ISSN 1060992X, Optical Memory and Neural Networks (Information Optics), 2015, Vol. 24, No. 1, pp. 36–47. © Allerton Press, Inc., 2015.
Methods to Evaluate the ThreeDimensional Features of Blood Vessels N. Yu. Ilyasova The Image Processing Systems Institute, Russian Academy of Sciences, Samara Korolev State Aerospace University (National Research University), Samara, Russia email:
[email protected] Received October 18, 2014; in final form, December 8, 2014
Abstract—The paper offers a mathematical model for imaging blood vessels. The model relies on divi sion of bloodvessel geometric characteristics into local parameters, which are calculated directly in the process of vessel tracing, and integral (global) features, which are used later for forming classifica tion features. Methods and algorithms to evaluate geometric features of threedimensional three structures are considered. The immunity of evaluation methods to various kinds of noise and possibil ity to cluster a bloodvessel sample are investigated experimentally. Keywords: blood vessels, geometric features DOI: 10.3103/S1060992X15010014
INTRODUCTION Being one of hightech applied sciences, today’s medicine keeps developing new effective methods of early disease detection. Last decades have seen a noticeable advance in the field of medical instrumentation. An important tool in medical imaging systems, the computeraided image analysis has improved diagnostic efficiency significantly. Ophthalmology and cardiology are the branches where information technologies are introduced most intensively. The research in these fields focuses on analysis of images of retinal and heart blood vessels: this kind of images holds important diagnostic information. The results of bloodvessel exam ination allow the physician to draw conclusions about not only the condition of a particular internal organ, but also the systemic disease (e.g. diabetes, polycythemia, anemia, and hypertension). The object of research in ophthalmology, the eyeground is a complex organ with a dense ramified bloodvessel system. Some pathology of this system can accompany such diseases as diabetic retinopathy [1] and hypertension [2]. The examination of the microcirculatory bloodstream of the eyeground is also used in treatment of systemic atherosclerosis, cerebral crisis, kidney malfunction etc. What is most valu able here is that important diagnostic information about the eyeground can be obtained with noninvasive aftereffectfree means. The availability of noninvasive examination and visualization turns the eyeground bloodvessel system into a most informative means for analyzing local microcirculatory bloodstream and evaluating the general hemodynamics. This simplicity of use made it possible to develop innovative meth ods of bloodchannel examination and introduce them in practice and accumulate much knowledge of how different diseases affect the condition of the circulatory system. The cardiocirculatory system is the object of study in cardiology. Having a relatively small number of blood vessels, it should be considered as a threedimensional object. Atherosclerotic lesions of blood ves sels are the cause of many diseases (atherosclerosis, hypertensive disease, ischemic heart disease, cerebral hemorrhage). So the comprehensive study of this kind of blood vascular systems is important for diagnosing many diseases of the man [3]. Vascular malformations are the key problem in today’s medicine. Though there’s been a considerable progress in detection and treatment of vascular diseases for the last few decades, the number of vascular lesion cases keeps growing (Semyonova, Branchevsky, Vodovozov, et al). Since the efficiency of vascular malformation treatment much depends on how exact the phase and severity of the disease are determined, one of important trend in disease detection and treatment is improvement of differential diagnostics methods, including prediction of incipient diseases and presymp tomatic diagnosis (Gavrilova, Kupeev, et al). 36
