# Automatic Tortuosity Estimation of Nerve Fibers and Retinal Vessels in Ophthalmic Images
## 1 Introduction  
As one of the most significant biomarkers, tortuosity of anatomical curvilinear organ has a close relationship with human diseases in medical images. In recent clinical research [1, 2], it is confirmed that some morphological changes of curvilinear structures, nerve fibers, retinal vessels, corneal, coronary, and optic nerve are important cues for diagnosis many diseases. The tortuosity measurement can be used for multiple imaging modalities. The tortuosity is an important way to evaluate architecture and morphology of vasculature, and can be used for skin diseases detection in optical coherence tomography angiography and photoacoustic imaging [3]. Higher densities and tortuosity geometries is also usually indicating the location of cancer-induced angiogenesis in contrast-enhance ultrasound imaging [4]. Human eye is a small organ and full of many curvilinear structures which includenerve fibers and vessels. Diagnosis of many eyes and systemic diseases [1; 5; 6] are usually tested by quantitative properties of such curvilinear structure tortuosity. How to measure the tortuosity and what is the best measurement metrics are major topics in their research. More specifically, ophthalmologists usually assess diabetic optic neuropathy [2] and hypertensive retinopathy [7] with tortuosity of corneal nerve fibers and retinal vessels, respectively. Figure 1 illustrates examples of nerve fiber and vessels with three-level tortuosity in corneal confocal microscopy (CCM) and retinal fundus image. Clinically, the tortuosity of these curvilinear structure is usually graded into three to five levels, or as normal/abnormal by ophthalmologists based on their experience. Nerve fibers and retinal vessels tortuosity are divided into different levels by the naked eye of clinicians after observing the CCM and retinal fundus image. However, the assessment made by clinicians is usually subjective and time consuming. Different clinicians will get different levels for the same image, and the same doctor also may give inconsistent decisions for the same image at a different time [1]. Therefore, it is desired to propose an objective, repeatable, highly adaptable and quantitative method to assess the tortuosity level of the curvilinear structure. So far, various automatic tortuosity measurement approaches [1; 2; 7] fall into five steps: first, original image usually is enhanced by filters in the vessel segmentation problems. The three most influential filters are the eigenvalue-based filter, the wavelet transform based filter and the local phase based filter. The typical eigenvalue-based filter proposed by [PERSON] et al. [8] is based on eigenvalues of the Hessian matrix. The isotropic undecimated wavelet transform has recently been used for vessel segmentation and showed good accuracy and computational efficiency [9]. Local phase is an important local feature that can measure structural information of an image. It has recently been shown that this information can be used to enhance vessels in a more precise way and produce promising segmentation results [10]. Second, a segmentation step is utilized to get the curvilinear structure image from original medical image [11; 12; 13]. Third, a centerline map is extracted from curvilinear structure image by a skeletonization method, and all the intersection points of the centerline are detected. Then, the centerline is broken into small centerline fragments according to intersection points. In some cases, curve fitting or parametric pre-processing is performed on the fragments for the breakpoints connection and next calculation. Finally, the tortuosity of each fragment is calculated by different metrics which are shown in Table 1.  
Figure 1: Corneal nerve fibers (**top**) and retinal vessels (**bottom**) with low, middle and high tortuosity levels. However, methods of the curvilinear tortuosity measurement usually lack of universally accepted definition, or standard specification [1; 2; 7].