Retinal Vessel Enhancement Based on Directional Field Jian Chena and Jie Tiana a Institute
of Automation, Chinese Academy of Sciences, Beijing, China; ABSTRACT
Motivated by the goal of improving detection of micro-vessels with low contrast, a new technique based on directional field is presented for enhancing vessels in retinal images. This technique consists of two steps: the estimation of directional field and the enhancement. We will make enhancement along the vascular direction and normalize the brightness of image in a single step. After estimating the directional field, the mean and variance values in a local neighborhood were calculated pixel-by-pixel, accordingly normalize and enhance the local neighborhood. In order to eliminate the artificial boundary between two adjacent areas, an anisotropic Gaussian kernel was introduced to weight the enhancement. The proposed method can obviously increase the contrast of retinal vessels. 20 retinal images were tested in our experiments, and the results demonstrate an effective vessel enhancement algorithm. Keywords: Directional field, vessel enhancement, retinal images, normalization, vessel segmentation
1. INTRODUCTION In ophthalmic practice, optic fundus photography technique has become a common procedure to diagnose vascular and nonvascular pathology. This is not only because of the development of computer science, but the fundus images can duly indicate a lot of diseases in early phase. The retina is the very unique region of the human body where the vascular condition can be directly observed in vivo, and the assessment of the characteristics of the vessels plays an important role in a variety of medical diagnoses and treatment. Many ophthalmic diseases, e.g. choroidal neovascularization,1 retinal artery occlusion,2 and glaucoma, reveal the retinal pathological changes, so do many world-wide non-ophthalmic diseases, such as diabetes mellitus, hypertension, arteriosclerosis, cardiovascular disease,3 and stroke. The main structures of retinal image are optic disc and vascular network.4 Automatic analysis techniques5678 instead of manual measurements have been widely used for the reason that the vascular network is very complex. Vessel extraction of retinal image is a significant procedure of diagnosing many diseases. However, it is difficult to automatically extract the micro-vessel because its low contrast (there is a wide range of vessels widths in retinal images9 ,10 from less than 1 pixel to more than 10 pixels. The small vessels which are of less than 2 pixels widths have generally low contrast. The vessels which have been affected by pathologies are also of low contrast). A enhancement of vessel before extraction can greatly improve the performance of vessel extraction algorithm. This paper proposed a vessel enhancement algorithm based on directional field (DF). The basic goal is to highlight the micro- low-contrast vessels. As you know, the majority of retinal vessel, especially the micro-vessel, have the property that its orientation and gray intensity vary slowly along the vessels, which makes it possible to obtain precise directional field of retinal image. If there exists a vessel in an area, the orientation of DF will point to the exact direction of vessel and the coherence of DF will be close to 1. On the other hand, if there does not exist a vessel, the orientation of DF will equally distribute in all directions or the coherence will be close to 0. Therefore, using directional field to enhance the retinal vessels would be a good choice. We will make enhancement along the vascular direction and normalize the brightness of image in a single step, and will highlight the vascular network using the directional Gabor filter. Further author information: (Send correspondence to Jie Tian) Jie Tian: E-mail:
[email protected], Telephone: 86-10-82618465
Medical Imaging 2008: Image Processing, edited by Joseph M. Reinhardt, Josien P. W. Pluim, 1 691422, (2008) Proc. of SPIE Vol. 6914, 1605-7422/08/$18 · doi: 10.1117/12.770041 Proc. of SPIE Vol. 6914 691422-1
2. METHODS The proposed technique consists of three steps, Estimation of Directional Field, Enhancement and Normalization and Gabor filtering. Estimation of DF: The orientation θ, coherence coh and strength str of directional field are estimated by the algorithm discussed in section 2.1. Enhancement and Normalization: The enhancement algorithm is applied to highlight the vessels. We use IE to denote the enhanced image masked by ROI of original image. Gabor filtering: The Gabor filter is capable of tuning the specific frequencies and orientations, thus allowing noise filtering and vessel enhancement in a single step. At the same time, the Gabor filter may break some vessels because the orientation θ of DF may be disordered at bifurcations.