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The Russian school of physiologists and morphologists studying blood circulation attach much impor tance to the circulatory system geometry (Gaikin, Aleksandrov, Shagal). In their opinion the vascular geometry is not random; rather it reflects the way the blood vessels adapt to diseases. The western medical school pays more attention to the quantitative analysis of dynamic parameters of bloodstream: linear and volume bloodstream velocities. The approach determines the development of instrumentation in this field (Blondal, Gilmore, Kotlar). The clinical application of such quantitative bloodstream analysis is limited by two factors: (1) complex and expensive technology and (2) interpretation of measurements is rather ambiguous. The use of computers and imaging systems and techniques made it possible to use automated image processing methods for diagnostic purposes, thus making the diagnostic faster and more reliable. The analysis of the local diameter of a blood vessel is a prevailing method at present. This approach can’t give sufficient information to understand the development of many diseases. Other statistical param eters such as diameter variations of the vessel (venous beading, arterial idiospasm), its curvature and sin uosity [3–12] are important clinical indications which help to estimate the current stage of the pathology and determine the probable course of the disease. It is very important to develop a new diagnostic method that would make it possible to surpass the abovementioned drawbacks and widen the current understand ing of pathologic processes of such disease as diabetic retinal angiopathy, which includes microangiopathy (associated with changes of eyeground blood vessels) and macroangiopathy (related to atherosclerotic processes in heart vessels). It is also important to develop new effective integral indications that would allow us to estimate the retinal vascular system condition using images of the eye ground and coronary heart vessels. This kind of indications can be used as a basis for designing automated diagnostic systems which would make it possible to standardize the diagnostic process and shorten the examination time [13, 14]. A single approach is offered for analysis of the two classes of images: microcirculatory system of the eye and cordial vascular system. The approach involves the estimation of a collection of geometric parameters of blood vessels which would facilitate further diagnostic analysis. THE MATHEMATICAL MODEL OF A BLOOD VESSEL SEGMENT A mathematical model to represent blood vessels is offered. The model uses the division of geometric characteristics of a vessel into local and integral (global) parameters. The local parameters are computed directly in the course of blood vessel tracing, and integral parameters are used to form classification fea tures [15]. To form local parameters, let us introduce such notions as characteristics of the vessel path and configurations of its boundaries, which we call the walls of the vessel: b b b b b ⎪⎧x1 = x1 (t), y1 = y1 (t ) ⎪⎧x1 (t ) = x(t ) − r(t )sin ϕ, , ⎨ b ⎨ b b b b ⎩⎪x2 = x2 (t), y2 = y2 (t ) ⎩⎪y1 (t ) = y(t ) + r(t )cos ϕ. We define the path parametrically by setting a central line γ(t) and function of the vessel thickness r(t) determined as the distance from the path to the boundary of the branch perpendicular to the route:
⎧γ (t ) = ( x(t), y(t)) , ∂γ ≠ 0 ⎪ ∂t ⎨ ⎪⎩r = r(t), 0 ≤ t ≤ Lv , where t is a natural parameter, the distance measured along the path from its origin; Lv is the length of the path. The central line of the vessel is a line formed by the middle points of the vessel transverse sections. This line can be represented as a continuous regular curve with natural parameterization. Then it is possible to uniquely determine such local parameters of the vessel as the direction of the path at each point ϕ(t), dx(t ) dt , local height f(t) (it is determined as the distance from the current point of the path to dy(t ) dt its projection onto line segment L going from the start point to the end point of the path), and curvature of the vessel k(t), k (t) = d 2 γ (t) dt 2 . These parameters are evaluated by the image in the course of vessel tracing and permit us to define such geometric characteristics of the vessel as the variation of the vessel tan ϕ(t ) = −
3
2 ⎛ d 2 y t ⎞ ⎡ ⎛ dy t ⎞ ⎤ 2 thickness S = r 2 − r 2 r , mean curvature K r = 1 1 + ⎢ ⎥ dt, straightness Pr = Lv L , ⎜ ⎟ Lv 0 ⎝ dx t2 ⎠ ⎣⎢ ⎜⎝ dx t ⎟⎠ ⎦⎥ configurations of the vessel path and walls. For describing the configurations of the vessel path and walls let us introduce the assumption about the harmonic behavior of the functions of direction and thickness:
∫
Lv
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ILYASOVA z r(t) t R x
y
Fig. 1. The threedimensional model of a vessel.
