Thickness-based binary morphological improvement of line intersectionsline intersections

The motivation of the efforts discussed so far in this chapter has been the efficient representation of arbitrary digital curves with a specific interest in closed object boundaries (e.g. human silhouettes).

However – similarly to section 2.3.3 –, we have investigated whether some of our findings can be exploited in other fields, as well. Accordingly, in this section, we present some results on the improvement of distorted digital line intersections appearing at the crossings of retinal vessels.

Namely, we have worked out a more precise discrete geometric approach to improve the quality of the intersections of digital curves extracted by skeletonization instead of using the simple definition of a junction given in section 3.6.

Skeletonization [114] (see also section 2.3.2) is a binary morphological operation to extract the centerline of an object. The skeleton is often found by thinning the object with removing pixels without affecting the general shape. The skeleton is a popular object representation, since pixel-wise observations can be used to characterize the spatial behavior of the objects. For example, junctions can be located as pixels with at least three 8-neighbors. However, the skeleton is usually distorted as it is illustrated in Figure 3.12, with having two junction pixels instead of a single one.

In such cases, the intersecting elements cannot be tracked properly later on.

(a) (b) (c)

Figure 3.12: Distortion of an intersection during skeletonization; (a) original image, (b) its skeleton, (c) desired skeleton.

Several approaches have been released to overcome this problem. Basically, these methods can be grouped as skeleton-based or global approaches. In the skeleton-based case, the distorted locations are found at the skeletonized images, while in the global case the intersections are tried to be localized in the original line drawing image. The usual drawback of a skeleton-based decision is that it needs a proximity parameter to differentiate between true intersections and enclosed segments connecting intersection points and the definition of this parameter is challenging for the whole image. Global approaches are usually based on corner detection [139, 140] inheriting the typical related problems, like too many/few corner candidates. However, their strength is that they try to locate and improve distorted intersections without altering the original image. We have encountered with this intersection distortion problem during the improvement of the skeleton of the retinal vascular system. As a specific field of automatic screening of diabetic retinopathy, the proper mapping of the vascular system has very important role in gathering information to diagnose diseases. For example, a proper traversal of the vessels gives information about how the thickness of the vessels’ changes, or about the artery/vein ratio. After applying a segmentation method [116], a binary image (see Figure 3.13) can be extracted with similarities to line drawing ones.

Figure 3.13: Binary vessel map of the retina.

In this section, we propose a global technique for suppressing distorted intersections in the

skeleton that can be considered as both a stand-alone or a complementary approach to others mentioned before. Our method is based on the separation of the thick and thin vessels of the complete vascular system (input image) before applying skeletonization. The main motivation behind this approach is to be able to extract the precise skeletonization of the thick vessels at the intersection, while the skeletonization of the thin components is executed separately. However, since the vessel system is split into two parts, a consecutive reconnecting step is needed to connect the thin skeletal elements to the thick ones.

In [21], we have given the theoretical model that is considered to measure the distortion as a function of the thickness of the intersecting elements. Here, for the intersection of two vessels, we consider two stripes bounded by pairs of parallel lines. That is, our model considers three parameters: the width of the intersecting stripes and the angle enclosed by them. To measure the degree of the distortion (DD) of the intersection, a natural error function can be defined as the distance between the desired and actual junction points. The theoretical calculations to determine this error for given width and angle parameters are omitted here; see [21] for more details. We have found that the degree of distortion is large for stripes having large width differences. This observation motivated the separation of the vessels based on their widths.

Our approach to improve the skeleton of binary vascular images is based on two major steps.

First, we split the vascular system into two parts containing the thick and thin vessel segments and perform the skeleton of the two sets. Then, we reconnect the components disconnected in the splitting step. The separation of the thick and thin vessels of the whole vascular system is performed by applying erosion steps that remove pixels having less than four 8-neighboring pixels.

