Introduction

Optical Coherence Tomography Angiography (OCTA) is a non-invasive imaging technique which provides high quality images from vessels of different tissues1,2,3. In comparison to other vascular imaging techniques such as Fluorescein Angiography (FA) or fundoscopy images, OCTA has no dangerous side effects of FA and has higher quality comparing to fundoscopy images4,5,6.

OCTA images, captured from retinal tissue, can show the retinal vessels precisely7,8. Several diseases, such as diabetic retinopathy or carotid artery occlusion, cause disturbance in retinal blood supply which results in a reduction in the retinal vascular density. This reduction can be viewed in retinal OCTA images and analyzed quantitatively by segmenting vessels9,10. This issue has been studied in several researches such as11,12,13.

Disturbance in the retinal blood supply, can also change the shape or size of FAZ (Fovea Avascular Zone) in retinal OCTA images14,15,16,17. FAZ, is a vascular free zone which can be seen as a black region usually in the center of OCTA images. Accurate information about the size or shape of the FAZ can be used as bio-markers for diagnosing some eye-related diseases18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36. However, computing accurate information about the FAZ, needs accurate segmentation of the FAZ from OCTA images.

Generally, FAZ segmentation methods are divided into two groups. The first group, consists of deep learning methods18,19,20,21,22,25,29,31,32,35,36. Usually, U-net architecture is used for segmenting the FAZ from OCTA images. Xie, et.al, exploited a network, named FAZ_NET for FAZ segmentation. The network consisted of 3 encoders, where each encoder extracted the respective features. Then by voting, the stable features were selected18. In22 a deep learning approach was proposed for joint segmentation of FAZ and retinal vessels. The segmentation of retinal vessels facilitates the process of FAZ segmentation especially in the OCTA images with low quality. An encoder based on Res-Net utilizing split attention mechanism was employed for FAZ segmentation. In25 a deep learning network based on encoder-decoder structure was proposed for the segmentation of FAZ. Two main components in this structure were encoder and decoder. Multi-scale features were extracted from the input OCTA image by the encoder and the result of the segmentation was refined using the decoder. In32 a segmentation framework for FAZ and vessels was proposed. The main blocks in this framework were an image transform network and a U-Net block. The first block helps in the segmentation of vessels using an unsupervised approach. The FAZ segmentation process was executed in a U-Net and was facilitated by the produced retinal vessel map. Convolutional Neural Network (CNN) has been exploited for 3D FAZ segmentation in27. Using the location and topology of FAZ as priors, a CNN structure has been used for FAZ segmentation. Despite the quality of deep learning approaches, these methods need large amounts of high quality OCTA images which are most of the time not available.

The second group of FAZ segmentation algorithms, consists of methods based on using morphological operators, region growing algorithms, active contours and edge detection methods23,24,28,29,30,33,34. In these methods, usually dilation and erosion filters have been used for segmenting the FAZ region. The methods can extract the FAZ without a need to have large high quality datasets. However, some of these methods cannot localize and extract the FAZ fully automatically. In37, the center of FAZ is determined manually. Also, the amount of region growing is controlled by the operator. Papers, such as28,30, localized the FAZ exactly in the center of the OCTA image, which is not always true. In30, LSM (Level Set Macro) has been used for FAZ segmentation. In28, Generalized Gradient Vector Flow (GGVF) active contour model accompanied with Otsu thresholding was used for FAZ segmentation. Diaz, et.al, used morphological closure and opening for segmenting the FAZ region. Then for removing the false candidates for FAZ, the largest segmented area was selected. This segmented area was then used as an initial seed for a region growing algorithm to segment the final FAZ. In33, a watershed algorithm was used for removing the over-segmented parts after applying region growing for FAZ segmentation.

Despite different methods which have been proposed for FAZ segmentation, still there is need for new algorithms which can segment the FAZ more accurately. Especially, deriving the algorithms which can localize and segment the FAZ automatically is very important. In this paper, we have proposed new algorithms for accurate segmentation of the FAZ without a need to determine the center of FAZ or the amount of region growing manually.