ˆ(t ) = A1 cos(ω1t + α1). We will consider only the segments of a vessel that corre rˆ(t) = A0 cos(ω0t + α 2 ), ϕ spond to the fundamental harmonic with the frequency proportional to the number of extremums of these functions according to the original definition of tortuosity in medicine. It means that we ignore insignifi cant changes that usually happen at the end of the vessel. So we can define the following geometric char acteristics of vessels that are convenient in visual diagnostics: the degree of bending of the vessel wall A0, tortuosity of the vessel wall ω0, the degree of bending of the vessel path A1, and tortuosity of the vessel path ω1. They are defined as amplitude (A0, A1 are thickness and path amplitudes) and frequency (ω0, ω1 are thickness and path frequency) characteristics of the thickness function and direction function the esti mation methods for which are given in [16–29]. To examine the threedimensional vessel structure, let us introduce a spatial model of vessels (Fig. 1) determined by the following parametric equation of the vessel path: ⎧ x = t, ⎪⎪y = A1 sin ω1t sin ωr t, ⎨ ⎪z = A1 sin ω1t cos ωr t, ⎩⎪r = r + A0 sin ω0t, where A1 is the amplitude of path variations, A0 is the amplitude of thickness variations, ω1 is the frequency of path variations, ω0 is the frequency of thickness variations, ωr is the frequency of path rotation. Diagnosing pathology usually involves inspecting the angiogram and, therefore, is rather a subjective process depending on the angle of view of the recorded Xray. The purpose of the spatial model of the vas cular system is to give visual geometric and topologic information allowing higher accuracy of measure ments and numerical estimates of the blood vessel geometry [30–35]. Papers [17, 22, 27] present vessel tracing algorithms and estimations of vessel global features based on the twodimensional model of vessels. The local parameters derived from the tracing of a branch in both two and threedimensional cases are the diameter and direction of the branch at each point. The global features (the mean diameter, straight ness, venous beading, amplitude and frequency of thickness variations, thickness tortuosity, amplitude and frequency of path bends) are computed using local parameters. Articles [22, 27] define these features for twodimensional structures and describe the difference for threedimensional models. Attention is also paid to such a specific feature of a threedimensional image as “frequency of rotation”, which char acterizes the departure of the path from the plane. METHODS TO EVALUATE THE FEATURES All the features considered in the paper are geometrical and computed by using conventional methods of mathematical analysis and appropriate numerical methods for finding distances and derivatives. The OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)
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discretization errors, quantization noise and input device noise result in the processed data also holding many errors and, therefore, the numerical methods giving rather inaccurate results. Using one of most popular features describing the varying shape of a vessel—the curvature as an example it is possible to show that the relative error of the feature grows indefinitely at points of vessel bending, the number of such points growing as the discretization interval decreases. Evaluating the second and firstorder derivatives with numerical methods and assuming for simplicity that vertical and horizontal discretization intervals
(
)
3
2 2 ⎡ ⎤ are equal, we get the following definition of the curvature: K r (x) = y1 + y32− 2y2 ⎢1 + y3 − y1 ⎥ , where 2h h ⎣ ⎦ y 2 = y(x), y1 = y(x − h), y3 = y(x + h), Δ y = h 2 is the greatest error in measuring the coordinates. With the derivatives being approximated by finite differences, the computation of the feature gives the relative error [36]: 3
δK r =
∑ ∂∂y ln K i =1
r
( y1, y2, y3 ) Δyi = 1 42 Kr h
i
1
(
)
3 2 2 ⎤
⎡ y3 − y1 ⎢1 + 2h ⎥ ⎣ ⎦
=
4Δy = 2 . y1 + y3 − 2y2 hy''
