Inertia-Based Vessel Centerline Extraction inRetinal Image

Jihene MalekElectronics and

MicroelectronicsLaborotoryUniversity of Monastir

Email: jihenemalek@yahoo.fr

Rached TourkiElectronics and

Microelectronics LaborotoryUniversity of Monastir

Abstract—Segmentation of blood vessels in retinal imagesallows early diagnosis of disease; automating this process imagesis an important and challenging task in medical analysis anddiagnosis. The main goal of our work is an extraction of thevessel centerline. The method is based on a tracking strategy, thevessel is assumed to be represented by a succession of spatiallyoriented cylinders, and its axis is obtained by the concatenationof the principal axes of inertia of the constituent cylinders. Thecenterline extraction is done by an iterative prediction estimationtracking technique based on a multi-scale analysis of imagemoments and on a shape model close to snakes. Experimentalresults are presented in Retinal images

Keywords — Retinal Image, centerline extraction, mo-ments of inertia, quantification.

I. INTRODUCTION

Retinal images of humans are widely used by theophthalmologists, play an important role in the detectionand diagnosis of many eye diseases [1] Therefore, automaticdetection of structures in retinal images is necessary, andamong them the detection of blood vessels is most important.The information about blood vessels, such as length, width,tortuosity and branching pattern, provides both informationon pathological changes and means to grade diseases severity[2]. An automatic analysis of fundus images would be ofgreat help to the ophthalmologist due to the small sizeof the symptoms and the large number of patients. Asprevious step, vessel assessment demands vascular treesegmentation from the background for further processing.Several works have been proposed in the detection of 2Dcomplex vessels network, among them. In [3], the concept ofmatched filter detection for retinal vessels segmentation wasproposed. The retinal vessels are segmented progressivelyin a two-stage region growing procedure [4]. In [5], FuzzyC-means clustering has been used for vessel classificationbased on intensity information. In[6], authors reducedretinal image noise using anisotropic diffusion filter with amultiscale response based on the eigenvectors of the Hessianmatrix. Hermandez Hoyos reported a method based on theeigenvectors of the Inertia matrix [7] [8] .In this paper, we propose a expansible skeleton methodfor vessel axis extraction. The algorithm is based on atracking strategy, which starts from a single point within thevascular segment to extract, and then iteratively estimates

the subsequent skeleton points. Point generation procedure isbased on two steps. First, the new point position is predicted.Then, the vessel local orientation is estimated by inertiamoment minimization for a small volume centered on thecurrent point. This predicted position is then corrected underthe influence of image forces and shape constraints. Theprincipal contribution of this approach compared to the othersconsists in the simplicity of the model initialization and inthe flexibility of its evolution.Section II gives the calculation of the moments of the imageand the expansible vessel skeleton method. Preliminary resultsare presented and discussed in Section III.

II. METHODS

The algorithm consists in the calculation of the gravitycenters and the local orientations of the cylinders representingthe vessel portions. It is based on mechanical properties ofthe tubular objects. Certainly, such an object turns aroundits principal axis more freely than around any other axis,because its moment of inertia in the direction of this axisis minimum. In order to apply this theory to the 2D retinalimages, an analogy between pixel intensities Ii and massesmi was established. The calculation of the moments of theimage began with the work of Hu [9].

A. Inertia tensor

The calculation of image inertia tensor is based on the factthat the image is considered as a solid of n particles. The ithparticle has mass mi. The moment of inertia of the system of Npoint whose distances Ri to the axis of rotation ∆ is given as :

I =N∑i=1

miR2i (1)

In the coordinates (Ox, Oy), ri =

[xiyi

]is the position of

the particle i, e∆ =

[αβ

]a unit vector corresponding to the

direction of the axis ∆ .

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IO∆ =∑i

(|ri| sin θi)2 (2)

=∑i

mi |ri × e∆|2

=∑i

mi (ri × e∆) (ri × e∆)

=∑i

mi [xiβi − yiαi] [xiβi − yiαi]T

=∑i

mix2iβ

2i +

∑i

miy2i α

2i − 2

∑i

mixiβiαiyi

(3)

Hence, we deducted the quantities Ixx and Iyy whichrepresents the moments of inertia with respect to the axesOx, Oy of this coordinate system. While the quantitie Ixy isnamed the products of inertia:

Ixx =N∑i=1

miy2i (4)

Iyy =N∑i=1

mix2i (5)

Ixy =N∑i=1

mixiyi (6)

The matrix of inertia associated to image is defined as:

I =

[Ixx -Ixy-Ixy Iyy

]Substituting the inertia values into this gives:

I =

[ ∑i mix2

i

∑i mixiyi∑

i mixiyi∑i miy2

i

]

The inertia matrix is a constant real symmetric matrix. Afterdetermining the inertia matrix, we determine its eigenvaluesand eigenvectors in the next section.