The proposed algorithms, have three steps:

  • In the first step, the FAZ region is localized. For this aim, two methods have been proposed. The first method is based on using morphological operators and the second method is based on low-rank estimation of the FAZ image in the embedded space. For the low-rank estimation method, the initial OCTA image is transformed into a 6-th order tensor using the patch Hankelization method with overlapped patches, then Tensor Ring (TR) decomposition with rank incremental is applied. The ranks will be increased until accurate localization of the FAZ achieved.

  • In the second step, the OCTA image is de-noised and smoothed using the method of38. Then the de-noised image is binarized and an active contour algorithm with the initial contour of FAZ which has been extracted in the previous step is used for the FAZ segmentation.

  • In the third step, the resulting segmented FAZ of the second step is refined. This is due to the fact that sometimes the segmented FAZ of the previous step contains extra growing parts which do not belong to the FAZ. Therefore, by removing these parts the final clean region is extracted.

Contributions of this paper can be summarized as

  • Transferring a 2-D OCTA image into a 6-th order tensor and applying low-rank TR decomposition for FAZ localization.

  • Using a de-noised and smoothed image after applying the method of38 for FAZ segmentation.

  • Proposing a new refinement method for removing the extra parts of the segmented FAZ.

The remaining of this paper is organized as follows: In Sect. 2, a short introduction to patch Hankelization and TR decomposition is presented. The proposed methods have been presented in Sect. 3. Finally, Sects. 4 and 5 are devoted to the simulation results and conclusion, respectively.

Patch Hankelization and tensor ring decomposition

Tensors are higher order arrays which are used for representing high dimensional datasets. Studies show that representing datasets in tensors, can preserve their structural and spacial information. However, many of actual datasets, such as 2-D OCTA images are not as high order to be represented in a tensor. In such a situation, several methods, known as tensorization, are used for deriving higher order datasets from lower order ones. Among different tesorization methods, Hankelization is one of the common approaches for deriving higher order datasets39,40,41. Generally speaking, simple Hankelization is a method for deriving a Hankel matrix from a vector. Recall that in a Hankel matrix, all of the elements in each skew-diagonal direction are the same. Modified versions of Hankelization, such as MDT (Multiway delay embedding Transform)39, patch Hankelization40 or overlapped patch Hankelization41 have also been proposed in different papers, for deriving higher order tensors.

Patch Hankelization with overlapped patches, has been used in this paper for transferring a 2-D OCTA image into a 6-th order tensor of size \(P \times P \times T_1 \times D_1 \times T_2 \times D_2\), where P is the patch size, \(T_i\)’s are the window sizes and \(D_i = P\times T_i(J_i/P - T_i + 1)\), where \(J_i\) is the size of the i-th mode of the matrix after reformatting it by putting the consecutive patches without overlap. For better understanding the patch Hankelization, please see Fig. 1 and41. In Fig. 1, each square shows a \(P\times P\) patch. In patch Hankelization with overlapped patches, the consecutive patches have overlap with each other.

Figure 1
figure 1

Patch Hankelization of a matrix. Each colored square is a patch of size \(P\times P\).

Full size image

A common way for analyzing a data resorted in a tensor, is decomposing that tensor. Tensor decomposition is to factorize that tensor into smaller or lower order tensors and matrices42. Tensor Ring (TR) decomposition is a widely used tensor decomposition method in which an N-th order tensor is decomposed into N third order interconnected core-tensors of size \(R_{n-1} \times I_n \times R_n\). \(R_n\)’s determine the size of core-tensors (with \(R_0 = R_N\)) and known as the rank of decomposition43 (see Fig. 2).

Figure 2
figure 2

Tensor ring decomposition of a 4-th order tensor. Each colored circle shows a third order core-tensor.

Full size image

Determining proper ranks, i.e., \({\mathbf {r}}=[R_1,R_2,...,R_N]\), is a challenging issue. Several methods selected fixed ranks in advance and as an input to the algorithm. However, other methods, preferred to increase the ranks during iterations until a desired accuracy was achieved39,40. In this paper, we used the rank incremental strategy, i.e., \(R_i\)’s are not set fixed but are increased during iterations until an accurate localization of FAZ is achieved. Note that TR decomposition and patch Hankelization have been previously used for image reconstruction (super-resolution)41,44,45. In this paper, they are not used for image reconstruction, but exploited for removing details and high rank information (which do not belong to the FAZ) and providing a situation for better localization and segmentation of the FAZ.

The proposed methods

The proposed methods have three steps which are discussed in details in the following subsections.

