# Automatic Tortuosity Estimation of Nerve Fibers and Retinal Vessels in Ophthalmic Images
## 1 Introduction
Several definitions have been proposed in the literature with an aim to automatically assess the tortuosity. [PERSON] et al. [2] compute the tortuosity based on frequency information such as the number of phase changes in the curvature. Some methods assess the arc length and chord length for the tortuosity estimation. The arc length is the pixel length between two endpoints of the centerline fragment, while the chord length is the linear distance between endpoints. [PERSON] et al. [7] multiply the sum of each ACR by the inflection points number for tortuosity assessment. [PERSON] et al. [18] assess the tortuosity by ratio of the arcs length to chords length (ACR). [PERSON] et al. [20] provides a 3-D tortuosity analysis of vessels and evaluate three ACR related tortuosity metrics for different types of abnormality detection. Other methods evaluate the tortuosity based on angle and curvature. [PERSON] et al. [24] measure tortuosity by total curvature and distance factor calculations for the femoral arteriograms. This measure is suitable for detection of centerline fragment tortuosity with high-frequency changes. [PERSON] et al. [14] differentiate simply tortuous retinal vessels from non-tortuours ones using total curvature and total squared curvature of vessel centerlines. [PERSON] et al. [16] measure the vessels curvature for diabetic-related disease detection by direction change and ACR. Some other methods utilize coding and mode technique to measure tortuosity. [PERSON] [25] encode the retinal vessels by slope chain code and then get the tortuosity. [PERSON] et al. [1; 26] take the advantage of combining results from different tortuosity metrics, and then leverage the multinomial regression method to select the best combination. In addition, the tortuosity evaluation still faces some problems to be solved, including human subjective errors, data variations, and metric differences. An image may be labeled by a clinician as the high tortuosity level while other clinicians as mild tortuosity level [1]. The analysis of the curvilinear organs is also a time-consuming task, since most of the analysis softwares still relies on the ophthalmologists manual measurement. It is worth noticing that the lack of available data limits the application of deep learning methods to tortuosity grading. Recently, a research [27] found that the performance of end-to-end tortuosity classification of corneal nerve images with AlexNet and VGG-16 did not exceed random accuracy on the tested dataset. The performance of conventional tortuosity measurements sometimes will also be affected by data differences, image pre-processing, and segmentation of curve structures. In order to make the tortuosity measurement more robust, the method is also invariant to scaling, rotation, and translation [6] in the clinical scene. Our previous work [28] evaluated the tortuosity assessment on the Retinex-based nerve fiber enhancement image. Therefore, an automatic and accurate measurement method is needed to evaluate the curvature of the curve. To tackle these problems, this paper introduces a new reliable framework for tortuosity assessment, which can conquer the human subjective error and metric variations. In addition, compared with the previous work [28], we evaluate the tortuosity assessment both on the nerve fiber and retinal vessel image based on the local phase tensor enhancement in order to verify the