Estimation of the geometric parameters of blood vessels is an important stage in the diagnosis of many cardiovascular diseases. In this work, we describe a method for estimating the diameter of blood vessels based on a modified local fan transform. We present experimental results that show in which way the accuracy of blood vessel estimation is affected by the noise-to-signal ratio in the image under analysis, vessel curvature radius, and the number of points and angles over which the averaging is done. The method is experimentally shown to be immune to various types of noise, structural complexity of the object, and variations in the vessel curvature radius.
variance of the brightness function f x, y . By analyzing these parameters as a function of angle, it is possible to detect the vessel in a given point, estimating its width and direction.
F (x0 , y0 , a , q, r) = D(x0 , y0 , a , q, r) = 1 a + q 2 r
т т f (x0 + t cos j ,y0 + t sin j )dtdj , Sq a - q 2 0 1 a + q 2 r
т т [f ( x0 + t cos j , y0 + t sin j ) - F (x0 , y0 , a , q, r)]2 dtdj ,
Sq a - q 2 0
(
The MLFT-aided method enables a local vessel direction to be evaluated. When compared with the LFT-aided approach, the MLFT method enables the radial profile with more pronounced minima to be obtained, with the local minima corresponding to bifurcation directions (Fig. 2).
Thus, for a vessel direction to be identified, an optimization problem of searching for minima needs to be solved for each angle-dependent radial profile for a specific radius [2]. The algorithm operates by analyzing a list of directions, which represents a vector for each radius, with its length being equal to the number of directions and its magnitude taking a unit value if a bifurcation is detected and being zero otherwise. The list of directions may contain regions of constant values equal to unit y.
This is the indication of detecting a bifurcation with larger-than-pixel width. In this case, the unit value is taken just at the central region's pixel.
The vessel width will be evaluated using an LFT and on the assumption that the vessel is in parallel with the OX-axis (Fig. 3). а)
b)
The fan transform of Eq. (
Fr , r Fr 2 , r d 2r
Fr 0, r Fr 2 , r sin
Fr 0, r Fr , r
Here, the angle a is such that the interval of integration intersects both boundaries of the vessel. Hence, the diameter can be evaluated as
In practice, diameter components d1 , d 2 often need to be evaluated. Given a circular window of radius r and a horizontal
vessel, the terms d1 and d 2 can be described by similar sets of equations:
(
With a real vessel having an arbitrary direction, the algorithm for width estimation is, at first, given the vessel direction defined by an angle a s , which is preliminarily calculated by the direction algorithm. With a specific value prescribed to the angle, the diameter components are evaluated as
F (a1 + a s , r) - F (p 2 + a s , r)
d1 = r (
- F (0 + a s , r) + F (a1 + a s , r) d2 = r
F (a 2 + a s , r) - F (3p 2 + a s , r)
F (0 + a s , r) - F (3p 2 + a s , r)
sin (a 2 - p )
- F (0 + a s , r) + F (a 2 + a s , r)
(
To enhance the accuracy of estimating the diameter components, a value averaged over N different angles will be considered. Below, an estimate for a single diameter component is given: d1 1 N r Ff k s , r Ff 2 s , r
N k 1,k c1, c1 Ff 0 s , r Ff 2 s , r Ff 0 s , r Ff k s , r
sin k
If Eqs. (
We analyzed in which way the accuracy of estimating the vessel diameter depends on the noise-to-signal ratio for additive white noise. Figure 5 shows the relative diameter estimation error as a function of the noise-to-signal ratio: 2 1 Ndˆi d 2 d 2 , where dˆ is the evaluated diameter, d is the real diameter measured on a test image, and r is supposed
N i1 to be not smaller than d sin 3 .
0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 2 0,04 0,09 0,18 0,26 0,40 0,52 0,72 0,87 1,12
noise/signal
Fig. 5. Error of width estimation vs. the noise-to-signal ratio. 0,16 0,14 0,12 0,1 0,08 0,06 0,04 0,02 0 2
The study conducted on synthetic images has shown the method for estimating local parameters to be immune against the additive noise. For instance, the error of estimating the local diameter was found to be not larger than 8% given the noise-tosignal ratio under 0.25.
