ABSTRACT A mathematical method of geometrical parameters estima-tion of a retina vessel fragment is presented. The application of the tracing procedure for the vessels geometric parame-ters extraction upon the images is proposed. To perform the estimation of a vessels thickness and direction a method, based upon the fan-beam transformation, was developed. Experimental studies of the global diagnostic parameters computation accuracy were conducted on the pathological retina images. The proposed method allows performing a differentiated analysis of an image to diagnose the retinal diseases.
1. INTRODUCTION
Although recent decades have seen obvious advances in diagnostics and treatment of ophthalmologic diseases more people are suffering from retinal impairments of vascular genesis. The efficiency of treatment of vascular retinal pa-thologies essentially decreases with the disease progress/ Thus, the modern research has been focused on the devel-opment of the maximally objective methods of diagnostics at the earliest disease stages and also on the ways to enhance the informativeness of the analysis.
Among the most frequent and prognostically unfavour-able diseases is diabetic retinopathy (DR). Because the early DR stages are marked by retinal vascular changes in abso-lute and relative diameter ratios of arteries and veins, growth of new vessels, increased vessel tortuosity, etc., the devel-opment of digital and computer technologies for studying the retinal vascular system may show promise in early DR diagnostics [1].
At present, the development of such technologies is as-sociated with improvement of systems for high-quality reti-nal image acquisition and development of methods for quan-titatively estimating the blood flow status [2]. Visible ophthalmologic changes in retina vessels produce an integral blood vessels characteristic, which allows to perform the accurate vascular pathology diagnostics.
The existing methods, that allows one to measure the vessel characteristics [3,4,5], could not be used to perform the analysis of a thinned vessels or vessels with the high cur-vature. For estimation of the vessels parameters commonly are used methods based on the approximation of an intensity
distribution curve [6,7] and methods based on application of a neural network [8]. But these methods are unstable in the presence of noise and disturbances like closely located ves-sels, works in a semi-automatic mode, use unreasonable models and applied on a narrow class of images.
In our previous papers [9,11] we described a computer system for early diagnostics of the blood vessels pathology. In this paper we introduce and discuss the methods to evalu-ate the characteristics of blood vessels for digital retinal im-age analysis, which allow the accuracy of vascular pathol-ogy diagnostics to be enhanced.
2. TRACING OF THE MICROVASCULAR SYSTEM
Previously we proposed to use a novel approach for ana-lyzing the retinal image via tracing of the vessel [9]. The method employs a scanning polar frame and allows us to calculate local vessel features (diameter and direction at each point). The automated tracing is performed via the assign-ment of the original and final points of scanning and takes into account the found direction of vessel in the current point (Fig. 1).
Figure 1 – The tracing procedure
Once the excerpt is filtered in order to diminish noise in the initial image the search for the directions is based on the analysis of sampling of image field pixels caught into the scanning polar frame. After the sampling filtration, which is necessary for the reduction of noise on the initial image, a function of gray-level intensity distribution of the region in-side the scanning frame is derived.
15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP
3. THE LOCAL FAN-BEAM TRANSFORMATION
The main idea of fan-beam transformation method (FBT) is to analyze the distribution of image gray-levels inside sec-tors, subject to radius, size and angle (Fig. 2) [10].
1 2
3
4 5
θ
α
R Δ
θ θ
θ θ
Figure 2 – Circular framework with various rotated sectors
used to perform FBT method To calculate the FBT in each sector we proposed to use
the next two formulas [10]: 2
0 0 0 02 0
( , , , , ) ( cos , sin )r
F x y r f x t y t dtdα θ
α θ
α θ ϕ ϕ ϕ+
−
= + +∫ ∫
[
]
2
0 0 0 02 0
20 0
( , , , , ) ( cos , sin )
( , , , , )
r
D x y r f x t y t
F x y r dtd
α θ
α θ
α θ ϕ ϕ
α θ ϕ
+
−
= + + −
−
∫ ∫
0 0( , )x y – base point,α – polar angle of sector, θ – angular size of sector, r – radius of sector. These formulas actually represent the average and dispersion of the retina image pixels intensity inside a sector.
Figure 3 – Vessels images samples
(bifurcation in the left, close located vessels in the right)
Figure 4 – The plots of FBT functions (upper – average, lower – dispersion)
for the cases presented in fig. 3 (bifurcation in the left, close located vessels in the right)
To find the angles corresponding to the directions in which a vessel enters and leaves the polar frame (see fig. 3) global minima of the fan-beam transformation function (see fig. 4) are obtained.
The procedure involved a sliding local approximation by a second-order polynomial. The search for minimums corre-sponding to the vessel centers is carried out analytically, with the subsequent analysis aimed at discarding false minimums.
In the course of tracing, the current point moves towards the vessel output from the frame. In the case of branching, it moves along the direction which is chosen from the earlier calculated one as the most close to the direct movement to the final point (fig. 5). The process is finished after the cur-rent point has approached the final one over a distance that is less than the step of tracing .
