582 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 2, FEBRUARY2015 Cardiac Fiber Unfolding by Semidefinite Programming Hongying Li ∗ , Marc C. Robini, Member, IEEE, Feng Yang, Isabelle Magnin, and Yuemin Zhu Abstract—Diffusion-tensor imaging allows noninvasive assess- ment of the myocardial fiber architecture, which is fundamental in understanding the mechanics of the heart. In this context, tractog- raphy techniques are often used for representing and visualizing cardiac fibers, but their output is only qualitative. We introduce here a new framework toward a more quantitative description of the cardiac fiber architecture from tractography results. The pro- posed approach consists in taking three-dimensional (3-D) fiber tracts as inputs, and then unfolding these fibers in the Euclidean plane under local isometry constraints using semidefinite program- ming. The solution of the unfolding problem takes the form of a Gram matrix which defines the two-dimensional (2-D) embedding of the fibers and whose spectrum provides quantitative information on their organization. Experiments on synthetic and real data show that unfolding makes it easier to observe and to study the cardiac fiber architecture. Our conclusion is that 2-D embedding of car- diac fibers is a promising approach to supplement 3-D rendering for understanding the functioning of the heart. Index Terms—Cardiac imaging, diffusion-tensor imaging, di- mensionality reduction, semidefinite programming, tractography. I. INTRODUCTION T HE human myocardium is composed of branched my- ocytes which are approximately 25 μm in diameter and 100 μm in length, and which are attached to each other by intercalated disks. Therefore, contrary to brain white matter tis- sue, the myocardium does not contain so-called “fibers” at the microscopic scale. However, cardiac myocytes form elongated structures with a preferential local orientation, which are of- ten regarded as fibers (or bundles of fibers) at coarse scales. The notion of cardiac fibers has been used since the end of the 19th century to study the structure of the myocardium [1]–[11], and cardiac fiber organization is fundamental in understanding the functioning of the heart [12], [13]. More recently, the exis- tence of fiber patterns in the myocardium has been further con- firmed by diffusion-tensor imaging (DTI) [14]; this technique measures water diffusion in the myocardium region and allows Manuscript received April 13, 2014; revised August 8, 2014; accepted September 18, 2014. Date of publication September 29, 2014; date of cur- rent version January 16, 2015. This work was supported by the French ANR under ANR-13-MONU-0009-01. Asterisk indicates corresponding author. ∗ H. Li is with the CREATIS, 69621 Villeurbanne cedex, France (e-mail: lihongying17@gmail.com). M. C. Robini, I. Magnin, and Y. Zhu are with the CREATIS, 69621 Villeurb- anne cedex, France (e-mail: marc.robini@creatis.insa-lyon.fr; isabelle.mag- nin@creatis.insa-lyon.fr; zhu@creatis.insa-lyon.fr). F. Yang is with the Department of Biomedical Engineering, Beijing JiaoTong University, 100044 Beijing, China (e-mail: feng.yang@bjtu.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2014.2360797 Fig. 1. Fiber tracts obtained from human cardiac DTI data (superimposed on the B0 volume). The tracts are located in the left ventricle and pass through the voxel pointed by the red arrow. noninvasive assessment of the human myocardial architecture [15]–[21]. In cardiac DTI, tractography produces a set of three- dimensional (3-D) curves which represent the paths followed by myocardial fibers, thereby facilitating the high-level percep- tion of the fiber architecture of the heart (the fibers are usually interpreted visually from 3-D rendering with a coloring scheme based on curve features such as local tangent directions [22]). The helical pattern of cardiac fibers can be successfully re- constructed from DTI tractography results [23], [24], but trac- tography is usually regarded as the final stage in representing the cardiac fiber architecture. In fact, up to now, few studies have focused on the quantitative description of fiber tracts, whether cardiac or neuronal. In [25], a clustering methodology was pro- posed to find correspondences across a fiber population obtained from cardiac DTI data. In [26] and [27], the authors used compu- tational techniques to quantify and compare the shape of fibers in the human brain. Finally, clustering approaches to classifying white matter fiber bundles were reported in [28] and [29]. We present here a framework for unfolding cardiac fibers in order to facilitate the description of their architecture. Our ap- proach consists in taking 3-D fiber tracts as inputs, unfolding them in the Euclidean plane, and extracting quantitative param- eters from the resulting two-dimensional (2-D) embedding. This new type of representation makes it easier to observe the relative positions of the fibers and allows a more accurate assessment of their organization. To motivate our idea, Fig. 1 displays four cardiac fiber tracts with the B0 volume in the background. These fibers pass through a same point, but even though they are few, their organization is difficult to apprehend. 0018-9294 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.