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The retinal fractal dimension (FD) is a measure of vasculature branching pattern complexity. FD has been considered as a potential biomarker for the detection of several diseases like diabetes and hypertension. However, conflicting findings were found in the reported literature regarding the association between this biomarker and diseases. In this paper, we examine the stability of the FD measurement with respect to (1) different vessel annotations obtained from human observers, (2) automatic segmentation methods, (3) various regions of interest, (4) accuracy of vessel segmentation methods, and (5) different imaging modalities. Our results demonstrate that the relative errors for the measurement of FD are significant and FD varies considerably according to the image quality, modality, and the technique used for measuring it. Automated and semiautomated methods for the measurement of FD are not stable enough, which makes FD a deceptive biomarker in quantitative clinical applications.

The blood vessels, as part of the human circulatory system, transport the blood with nutrition and oxygen and remove the waste throughout the body. The development of the vascular system is not a random process but follows a set of optimization principles, such as the minimum friction between the blood flow and the vessel wall, the optimal heart rate to achieve proper blood supply, and the shortest transportation distance [

One of the biomarkers that could describe changes in microvasculature due to the disease progression is the fractal dimension (FD). The theory of FD was first introduced by Mandelbrot in 1983 [

In many clinical studies, the fractal dimension has shown its potential in characterizing the growth of neurons, tissues, and vessels. Firstly, the fractal describes growing progression of the neuron cells by quantifying their complex dendrites. For instance, Ristanović et al. [

However, we found conflicting findings in different clinical studies. Some literature reports a higher FD in images of a patient group with a late stage of proliferative diabetic retinopathy compared to a healthy control group [

Of course, all the above-mentioned studies had different setups. Not only the number of patients but also the cameras used in data acquisition in each study were different. Therefore, the images’ resolution, illumination, and quality varied across studies. Moreover, the computer software which semiautomatically does the optic disc detection, vessel segmentation, vessel skeletonization, and the fractal computation was also different in each study. Finally, the region of interest for FD calculation was not the same for all studies. These different experimental settings, therefore, may be the reasons of conflicting findings in each study.

In that case, it is worth to investigate the reliability of the FD measurement, since the measurement itself might not be stable enough to provide reliable results. Previously, few works analyzed the stability and the reliability of FD measurements. Wainwright et al. [

In our previous study [

The paper is organized as follows: in Section

In this section, we introduce the public retinal image datasets and the test image dataset that were used in the stability studies. We used three datasets: MESSIDOR, DRIVE, and a test dataset including images captured by five different cameras.

The MESSIDOR public dataset [

The DRIVE dataset [

In order to investigate the variation of FD computed on the images acquired by different cameras, we established a new dataset which consists of the retinal images captured by 5 different fundus cameras on 12 young healthy volunteers. The 5 fundus cameras were installed in the Ophthalmology Department of the Academic Hospital Maastricht (AZM) in Netherlands. The volunteers are young students with 20 to 25 years of age. The retinal photographs were taken on the left eye of every subject 5 times with each camera, both fovea centered and optic disc centered (120 images in total).

The 5 cameras are 3nethra Classic, Canon CR-1 Mark II, Nidek AFC-230, Topcon NW300, and EasyScan. The 3nethra Classic (Forus, India) provides color fundus images with size of 2048

In this section, we introduce the pipeline and methodologies, which are used to compute the fractal dimension from a fundus image. The pipeline involves 6 steps (see Figure

The pipeline for calculating the fractal dimension from a color fundus image.

After the image local normalization, we apply 3 state-of-the-art vessel segmentation methods on color retinal images and one particular segmentation method on the SLO images to obtain the vessel probability maps (soft segmentation). Afterwards, a threshold value is applied to the obtained vessel probability maps in order to construct binary segmentations (hard segmentations). At the same time, we automatically determine the region of interest for FD calculation by detecting, segmenting, and parameterizing the optic disc and the fovea. Finally, the fractal dimension is calculated on the binary vessel segmented images within a circular ROI using 3 classic FD measurements. In the following section, each step of the pipeline is introduced in detail.

