Int J Comput Vis (2013) 105:109–110DOI 10.1007/s11263-013-0650-z

Mathematical Methods for Medical Imaging

Xavier Pennec · Sarang Joshi · Mads Nielsen

Published online: 13 August 2013© Springer Science+Business Media New York 2013

Computational anatomy is an emerging discipline at the inter-face of geometry, statistics and image analysis which aimsat modeling and analyzing the biological shape of tissuesand organs. The goal is to estimate representative organanatomies across diseases, populations, species or ages, tomodel the organ development across time (growth or aging),to establish their variability, and to correlate this variabil-ity information with other functional, genetic or structuralinformation.

Geometry, and especially differential geometry is a nat-ural foundation of the shape modeling. Dealing with data asextracted from medical images, makes the necessity of han-dling statistical modeling. Hence, the mathematical founda-tion of computational anatomy, seeks to unify statistics, andgeometry. The aim, that methods may serve the computa-tional anatomy emphasized the need for numerical methods.

D’Arcy W. Thomson (1860–1948) used transformationsof the underlying space to equalize the form of hand drawnbiological objects. In some examples factoring out an affinetransformation, in other examples projective transforma-tions, and in a single example a non-linear transformationcontaining singularities when transforming a Scarus Sp. intoa Pomacanthus.

David G. Kendall created, used originally in archeologyto argue if historic landmarks was more than accidentallyaligned, what is now called Kendall’s shape space, by fac-toring out similarity transformations of a labeled point sets.

X. PennecINRIA Sophia-Antipolis, Sophia Antipolis, France

S. JoshiSCI, University of Utah, Salt Lake, UT, USA

M. Nielsen (B)University of Copenhagen, Copenhagen, Denmarke-mail: madsn@diku.dk

He introduced Procustes analysis and the invariant quotientmetric to equip the shape space with a metric and a measure.

Timothy F. Cootes and co-workers introduced the activeshape models, ignoring the intrinsic curvature of the space ofProcustes-aligned shapes, used linear statistics and therebymade computations straightforward. This has had a immenseimpact on the computational modeling with now close to5,000 citations.

Much work has followed these contributions in creatingshape spaces equipped with (invariant) metrics, using trans-formations to align shapes, equipping the group of transfor-mations with metrics, etc. and to reformulate mathematicaltheories in a computational tractable manner.

This special issue follows this line of research. Five papershave been included in this special issue. Of these, three papersdeal with diffeomorphic mappings and metric constructionsof these (originating from the Large Diffeomorphic MetricMapping (LDDMM) framework), one with a novel shapedescriptor, and one with making statistics in curved spaces.

The paper by Lorenzi et al. “Geodesics, parallel transport& one-parameter subgroups for diffeomorphic image reg-istration” underpin the mathematical rigor of using the one-parameter sub-groups of the LDDMM framework. These sta-tionary velocity fields show to be of good approximation tothe LDDMM for smaller deformations while larger inter-subject registrations have substantially different geodesicsthan the full LDDMM implementations.

The paper by Günther et al. “Flexible shape matchingwith finite element based LDDMM” decouples the currentand deformation discretization by using conforming adap-tive finite elemts. This leads to more flexibility in the formu-lation as illustrated for example by incorporating multiplescales.

The paper by Modin et al. “Geodesic warps by confor-mal mappings” studies the shape equivalence under planar

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conformal mappings. The space of planar conformal map-pings is equipped with a metric and a numerical discretiza-tion is developed. Computational examples are illustrated onartificial examples.

The paper by Zeng et al. “Teichmüller shape descriptorand its application to Alzheimer’s disease study” studies thefamily of non-intersecting closed 3D curves. It is appliedto MRI data form the Alzheimer’s Disease Neuroimag-ing Initiative (ADNI) distinguishing normal controls fromAlzheimer’s patients by the shape of the cortical surface. TheTeichmüller space is a quotient space of conformal mappings;two surfaces in between which a conformal mapping existshave the same representation in Teichmüller shape space. Thespace is equipped with a metric. By solving a Ricci flow, theTeichmüller representation may be found.

The paper by Fletcher et al. “Geodesic regression and thetheory of least squares on Riemannian manifolds” introduces

the geodesic regression as a mapping between a real-valuedparameter to be regressed to a manifold-valued dependentrandom variable. The key idea is that regressed curves mustbe geodesics on the manifold. Existence, uniqueness, andmaximum likelihood criteria are developed. Also here brainshape is analyzed: the relation of the shape of corpus callosumto age.

All these are aspects of the fundamental property thatshapes do not live in Euclidean space. Since the pioneer-ing work of D’Arcy Thomson equaling shapes by transfor-mations, David Kendall showing that shapes represented aslabeled landmarks live in complex projective spaces, andTimothy Cootes reformulating this in a computational simplemanner, the work has continued developing representationsand appropriate metrics to allow for statistical modeling. Thisspecial issue continues this development bridging statistics,geometry, and computational science.

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