QUASI-ISOMETRIC LENGTH PARAMETERIZATION OF CORTICAL SULCI: APPLICATION TO HANDEDNESS AND THE CENTRAL SULCUS MORPHOLOGY Olivier Coulon 1,2 , Julien Lef` evre 1,2 , Stefan Kl ¨ oppel 3 , Hartwig Siebner 4 , Jean-Franc ¸ois Mangin 5 1 Aix-Marseille Universit´ e, CNRS, LSIS UMR7296, Marseilles, France 2 Institut de Neurosciences de la Timone UMR 7289, Aix Marseille Universit´ e , CNRS, Marseille, France 3 Freiburg Brain Imaging, Department of Psychiatry und Psychotherapy; University Medical Center Freiburg, Freiburg, Germany 4 Danish Research Centre for Magnetic Resonance, Copenhagen, Denmark 5 CEA, UNATI, Neurospin, Gif-sur-Yvette, France ABSTRACT We present in this paper a method to perform a length param- eterization of cortical sulcus meshes. Such parameterization allows morphological features to be localized in a normalized way along the length of the sulcus and can be used to per- form population studies and group comparisons. Our method uses the second eigenfunction of the Laplace-Beltrami opera- tor, and the resulting parameterization is quasi-isometric. The process is validated on the central sulci of a set of subjects and its efficiency is demonstrated by quantifying morphological differences between left and right-handed subjects. Index Terms— cortical sulci, Laplace-Beltrami operator, mesh parameterization, morphometrics, handedness 1. INTRODUCTION Cortical sulci are the main macroscopic landmarks on the surface of the brain. Although their relationship with func- tions is still a open and debated topic, it has been shown that their morphology can provide biomarkers of functional specificities such as handedness [1, 2], or pathologies [3]. Various measurements and representations of sulci have been used to study their morphology, such as sulcal lines [4, 5] or meshes[1, 2, 6, 7, 8]. In previous work we proposed a method to parameter- ize meshes representing cortical sulci [1]. In particular this method provides a normalized localization along the length of the sulcus, which has been used for the localization of depth asymmetries of the central sulcus [1, 6] or the superior tem- poral sulcus [9], for the automatic detection and localization of the hand-knob [7], a morphological marker of the primary motor area of the hand [10], or for inter-species comparisons [11]. Any point on the sulcus is localized with a value be- tween 0 and 100 relative to its position between the dorsal This work has been supported by the Agence Nationale de la Recherche (ANR MoDeGy, ANR-12-JS03-001-01). and ventral extremities. Using such localization, sulcal shape descriptors such as depth curve [1] and sulcal profile [7] have been produced and used for group comparisons. This length parameterization relies on a two-step procedure: first the two extremities need to be detected on a mesh that describes the geometry of the sulcus; then this mesh is parameterized by means of a heat propagation process with the two extremi- ties acting as sources. Problems with this approach [1] are that: i) the algorithmic procedure to detect the extremities is costly and not entirely reliable (using geometric features such as curvature, and position and distance criteria); ii) the heat propagation process converge to a parametrization that is not isometric, and induces a higher density of coordinates towards the extremities. This is a problem since we use the coordinate for localization and depending on the position along the sul- cus a small displacement could correspond to a larger change in coordinates. In this paper we propose a method that solves both prob- lems at once using the Fiedler vector, i.e. the second eigen- function of the Laplace-Beltrami operator [8]: extremities are detected using maxima of the Fiedler vector, which is a very fast and robust approach, and the initial parametrization in- duced by the Fiedler vector is then transformed to become quasi-isometric. After describing the method in section 2 we demonstrate its efficiency in section 3 by applying it to the left and right central sulci of left and right-handed subjects, and showing quantitative morphological differences between the two populations that had been previously reported quali- tatively [2] . 2. DATA AND METHODS 2.1. Data and preprocessing We examined 34 healthy adult subjects, divided in two groups of 18 right-handed subjects and 16 left-handed sub- jects matched in age. High-resolution structural MRI was