The fractal dimension is usually calculated on a vessel binary map, where pixel intensity of 1 is considered as vessel and 0 as background. Generally manual vessel annotations provided by the human observers have better quality than automatic vessel segmentation techniques. Additionally, for large volume clinical studies, an automatic vessel segmentation program is needed for the vessel detection. In our study, we investigated three vessel segmentation methods for extracting the vessels from RGB retinal images, Frangi’s vesselness method, Soares’ method, and Zhang’s method, and the BIMSO method for SLO images.

Frangi’s vesselness is a multiscale vessel enhancement method proposed by Frangi et al. [

(a) An original image from the DRIVE database; (b)–(d) the vessel probability maps generated by the methods of Frangi et al. [

Soares’ segmentation is a supervised method for vessel enhancement proposed by Soares et al. [

Afterwards a supervised Gaussian Mixture Model (GMM) classification method is used to classify the pixels into vessel or background using the obtained features. The output is a probability map indicating the likelihood for a pixel being a vessel (shown in Figure

Zhang’s method is based on describing the image as a function on an extended space of positions and orientations [

BIMSO method is a brain-inspired multiscale and multiorientation technique proposed by Abbasi-Sureshjani et al. [

In this subtask, the fractal dimensions were calculated in different circular regions with various radii around the fovea and optic disc (OD) centers. For fovea centered images, the regions of interest were centered at the fovea centralis with radii of 4, 5, and 6 times the optic disc radius

Optic Disk detection is done using the method proposed by Bekkers et al. [

Optic disk segmentation is performed after locating the OD centralis. The segmentation is done within a small patch of an enhanced OD to detect its circular boundary. On a regular RGB fundus image, the OD region has higher color differences than the background region. For instance, the tissue and vessels inside the disc have greater yellow-blue color difference than the background vessel and tissue (see Figure

The RGB color difference between the pixels inside and outside the optic nerve head region.

The color derivatives of an RGB image are computed using the Gaussian color model proposed in [

After the enhancement, the OD boundary becomes stronger and the potential interferences caused by the edge of vessels are suppressed and a simple zero crossings of the Laplace operator is used for OD edge detection. After that, an ellipse is fitted to the detected boundary positions and the major and minor radius are obtained. Finally, the OD radius

Fovea center detection is done within a ring area around the optic disc center. As mentioned earlier, the average distance between the fovea centralis and the optic disc centralis is about

The fractal dimension is a measurement which quantifies the highly irregular shape of fractals or fractal objects. An important property of the fractal objects is their self-similarity over different scales or magnifications. This means that at different scales a same pattern with different sizes can be observed, such as trees, snowflakes, and river systems. This self-similar property can be described by the following formula:

Based on the above relation between measurements in different scales, a box-counting method is introduced to do a simple, fast estimation of the fractal dimension

In this paper, we are mainly interested in three fractal methods that are widely used in the literature: the box dimension

Box dimension (

So, in the image domain, the measurement

Information Dimension

Correlation dimension

In this section, we present our stability analysis of the fractal methods in terms of the choice of manual annotations, different segmentation methods, various regions of interest, the accuracy of the segmentation method, and different imaging modalities. To study the variation of FDs, we use the relative error (RE) with respect to the binary images annotated by Observer 1 as the reference. The RE is obtained using

We also obtained the intergroup and intragroup fractal dimension

The mean and standard deviation of FD values (

DR grade | Number of images | ROI: full FOV | ROI: | ||||
---|---|---|---|---|---|---|---|

Mean | SD | RSD | Mean | SD | RSD | ||

R0 | 546 | 1.3864 | 0.0324 | 2.34% | 1.3285 | 0.0316 | 2.38% |

R1 | 153 | 1.3852 | 0.0345 | 2.49% | 1.3317 | 0.0304 | 2.28% |

R2 | 247 | 1.3781 | 0.0364 | 2.64% | 1.3215 | 0.0384 | 2.91% |

R3 | 254 | 1.3869 | 0.0384 | 2.77% | 1.3276 | 0.0375 | 2.82% |

| |||||||

Total | 1200 | 1.3846 | 0.0350 | 2.52% | 1.3273 | 0.0343 | 2.59% |

Box plots of the fractal dimensions

The results of multiple one-way ANOVA tests are shown in Table

Comparison between FD values in different DR groups (ANOVA test).