20 25 30 35 40 45 50 55
60 беск
20 25 30 35
40 45 50
55
60 беск b)
The average vessel diameter estimate as a function of vessel curvature is shown in Fig. 6а. As test vascular images, we utilized the images of an annular sector 11 pixels in width and inner circle radius ranging from 20 to 60 pixels. A same -width straight-line segment (of infinite curvature radius) was also analyzed in order to determine an "estimated width". From the above plots, the error of width determination is seen to increase with increasing vessel curvature. Figure 7 depicts in which way t he average vessel diameter estimate depends on the number of averaging points (see section 4, Fig. 4), given the signal-to-noise ratio equal to 25. The experimental results have shown that with the number of averaging points increasing to 13, the error falls from 0.1 till 0.02.
a) 0,25 0,2 0,15 0,1 0,05 0 2 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0
2 0,08 0,06 0,04 0,02
0 3 4 5 6 7 8 9 10 11 12
13
Figure 8 shows the diameter estimation error against the number of averaging angles (scanning sector size) with the r.m.s. signal-to-noise ratio, respectively, equal to 15 and 25. The experimental study showed that the greater is the volume of data used for averaging, the more reliably is the width estimation due to additional filtering of noise.
The experimental study showed that the estimation error can be essentially reduced by performing the averaging over a designated circumference sector of a local fan transform. As a disadvantage, we can mention that this approach is sensible to the highly curved vessels and the inability of the algorithm to be adjusted to the vessel and background brightness.
The directions were evaluated in a noisy 1024 x 1024 image (Fig. 9) with bifurcations uniformly located along the x-and yaxes. The general number of objects was 2,304.
A comparative study of the following methods for vessel direction and bifurcation identification was conducted: a direct method for direction identification (KM method), a method of a local discrete Radon transform, a method of a local discrete fan transform (LDFT) and a modified LFT [2]. The worst results were demonstrated by the algorithm based on the Radon transform variance estimation, because a classical Radon transform is unable to discern two opposite directions, thus leading to numerous cases of false recognition and the increased number of missed-out objects.
The analysis showed that compared to other methods, the MLFT provides the least error of bifurcation angle estimation and the least error of false bifurcation recognition, is able to recognize correctly a larger proportion of bifurcations and the most stable to white noise.
Estimation of local blood vessel parameters used to form diagnostic features is a major problem of modern medicine, enabling an early diagnosis of various vascular pathologies. In this work, we have proposed a method for vessel diameter measurement that exploits a local fan transform. The method is based on the Radon transform, which is modified in such a way that bifurcations, crossovers, and terminations of vessels can be efficiently analyzed in the presence of interfering factors, including spots and close vessels. By analyzing the average brightness and variance of the radial function against the angle, vessel's width and direction can also be estimated and bifurcation points identified. To enhance the robustness, the transfor m is performed for a range of radii. The developed algorithm is stable to noise and disturbances, enabling bifurcations, crossovers, and terminations of vessels to be analyzed with high efficiency in the presence of interfering factors. The results of experiments have been discussed, showing in which way the accuracy of vessel width estimation is affected by the noise-to-signal ratio in the image under analysis, the vessel curvature, and the number of points and angles of averaging. The proposed method has been experimentally confirmed to be stable to various types of image noise.
The worst results have been shown by the estimation technique based on the Radon transform variance estimation because a classical Radon transform is unable to discern between two opposite directions, leading to a large number of false identifications and increased number of missed objects.
This work was partially supported by the Ministry of education and science of the Russian Federation in the framework of the implementation of the Program of increasing the competitiveness of SSAU among the world’s leading scientific and educational centers for 2013-2020 years; by the Russian Foundation for Basic Research grants (# 15-29-03823, # 15-29-07077, # 16-41630761; # 16-29-11698); by the ONIT RAS program # 6 “Bioinformatics, modern information technologies and mathematical methods in medicine” 2016 -2017.