This allows the conclusion that the relative error of the feature grows indefinitely at the vessel bending points. Therefore, inaccurate input data make the use of conventional definitions of mathematical analy sis and numerical methods for figuring the features unreasonable. For this reason it is necessary to develop noiseresistant algorithms for evaluating the features. It is also necessary to use global characteristics (which are derived from a large body of input data using the whole diagnostic image) rather than local geo metric parameters (local curvature, local direction, etc.) to make the evaluation of diagnostic features more accurate. In both two and threedimensional case a small enough segment of the vessel can be considered as a portion of a cylinder. The projection of a cylinder onto a plane not perpendicular to its axis is a figure whose transverse size is equal to the diameter of the cylinder (except the end portions). That is why all thickness features of a branch (the mean diameter, venous beading, thickness amplitude, thickness fre quency, (wall) thickness tortuosity, and straightness) look the same for both two and threedimensional tree structures [22] (see Table 1). Let us define the vessel features for threedimensional vascular systems that supplement the vessel fea tures introduced for twodimensional systems. Let us first introduce the path features for a vessel segment (the amplitude and frequency of path variations) for a 3D vascular tree structure and also the feature describing the offplane departure of the path, the frequency rotation. The Amplitude of Path Variations (for a 3D Tree) To estimate the path variation amplitude, let us introduce the function of distance of a path point from axis Ox (Fig. 1): R(x) = y (x) + z (x). From the parametric equation of the path we get R(x) = A sin ω x . It follows from this expression that the mean value is R(x) = 2 A π and the path variation amplitude is A1 = πR 2. 2
2
The Rotation Frequency of the Path (for a 3D Tree) To calculate the frequency of oscillation of the path ω1 in the threedimensional case, we will use the Fourier transform. The function R(x) is periodic, but not sinusoidal, therefore, to improve accuracy use 2 the following approach: there is the Fourier transform from the centered function R (x), where R 2(x) = (1 2) A12(1 − cos 2ω1x). Then, we obtain the estimate of the frequency of the track:
ω1 = 1 arg max 2 m
N −1
∑
−2πi mn 2 Rn e N
, where Rn2 = −(1 2) A12 cos 2ω1t n.
n =0
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Table 1. Vessel global features Mean diameter
N
D = 2r = 2 N
∑ r(t n )
n =1
Venous beading
S = r2 − r 2 r
Path straightness
Pr = Lv L =
N −1
2 2 2 2 ∑ ( x n − x n+1 ) + ( y n − y n+1 ) ( x1 − x N ) + ( y1 − y N )
n −1
Thickness variation amplitude
A0 = 2r − 2r
Thickness variation frequency
ω0 = 2π arg max R(m), R(m) = N 1
2
2 N −1
∑ r(t n) exp
n =0
(−i 2πNnm)
Thickness tortuosity
I 0 = A0ω0
Path amplitude
⎛ ln I 1 + 1 + I 12 ⎞ N ⎟ , f = N −1 f (t ), A1 = 2 fE(k) ⎜⎜1 + ∑ n ⎟ 2 ⎜ ⎟ I1 1 + I1 n −1 ⎝ ⎠ where E(k) is elliptical integral of the second kind
)
(
Path tortuosity
I1 is determined from the equation 2
Pr = (2 π) 1 + I 1 E (k), k = I 1
Path frequency (for a 2D tree)
ω1 = I 1 A1
Path frequency (for a 3D tree)
ω1 = 1 arg max 2 m
Path rotation frequency
2
ωr = ω1 −
N −1
∑
2
Rn e
−2πi mn N
n =0
2
1 + I1
2 2 , Rn = − 1 A1 cos 2ω1t n 2
(
k 2(1 + A12ω12 ) , 2 A1
k = E −1 πPr 2 1 + A12ω12
)
The Path Rotation Frequency Theoretically the rotation frequency of the path can be evaluated in the same way as the frequency of the path: ωr = (1 2)arg max ℑ(sin ϕ(x)), where ϕ(x) is the polar angle of the projection of a path point on the yOz plane. However, in most cases real blood vessels are described by a lowfrequency function of the path, which can lead to a large error of the evaluation of the feature. For this reason we will use the follow ing approach. Straightness Pr of a curve (see the parametric equation of the path above) can be found as 2 2 Pr = (2 π) 1 + A1 ω1 E (k), where k 2 = A12 (ω12 − ω2r ) (1 + A12ω12 ), E(·) is the full elliptical integral of the sec ond kind. Clear that when ωr = 0, we obtain the formula for the flat case. Using the obtained estimates
Pr, A1, ω1, we get the estimation of the rotation frequency of the path: ωr = ω12 − (k 2(1 + A12ω12 ) A12 ), where 2 −1 2 2 k = E (πPr 2 1 + A1 ω1 ), E–1(·) is the inverse function of the secondkind elliptical integral. So the algorithm for evaluating the feature consists of three steps:
(1) determining k by solving the equation for k numerically:
E (k) = πPr 2 1 + A1 ω1 . 2
2
(1)
(2) making the estimation using the formula: ωr = ω1 − (k (1 + A1 ω1 ) A1 ). 2
2
2
2
2
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Path
Sample Estim.
