The retinal fundus is the only part of the body where blood vessels can be observed directly and noninvasively, allowing assessment of the effects on the vasculature of both ophthalmic and systemic diseases, such as hypertension and diabetes. Ophthalmologists generally examine patients’ eyes using an ophthalmoscope. It is flexible and the examination can be done in real time; however, the findings can only be stored in the form of drawing. Fundus photographs are widely and frequently obtained for diagnostic records and longitudinal comparisons. Fundal photography is well suited for mass screening of eye diseases, such as glaucoma , because of its simplicity and low cost.
Because the number of qualified professionals and their time are limited, computerized analysis and quantitative assessment of fundus images can be valuable. There have been numerous studies regarding computerized image analysis of these images. In this section, some of the methods for segmentation of retinal structures using CA with statistical or mathematical models are briefly introduced.
Models for Segmentation of Optic Disc and Vessels on Fundus Images The
optic nerve head, also called the optic disc, is one of the main structures in the retina. It is the site where retinal ganglion cell axons converge to form the optic nerve. It is also the entry site for vessels supplying the retina. It is generally the brightest region in a fundus photograph, as shown in Fig. 3.41, and serves as a landmark. Localization and segmentation of the optic disc is essential in computerized
Fig. 3.41 Major anatomical structures observed in a fundus photograph
analysis for the diagnosis of glaucoma, vessel tracking, and other purposes. A number of research groups have proposed computerized methods for localization and segmentation, including those using simple thresholding, deformable models based on brightness and edge information, and pixel classification based on image features, such as gray level, edge characteristics, and texture.
One method that uses an SSM for segmentation of the optic disc was proposed by Li et al. . It consists of building a point distribution model based on the idea of an ASM . For disc boundary detection, 48 landmark points, 34 of which are evenly spread around the optic disc boundary, and the rest of which are on the main blood vessels inside the disc, are selected. Figure 3.42 illustrates the landmark points. The shape model was constructed by using the landmarks of eight training cases and applying PCA. In their study, only the four largest eigenvectors were used, which represent about 93% of the total variance of the training shapes. In the application of the model to segmentation, the method consists of initialization, matching point detection, and shape parameter update. The disc center location, which is detected on the basis of the similarity to multi-scale eigendiscs, and the mean shape (of the training cases) are used to initialize the model. For each landmark, its matching point is searched by the first derivative of the intensity distribution along the normal profile of the model. At the disc margin, a single pulse can be observed, while a negative pulse followed by a positive pulse can be found at blood vessels. In this study, two aspects were proposed for improving the original ASM: inclusion of self-adjusting weight and exclusion of misplaced landmark points in the update of shape parameters. Weights were adjusted by whether or not a matching point was detected and how far it was from the model landmark point. If it was too far, the point was not included in the parameter update. Landmarks were iteratively updated until the error was converged.
Li et al. also used ASM for the purpose of detecting the fovea, which is located near the center of the retina and is responsible for high-resolution central vision. In
Fig. 3.42 Landmark points in ASM 
fundus images, it is observed as a dark brown circular/oval region located about two disc diameters temporal to the disc (Fig. 3.41). In their method , 30 landmarks are placed on a main vessel arch that traverses the optic disc, which are eventually fit by a parabola to estimate the location of the fovea. The model construction and fitting are processed in the same manner as for the disc itself. An example of the landmarks and the fitted parabola is illustrated in Fig. 3.43. Despite the small number of training cases, the results of a small dataset were generally satisfactory using the modified ASM.
Fujita et al.  used probabilistic models for localization of the optic disc. The idea is based on the fact that the disc is visible as a bright oval region, slightly brighter in the temporal half than in the nasal half (Fig.3.41), and that the central retinal artery and vein emerge from the disc and branch out to supply and drain the retina, respectively. Using training cases, an intensity model and a vessel likelihood model were created by registering disc centers and averaging the intensity images and vessel detection images, as shown in Fig. 3.44. In the localization step, the vessel score at each pixel is determined by shifting the vessel likelihood model and multiplying it with a test image. At the pixel with the highest score, local matching with the intensity model is performed for refinement. The method, in general, works well on the test databases that are publicly available; however, it tends to fail on images in which the optic disc and the major vessels are partially visible and on images with large abnormalities.
The blood vessels are another important anatomical feature in the retina. Since the blood vessel network in individuals is highly distinctive, it is used for personal
Fig. 3.43 Landmark points placed on a main vessel arch and a parabola fitted to the landmark points 
Fig. 3.44 An intensity model (a) and a vessel likelihood model (b) learned from a training dataset 
identification, and sequential examination images are often stored and registered to look for changes over time. In terms of disease diagnosis, vessel caliber is important quantitative information because of its association with cardiovascular disease risk. Therefore, not only is the detection of vessels essential but accurate determination of vessel diameter is also important. A large number of computerized methods for retinal vessel segmentation have been proposed. Some of these methods use mathematical models for vessel segmentation, tracking, and measurement.
In principle, the models are based on a Gaussian function. Chaudhuri et al.  proposed a matched filter that is shaped as a Gaussian curve corresponding to the
Fig. 3.45 An example of bright lines observed along the central axis of a blood vessel (arrows) caused by the reflection of a flash of light
cross section of a vessel. The Alter is rotated through 12 angles, and the maximum response is used for segmentation. Similarly, an amplitude-modified second-order Gaussian filter was proposed by Gang et al.  for detection and measurement of vessels. A dual Gaussian model was used by several groups [82, 99, 175, 183, 214] to take into account the central light reflex, (Fig. 3.45), which appears as a bright line along the axis of blood vessels caused by the reflection of a flash from a fundus camera. It is often more apparent in arteries than veins and may affect vessel segmentation. The model can be constructed by the subtraction of two Gaussian functions, in which one with a smaller о, proportional to the width of the central light reflex, is subtracted from one with a larger о, proportional to the width of a vessel. An example of such a model is shown in Fig. 3.46. These models are used for segmentation, measurement of vessel width and length, and classification of arteries and veins. To control the model with a smaller number of parameters, a multiresolution Hermite polynomial model was proposed , which is represented as
where G corresponds to the Gaussian function. The shape is quite similar to a dual Gaussian model. When a = 0, H = G and the two peaks get further apart as a becomes larger. By using the models that take into account the central light reflex, segmentation results are generally better than those using a simple Gaussian model.
Fig. 3.46 Dual Gaussian model constructed by the subtraction of two Gaussian functions