B. The eigenvector and eigenvalue of the inertia matrix

Given inertia matrix I:

I =

[a11 a12

a21 a22

]Principal axes and principal values of matrix I can beobtained by diagonalizing I. That is, by finding eigenvalues(i.e., principal values λ1 , λ2) and eigenvectors (i.e., principalaxe e1 and e2) Principal axes and principal values can be

found by finding a suitable rotation matrix A and performingthe similarity transformation:

ID = AIA−1 =

[λ1 00 λ2

]The vector e1 is an eigenvector with eigenvalue λ1 if

Iei = λei (7)

To solve the above equation for the eigenvectors and eigen-values of the matrix M, we can rewrite it in the form :

(I− λiI) ei = 0 (8)

Where I is the n × n identity matrix and 0 is a vectorof zeros. One way to find an eigenvector is to note that itmust be orthogonal to each of the rows of the matrix (I− λiI).

C. Expansible vessel skeleton

The vessel axis extraction is acquired by an expansibleskeleton method. The notion of expansible skeleton hasbeen inspired by [10]. The extraction is based on a trackingstrategy, which starts from a point belonging to the vessel,and then iteratively estimates the subsequent axis points.Point generation is a process that involves two steps. First,a prediction step in which the position of the new point isobtained, based on the the computation of image momentswithin a circular sub-surface called analysis cell, centered inthe current point Fig.1. This position is then corrected underthe influence of image forces and shape constraints of theaxis, which is the step of estimation. The new corrected pointis taken as the starting point for the next iteration and theprocess is repeated. The result of this algorithm is a set ofdiscrete points that define the vessel axis. These points areinterpolated using a B-spline curve. .

Figure 1. Analysis cell centered in the current point.

1) Prediction: Let pi be the point added to the axis at the tiiteration, the position of the next point is predicted by applyinga ”constant” velocity displacement of pi along the vessellocal orientation. The local orientation is estimated by inertiamoment minimization for a small cell centered on the currentpoint. To determine the local orientation, the eigenvalues of thematrix of inertia are calculated. The vessel local orientation isdefined by the eigenvector ei with smallest eigenvalue λi atthe center of the cell. However, to predict the position of the

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next point, we need not only to the orientation provided bythe vector e, but also the displacement direction, which givenby the sign s calculated as follows:

Si =(eivi)|ei||vi|

(9)

The position vector of the new predicted point is defined as:

x̂i+1 = xi + µSiei (10)

Where µ is the amplitude of displacement. The followingfigure shows the skeleton points detected along a blood vessel:

Figure 2. Positioning the skeleton points of the vessel.

2) Estimation: The inertial approach is a simple and ef-fective method for the automatic extraction of vessel axis. Itis much less noise-sensitive than the differential methods andgives satisfactory results even with very small and low contrastvessels. However, in the case where there are bifurcations orcurvatures that cause significant problems Fig.3 [11] becausethe predicted point may not be in the correct position. For this,a correction mechanism must be executed in the estimationstep.

Figure 3. Point needing correction.

We estimated the final position by adding a correctionto the predicted position x̂i+1. We exerted to the predictedpoint two types of forces: external and internal. We proposedtwo external forces, that attract predictive point to the localmaximum intensity point and to the local gravity center of acell centered at x̂i+1 respectively. The total displacement ofthose forces is defined as:

∆exti+1 = ∆max

i+1 + ∆gci+1 (11)

Where :∆gci+1 = −η

(xgci+1 − x̂i+1

)is the displacement due to the

attraction force exerted by the local gravity center xgci+1 ,

weighted by the coefficient η .

∆maxi+1 = −κ

(xmaxi+1 − x̂i+1

)is the displacement due to the

attraction force exerted by the local maximum intensity pointxmaxi+1 , controlled by κ

In order to limit oscillations and reduce the noise sensitivity,the internal forces are used to impose smoothness and conti-nuity constraints. These forces produce the internal constraintsover the first and second order discrete derivatives [12]. Theexpression of the total displacement due to the internal forcesis:

∆inti+1 = −αa (x̂i+1 − xi)− βa (xi−1 − 2xi + x̂i+1) (12)

Two coefficients dictate the physical characteristics of theaxis :αa controls its elasticity while βa controls its flexibility.Hence, the estimated new point position is :

xi+1 = x̂i+1 + ∆exti+1 + ∆int

i+1 (13)

Fig.4 shows the result of extracting the skeleton afterprediction-estimation process

Figure 4. Vessel axis extracted after correction.

III. RESULTS END DISCUSSION

The retinal vessel tree is composed of several ramifications.So, the central axis of vessels contained bifurcation points thathave more than one predicted point, which present a problemin the extraction processes. However, region bifurcation pointscharacterized by a large variation in the local radius ofthe blood vessel, bifurcation angle and the local orientationvariation of the vessel. Hence, we needed to detect thesevariations over all axis points extracted.