41
ω1
A1 δ%
Sample
Cal.
δ%
Sample
Cal.
δ%
0.9
0.879
2.31
1.55
1.527
1.47
1.60
1.599
0.04
0.95
0.929
2.20
1.52
1.533
0.84
0.82
0.861
5.01
0.88
0.861
2.13
1.55
1.526
1.55
0.40
0.389
2.75
1.2
1.230
2.52
0.55
0.523
4.85
0.70
0.718
2.58
1.1
1.107
0.64
0.5
0.502
0.5
0.35
0.343
1.86
1.05
1.058
0.76
0.3
0.294
1.83
0.39
0.394
0.95
Fig. 2. The results of testing the estimation methods for vessel path features (ω0 = 9, A0 = 0.1).
The approach has a drawback. The integral of the second kind is known to reach its maximum at k = 0: E max = E(0) = π 2. Then equation (1) can’t be solved for k when the estimation of straightness takes a cer tain value: 2 2 2 2 (2) Pr = (2 π) 1 + A1 ω1 E max = 1 + A1 ω1 . In this event we go to third step. 3) Checking the estimation of straightness. If the estimation is if excess of the value computed by for mula (2), the rotation frequency of the path is evaluated by using the expression: N −1
ω r = arg max m
∑ sin ϕ(t )e n
− 2π i nm N
2
.
n=0
TESTING THE EVALUATION OF THE DIAGNOSTIC FEATURES OF THE 3D VASCULAR MODEL Test threedimensional images consisting of paths whose central lines and walls have preset frequencies and amplitudes and the rotation frequency is also preset (Figs. 2, 3) were used in the experiments. The goal was to test how accurate the diagnostic features of a vessel such as the amplitude and frequency of varia tions of the vessel path and thickness, and the rotation frequency are computed. Given 100 measurement points, the average relative error in the computation of the path features was about 2%, and in computa tions of the vessel thickness features was about 1%. The effect of noise on the quality of the estimations was investigated. With the feature space being clustered, the relationship between the noise and clustering errors was also studied. Additive white Gaussian noise and pulse noise were considered in the tests. A sample of realizations was taken for each noise level. Then a vector of relative errors of feature estimations (the average error of a feature estimation and the boundaries of its confidence interval) was computed for each sample. We define the degree of noise as the noisetosignal ratio: d 2 = Dv Ds , where Dv is the variance of noise and Ds is the variance of signal. Euclidean norm D = D the both kinds of variance.
2
= D x2 + Dy2 + Dz2 is taken as the numerical value for
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ILYASOVA ω0
A0
Vessel fragment Sample
Estim.
δ%
Sample
Estim.
δ%
1
1.002
2.30
0.3
0.295
1.60
2.5
2.497
0.11
1.32
1.329
0.68
3
3.001
0.04
3
2.962
1.26
2
1.997
0.14
6
6.004
0.06
3.5
3.503
0.09
9
9.104
1.15
1.5
1.496
0.23
12
12.06
0.47
Fig. 3. The results of testing the estimation methods for vessel thickness features (ω1 = 0.5, ωr = 0.5, A1 = 1).
(a)
δ 0.30 0.25 0.20 0.15 0.10 0.05 0
0.2
0.4
0.6 0.8 1.0 Noise/Signal
(b)
δ 0.6 0.5 0.4 0.3 0.2 0.1 0
0.2
0.4
0.6 0.8 1.0 Noise/Signal
(c)
δ 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0
0.2
0.4
0.6 0.8 1.0 Noise/Signal
Fig. 4. The error in estimating the path variation amplitude (A1) versus the signaltonoise ratio: (a) additive noise; (b) pulse noise; (c) after application of the filter (R = 5).