FAZ localization

Two methods have been proposed for FAZ localization: 1-Morphological based method, 2-Low-rank based method, where each method has been discussed in details in the following two sub-sections.

Morphological based method

For FAZ localization using morphological filters, the initial OCTA image is de-noised and smoothed using the algorithm of38 (https://github.com/csjunxu/PGPD-ICCV2015). The resulting de-noised image is then fed into a Canny edge detector algorithm (https://github.com/Digital-Image-Processing-kosta/Canny-edge-detection-algorithm/tree/master) with threshold (\(thr_{edge}\)) which results in a binary image \({\mathbf {X}}_{can}\). Then a morphological filter using a disk with radius \(r_{morph}\) is applied to \({\mathbf {X}}_{can}\). The resulting image, named \({\mathbf {X}}_{morph}\), will be a binary image which can be used for localizing the FAZ region. For this aim, the inversion of \({\mathbf {X}}_{morph}\) is computed as \({\mathbf {X}}_{bin}=1-{\mathbf {X}}_{morph}\). The resulting \({\mathbf {X}}_{bin}\) is used as an initial contour for FAZ. However, sometimes, \({\mathbf {X}}_{bin}\) contains several extra non-zero pixels which do not belong to the FAZ region. For removing these false non-zero pixels, the following rule is applied: for an \(N\times M\) image and (i, j)-th non-zero pixel of \({\mathbf {X}}_{bin}\), if

\((i-N/2)^2+(i-M/2)^2 <0.05\times ((N/2)^2+(M/2)^2)\), \({\mathbf {X}}_{bin}(i,j)\) is kept, otherwise, it is put to zero.

The resulting binary image, \({\mathbf {X}}_{bin}\), localizes the FAZ and will be used as an initial contour for FAZ segmentation. The main difference of this localization method comparing to the method of24, is that in the proposed morphological FAZ localization method, the de-noised image is used for canny edge detection. In addition, instead of selecting the largest part or using the geometrical structure of the FAZ, only small parts of the non-zero elements which are mostly close to the center of the image have been selected.

The pseudo-code of the morphological based FAZ localization has been listed in Algorithm. 1.

Algorithm 1
figure a

Morphological based FAZ localization

Full size image

Low-rank based method

In the proposed low-rank based method, a 2-D OCTA image is first patch Hankelized with overlapped patches, as:

$$\begin{aligned} \underline{{\mathbf {X}}}^H = H({\mathbf {X}},P,O), \end{aligned}$$
(1)

where, \({\mathbf {X}}\) is the original 2-D OCTA image, \(\underline{{\mathbf {X}}}^H\) is the resulting 6-th order tensor after Hankelization, P is the patch size and O is the overlap of consecutive patches. Operator H denotes the patch Hankelization of a matrix. The resulting higher order tensor is decomposed by TR decomposition with rank vector \({\mathbf {r}}\) which results in \(\underline{{\mathbf {X}}}^H_{est}\). The low-rank OCTA image is reconstructed by de-Hankelizing \(\underline{{\mathbf {X}}}^H_{est}\) as (For more information about de-Hankelization please see40):

$$\begin{aligned} {\mathbf {X}}_{low} = H^{-1}(\underline{{\mathbf {X}}}^H_{est}), \end{aligned}$$
(2)

where \({\mathbf {X}}_{low}\) is the resulting low-rank matrix after de-Hankelizing \(\underline{{\mathbf {X}}}^H_{est}\) and \(H^{-1}\) is the de-Hankelization operator. Note that, by patch Hankelizing a matrix, a 6-th order tensor achieved, while by de-Hankelizing a 6-th order tensor, a two dimensional matrix is achieved. The resulting matrix \({\mathbf {X}}_{low}\) is then binarized with threshold \(thr_1\), and a binary matrix \({\mathbf {X}}_{bin}\) is calculated as

$$\begin{aligned} {\mathbf {X}}_{bin} = 1-binarize({\mathbf {X}}_{low}, thr_1). \end{aligned}$$
(3)

The above procedure is repeated and in each iteration the rank vector \({\mathbf {r}}=[R_1,R_2,...,R_N]\) is increased by 1. For stopping the procedure, the number of non-zero elements of \({\mathbf {X}}_{bin}\) in each iteration is calculated. If for 5 iterations (not necessary consecutive) the number of non-zero elements exceeds a limit, the procedure is stopped. In an especial situation, when the number of non-zero elements of \({\mathbf {X}}_{bin}\) highly exceeds the limit and the iteration number is more than 3 or 4, the procedure stopped. For speeding up the algorithm, if after 5 iterations, the number of non-zero elements of \({\mathbf {X}}_{bin}\) is still equal to zero, \(thr_1\) is increased.