Figure 5 – An example of a resulting trace
4. GEOMETRICAL PARAMETERS ESTIMATION
The local geometrical features involve the route (the locus of medial points), as well as the distribution of the branch di-ameter and the direction along the route. These are calculated immediately from the image using an algorithm for vessel tracing – central line tracing. In order to determine a set of geometrical parameters and enhance the measurement accu-racy we have developed a mathematical model of a vessel branch (Fig.6)
Figure 6 – Mathematical model of a branch fragment
The parameters of the model are defined by the follow-
ing functions: )(txx = , )(tyy = , )(trr = , vLt ≤≤0 , where )(tx , )(ty are differentiable functions defining a cen-
15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP
ter line hereafter called the route; )(tr is the branch thick-ness function (the distance from the route to the vessel boundary reckoned along the perpendicular to the route); t is the distance from the route initial point measured along the route; and vL is the route length.
This model uniquely define the local features calculated immediately from the image. The global features include the average diameter, linearity, beading, thickness variation am-plitude, thickness variation frequency, thickness tortuosity, route variation frequency, route variation amplitude, and route tortuosity. These serve to characterize the entire vessel on the whole and are later used as diagnostic features.
The vessel thickness variation amplitude 0A character-izes the deviation of the vessel walls from a straight line
and is defined as 2 20 2 2A r r= − where r is the vessel
average radius and 2r is the radius mean square. The vessel thickness variation frequency 0ω character-
izes a change in the wall direction per unit length and is de-
fined as: 0 02 mNπω = . 0 arg max ( )
1m R m
m N⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠< <
– is the num-
ber of maximal value of Fourier-spectrum 1
0
2( ) ( ) expN
nn
nmR m r t iNπ−
=
⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠∑ of the thickness function.
The route tortuosity 1I characterizes the rate of change of the route function at a selected segment, which is ap-proximated by a harmonic function of amplitude and fre-
quency is derived from 21
2 1 ( )P I E kπ
= + ⋅
21 11k I I= + , where P is the branch linearity, and ( )E k
is the total elliptic integral of the 2nd kind. The route variation amplitude 1A characterizes the de-
gree of deviation of the route trajectory from the straight line
and is defined as ( ) ( )1 2 2
1 1 1 1
2 ( )
1 ln 1 1
f E kAI I I I
⋅=+ + + +
where 1I is the route tortuosity, f is the vessel average height.
The vessel fragment route variation frequency 1ω char-acterizes how often the direction of the branch is changed per unit length and is defined as 1 1 1I Аω = .
5. RESULTS AND DISCUSSION
Given below are the experimental studies of the computation accuracy of the following four global parameters: route variation amplitude, route variation frequency, thickness variation amplitude, and thickness variation frequency. In studying the route parameters, images of ideal routes with their trajectories defined by a sinusoidal function of different frequency and amplitude were generated (Table 1).
The error in determining the route frequency is caused by the error introduced by the branch tracing algorithm. In
studies of the thickness parameters, images of ideal routes with trajectories of their boundaries defined by a sinusoidal function of different frequency and amplitude were generated (Table 2). The error in constructing the parameter estimates is caused by the effect of route image discretization.
The studies conducted have shown that the above-discussed features can be used for assessment of the general retinal pathology.
Table 1. The results of the experimental studies of the methods for estimating the route parameters on test images
Thickness Amplitude Frequency Ideal Estimated Ideal Estimated
7.0 7.041 9.5 9.482 To analyze retinal images in this paper were presented
an approach, based on application of the fan-beam trans-formation method. The performance of the algorithm for estimating the vessel local geometrical features was checked on modeled and original images of the optic disc region of a retina with the diabetic retinopathy.
In the course of studies, we examined how the algo-rithm parameters affected the accuracy of vessel detection at the optic disc edge. Out of all the experiments conducted, below we discuss only experiments concerned the impact of noise on the direction estimation accuracy in 50 test images (Fig. 7).
Figure 7 – The MSE of the direction estimation vs. noise
level calculated on two different image sets ( Image type 1 and Image type 2), Method 1 – previously developed
15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP
The research has shown that the new method increases the accuracy of an estimation of vessel directions as com-pared with previously developed methods. The results of the parameters estimation on a natural retina image are pre-sented in fig. 8
Figure 8 – The result of the vessels directions detection
The obtained results indicate the probability to use the method to determine the vessels direction on the edge of the OD region. So the evaluation of the type of the pathology becomes more stable to additive noise.
The methods for estimation of the geometrical parame-ters, that are discussed is this paper have formed the basis for a computerized system for measuring the geometric parameters of biomedical images. The system allows the objective quantitative results to be derived and extends the capabilities of the existing medical methods.
Introduction of the developed methods into medical use will enhance its capabilities and allow automatic diag-nostics of some diseases and monitoring of pathological retinal changes on the basis of objective quantitative data. Acknowledgements: This work was financially supported by the U.S. Civilian Research and Development Foundation (CRDF) and the Ministry of Education of the Russian Fed-eration international grant under Basic Research and Higher Education (BRHE) program (№ RUX0-014-SA-06); by the Russian Foundation for Basic Research grant (№ 06-07-08006, № 07-08-96611); by the “Fundamental Sciences for Medicine” program.
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