The Effect of Additive Noise Let us consider a statistically independent sample of normal distribution ν1,..., ν k ≈ N (0, σ 2 ). The sam ple is a sequence of white noise observations. If we add it to the original multidimensional signal, we get a model of noisecorrupted signal f i n = xin + ν in, i = 1, k, n = 1,4. Four models presented in Table 2 were used to generate the realizations. The models were chosen to provide visual distinguishability of spatial struc tures. 25 spatial structures were generated for each model by using 400 path points at 0.025 intervals. So, 100 realizations were formed for each noisetosignal ratio. Fig. 4a shows the relationship between the error of the amplitude estimation and noisetosignal ratio. The dashed line stands for the boundaries of the confidence interval. The estimation method for this feature proved quite resistant to additive noise: the average relative error just exceeds a 20% threshold even if the signal and noise have the same intensity with the upper boundary of the confidence interval extending to 24%. Should we set a particular critical value (e.g. 5%) for the error, we can say that with the signaltonoise ratio being less than 0.13, the chances that the greatest estimation error exceeds the critical value are 0.05. The aver age error will not exceed the critical value if the noisetosignal ratio is 0.19. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)
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Table 2. Test bloodvessel models Model number
Amplitude of the path
1 2 3 4
1 2 3.1 4
Frequency of the path 0.5 1 1.652 2.54
Rotation frequency
Amplitude of thickness variations
Frequency of thickness variations
The mean value of thickness
0.5 2.14 2.14 1.4
3 1 3 3
1.32 1 1.32 2
4 4 4 4
Table 3. The initial models of cluster centers Feature
Model 1
Model 2
Model 3
Path amplitude
1.02
3.93
2.77
Path frequency
0.53
1.54
1.41
Path rotation frequency
0.52
2.62
1.67
Path straightness
1.2
8.1
4.31
Path tortuosity
0.58
6.05
3.83
Mean diameter
8.47
8.41
8.47
Venous beading
0.49
0.5
0.48
Thickness variation amplitude
2.9
2.96
2.88
Thickness variation frequency
1.34
2.12
1.44
Thickness tortuosity
3.89
6.31
4.15
The Effect of Pulsed Noise In the general case the model describing the pulsed noise implies that signal measurements are substi tuted by random or fixed values with some probability. In the experiment the pulsed noise caused a three dimensional point to move over distance R with probability p = 0.05, the movement happening in a ran dom direction. The models presented in Table 2 were used to generate the test spatial structures. To gen eralize the analysis (to be able to compare the test results for different kinds of noise), ratio pDv (1 − p)Ds is taken as the noisetosignal ratio for pulsed noise. Fig. 4b gives the error of estimating the path variation amplitude. The examination of this plot showed that pulsed noise has a more considerable effect on the estimation quality than additive noise. An error of 20% is observed with the noisetosignal ration being as low as 0.5. That is why a median filter [37] was applied to increase the feature estimation accuracy. The experiments allowed the conclusion that the filter works best when the window size R is 5. Fig. 4c presents the error as a function of noise level after application of the median filter. It is seen that the use of the filter improves the feature estimation quality