Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms
Abstract
1. Introduction
- The math-inspired metaheuristic algorithms of CSA, TSA, AOA, GNDO, GOBC-PA, and SCA are implemented for clustering and retinal vessel segmentation.
- The statistical analyses indicate that the considered math-inspired algorithms are able to produce effective results despite their simple algorithmic structures.
2. Materials and Methods
2.1. Circle Search Algorithm
Randomly create an initial population by, |
Determine the parameter |
Cycle = 1 |
WHILE |
Calculate the values of the variables , and |
Calculate the value of |
Update by using, |
If Updated solutions are out of the boundaries |
Set the solutions equal to the boundaries |
Calculate the fitness value of |
End |
Evaluate with the current best solution |
Update and |
Cycle = Cycle + 1 |
END |
2.2. Tangent Search Algorithm
Randomly create uniformly distributed initial population, |
Cycle = 1 |
WHILE |
Apply Switch procedure for each according to Pswitch |
If : Intensification phase |
Produce a random local walk by, |
Exchange some variables of the obtained solution with the related variables of the |
current optimal solution by to produce new possible solutions |
Check the boundaries of the recently produced solutions |
Repair the overflowed solutions |
Else : Exploration phase |
Apply to each variable with a probability equal to 1/D |
in order to expand the research capacity |
End |
Apply Escape Local Minima procedure according a given probability value called Pesc |
If : Escape local minima phase |
Randomly select one of the current solutions and apply selection procedures of |
End |
Replace with a randomly selected solution having lower fitness value |
Cycle = Cycle + 1 |
END |
2.3. Arithmetic Optimization Algorithm
Randomly create an initial population consisting of positions of the solutions, |
Cycle = 1 |
WHILE |
Calculate the fitness value of each solution in the population |
Determine the best solution so far |
Update the Math Optimizer Accelerated (MOA) value |
Update Math Optimizer probability (MOP) value |
For |
For |
Produce a random values between [0, 1] for the conditions of r1, r2 and r3 |
If : Exploration phase |
If |
Apply Division math operator (D “”) and update the position of |
solution i |
Else |
Apply Multiplication math operator (M “”) and update the position of |
solution i |
End if |
Else |
If : Exploitation phase |
Apply Substraction math operator (S “”) and update the position of |
solution i |
Else |
Apply Addition math operator (A “”) and update the position of |
solution i |
End if |
End if |
End |
End |
Cycle = Cycle + 1 |
END |
2.4. Generalized Normal Distribution Optimization Algorithm
Randomly create an initial population by, and |
Calculate the fitness value of each solution and determine the best solution as |
Cycle = 1 |
WHILE |
For |
Produce a random number () in the interval of [0, 1] |
If : Local exploitation strategy |
Select the current optimal solution and calculate the mean position M |
Compute the generalized mean position , generalized standard variances |
