Directional and non-directional representations for the characterization of neuronal morphology Burcin Ozcan a , Demetrio Labate a , David Jim´ enez b , Manos Papadakis a a Department of Mathematics, University of Houston b Department of Mathematics, University of Costa Rica ABSTRACT The automated reconstruction of neuronal morphology is a fundamental task for investigating several problems associ- ated with the nervous system. Revealing the mechanisms of synaptic plasticity, signal transmission, network connectivity and circuit dynamics requires accurate quantitative analyses of digital three-dimensional reconstructions. Yet, while many commercial and non-commercial software packages for neuronal reconstruction are available, these packages typically provide limited quantitative information and require a significant manual intervention. Recent advances in applied har- monic analysis, especially in the area of multiscale representations, offer a variery of techniques and ideas which have the potential to dramatically impact this very active field of scientific investigation. In this paper, we apply such ideas for (i) the derivation of a multiscale directional representation from isotropic filters aimed at detecting tubular structures and (ii) the development of a multiscale quantitative measure capable of distingushing isotropic from anisotropic structures. We showcase the application of these methods for the extraction of geometric features used for the detection of somas and dendritic branches of neurons. 1. INTRODUCTION The automated morphological reconstruction of neurons imaged by optical or fluorescent microscopy is a challenging problem for which many algorithms and software packages have been proposed over many years. 6 However, even the most sophisticated of such packages have serious limitations, due to the difficulty of processing accurately complex three- dimensional data sets which are affected by many sources of noise and signal degradation. The main task of a typical algorithm for the automated reconstruction of neuronal morphology consists of segmenting the somas and the axonal and dendritic branches of neurons in a way which is suitable for quantitative analysis and computational modeling. A variety of methods have been employed to carry out this task. The purpose of this paper is to illustrate the application of some powerful and innovative methods emerged in the area of multiscale representations during the last decade and whose impact in the applied science has not been fully exploited yet. Such methods include directional multiscale representations such as curvelets, 2 shearlets 10 and wavelets with composite dilations, 5 which are designed to encode data containing anisotropic features with higher efficiency than traditional multiscale systems; they also include isotropic wavelets 11, 12 which provide special rotation covariance properties. Thanks to their ability to capture the geometry of high-dimensional data with high efficiency, this new generation of multiscale methods can be especially useful for the extraction of the intrinsic geometric features of neurons. In particular, we will illustrate two specific applications of this ideas to (i) the detection of tubular structures and (ii) the characterization of local isotropy. The detection of tubular structure, in particular, is motivated by the to task of segmenting the axonal and dendritic branches of neurons. To facilitate this problem, we introduce a novel directional representation derived from isotropic filters which acts as a two dimensional Laplacian at the direction of the gradient of the intensity level of the image. This representation is applicable to images which contain tubular structures along with more isotropic ones. A similar ideas was used by one of the authors in a face recognition application to identify face components 4 such as eyes, nose and mouth. In this case, the singularity curves are used to model the edge boundaries associated with those face components. Since our representation acts locally, it is able to pick derivatives at the direction of the image intensity gradient which is the direction of the singularity curves. Note that our new directional representation (Theorem 2.2, item 3, below) differs from other directional representations appeared in the literature such as shearlets and curvelets since it does not have predetermined directional subbands. In E-mail: BO: bozcan@math.uh.edu, DL: dlabate@math.uh.edu, DJ: djimenez81@gmail.com, MP: mpapadak@math.uh.edu