noticeably – with the noisetosignal ration being 1, the error is less than 10%. Fig. 5 helps to compare the effect of noise on the feature estimation accuracy in three different experimental conditions: 1) additive noise, 2) pulsed noise, and 3) use of filtering to lessen the effect of pulsed noise. Table (a) presents experimental observations when the level of noise is high (the noisetosignal ration is 1), Table (b) holds experimental data for the medium level of noise (the noiseto signal ration is 0.3). The shades of grey indicate the estimation quality (relative error) for different kinds of feature: the dark grey stands for poorest quality, the light grey for medium quality, and the white for highest quality. The possibility to assign the threedimensional structure to one of known classes was investigated to understand the possibility of using the geometric features under consideration for diagnostic purposes. A clustering algorithm based on the kintergroupmeans algorithm [38] was used. Vectors of feature estimations serve as a feature space. A few predefined models (Fig. 6) were used to generate samples for clustering, the models themselves being the initial centers of clusters. Table 3 gives feature estimations for realizations formed from these models. The models were chosen to provide visual distinguishability of spa OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)
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ILYASOVA a:
Condition 1
Condition 2
Condition 3
Error
max
avg
min
max
avg
min
max
avg
min
Path amplitude
0.242
0.213
0.185
0.477
0.431
0.385
0.12
0.08
0.06
Path frequency
0.056
0.031
0.014
5.028
2.476
0.001
0.822
0.261
0.001
Path rotation frequency
0.058
0.046
0.012
0.1924
0.1535
0.1147
0.1166
0.0833
0.05
Path straightness
0.252
0.171
0.121
3.355
3.422
3.313
0.776
0.484
0.337
Path tortuosity
0.301
0.254
0.211
4.833
2.963
0.515
0.362
0.187
0.001
Mean diameter
0.037
0.025
0.01
0.037
0.023
0.016
0.021
0.015
0.004
Venous beading Thickness variation amplitude Thickness variation frequency
0.146
0.091
0.035
0.452
0.378
0.305
0.122
0.097
0.054
0.132
0.088
0.044
0.436
0.384
0.303
0.137
0.091
0.045
0.0230
0.0115
0.001
0.0646
0.0323
0.001
0.0034
0.0017
0.001
Thickness tortuosity
0.132
0.093
0.044
0.436
0.384
0.333
0.126
0.082
0.046
b:
Condition 1
Condition 2
Condition 3
Error
max
avg
min
max
avg
min
max
avg
min
Path amplitude
0.0803
0.0685
0.0576
0.1501
0.1265
0.1030
0.0091
0.0066
0.0041
Path frequency
0.0375
0.0188
0.0001
1.001
0.5005
0.0001
0.0001
0.0003
0.0001
Path rotation frequency 0.0358
0.0185
0.0012
0.1024
0.0645
0.0267
0.0104
0.0052
0.0002
Path straightness
0.0674
0.05
0.0174
1.4988
1.2494
1.0001
0.051
0.0255
0.0001
Path tortuosity
0.1032
0.0851
0.067
1.6734
0.8367
0.0001
0.0002
0.0001
0.0003
Mean diameter
0.0146
0.0095
0.0044
0.015
0.01
0.005
0.0028
0.0017
0.0006
Venous beading Thickness variation amplitude Thickness variation frequency Thickness tortuosity
0.0536
0.0363
0.019
0.1343
0.1121
0.0898
0.0167
0.0083
0.0001
0.0506
0.0337
0.0168
0.1424
0.1118
0.0812
0.0071
0.0036
0.0003
0.0034
0.0017
0.0001
0.032
0.016
0.0001
0.0024
0.0012
0.0002
0.0506
0.0337
0.0168
0.149
0.017
0.0853
0.0167
0.0083
0.0001
Fig. 5. The relative errors of feature estimations for NSR = 1 (Table a) and NSR = 0.3 (Table b). Additive noise (condi tions 1), pulsed noise (conditions 2), using the median filter (R = 5) to diminish the effect of pulsed noise (conditions 3).