For removing any possible non-zero pixels of \({\mathbf {X}}_{bin}\) which do not belong to the FAZ area, the nearest non-zero pixel of \({\mathbf {X}}_{bin}\) to the center of image is extracted and named as \(x_c\). Then, non-zero pixels whose distances to \(x_c\) are less than a limit are kept and other elements are put to zero. The resulting \({\mathbf {X}}_{bin}\) will be used as an initial contour for FAZ segmentation. The details of the low-rank based FAZ localization method are listed in Algorithm. 2.

Algorithm 2
figure b

Low-rank FAZ localization

Full size image

FAZ segmentation

For segmenting the FAZ, the original OCTA image is first de-noised and smoothed, using the method of38. The resulting de-noised image is binarized as:

$$\begin{aligned} {\mathbf {X}}^{org}_{bin} = 1-binarize({\mathbf {X}}^{org}_{den}, thr_2), \end{aligned}$$
(4)

where, \({\mathbf {X}}^{org}_{den}\) is the de-noised image, \(thr_2\) is the threshold for binarization and \({\mathbf {X}}^{org}_{bin}\) is the resulting binary image. Then an active contour algorithm with the initial contour \({\mathbf {X}}_{bin}\) (computed in the previous step) is applied to \({\mathbf {X}}^{org}_{bin}\) for FAZ segmentation, as

$$\begin{aligned} {\mathbf {X}}_{faz} = AC({\mathbf {X}}^{org}_{bin},{\mathbf {X}}_{bin}), \end{aligned}$$
(5)

where \({\mathbf {X}}_{faz}\) is the segmented FAZ and AC denotes the active contour algorithm. Our experiments showed that applying active contour directly to the de-noised image (instead of the binarized image), results in an inaccurate segmentation, that’s why for the initial FAZ segmentation, active contour algorithm applied to the binarized image.

FAZ refinement

The extracted FAZ of the previous step, i.e. \({\mathbf {X}}_{faz}\), sometimes, contains extra growing parts, which do not belong to the actual FAZ region. These extra regions usually have root-like structures which grow around of the main body of the FAZ. The extra parts appear due to the missed vessels around the FAZ which create small fractions in the FAZ boundaries. Therefore, they usually connect to the FAZ body with thin connections. For removing these extra parts, first, their thin connections to the main body of FAZ should be removed, as: \({\mathbf {X}}_{faz}(i,j)\) will put equal to zero if:

\({\mathbf {X}}_{faz}(i-1,j)+{\mathbf {X}}_{faz}(i+1,j)=0\) or

\({\mathbf {X}}_{faz}(i,j-1)+{\mathbf {X}}_{faz}(i,j+1)=0\) or

\({\mathbf {X}}_{faz}(i-2,j)+{\mathbf {X}}_{faz}(i+2,j)=0\) or

\({\mathbf {X}}_{faz}(i,j-2)+{\mathbf {X}}_{faz}(i,j+2)=0\) or

\({\mathbf {X}}_{faz}(i-1,j-1)+{\mathbf {X}}_{faz}(i+1,j+1)=0\) or

\({\mathbf {X}}_{faz}(i-1,j+1)+{\mathbf {X}}_{faz}(i+1,j-1)=0\).

Due to the root-like structure of extra growing parts, using the above rule, their thin connections to the main body of the FAZ will be removed. The procedure can be repeated several times to completely remove the connections between the extra parts and the FAZ main body. Then, again active contour with the initial contour \({\mathbf {X}}_{bin}\) is applied to \({\mathbf {X}}_{faz}\), as

$$\begin{aligned} {\mathbf {X}}_{faz} = AC({\mathbf {X}}_{faz},{\mathbf {X}}_{bin}). \end{aligned}$$
(6)

Since the connections between extra growing parts and the main body of FAZ have been removed, the new segmented FAZ no more contains the extra parts and the final clean FAZ is recovered. Finally, for smoothing the boundaries of the segmented FAZ, a dilation operator using a disk filter with radius 2 is applied.

