Ординатура / Офтальмология / Английские материалы / Automated Image Detection of Retinal Pathology_Jelinek, Cree_2009

angiograms) the sign of dot products would be switched so that the dot product of the gradient and normal would be negative on the left boundary and positive on the right.

Using this equation offers two advantages over the previous. First, it is now impossible during the active contour process for the sides of the ribbons to cross over and switch to the opposite side. Doing so would result in less energy as the side with a negative dot product would subtract a positive dot product, reducing the overall energy. The second advantage involves vessels that are in close proximity to each other. It often happens that two vessels appear side by side in an image and this equation prevents the boundary from moving to the nearest boundary of a nearby vessel.

9.8.4Cross section-based B-spline snakes

All of the ribbon snakes discussed thus far are based on aligning ribbon object edges along areas of maximum intensity gradient, with an additional criteria that the direction of the gradient is close to perpendicular to the ribbon edge. These properties are represented in the objective function that we wish to maximize. For these reasons, a ribbon snake can be thought of as “edge-based” where a good fit to a vessel is determined by the magnitude and direction of the image gradient along the ribbon edges. Another approach is to determine a good fit based on an alternate criteria. One such criterion developed is the use of matched filters in conjunction with snakes.

In this research, the Amplitude Modified Second Order Gaussian (AMSOG) filter (discussed in Section 9.5) was selected based on its applicability for width change detection. This filter has been demonstrated as effective at locating and measuring vessel widths [29]. It also provides a precise definition of a vessel width based on the parameter s given in Equation 9.3.

Given Equation 9.2 for the filter shape, an object that could utilize this filter in an active contour method was created by using both the Ribbon and the NORibbon as discussed in Section 9.7.2. These objects will be referred to as an AMSOG Ribbons and AMSOG NORibbons respectively and in more general terms are referred to as “cross section” snakes. Instead of fitting ribbon edges to the contours of maximum gradient, the idea is to position a ribbon-like object based on how closely its crosssectional shape fits the image cross section.

The AMSOG Ribbon object is initialized by solving for the parameter s using Equation 9.3 based on the original vessel tracing width. This is done by calculating the value for s based on a sampled cross section centered at the appropriate location. For the ribbon, this point is on the central axis. For the NORibbon, it is at the offset point. The response of this cross section and filter is calculated with the filter center and the cross section center aligned. This response contributes to the new objective function, that is the new energy equation of the AMSOG NORibbon as shown by the

where f (si) denotes the filter (i.e., Equation 9.2), si is the value of the parameter sigma at point i, I(pi(x;y)) is the cross section in the normal direction centered at pi

and represents the operation of calculating the filter response. For a ribbon,

and for a NORibbon,

where hi and oi are the normal and offset at point i.

Maximizing the energy in Equation 9.26 yields an optimal solution of either ~s for AMSOG Ribbons or s~i and ~oi for AMSOG NORibbons.

9.8.5B-spline ribbon snakes comparison

Several ribbon models have been discussed in an effort to produce vessel boundaries that are smooth and continuous and from which a vessel width can be determined. Ribbon snakes are initialized with the results from the tracing algorithm but all then use the original image intensity structure to refine the final boundary locations. Each snake differs based on the ribbon it uses and its energy equation. Three-edge-based ribbon snakes, namely ribbon, DWibbon, and NORibbon were presented. Additionally, a new, novel methodology referred to as cross section-based ribbon snakes was presented. Two cross section based snakes, the AMSOG Ribbon and the AMSOG NORibbon, were presented. Each ribbon method differed based on the parameters that define it and each snake differs in the ribbons parameters that are held constant or allowed to vary within the “snake” framework. These differences are summarized in Table 9.1.

All these methods address the identified shortcomings of the tracing algorithm, particularly in the areas of continuity and smoothness. Empirical evidence not reported in this chapter has suggested that the boundaries generated by the AMSOG NORibbon are more applicable for change detection.