and penalty factor |
Apply the local exploitation strategy to calculate and |
Else: Global exploration strategy |
Apply the global exploration strategy |
End if |
End for |
Cycle = Cycle + 1 |
END |
2.5. Global Optimization Method Based on Clustering and Parabolic Approximation
Randomly create an initial population |
Set the number of cluster size as and maximum epoch size as |
Calculate the fitness value produced by the objective function |
For |
Determine the cluster centers |
Determine the membership matrix of via clustering |
Sort the cluster centers with ascending order in accordance with the fitness values |
Select the first ceil cluster centers as () |
For |
Determine the H set which represent the members of taken from depending |
on |
If |
Determine the parabola coefficient matrix () |
Find the coefficients of approximated parabola () |
End if |
End for |
For |
Determine the vertex of parabola |
If , the parabola can be assumed as concave and keep |
Else if , the parabola can be assumed as convex, and if is better |
than then replace with |
Else if is not a number or infinite then keep |
End if |
End for |
Randomly create a new population around the |
Randomly create new population around the current best two solutions |
Randomly create a new population |
Determine as |
Calculate the fitness value of the objective function by using the new population |
Determine as the best solution of |
If , can be defined as the global minimum |
End if |
End for |
2.6. Sine Cosine Algorithm
Randomly create an initial population consisting a set of search agents, |
Cycle = 1 |
WHILE |
Calculate the fitness value of each search agent in the population |
Determine the best search agent so far and store it in a variable as destination agent |
Update the random parameters and |
Update the position of each search agent in the current population |
Store the current destination point as the best search agent obtained so far |
Cycle = Cycle + 1 |
END |
3. Results
3.1. Segmentation Performance
3.2. Statistical Analysis
3.3. Convergence Analysis
4. Discussion
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Algorithm | Control Parameters |
---|---|
CSA |
|
TSA |
|
AOA |
|
GOBC-PA |
|
SCA |
|
GNDO |
|
Sensitivity | Specificity | Accuracy | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Image | CSA | TSA | AOA | GNDO | GOBC-PA | SCA | CSA | TSA | AOA | GNDO | GOBC-PA | SCA | CSA | TSA | AOA | GNDO | GOBC-PA | SCA |
1 | 0.8375 | 0.8745 | 0.8375 | 0.8375 | 0.8375 | 0.8375 | 0.9749 | 0.9706 | 0.9749 | 0.9749 | 0.9749 | 0.9749 | 0.9644 | 0.9640 | 0.9644 | 0.9644 | 0.9644 | 0.9644 |
2 | 0.9102 | 0.9102 | 0.9387 | 0.9387 | 0.9387 | 0.9387 | 0.9698 | 0.9698 | 0.9655 | 0.9655 | 0.9655 | 0.9655 | 0.9650 | 0.9650 | 0.9635 | 0.9635 | 0.9635 | 0.9635 |
3 | 0.9092 | 0.9092 | 0.7436 | 0.6715 | 0.6715 | 0.6715 | 0.9487 | 0.9487 | 0.9694 | 0.9756 | 0.9756 | 0.9756 | 0.9470 | 0.9470 | 0.9509 | 0.9452 | 0.9452 | 0.9452 |
4 | 0.9464 | 0.9278 | 0.9464 | 0.9464 | 0.9464 | 0.9464 | 0.9615 | 0.9648 | 0.9615 | 0.9615 | 0.9615 | 0.9615 | 0.9606 | 0.9626 | 0.9606 | 0.9606 | 0.9606 | 0.9606 |