Considering the above discussions, the two methods for FAZ segmentation can be summarized as:

  • Morphological based method whose steps are: 1-FAZ localization using morphological operators, 2- FAZ segmentation and 3- FAZ refinement.

  • Low-rank based method whose steps are: 1-FAZ localization using low-rank estimation, 2- FAZ segmentation and 3- FAZ refinement.

The complete block-diagrams of the proposed methods have been illustrated in Figs. 3 and 4.

Figure 3
figure 3

Block-diagram of the proposed FAZ segmentation algorithm using morphological filters.

Full size image
Figure 4
figure 4

Block-diagram of the proposed FAZ segmentation algorithm using low-rank estimation.

Full size image

Simulation results

In this section, the performance of the proposed methods for FAZ segmentation have been extensively tested and compared with other existing methods. For this propose, the OCTAGON dataset13,24 (http://www.varpa.es/research/ophtalmology.html#octagon) has been exploited. The dataset contains 69 (\(3\times 3\) and \(6 \times 6\)) retinal OCTA images of diabetic patients and 144 (\(3\times 3\) and \(6 \times 6\)) retinal OCTA images of healthy people. The size of images is \(320 \times 320\) pixels and have taken from superficial and deep layers of retina. Each OCTA image of diabetic patients contains a binarized ground-truth FAZ mask. Also, each image of healthy individuals, contains two ground-truth masks for FAZ, one segmented manually and one segmented by an expert. The healthy people were in 6 age ranges, 10-19, 20-29,30-39, 40-49, 50-59 and 60-69 years.

The proposed method has been compared with the methods of24 (https://github.com/macarenadiaz/FAZ_Extraction) (named OCTA-Diaz),37 (named QOCTA),46 (named OCTAVA), a deep learning method for FAZ segmentation22 (https://github.com/lkpengcs/FARGO) (named FARGO) and a GUI used for FAZ segmentation (https://github.com/qnn122/FAZSEG-MATLAB), named FAZSEG. In the first simulation, the performances of the algorithms have been compared for FAZ segmentation of diabetic patients. For an accurate comparison, the following criteria have been used:

$$\begin{aligned} Dice = 2\frac{{\mathbf {X}}_{faz} \cap {\mathbf {X}}_{ref}}{{\mathbf {X}}_{faz} + {\mathbf {X}}_{ref}}, \end{aligned}$$
(7)
$$\begin{aligned} Jaccard=\frac{{\mathbf {X}}_{faz} \cap {\mathbf {X}}_{ref}}{{\mathbf {X}}_{faz} \cup {\mathbf {X}}_{ref}}, \end{aligned}$$
(8)

where, \({\mathbf {X}}_{faz}\) is the final segmented FAZ and \({\mathbf {X}}_{ref}\) is the ground-truth FAZ mask. For the proposed method, the patch size and the overlap of patches were selected as \(P=10\) and \(O=6\), respectively. For the low-rank FAZ segmentation, \(thr_1\) was set to 0.1 and increased to 0.15 if the number of non-zero elements of \({\mathbf {X}}_{bin}\) was zero after 5 iterations. For the morphological based FAZ segmentation, the radius of dilation operators varies between 5 to 16 and \(thr_{edge}\) was set equal to 0.025. For both proposed methods, \(thr_2\) varies between 0.08 to 0.2. Dice and Jaccard indexes of the segmented FAZ’s for diabetic patients have been presented in Table 1. The results have been reported separately for \(3\times 3\) and \(6\times 6\) superficial and deep images. The proposed-morph refers to the proposed method using morphological operators and the proposed-low refers to the proposed method using low-rank estimation. As the results show, the proposed methods have higher performances comparing to the other existing methods.

Table 1 Dice and Jaccard indexes of the segmented FAZ’s using different methods.
Full size table

The methods have also been compared for FAZ segmentation of healthy individuals. Dice and Jaccard indexes of the results have been presented in Table 2. For the morphological based method, \(thr_{edge}\) has been set to 0.025, however, for some cases it was decreased to 0.015 or 0.005. Other parameters were the same as the diabetic cases. As the results show, in most of the cases, the proposed methods have the best performances. Only for \(3 \times 3\) superficial images, the proposed methods have slightly lower performances comparing to the method of24.