Taking three points pi, pi−1, pi+1 belong to an axis, thefirst condition is that |xi − xi−1| ≥ σ and |xi+1 − xi| ≥ σwhere σ is the local radius of the cell analysis centeredin pi. there is a strong variation of the local orientation ifCV aOr (ei+1, ei−1) = 1.

CV aOr (e1, e2) =

{1 if arccos

(e1.e2|e1||e2|

)> θ

0 else(14)

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Where θ a threshold value.A local variation in the size of the vessel is detected if the

variable CV aSiz = 1 .

CV aSiz (σ1, σ2) =

{1 if

(max(σ1,σ2)min(σ1,σ2)

)> υ

0 else(15)

The variation of the local shape of the vessel is distinguishedwhether the mass of the particle belonging to the vessel isgreater than a threshold γ .

CCir (x, σ) =

{1 if mcir(x, σ) > γ

0 else(16)

After detecting all possible variations, a bifurcation point isdetected if the variable CBif satisfies the following conditions:

CBif =

1 if (CV aSiz = 1 and CV aOr = 1)

1 or (CV aSiz = 1 and CCir = 0)

1 or (CV aOr = 1 and CCir = 0)

0 else

(17)

Figure 5 shows the retinal vascular tree detected. We cansay that the skeleton expansible method can give acceptableresults. This method has two limitations that are the choice ofthe initialization point of the vessel which has a great influenceon the result, and the choice of cell size analysis.

(a) (b)

Figure 5. retinal vascular tree axis detection.

IV. CONCLUSION

In this paper, we presented a method which allows extract-ing the centerline of a single blood vessel and subsequentlyextracting the entire retinal vascular tree. First, we determinedthe inertia matrix of the image and their eigenvalues andeigenvectors. Then the skeleton is extracted by an iterativepredictionestimation tracking technique. Finally, we presentedthe method used to detect the points of bifurcations.

REFERENCES

[1] Rawi M. A., Qutaishat M. and Arrar M, An improved matched filter forblood vessel detection of digital retinal images, Computers in Biologyand Medicine. vol. 37(2), pp. 262267, 2007.

[2] N. Patton, T. M. Aslam, T. MacGillivray, I. J. Dearye, B. Dhillon,R. H. Eikelboom, K. Yogesan, and I. J. Constable. Retinal imageanalysis: Concepts, applications and potential, Progress in Retinal andEye Research, vol. 25, pp.99127, 2006.

[3] S. Chaudhuri, S. Chateterjee, N. katz, M. Nelson, and M. Goldbaum,Detection of blood vessels in retinal images using two-dimensionalmatched filters, IEEE Trans. Med. Imag., vol. 8, no. 3, pp. 263269,1989.

[4] M. E. Martinez-Perez, A. D. Hughes, A. V. Stanton, S. A. Thom, A. A.Bharath, and K. H. Parker, Scale-space analysis for the characterizationof retinal blood vessels, in Medical Image Computing and Computer-Assisted Intervention, C. Taylor and A. Colchester, Eds. Springer: NewYork, 1999, vol. 16794, Lecture Notes Comput.Sci., pp. 9097, 1999.

[5] Y. Tolias and S. Panas, A fuzzy vessel tracking algorithm for retinalimages based on fuzzy clustering, IEEE Trans. Med. Imag., vol. 17, pp.263-273, 1998.

[6] M. Ben Abdallah, J. Malek, K. Krissian, R. Tourki, An automated vesselsegmentation of retinal images using multiscale vesselness, Systems,Signals and Devices (SSD), 8th International Multi-Conference onSystems, Signals and Devices, 2011.

[7] M. Hernndez-Hoyos, M. Orkisz, J. P. Roux, P. Douek, Inertia-Basedvessel axis extraction and stenosis quantification in 3D MRA Images,In CARS’99 Computer Assisted Radiology and Surgery. H.U. Lemke,M.W. Vannier, K. Inamura and A.G. Farman Eds. Amsterdam, pp. 189-193, 1999.

[8] M. Hernandez-Hoyos, M. Maciej Orkisz, P. Puech, C. Mansard-Desbleds, P. Douek, E. M Isabelle, Computer-assisted Analysis of Threedimensional MR Angiograms, RadioGraphics, vol. 22, pp. 421436,2002.

[9] Hu M.K.Visual Pattern recognition by moment invariants. IRE Transac-tions on Information Theory,Vol.8, No.2, 1962,

[10] F. Angella, O. Lavialle, P. P. Baylou, Adeformable an expansible tree forstructure recovery, IEEE International conference on image processing,Chicago, Illinois, pp. 5-8, 1998

[11] B. Nazarian, C. Chedot, J. Sequeira, Automatic reconstruction of irreg-ular tubular structures using generalized cylinders. Revue de CFAO etdinformatique graphique, Vol. 11, pp. 11-20, 1996

[12] M. Kass, A. Witkin, D. Terzopoulos, Snakes: Active contour models:International Journal of Computer Vision, Vol. 1(4): 321331, 1988.

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