DR grade | Mean difference | Std. error | | 95% confidence interval | |||
---|---|---|---|---|---|---|---|

Lower bound | Upper bound | ||||||

ROI: full FOV | R0 | R1 | 0.00123 | 0.00319 | 0.981 | −0.0070 | 0.0094 |

R2 | 0.00834 | 0.00267 | | 0.0015 | 0.0152 | ||

R3 | −0.00046 | 0.00265 | 0.998 | −0.0073 | 0.0063 | ||

| |||||||

ROI: full FOV | R1 | R2 | 0.00711 | 0.00358 | 0.195 | −0.0021 | 0.0163 |

R3 | −0.00169 | 0.00356 | 0.965 | −0.0109 | 0.0075 | ||

| |||||||

ROI: full FOV | R2 | R3 | −0.00879 | 0.00311 | | −0.0168 | −0.0008 |

| |||||||

ROI: | R0 | R1 | −0.00324 | 0.00313 | 0.730 | −0.0113 | 0.0048 |

R2 | 0.00696 | 0.00263 | | 0.0002 | 0.0137 | ||

R3 | 0.00086 | 0.00260 | 0.987 | −0.0058 | 0.0076 | ||

| |||||||

ROI: | R1 | R2 | 0.01020 | 0.00352 | | 0.0011 | 0.0193 |

R3 | 0.00410 | 0.00350 | 0.646 | −0.0049 | 0.0131 | ||

| |||||||

ROI: | R2 | R3 | −0.00610 | 0.00306 | 0.191 | −0.0140 | 0.0018 |

The comparison of FD between two human observers and different vessel segmentation methods by considering Observer 1 as reference.

Method | Box dimension ( | Information dimension ( | Correlation dimension ( | ||||||
---|---|---|---|---|---|---|---|---|---|

Max | MRE | | Max | MRE | | Max | MRE | | |

Observer 2 | 7.1% | 2.0% | | 6.7% | 1.9% | | 6.2% | 1.8% | |

Frangi [ | 9.3% | 4.3% | 0.8035 | 9.4% | 4.3% | 0.8802 | 9.4% | 4.3% | 0.6990 |

Soares [ | 8.7% | 2.9% | 0.4926 | 8.7% | 3.0% | 0.7339 | 8.9% | 3.0% | 0.8657 |

Zhang [ | 7.4% | 3.9% | 0.4950 | 7.4% | 3.8% | 0.8506 | 7.3% | 3.8% | 0.691 |

It means that even if the FDs are calculated on vessel maps annotated by human observers, the methods cannot produce stable values for diagnosis, which makes fractal dimension measurement useless. In addition, Figure

The box dimension values using the manual segmentation by two observers for all patients.

The maximum errors of the box dimension for the three segmentation techniques are 9.32%, 8.70%, and 7.37%, respectively. The average errors are 4.29%, 2.88%, and 3.97%, which are significantly compared to the RSD values. These values suggest that using an automatic segmentation would induce a large error in fractal calculation. In addition, the very high

The box dimension values using different segmentation methods for all patients.

The comparison of

Method | Radius | Max | MRE | |
---|---|---|---|---|

ROI1 | | 3.8% | 2.4% | <0.01 |

ROI2 | | 1.0% | 0.4% | <0.01 |

ROI3 | | Reference |

The mean relative errors for the 20 images with respect to the reference ones are shown in Table

FD variation against vessel segmentation accuracy.

Threshold | |||
---|---|---|---|

| | | |

Average of vessel segmentation accuracy | 63.74% | 77.95% | 63% |

Average of FD variation | 14.44% | 10.42% | 25% |

The mean relative error of fractal dimension against the quality of vessel segmentation based on MCC.

The retina of one subject captured by different cameras: (a) 3nethra, (b) Canon, (c) Nidek, (d) Topcon, and (e) EasyScan.

First we compare the variation among different cameras, where the box dimensions of 12 subjects are shown in Figure

Finally, we investigate the repeatability of different cameras by comparing the FDs of different acquisitions of one subject. The repeatability is measured as the standard deviation of the fractals calculated on 5 acquisitions of the same subject divided by the average of them. As we can see from Table

The mean relative error of FD for repeated acquisitions in different cameras.