Table 2 Dice and Jaccard indexes of the segmented FAZ’s using different methods.
Full size table

The methods have been compared visually in Table 3. The resulting segmented FAZ of the algorithms have been illustrated for \(3 \times 3\) and \(6 \times 6\) deep and superficial images of diabetic patients. Dice and Jaccard indexes of each image with respect to the ground-truth mask have been reported beneath each image. The results show the higher performance of the proposed methods comparing to the other methods.

Table 3 Visual comparison of different algorithms for FAZ segmentation of \(3 \times 3\) superficial, \(3\times 3\) deep, \(6 \times 6\) superficial and \(6 \times 6\) deep OCTA images.
Full size table

The accuracies of the methods for FAZ localization have been compared in Table 4. The methods of37 and46 are semi-automatic methods, in which the center of the FAZ is determined by user. However, the other methods, localize the FAZ automatically. The method of24 have localized the FAZ correctly in all of diabetic patients, but has missed the FAZ in four \(6 \times 6\) superficial OCTA images. The main reason was that the FAZ region in the \(6 \times 6\) OCTA images is relatively small comparing to the size of the image. The proposed method using morphological filters has localized the FAZ correctly in all of diabetic patients, but has missed the FAZ in one \(6 \times 6\) superficial image. However, the proposed method based on low-rank estimation has localized the FAZ region correctly in all of diabetic and healthy cases. This shows the superior performance of the proposed methods for FAZ localization.

Table 4 Accuracies of different methods in FAZ localization for diabetic and healthy individuals.
Full size table

For better understanding the effectiveness of the low-rank estimation for FAZ localization, the resulting \({\mathbf {X}}_{bin}\) for different ranks have been illustrated in Table 5. The threshold \(thr_1\) for deriving binary images has been set equal to 0.1. It is clear that, by increasing the rank, a more accurate contour of FAZ is achieved. The results show that non-zero pixels of \({\mathbf {X}}_{bin}\) are mostly related to the FAZ region. In other words, the pixels which do not belong to the FAZ region are mostly equal to zero. To have a better clarification, the binary images of Table 5 have been compared with the binary images presented in Table 6. The binary images in Table 6 have been derived by directly binarizing the original image using different thresholds. By comparing the results, it is clear that the binary images which have been provided by directly binarization of the original image, contains several non-zero pixels which do not belong to the FAZ region. Even by reducing the threshold, these extra pixels could not be removed completely, which can result in an incorrect localization of the FAZ region. While in the low-rank based method, the number of non-zero pixels which do not belong to the FAZ region is extremely limited and an accurate contour of FAZ can be localized. This is due to the fact that by applying low-rankness, the details of the images, which are mostly belong to the vessels are removed. This resulting low-rank image can be easily binarized into an image whose non-zero pixels mostly belong to the FAZ area.

Table 5 The resulting \({\mathbf {X}}_{bin}\) for different ranks.
Full size table
Table 6 Resulting binary images achieved by binarizing the original image using different thresholds.
Full size table

The FAZ localization performances of the two proposed methods have been compared for a \(6 \times 6\) superficial OCTA image of a healthy case. This selected OCTA image, is the only case (in the OCTAGON dataset) in which the proposed morphological based method could not localize the FAZ, while the low-rank based method has been localized the FAZ correctly. For the morphological based method, the edge-detector threshold was set to 0.025 and the radius of the dilation filter was set equal to 3. The results have been illustrated in Table 7. This shows that the low-rank based method is more successful for FAZ localization comparing to the morphological based method. In addition, the low-rank based method, does not need determining the threshold for the edge detection and the radius of the morphological filter, which are needed for the morphological based method. In the low-rank based method, the patch size and overlap of the patches can be selected the same for all of the images (as selected \(P=10\) and \(O=6\) for all of the simulations). Also the binarization threshold \(thr_1\) can be selected the same for all of the simulations. However, the low-rank based method, needs patch Hankelization and TR decomposition, which increase the computational burden and simulation time comparing to the morphological based method. The averaged computational time of each method for the FAZ segmentation from OCTA of diabetic patients have been presented in Table 8. The results show that the morphological based method is a faster algorithm, however, the low-rank based method is more successful in FAZ localization.