5 | 0.7867 | 0.8842 | 0.8842 | 0.7867 | 0.7867 | 0.7867 | 0.9777 | 0.9730 | 0.9730 | 0.9777 | 0.9777 | 0.9777 | 0.9629 | 0.9674 | 0.9674 | 0.9629 | 0.9629 | 0.9629 |
6 | 0.7931 | 0.8631 | 0.6963 | 0.7931 | 0.6963 | 0.6963 | 0.9661 | 0.9761 | 0.9718 | 0.9661 | 0.9718 | 0.9718 | 0.9526 | 0.9726 | 0.9450 | 0.9526 | 0.9450 | 0.9450 |
7 | 0.7380 | 0.7911 | 0.7380 | 0.7911 | 0.6690 | 0.7380 | 0.9774 | 0.9740 | 0.9774 | 0.9740 | 0.9808 | 0.9774 | 0.9610 | 0.9631 | 0.9610 | 0.9631 | 0.9557 | 0.9610 |
8 | 0.4576 | 0.6690 | 0.5586 | 0.5586 | 0.4576 | 0.4576 | 0.9869 | 0.9792 | 0.9833 | 0.9833 | 0.9869 | 0.9869 | 0.9232 | 0.9573 | 0.9442 | 0.9442 | 0.9232 | 0.9232 |
9 | 0.6973 | 0.6973 | 0.6973 | 0.5889 | 0.5889 | 0.6973 | 0.9723 | 0.9723 | 0.9723 | 0.9773 | 0.9773 | 0.9723 | 0.9505 | 0.9505 | 0.9505 | 0.9375 | 0.9375 | 0.9505 |
10 | 0.8994 | 0.8627 | 0.8627 | 0.8089 | 0.8089 | 0.8089 | 0.9705 | 0.9742 | 0.9742 | 0.9780 | 0.9780 | 0.9780 | 0.9671 | 0.9681 | 0.9681 | 0.9673 | 0.9673 | 0.9673 |
11 | 0.9591 | 0.9264 | 0.8795 | 0.8795 | 0.8795 | 0.8795 | 0.9558 | 0.9619 | 0.9682 | 0.9682 | 0.9682 | 0.9682 | 0.9559 | 0.9601 | 0.9629 | 0.9629 | 0.9629 | 0.9629 |
12 | 0.6664 | 0.6664 | 0.7663 | 0.5488 | 0.5488 | 0.6664 | 0.9800 | 0.9800 | 0.9755 | 0.9842 | 0.9842 | 0.9800 | 0.9507 | 0.9507 | 0.9598 | 0.9315 | 0.9315 | 0.9507 |
13 | 0.8956 | 0.9273 | 0.8956 | 0.8461 | 0.8461 | 0.8956 | 0.9547 | 0.9498 | 0.9547 | 0.9599 | 0.9599 | 0.9547 | 0.9508 | 0.9485 | 0.9508 | 0.9513 | 0.9513 | 0.9508 |
14 | 0.7149 | 0.9309 | 0.7149 | 0.6156 | 0.6156 | 0.7149 | 0.9823 | 0.9632 | 0.9823 | 0.9857 | 0.9857 | 0.9823 | 0.9604 | 0.9618 | 0.9604 | 0.9485 | 0.9485 | 0.9604 |
15 | 0.7816 | 0.7816 | 0.7816 | 0.7257 | 0.7257 | 0.7257 | 0.9714 | 0.9714 | 0.9714 | 0.9746 | 0.9746 | 0.9746 | 0.9598 | 0.9598 | 0.9571 | 0.9571 | 0.9571 | 0.9571 |
16 | 0.8776 | 0.9493 | 0.8776 | 0.8776 | 0.8776 | 0.9252 | 0.9734 | 0.9655 | 0.9734 | 0.9734 | 0.9734 | 0.9693 | 0.9668 | 0.9646 | 0.9668 | 0.9668 | 0.9668 | 0.9667 |
17 | 0.6654 | 0.7488 | 0.7488 | 0.6654 | 0.5718 | 0.6654 | 0.9807 | 0.9770 | 0.9770 | 0.9807 | 0.9845 | 0.9807 | 0.9540 | 0.9610 | 0.9610 | 0.9540 | 0.9412 | 0.9540 |
18 | 0.8025 | 0.8727 | 0.8025 | 0.8025 | 0.8025 | 0.8025 | 0.9710 | 0.9662 | 0.9710 | 0.9710 | 0.9710 | 0.9710 | 0.9573 | 0.9597 | 0.9573 | 0.9573 | 0.9573 | 0.9573 |
19 | 0.8810 | 0.9299 | 0.9299 | 0.8810 | 0.8810 | 0.8810 | 0.9659 | 0.9607 | 0.9607 | 0.9659 | 0.9659 | 0.9659 | 0.9593 | 0.9586 | 0.9586 | 0.9593 | 0.9593 | 0.9593 |
20 | 0.6961 | 0.8870 | 0.8030 | 0.6961 | 0.5882 | 0.6961 | 0.9714 | 0.9591 | 0.9650 | 0.9714 | 0.9770 | 0.9714 | 0.9450 | 0.9547 | 0.9450 | 0.9450 | 0.9291 | 0.9450 |
Mean | 0.7958 | 0.8505 | 0.8051 | 0.7630 | 0.7369 | 0.7716 | 0.9706 | 0.9679 | 0.9711 | 0.9734 | 0.9747 | 0.9730 | 0.9557 | 0.9599 | 0.9578 | 0.9547 | 0.9515 | 0.9554 |
Sensitivity | Specificity | Accuracy | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Image | CSA | TSA | AOA | GNDO | GOBC-PA | SCA | CSA | TSA | AOA | GNDO | GOBC-PA | SCA | CSA | TSA | AOA | GNDO | GOBC-PA | SCA |