9.9Vessel Width Change Detection

A prerequisite in the task of detecting change is the ability to account for projective geometry differences between two images by aligning the images through a process known as registration. By registering two images, transformations are created that can be used to convert images to a common scale space. These transformations ensure that the distance and direction between any two corresponding points in transformed images are equal. Without this ability to register and transform images, it would not be possible to compare distances and hence detect changes in width. The DBICP Registration algorithm provides this needed capability to reliably and accurately register almost any set of fundus images that contain detectable features [70].

Once registered, differences in measurements between two images may be attributed to the variation in the method of estimation of vessel width from image to image. This stochastic variation is influenced by multiple factors such as variable focus, variable illumination, and physiological changes from pathologies or even the pulsation of vessels caused by the beating of the heart. These variations result in vessel boundary estimation that is erroneous. This section presents a method that addresses these stochastic challenges by presenting decision criteria designed to account for this uncertainty.

9.9.1Methodology

Any approach at width change detection would need to follow three basic steps. The first step is to find the vessels in each image with the boundaries being identified to sub-pixel accuracy. Second is the step of transforming all vessels into the same coordinate system and identifying corresponding vessel pieces. Last is the ability to measure to sub-pixel accuracy the vessel widths and to identify changes in width over time. These requirements will be discussed in the following sections. A final step not mentioned is displaying the detected differences. The object of detecting the differences is to draw the attention of the physician to the regions that appeared to have changed. A method to call out the regions of change that does not obstruct the observer’s view of the region of change in the original image is needed. To accomplish this, a box is drawn around vessel segments that are determined to have changed as can be seen in sample change detection results in Figure 9.9.

9.9.1.1 Vessel detection

The first requirement in the detection of vessel width change is to be able to accurately and consistently identify vessels in retinal images. This includes the accurate location and continuous definition of the vessel boundaries. The AMSOG NORibbon discussed in Section 9.8.4 is used because it was empirically determined to identify more repeatable continuous vessel boundaries than other methods.

9.9.1.2 Image registration and determining corresponding vessels

The second requirement in the detection of vessel change is the ability to accurately register images. By using the DBICP registration algorithm, it is possible to determine transformations that can be used to accurately align both images as well as results generated from these images. Once aligned it is then possible to determine corresponding vessel cross sections and directly compare their widths. However, finding corresponding vessels is a challenge, particularly in the cases where a vessel has disappeared or has failed detection in one image but is detected in another. Thus this possibility must be considered when determining corresponding vessel sections.

When a vessel detected in one image coincides with a vessel in another image, then the vessels in both images are accepted as being the same. If a vessel is within some distance d of another vessel, such as one vessel width, then an additional constraint needs to be applied to ensure they are the same vessel. This additional constraint is a check to see if both vessels are generally progressing in the same direction. If this condition is not met, then it is assumed that the vessels are different. If no other vessel is found, then it can be concluded that the corresponding vessel has either escaped detection or does not exist. In this case, the image that is missing the vessel is once again consulted to determine if the vessel is truly not there or if it was a failed detection. Only after this second attempt to test for the presence of the vessel is the vessel considered as being not present.

9.9.1.3 Width change detection

Once vessels are detected, the boundaries identified, and the images registered, it is then possible to identify corresponding vessel points and compare widths to identify width change.

9.9.1.4 Comparison of widths

All widths for a given vessel segment are measured from two points on opposite sides of a vessel. These two points are determined from the B-spline ribbons representing the detected vessel boundaries. The measured width between them is along a line that is guaranteed to be perpendicular to the vessel orientation. A separate algorithm determines corresponding cross sections between images. Once corresponding points on corresponding vessels are discovered, widths can be compared directly to identify differences. By following these steps, there are no problems caused by differences in scale, discontinuities or discretization of the boundaries or cross section angle; and there are no issues with how a cross section’s end points are defined.

In determining vessel width change, it becomes necessary to define what constitutes change. What difference between corresponding vessel regions should be construed as change? Other research has shown that the “normal” expected difference between vessels over time caused by the cardiac cycle is approximately 5% (4.8% in [71], 5.4% in [45]). Thus this serves as the basis for comparison of two vessels

— any vessel portions that exhibit more than this change should be flagged as sites where differences exist.