Table 7 Comparing the FAZ localization abilities of the proposed methods for FAZ localization of a \(6\times 6\) superficial OCTA image.
Full size table
Table 8 Averaged computational times (sec) of the proposed methods for FAZ segmentation from OCTA of diabetic patients.
Full size table

The effectiveness of the refinement step for removing extra growing regions has been investigated in Table. 9. The original OCTA image in this Table is a \(3 \times 3\) superficial OCTA image of a healthy individual. As the results show, the initial segmented FAZ region, contains several root-like parts which do not belong to the FAZ region. For the refinement of the image, the thin connections of these extra parts to the body of FAZ region are removed. Finally, by applying active contour to the binary image whose thin connections have been removed, the final clean FAZ has been segmented.

Table 9 The FAZ refinement procedure.
Full size table
Figure 5
figure 5

Dice and Jaccard indexes of the segmented FAZ using the two proposed methods in terms of the binarization threshold (\(thr_2\)).

Full size image

The effect of selecting binarization threshold (\(thr_2\)) on the final results have been investigated in Fig. 5. For this purpose, the proposed methods have been applied for FAZ segmentation of an \(6 \times 6\) superficial image of a diabetic patient. The binarization thresholds for the proposed methods varied from 0.08 to 0.24 and Dice and Jaccard indexes of outputs have been computed and illustrated. As the results show, selecting too small or two large threshold, will result in a low segmentation quality. However, for an acceptable range for the threshold (\(0.11-0.19\)), the performances of the methods are relatively stable.

Finally, the performances of the two proposed methods have been investigated for FAZ segmentation of a larger OCTA dataset, named OCTA-50020 https://ieee-dataport.org/open-access/octa-500. The OCTA-500, contains 500 3-D OCTA images from different patients. For each 3-D images, a 2-D OCTA image from ILM to OPL and a corresponding FAZ mask are available. The dataset contains 200 \(3 \times 3\) images of size \(304 \times 304\) pixels and 300 \(6 \times 6\) images of size \(400 \times 400\). Among the 300 \(6 \times 6\) images, 9 images could not been analyzed due to a technical issue in de-noising. For the proposed morphological based method, the \(r_{morph}\) was selected 10 for \(3 \times 3\) images and 5 for \(6 \times 6\) images. In the proposed low-rank based method, the patch size and the overlap of patches have been selected 10 and 6, respectively. Binarization threshold (\(thr_2\)) varies from 0.05 to 0.23. For the low-rank based method, for most of the cases, \(thr_1\) was selected to 0.1 and only for a few cases changed to 0.15. The averaged Dice and Jaccard indexes for the FAZ segmentation of the 2-D OCTA images of this dataset using the proposed methods, have been reported in Table. 10. The results confirmed the ability of the proposed methods for FAZ segmentation of this large dataset.

The accuracies of the proposed methods for FAZ localization of the OCTA-500 dataset have also been reported in Table. 11. It is clear that both methods are effective in FAZ localization, while the low-rank based method is more successful comparing to the morphological based method.

Table 10 Dice and Jaccard indexes for the segmented FAZ, using the proposed methods, for the images of the OCTA-500 dataset.
Full size table
Table 11 Accuracies of the proposed methods in the FAZ localization of the OCTA-500 dataset.
Full size table

Based on the simulation results, the advantages of the proposed methods comparing to the other existing methods can be summarized as:

  • Localizing the FAZ region more accurately.

  • Providing higher Dice and Jaccard indexes (in most cases) which show a more accurate FAZ segmentation.

  • No need to determine the initial FAZ contour manually.

  • No need to separate the true FAZ region from other incorrectly detected regions using the FAZ geometrical shape or selecting the largest part.

Conclusion

New methods for FAZ segmentation from retinal OCTA images have been proposed and widely tested. The proposed methods have three steps. The first step dedicated to FAZ localization and deriving the initial contour of FAZ. For this aim, the first method used morphological operators and edge detection and the second one exploited TR decomposition and low-rank estimation. In the second step, both algorithms used active contour for FAZ segmentation. Finally, in the third step, the segmented FAZ was refined and the extra parts which do not belong to the main body of FAZ were removed. Simulation results confirmed the performance and effectiveness of the proposed methods in FAZ localization and segmentation.