1 | 0.5165 | 0.5668 | 0.5165 | 0.5165 | 0.5165 | 0.5165 | 0.9747 | 0.9700 | 0.9747 | 0.9747 | 0.9747 | 0.9747 | 0.9298 | 0.9374 | 0.9298 | 0.9298 | 0.9298 | 0.9298 |
2 | 0.5319 | 0.6246 | 0.4807 | 0.4807 | 0.4807 | 0.4807 | 0.9729 | 0.9656 | 0.9767 | 0.9767 | 0.9767 | 0.9767 | 0.9411 | 0.9488 | 0.9331 | 0.9331 | 0.9331 | 0.9331 |
3 | 0.7501 | 0.6057 | 0.4885 | 0.4885 | 0.4885 | 0.4885 | 0.9605 | 0.9696 | 0.9755 | 0.9755 | 0.9755 | 0.9755 | 0.9500 | 0.9414 | 0.9225 | 0.9225 | 0.9225 | 0.9225 |
4 | 0.4191 | 0.5752 | 0.5752 | 0.4191 | 0.4191 | 0.2740 | 0.9625 | 0.9523 | 0.9523 | 0.9625 | 0.9625 | 0.9735 | 0.8924 | 0.9244 | 0.9244 | 0.8924 | 0.8924 | 0.8033 |
5 | 0.7685 | 0.7261 | 0.7487 | 0.7261 | 0.7261 | 0.7038 | 0.9561 | 0.9617 | 0.9588 | 0.9617 | 0.9617 | 0.9649 | 0.9453 | 0.9454 | 0.9456 | 0.9454 | 0.9454 | 0.9451 |
6 | 0.7269 | 0.7741 | 0.7528 | 0.7269 | 0.7269 | 0.7269 | 0.9664 | 0.9602 | 0.9633 | 0.9664 | 0.9664 | 0.9664 | 0.9497 | 0.9495 | 0.9500 | 0.9497 | 0.9497 | 0.9497 |
7 | 0.8059 | 0.7076 | 0.7300 | 0.7076 | 0.7076 | 0.7300 | 0.9624 | 0.9740 | 0.9718 | 0.9740 | 0.9740 | 0.9718 | 0.9536 | 0.9526 | 0.9537 | 0.9526 | 0.9526 | 0.9537 |
8 | 0.7304 | 0.7546 | 0.6654 | 0.6654 | 0.6654 | 0.7012 | 0.9723 | 0.9686 | 0.9790 | 0.9790 | 0.9790 | 0.9757 | 0.9560 | 0.9557 | 0.9527 | 0.9527 | 0.9527 | 0.9551 |
9 | 0.7328 | 0.7604 | 0.7328 | 0.7604 | 0.7604 | 0.7843 | 0.9762 | 0.9738 | 0.9762 | 0.9738 | 0.9738 | 0.9712 | 0.9565 | 0.9578 | 0.9565 | 0.9578 | 0.9578 | 0.9583 |
10 | 0.5982 | 0.6457 | 0.5982 | 0.5982 | 0.5982 | 0.5982 | 0.9718 | 0.9673 | 0.9718 | 0.9718 | 0.9718 | 0.9718 | 0.9355 | 0.9406 | 0.9355 | 0.9355 | 0.9355 | 0.9355 |
11 | 0.7971 | 0.7005 | 0.7701 | 0.7396 | 0.7396 | 0.7396 | 0.9705 | 0.9796 | 0.9734 | 0.9765 | 0.9765 | 0.9765 | 0.9612 | 0.9589 | 0.9613 | 0.9609 | 0.9609 | 0.9609 |
12 | 0.8294 | 0.8080 | 0.7899 | 0.7899 | 0.7899 | 0.7899 | 0.9504 | 0.9578 | 0.9630 | 0.9630 | 0.9630 | 0.9630 | 0.9443 | 0.9487 | 0.9511 | 0.9511 | 0.9511 | 0.9511 |
13 | 0.6632 | 0.6261 | 0.6465 | 0.6632 | 0.6632 | 0.6465 | 0.9630 | 0.9691 | 0.9659 | 0.9630 | 0.9630 | 0.9659 | 0.9421 | 0.9404 | 0.9416 | 0.9421 | 0.9421 | 0.9416 |
14 | 0.6804 | 0.7087 | 0.6504 | 0.6804 | 0.6504 | 0.6804 | 0.9756 | 0.9728 | 0.9783 | 0.9756 | 0.9783 | 0.9756 | 0.9535 | 0.9549 | 0.9513 | 0.9535 | 0.9513 | 0.9535 |
15 | 0.7380 | 0.6943 | 0.5867 | 0.5867 | 0.5867 | 0.5867 | 0.9667 | 0.9705 | 0.9777 | 0.9777 | 0.9777 | 0.9777 | 0.9527 | 0.9511 | 0.9403 | 0.9403 | 0.9403 | 0.9403 |
16 | 0.7128 | 0.7128 | 0.7128 | 0.6490 | 0.6490 | 0.6490 | 0.9686 | 0.9686 | 0.9686 | 0.9745 | 0.9745 | 0.9745 | 0.9499 | 0.9499 | 0.9499 | 0.9454 | 0.9454 | 0.9454 |
17 | 0.7600 | 0.7280 | 0.7600 | 0.7280 | 0.7280 | 0.6944 | 0.9727 | 0.9755 | 0.9727 | 0.9755 | 0.9755 | 0.9782 | 0.9596 | 0.9586 | 0.9596 | 0.9586 | 0.9586 | 0.9568 |
18 | 0.8184 | 0.6601 | 0.5614 | 0.6601 | 0.6601 | 0.6601 | 0.9452 | 0.9563 | 0.9619 | 0.9563 | 0.9563 | 0.9563 | 0.9418 | 0.9411 | 0.9334 | 0.9411 | 0.9411 | 0.9411 |