We repeat the procedure recursively until no more changes are found in the image. By this process, thin blood vessels are erased with the thick vessel subsystem preserved. Then, the thin vascular subsystem can be trivially generated by subtracting the thick one from the original binary image.

The result of this splitting procedure is shown in Figure 3.14 for the input binary vascular system depicted in Figure 3.13.

(a) (b)

Figure 3.14: The result of the splitting step; (a) the thick vessels subsystem, (b) the thin vessels subsystem.

We note that our approach for erosion is slightly different from the classic morphological op-eration. Namely, our approach preserves all the vessels above a given width to guarantee that the thick vessels remain connected. After successfully splitting the vascular system into two parts, we perform the skeletonization of the thick and thin subsystems. For this purpose, any skeletonization algorithm can be used. In our implementations, we considered the one recommended by Deutsch [141].

When taking the union of the skeletons of the thick and thin vessel subsystems, several discon-tinuities (gaps) remain to be filled in. This phenomenon is also depicted in Figure 3.15. To fill in such gaps, we perform direction estimations at the corresponding endpoints of the thin skeleton to connect them to the thick skeleton subsystem. The direction estimation is performed by following the classic recommendation of discrete geometry [122] for the calculation of a tangent at the given endpoint. That is, the reconnecting line is calculated from the endpoint and the pixel on the skeleton of thin vessel having distance of P pixels from the endpoint. The value of P is reduced iteratively from a threshold until a successful reconnection is found. There is a wide range of vessel widths from (less than) a pixel to a maximum width. Since direction estimation may miss the thick vessel, a parameterM for the maximum allowed reconnection distance between the endpoint and a thick vessel is considered. M should be set to the half of the maximum vessel width. We accept the thin vessel to be connected to the potential thick vessel, if the thick skeletal pixel and the thin endpoint pixel can be connected in this way. If the algorithm is not able to find a thick skeletal pixel for a thin endpoint, then it resumes the skeleton extracted from the whole vascular system for that location.

(a) (b)

Figure 3.15: The reconnection step to fill in the gaps between the skeletons of the thick and thin vessels; (a) a gap to be filled in, (b) result after reconnecting the separate skeletons.

For our experimental tests we considered a database of 130 vessel intersections extracted from the 19 binary vascular images of the database DRIVE described in section 1.5.6. At the resolution of the images in DRIVE the corresponding vessel widths are between 1 and 10 pixels. Considering this dataset, the parameters of our algorithm have been adjusted for optimal performance as: max-imum allowed reconnection distance M =5, distance from the endpoint for direction estimation P =6 pixels.

To see the improvement, we have compared the skeleton resulted by our approach with the classic skeleton extracted from the original intersection image without any splitting/reconnection step. For a quantitative comparison, we considered the degree of distortion term DD defined before. From the 130 intersections, our algorithm generated 47 different skeletons than classic skeletonization. That is, basically in 36% of the cases the segments had sufficiently different widths for a reliable split. Comparing the results with respect to the DD term, we got that our algorithm gave better results for 29 intersections. In 14 cases, the detected junction points were different, butDDremained the same. There were 2 intersections, where our algorithm gave worse

results than classic skeletonization. In these cases, the thin vessels can be considered rather as circular segments than linear ones causing a failure for direction estimation. Considering all the 130 intersections, our algorithm reduced the totalDD error from 140 to 72 pixels with providing a 48% suppression of distortion for these cases. Our results are also summarized in Table 3.4.

Splitable intersections (47/130)

Skeletonizaton: Classic Proposed Total DD error (pixels): 140.0 72.0 Average DD error (pixels): 3.11 1.6 Distortion decreased to: 51.4286%

All intersections (130/130)

Unsuccessful splitting 85

Better results with proposed method 29 Same results with proposed method 14 Worse results with proposed method 2

Table 3.4: Improvement of the proposed method against classic skeletonization for 130 vessel intersections.

Chapter 4

Combining algorithms for automatic