Directional ratio based on parabolic molecules and its application to the analysis of tubular structures Demetrio Labate, Burcin Ozcan, Manos Papadakis Department of Mathematics, University of Houston ABSTRACT As advances in imaging technologies make more and more data available for biomedical applications, there is an increasing need to develop efficient quantitative algorithms for the analysis and processing of imaging data. In this paper, we intro- duce an innovative multiscale approach called Directional Ratio which is especially effective to distingush isotropic from anisotropic structures. This task is especially useful in the analysis of images of neurons, the main units of the nervous systems which consist of a main cell body called the soma and many elongated processes called neurites. We analyze the theoretical properties of our method on idealized models of neurons and develop a numerical implementation of this approach for analysis of fluorescent images of cultured neurons. We show that this algorithm is very effective for the detection of somas and the extraction of neurites in images of small circuits of neurons. 1. INTRODUCTION One of the main problems in image analysis is the extraction of shape characteristics. This problem is especially challeng- ing in biomedical imaging, where the objects of interest are frequently very complex, have a large variability in size and the images contrast is often highly irregular. In this paper, we are concerned with images of neuronal networks acquired using laser scanning microscopy. As each neuron consists of a blob-like cell body, called the soma, and many elongated processes called the neurites (including one axon and multiple dendrites), a first essential imaging task is to distinguish two classes of shapes: a class of rather isotropic shapes associated with the somas and a class of highly anisotropic shapes associated with the neurites. Needless to say, any notion of isotropy vs. anisotropy depends on scale at which the structures of interest are being considered. For instance, a thick tube can be regarded within a small image patch as an isotropic structure, but, if is seen as a part of a bigger patch, its anisotropy becomes evident. As a last note, we remark that an additional feature plays a significant role in such images: the directionality of an object, which can be intuitively described as the direction of the ”principal” axis, if any, of the restriction of an image within a certain image patch. Again, intuitively speaking, all these features taken together can use to characterize anisotropy. In order to distinguish anisotropic objects, such as vessel-like structures, from isotropic ones, e.g disk-like shaped objects, we introduce the notion of Directional Ratio which is a precise measure of anisotropy of a structure within a certain scale. This new concept can be defined for every transform or analysis operator induced by families of directional atoms. Although the following presentation is restricted to the 2-D setting, all concepts and development of the theory can be extended verbatim to 3-D. Furthermore, directional ratio can be computed for any image. However, our theoretical analysis of this new concept will be limited to a class of idealized images which we call cartoon neurons and are defined as functions of the form χ A , where A is a subset of R 2 (or R 3 ) containing a union of rectangular and blob-like regions. 2. A MULTISCALE MEASURE OF LOCAL ISOTROPY AND ANISOTROPY We start by introducing the following definition of local isotropy. Let B(x, r) be a ball centered at x of radius r > 0. DEFINITION 2.1. Let A ⊂ R 2 . If x ∈ A we say that the set A is locally isotropic at location x and at scale s > 0 if B(x, s/2) ⊆ A. Obviously the scale at which the isotropy is observed is not unique. For example, let A be a ball of radius S > 0. Then the ball at its center is locally isotropic at any radius up to scale 2S. Our definition above attempts to establish a connection E-mail: BO: bozcan@math.uh.edu, DL: dlabate@math.uh.edu, MP: mpapadak@math.uh.edu