19 | 0.5350 | 0.5350 | 0.5350 | 0.5350 | 0.4334 | 0.4334 | 0.9399 | 0.9399 | 0.9399 | 0.9399 | 0.9456 | 0.9456 | 0.9131 | 0.9131 | 0.9131 | 0.9131 | 0.8954 | 0.8954 |
20 | 0.5916 | 0.6589 | 0.5916 | 0.5916 | 0.5916 | 0.5916 | 0.9595 | 0.9521 | 0.9595 | 0.9595 | 0.9595 | 0.9595 | 0.9265 | 0.9319 | 0.9265 | 0.9265 | 0.9265 | 0.9265 |
Mean | 0.6853 | 0.6787 | 0.6447 | 0.6356 | 0.6291 | 0.6238 | 0.9644 | 0.9653 | 0.9681 | 0.9689 | 0.9693 | 0.9697 | 0.9427 | 0.9451 | 0.9416 | 0.9402 | 0.9392 | 0.9349 |
Algorithms | Better Than | |
---|---|---|
DRIVE (Figure 1b) | TSA | - |
AOA | TSA (3.50 × 10−3) | |
SCA | TSA (2.4621 × 10−4) | |
CSA | TSA (2.2337 × 10−5) AOA (4.1790 × 10−4) SCA (2.2337 × 10−5) | |
GNDO | TSA (1.8307 × 10−5) AOA (3.6315 × 10−4) SCA (1.8307 × 10−5) | |
GOBC-PA | TSA (1.0269 × 10−5) AOA (2.4155 × 10−4) SCA (1.0269 × 10−5) | |
STARE (Figure 2a) | TSA | - |
AOA | - | |
SCA | TSA (1.2335 × 10−4) AOA (1.70 × 10−3) | |
CSA | TSA (2.2337 × 10−5) AOA (4.1790 × 10−4) SCA (2.2337 × 10−5) | |
GNDO | TSA (1.4191 × 10−5) AOA (3.0345 × 10−4) SCA (1.4191 × 10−5) | |
GOBC-PA | TSA (1.0269 × 10−5) AOA (2.4155 × 10−4) SCA (1.0269 × 10−5) |
DRIVE (Figure 1b) | STARE (Figure 2a) | ||
---|---|---|---|
CSA | Minimum MSE | 0.6631 | 0.5953 |
CPU Time (seconds) | 2.8103 | 3.4689 | |
TSA | Minimum MSE | 0.7077 | 0.6189 |
CPU Time (seconds) | 0.4152 | 0.4869 | |
AOA | Minimum MSE | 0.6803 | 0.6138 |
CPU Time (seconds) | 2.6651 | 3.4248 | |
GNDO | Minimum MSE | 0.6576 | 0.5918 |
CPU Time (seconds) | 5.2565 | 6.6776 | |
GOBC-PA | Minimum MSE | 0.6562 | 0.5920 |
CPU Time (seconds) | 6.4216 | 8.3765 | |
SCA | Minimum MSE | 0.6687 | 0.5996 |
CPU Time (seconds) | 2.6980 | 3.4260 |
Elapsed Time for NFEs (seconds) | |||
---|---|---|---|
Algorithms | Total NFEs | DRIVE (Figure 1b) | STARE (Figure 2a) |
CSA | 1010 | 2.1720 s (77.28% of the total CPU time) | 2.6960 s (77.72% of the total CPU time) |
TSA | 110 | 0.2820 s (67.92% of the total CPU time) | 0.3440 s (70.64% of the total CPU time) |
AOA | 1010 | 2.0740 s (77.82% of the total CPU time) | 2.6720 s (78.02% of the total CPU time) |
GNDO | 2000 | 3.9220 s (74.61% of the total CPU time) | 5.1320 s (76.85% of the total CPU time) |
GOBC-PA | 2710 | 5.0150 s (78.09% of the total CPU time) | 6.5470 s (78.16% of the total CPU time) |
SCA | 1010 | 2.0750 s (76.91% of the total CPU time) | 2.6120 s (77.20% of the total CPU time) |
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Çetinkaya, M.B.; Adige, S. Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms. Appl. Sci. 2025, 15, 5693. https://doi.org/10.3390/app15105693
Çetinkaya MB, Adige S. Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms. Applied Sciences. 2025; 15(10):5693. https://doi.org/10.3390/app15105693
Chicago/Turabian StyleÇetinkaya, Mehmet Bahadır, and Sevim Adige. 2025. "Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms" Applied Sciences 15, no. 10: 5693. https://doi.org/10.3390/app15105693
APA StyleÇetinkaya, M. B., & Adige, S. (2025). Retinal Vessel Segmentation Using Math-Inspired Metaheuristic Algorithms. Applied Sciences, 15(10), 5693. https://doi.org/10.3390/app15105693