Camera | Image modality | Image size | FOV | Max RSD | Mean RSD |
---|---|---|---|---|---|

3nethra | RGB | | 40° | 2.60% | 1.25% |

Canon | RGB | | 45° | 1.64% | 0.69% |

Topcon | RGB | | 45° | 3.86% | 1.41% |

Nidek | RGB | | 45° | 2.20% | 0.94% |

EasyScan | SLO | | 45° | 3.68% | 1.25% |

Average | — | — | — | 2.80% | 1.11% |

In previous studies, fractal dimension is considered as a potential biomarker for disease detection. However, conflicting findings were found in different literature. Therefore, we examined the reliability of three classic fractal measurements for their use in clinical study applications. We divided our experiments into six studies, which we will discuss in the remainder of this section.

In our first and second studies, we investigated intergroup and intragroup variability of FD methods using the MESSIDOR dataset. Also, we studied intraobserver variation using ground truth segmentation from the DRIVE dataset. The experimental results show that, even with ground truth vessel maps, the fractal dimensions are not reliable. The RSD of

The vessel annotations of 2 human observers. The major difference is the missing of small vessels, as indicated by the red circles; (a) Observer 1

In the third study, we investigated the influence of automatic segmentation method on FD computations. We examined the FD on the vessel maps produced by three different vessel segmentation methods on the same imaging modality (RGB fundus images). The results show that the FDs calculated with various segmentations have significant differences compared to the values calculated using the annotations by Observer 1. In addition, the statistical tests show that the FDs were not associated with those computed from ground truth images. Therefore, the FD computed by automatic computer software might not be reliable, as was the case in the studies from [

In the fourth study, we investigated the variation of FD calculated within different regions of interest centered at the fovea centralis. This study is motivated by the fact that, in clinical retinal photography, the actual captured area on the retina is not always the same because of eye motion. The result shows that FDs calculated in 3 different ROIs are associated with each other, with

In the fifth study, we investigated the influence of the accuracy of vessel segmentation methods on the fractal measurements. Most vessel segmentation methods need a threshold value to convert the vessel probability map into a vessel binary map. This threshold value also affects the accuracy of the segmentation. In this study, we computed the FD on vessel binary segmentations using different thresholds (MCC ranged from 61% to 78%). As expected, the computed FD values become closer to the ones obtained from manual segmentations when segmentation accuracies increase (with respect to manual segmentation). Moreover, the variation decreases faster when the segmentation accuracy is higher than 75%. Therefore, a proper thresholding technique is required to obtain a stable FD measurement.

Finally, in the sixth study, we compared the FDs calculated on images acquired by different fundus cameras. The result shows that the variations of FD are significant when different cameras are used. These five cameras use different flashing systems resulting in different contrast and tissue reflections. Finally, the image sizes and resolutions are different, so the details of retina captured by these cameras are also not identical. Moreover, some cameras were easier to operate (e.g., via autofocus), resulting in more consistent image quality. The comparison result shows that, in general, the FD of the same subject using different cameras has significant differences. The differences in terms of image properties cause significant variations as we see from the results.

Besides the variation between cameras, we also investigated the repeatability of the FD measurement on the same subject using the same camera. The slight differences among multiple acquisitions on the same patient with the same camera are caused by variation in image quality, for example, caused by eye motions (blurry image), weak flashing/illumination, or incorrect focusing. The results show that the 5 cameras generally produce 1.11% variation between multiple photographs.

Our experiments suggest that the classic fractal dimensions must be calculated under very strict conditions, and tiny changes on the images and vessel segmentation can cause significant variations. The vessel segmentation method must be very carefully chosen, the region of interest in all images must be equally set for the FD calculation, and an optimal threshold value for creating a high accuracy binary vessel segmentation map is required. For future studies, FD’s high sensitivity to the segmentation methods and thresholding techniques will be addressed by measuring FD directly from the vessel probability maps.

The authors declare that there are no competing interests regarding the publication of this paper.

The work is part of the Hé Programme of Innovation Cooperation, which is financed by the Netherlands Organization for Scientific Research (NWO) (Dossier no. 629.001.003).