However, comparing widths using this value as an absolute threshold does not take into account the uncertainty in the vessel segmentation, the uncertainty in identified boundaries, and the variability caused by inter-image inconsistencies. This uncertainty and variability can be caused by a variety of factors such as the presence of pathologies or differences in contrast caused by varying illumination and/or varying focus. The method presented in this chapter takes into account this 5% cardiac change and was designed mindful of these considerations.

9.9.2Change detection via hypothesis test

A method for testing for width change is based on the generation of statistics for each width measurement and conducting a hypothesis test [72; 73]. This is done as an attempt to consider in a stochastic framework the uncertainty and variability of the images and the detected vessel boundaries from those images. A hypothesis test is a standard statistical inference that uses probability distributions to determine if there is sufficient support for a statement or “hypothesis.” In this case, we want to determine if there is sufficient evidence to believe that the two measured widths are the same (within 5%) or different.

In order to determine statistics that could be used in a hypothesis test, the width measurements are modeled in a manner that can be interpreted as a probability distribution. To do this, the image is sampled across a blood vessel cross section from which the width measurement is determined. This sampling is a 1D signal and the derivative of this signal is computed. The resulting signal is treated as two probability density functions (PDF) of two random variables, with the center of the density function being located at the vessel end points and the limits of the pdf defined at the zero crossing or at the first relative minima. From these functions, values for the standard deviation, s, can be determined. This idea is illustrated in Figure 9.8. The average values for the standard deviation of each width’s end points can be used to provide a value that can be used in a hypothesis test.

In hypothesis testing, we are trying to determine if the probability distribution of two separate measurements are the same. This is done based on the measurement’s mean and standard deviation. The first step of the hypothesis test is to identify the hypothesis — that is to identify what exactly it is we are trying to prove. If the hypothesis test is successful, it means that there is substantive support for the hypothesis at a specific significance level. The significance level is a parameter that is used to help control Type 1 errors. (A Type 1 error is defined as rejecting the hypothesis when it is true, i.e., false negatives). As the significance level increases, the number of rejections decreases, decreasing the number of false negatives. The effect of this is an increase in the number of positives, both true and false.

In setting up a hypothesis test, the negation of the hypothesis is termed the null hypothesis (H0) and the hypothesis itself is termed the alternative (Ha). Mathematically, a hypothesis test is set up as shown below in Equation 9.29. The values of m1 and m2 are the mean values of the measurements with D0 indicating the amount of difference in the measurement allowed by the hypothesis test. If the difference in the measurement is greater than this value, the null hypothesis is rejected. A value

FIGURE 9.8

Two measured widths, w1 and w2 are illustrated. For each width measurement, the derivatives of the image intensity along the cross section on which the width is measured result in two probability density functions centered at the end points. From these, values for the standard deviation of the width measurement can be estimated.

of D = 0 indicates that there is no tolerance for a difference in the measurements — both measurements are expected to be the same.

To conduct the hypothesis test, a test statistic, z, is computed as follows. Note that:

q

s12 + s22

This statistic is compared against a value, za , that would be expected from a normal distribution at a certain significance level, a.

The goal of this method is to determine if vessels have changed (where we have defined change as a difference of more than 5%). However we only want to arrive at this conclusion when there is strong evidence to support it. Thus the hypothesis is that changed vessels are those that show more than a 5% change. In Equation 9.29 the values of m1 and m2 are the two widths that are to be compared. Replacing this with dw, the amount of measured change, (i.e., the difference between wi and w2), then the hypothesis is that there is change when dw > 0:05wmax where wmax is the maximum of the two widths and D0 in Equation 9.29 is replaced with 0:05wmax.

So, Equation 9.29 becomes

continuous ribbon-like blood vessel models using B-splines were described, whose purpose was to continuously define a blood vessel and allow the identification of an orthogonal cross section for accurate vessel width determination. These ribbons then utilized a snake algorithm that allowed for further refinement of the vessel boundary locations. Once final blood vessel boundary locations are determined, it is then possible to transform images into common coordinate systems through the process of registration. Once registered, widths can be compared utilizing the described hypothesis test framework, attributing any change of 5% or less to normal vessel changes caused by the cardiac rhythm.

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