Fig. 12.2 Tortuosity measured using different sampling intervals along two typical vessels

Since tortuosity is additive, it is clear that it is the tortuosity per unit length, rather than tortuosity itself, that is the actual metric of interest. Therefore, Fig. 12.2 plots the tortuosity divided by the length of the vessel (in pixels).

12.5 Tracing Retinal Vessels

A common approach to tracing linear features (viz., thin objects across which the image presents an intensity maximum in the direction of the largest variance, gradient, or surface curvature (i.e., perpendicular to the linear feature)) in an image is to segment the image and perform skeletonization, after some initial preprocessing. Typically, an image will be hampered by noise (inevitable statistical fluctuations as well as other irrelevant structures), poor resolution, low contrast, and background gradients (nonuniform illumination). Although prevention is better than cure, to some extent these artifacts can be minimized by image processing operations such as (nonlinear) smoothing [47], deconvolution, shading correction, and morphological filtering [48]. Comparison of images often calls for histogram equalization or histogram matching.

Segmentation is a challenging process in all but the very simplest images. Methods for segmentation can be roughly categorized as region-based and boundarybased approaches. Region-based segmentation is usually implemented by some form of (adaptive) thresholding. However, intensity thresholding, while commonly used for its simplicity and efficiency, is generally known to be one of the most errorprone segmentation methods. Boundary-based methods include edge-detecting and subsequent linking, boundary tracking, active contours (see Chapter 4), and

watershed segmentation. The next step would be to extract the centerlines, for which various skeletonization algorithms have been proposed [48]. The process is very sensitive to noise, and generally results in a number of errors such as spurious gaps and branches (spurs) and ambiguities (especially in 2-D) such as spurious loops and crossings. Various filling and pruning strategies must then be employed to try to rectify these retrospectively.

An alternative approach, which circumvents the problems inherent in segmentation and skeletonization, is to obtain the centerlines directly from the grayscale images by applying a Hessian [49–52] or Jacobian [53] based analysis of critical points, using matched or steerable filters [54] or by nonmaximum suppression [55, and Chapter 8].

The Hessian is a generalization of the Laplacian operator; it is a square matrix comprising second-order partial derivatives of the image, and can therefore be used to locate the center of a ridge-like structure. Specifically, the local principal ridge directions at any point in an image are given by the eigenvectors of the secondderivative matrix computed from the intensity values around that point.

The Hessian of an intensity image can be obtained at each point by computing

and g(· ; t) is a Gaussian function with variance t, f is an image, (x,y) is a pixel location, and represents the convolution operation. The partial derivatives can be computed by convolving the image f with a derivative-of-Gaussian kernel. Due to the symmetry of this matrix, the eigenvectors are orthogonal, with the eigenvector corresponding to the smaller absolute eigenvalue pointing in the longitudinal direction of the ridge. The scale-normalized determinant of the Hessian has better scale selection properties than the more commonly used Laplacian operator [56]. The Hessian values can be incorporated into a boundary tracking algorithm and linked using the so-called live-wire segmentation paradigm [57–59].

12.5.1 NeuronJ

This latter approach is the method of semiautomated tracing employed by NeuronJ [49], a plugin for ImageJ, which was developed to identify and trace neurons with limited user intervention but which we have used equally effectively to trace the centerlines of retinal blood vessels. The user selects a starting point and the search algorithm finds the optimal paths from that point to all other points in the image (on the basis of their Hessian “vesselness” values), where “optimal” means having a globally minimal cumulative cost according to a predefined function. The paths can

Fig. 12.3 Use of a tracing sampling interval of (a) 5 and (b) 10 in tracing a typical segment of a vessel

be displayed in real time as the user moves the cursor towards the end of the object of interest, until the presented path starts to deviate too much from what is considered the optimal tracing by the user. The tracing can then be anchored up to that point by a single mouse click, after which the algorithm proceeds by presenting optimal paths from that point. The process is iterated until the entire object has been traced. In cases where the user is not completely satisfied with the paths presented, which may sometimes occur in regions with very low contrast, it is possible to switch to manual delineation.

Before tracing can begin, several parameters must be selected. We chose parameter settings based on test tracings. For our images, we used a sampling interval of 5 (viz., 1 out of every 5 pixels along the tracings is used). Figure 12.3a shows a tracing with a sampling interval of 5, and Fig. 12.3b uses a sampling interval of 10. The former produces a smoother and better fit to the vessel, while the latter produces a more jagged centerline, which would artifactually result in a higher tortuosity. However for narrow vessel segments which are straight, or nearly so, a larger sampling interval performs better at finding the centerline. Figure 12.4a shows a nearly straight segment of a vessel using a sampling interval of 5. (Short lines are shown extending from each subsegment for clarity). In Fig. 12.4b, with a sampling interval of 10, the centerline tracing more closely follows the straight vessel. This is a consequence of NeuronJ using integer coordinates (taken as the centers of the pixels); representing lines other than those along the horizontal, vertical or at 45◦ to the grid can introduce subpixel errors which are more significant the smaller the subsegments. (We will return to this point later.)

The tracings are generally smoothed prior to sampling using a moving-average filter; we used a smoothing half-length of 5, namely a filter length of 11. We found that selecting a smoothing half-length lower than the sampling interval tends to produce more jagged lines, while a smoothing half-length larger than the sampling interval makes it difficult to follow sharply bending vessels.

Fig. 12.4 Use of a tracing sampling interval of (a) 5 and (b) 10 in tracing a straight segment of a vessel. To highlight the error on straight lines, each segment has been extrapolated to show the angle it forms with the next

Fig. 12.5 Showing the choice of tracing at bifurcations (see magnified areas) in a particular image (MA Originaux 33)

Typically, there are four long blood vessels which emerge from the optic disk in a retinal image – two arterioles and two venules. We identified the (oxygenrich) arterioles as the redder vessels (they also tended to have smaller diameters and higher tortuosity than the venules) and traced two of them (an “upper” and a “lower” arteriole) from each image, starting where the vessels leave the optic disk. At a bifurcation, we selected the ongoing branch at the shallower branching angle, which generally corresponded to the thicker, longer branch (Fig. 12.5). The digitized coordinates of the vessel centerlines were then exported to an Excel file.

The quantization of the digitized coordinates to integers when tracing in NeuronJ is an unwanted limitation. It introduces an error that is particularly noticeable for