(a)
(b)
(c)
Fig. 6. The initial vessel models. (a) model 1 ( A = 1; ω = 0.5; ωr = 0.5; A0 = 3; ω0 = 1.32); (b) model 2 ( A = 4; ω = 1.4; ωr = 2.54; A0 = 3; ω0 = 2); (c) model 3 ( A = 2.5; ω = 1.5; ωr = 1.5; A0 = 3; ω0 = 1.4). OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)
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tial structures. Figure 6 shows the experimental observations obtained for each of the three models. To carry out the clustering, 150 test realizations were generated by randomly changing the parameters of the initial models. The experimental results showed that 137 of 150 generated vessel images were clustered correctly, 13 test images (2 for model 1, 4 for model 2, 7 for model 3) failed to be clustered (i.e. the initial model for a particular test image was not determined correctly). It means that the geometric features under consideration can quite be used for diagnostic purposes. CONCLUSIONS A new generalized mathematical model is offered to facilitate the visualization of the retinal and car diac vascular systems. The model allows us to formalize the description of geometric parameters, evaluate them and form diagnostic features. A basic set of bloodvessel geometric parameters is defined that includes estimations of local directions and thicknesses of vessels. The set underlies a method that allows the information important for diagnostic purposes to be selected from blood vessel patterns. Methods and algorithms of evaluating geometric parameters of 3D vascular systems are presented. The effect of additive and pulsed noise on the quality of diagnostic feature estimations was investigated. Model bloodvessel images were used to perform the clustering of the feature space with the effect of noise on the clustering efficiency being examined. The experimental results confirmed the functional stability of the methods. With the noisetosignal ration being no greater than 0.2, the features can be estimated to an average pre cision of no greater than 5%. The experiments showed that pulsed noise has the largest effect on the effi ciency of feature estimations. It is necessary to use a median filter with a window size of 5 to reduce the effect of pulsed noise and increase the estimation accuracy. The test clustering of features was done to show that it is quite possible to use the features for determining the condition of vascular systems. With three visually distinguished model structures of a blood vessel being used as clustering centers, the classi fication error was 8%. ACKNOWLEDGMENTS The work was supported by the Ministry of Education and Science of the Russian Federation; RFBR grants 120100237a, 140100369a, 140797040p; the 20132014 Program “Bioinformatics, modern information technologies and mathematical methods in medicine” of the basic research of DNIT of RAS. REFERENCES 1. Teng, T., Progress towards automated diabetic ocular screening: a review of image analysis and intelligent sys tems for diabetic retinopathy, Medical & Biological Engineering & Computing, Teng, T., Claremont, D., and Lefley, M., Eds., 2002, vol. 40, no. 1, pp. 2–13. 2. Foracchia, M., Extraction and quantitative description of vessel features in hypertensive retinopathy fundus images, Book Abstracts / 2nd International Workshop on Computer Assisted Fundus Image Analysis, Foracchia, M., Ed., 2001, p. 6. 3. Hiroki, M., Tortuosity of the white matter medullary arterioles is related to the severity of hypertension, Cere brovascular Diseases, Hiroki, M., Miyashita, K., and Oda, M., Eds., 2002, vol. 13, no. 4, pp. 242–250. 4. Bribiesca, E., A measure of tortuosity based on chain coding, Pattern Recognition, 2013, vol. 46, issue 3, pp. 716–724. 5. Cheung, C.Y., Retinal vascular tortuosity, blood pressure, and cardiovascular risk, Ophthalmology, Cheung, C.Y., Zheng, Y., Hsu, W., Lee, M.L., Lau, Q.P., Mitchell, P., Wang, J.J., Klein, R., and Wong, T.Y., Eds., 2011, vol. 118, no. 5, pp. 812–818. 6. Martin Rodriguez, Z., Improved characterisation of aortic tortuosity, Medical Engineering & Physics, Martin Rodriguez, Z., Kenny, P., and Gaynor,L., Eds., 2011, vol. 33, no. 6, pp. 712–719. 7. Bullitt, E., Analyzing attributes of vessel populations, Medical Image Analysis, Bullitt, E., Muller, K.E., Jung, I.,Lin, W., and Aylward, S., Eds., 2005, vol. 9, no. 1, pp. 39–49. 8. Sasongko, M.B., Alterations in retinal microvascular geometry in young type 1 diabetes, Diabetes Care, Wang, J.J., Donaghue, K.C., Cheung, N., Jenkins, A.J., BenitezAguirre, P., and Wang, J.J., Eds., 2010, vol. 33, no. 6, pp. 1331–1336. 9. Johnson, M.J., Robust measures of threedimensional vascular tortuosity based on the minimum curvature of approximating polynomial spline fits to the vessel midline, Medical Engineering & Physics, Johnson, M.J. and Dougherty, G., Eds., 2007, vol. 29, no. 6, pp. 677–690. 10. Ilyasova, N.Yu., Methods for digital analysis of human vascular system. Literature review, Computer Optics, 2013, vol. 37, no. 4, pp. 511–535, ISSN 01342452. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)
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