline is as follows: We will first study the abstract mathematical structure of the problem, precisely defining the space of densities and the space of transformation. We will also study the set of transformations that leave the density unchanged. We will see that the explicit characterization of this set of transformations plays a critical role in understanding the geometric structure of the density registration problem. We will then introduce the general (regularized) density matching problem and present efficient numerical algorithms for several specific choices of regularizers. Finally, we will present the before mentioned application to model breathing as imaged via 4D respiratory correlated computed tomography.
For the entire collection see [Zbl 1428.92004].
##### MSC:
 92C55 Biomedical imaging and signal processing 58D17 Manifolds of metrics (especially Riemannian) 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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