2.1 Estimation of Directional Field There are two ways to estimate the directional field from a retinal image. One is called directional mask approach which is of poor accuracy with limited number of fixed possible directions. The other is called gradient-based method11 .12 This method theoretically can provide arbitrary directions but is more complicated than the first one. An accurate directional field is needed to increase the overall accuracy when extracting the retinal vessels, so we choose the latter in this paper. In the next context, three parameters refer to DF will be estimated, they are the direction θ, the coherence coh and the strength str. The direction θ is perpendicular to the gradient, so we can estimate θ from gradient. The gradient is defined as follows: ∂I(x,y) ∂I(x, y) Gx (x, y) ∂x , = sign ∂I(x,y) Gy (x, y) ∂x ∂y where I(x, y) represents the gray-scale image. The first element of the gradient vector has been chosen to always be positive because opposite directions of gradient indicate equivalent θ. Note that in a retinal image, there is no difference between a local vascular orientation of 90◦ and 270◦, since the vessels oriented at 90◦ and the vessels oriented at 270◦ in a local neighborhood cannot be differentiated from each other. Gradients cannot directly be averaged in some local neighborhood since opposite gradient vectors will then cancel each other, although they indicate the same vascular orientation. For example, the unit gradient vectors oriented at 90◦ and 270◦ are indicate equivalent θ at 0◦ , but the two vector will cancel each other if directly averaged. A solution to this problem is proposed by squaring the gradient vectors considered as complex numbers before averaging. With this solution, the angle of gradient vector is doubled and the length is squared, therefore, the opposite gradient vectors (e.g., 90◦ and 270◦ ) will point to the same direction (180◦ ). As a consequence, the opposite gradients will reinforce each other while the perpendicular ones will cancel. The gradient vectors are first estimated using Cartesian coordinates, in which a gradient vector is given by [Gx Gy ]T . For doubling the angle and squaring the length, the gradient vector is converted to polar coordinates, in which it is given by [Gρ Gϕ ]T . This conversion is given by: Gρ G2x + G2y , = Gϕ tan−1 (Gy /Gx ) then the squared gradient vectors [Gs,x Gs,y ]T is given by: 2 2 Gρ cos(2Gϕ ) Gx − G2y Gs,x = = . Gs,y G2ρ sin(2Gϕ ) 2Gx Gy Next, the average squared gradient [Gs,x Gs,y ]T can be calculated as follows: Gs,x Gs,x W = Gs,y W Gs,y where W is the window of averaging filter. This operation will reinforce each other when gradient vectors are opposite.
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After averaging, the gradient vectors have to be converted back to their single angle representation. The vascular orientation is then perpendicular to the direction of the average gradient vector. Now, the average gradient direction Φ, with − 12 π < Φ < 12 π, is given by: Φ=
1 (Gs,x , Gs,y ) 2
⎧ ⎨ tan−1 (y/x) tan−1 (y/x) + π (x, y) = ⎩ tan−1 (y/x) − π
x≥0 x VT H ⎨ ch if Vij < VT L f (V (i, j)) = cl ⎩ c0 · Vij c1 otherelse where VTH , VTL are two thresholds of variance and VT H >VT L , ch is a comparatively large constant while cl is small one. The purpose of this weighting function is to tune the rate of enhancement. If the variance is greater than the high threshold, we will limit its enhancement. On the other hand, if the variance is lower than low threshold, we can judge that the region does not have vessels, then smooth this region. Between these two situations, c0 and c1 are used to control the weighting function, thus providing the required measure. In order to obtain better results, we used anisotropic Gaussian kernel instead of isotropic one. See Fig. 1. The orientation of Gaussian kernel is completely determined by the local vascular orientation. Along with the main direction of Gaussian kernel, with increasing distance to the center the value of Gaussian kernel dropped slowly while the value decline fast along the perpendicular to the main direction. The anisotropic Gaussian kernel is defined as follows: 1 x2 y2 1 exp(− [ θ2 + θ2 ]) g(x, y) = 2πσ1 σ2 2 σ1 σ2 xθ = xcosθ + ysinθ yθ = −xsinθ + ycosθ where θ denotes the orientation of Gaussian kernel, and σ1 and σ2 are the standard deviations of the Gaussian kernel. If the coherence of DF is close to 1, the enhancement is distributed mainly at the direction of vessel. On the other hand, if the coherence is 0, the enhancement is equally distributed in all directions. Therefore, the signals of vessels are enhanced but the noise is suppressed. This can be tuned by the ratio of σ1 by σ2 . The DF is shown in Fig. 3. The sizes of windows used in this subsection are all 9 × 9 pixels.
2.3 Gabor Filtering The configurations of two parallel edges of a vessel with well-defined orientation in a retinal image provide useful information which helps in removing undesired noise and preserving the true vessels. In order to reduce the noise and highlight the vessels of a retinal image, a directional filter is good choice. We can filter image along with its directional field. If I is the original image and θ is the DF, this operation can be defined as follows: If = dir filt(I, θ), 5 Proc. of SPIE Vol. 6914 691422-5
Figure 3. Directional field. (a) Gray-scale coded directional field. (b)-(d) Line-matrix representation.
there are many directional filters in literatures, such as order-statistics filter, Gabor filter, and nonliner diffusing filter13 and so on. The Gabor filter1415 is capable of tuning the specific frequencies and orientations, thus allowing noise filtering and vessel enhancement in a single step. Therefore, it is appropriate to use Gabor filters to remove the noise and preserve true vessels. Gabor filter is a bandpass filter, and it is defined as follows: 2 yθ2 1 xθ + 2 ·exp(jaxθ ) h(x, y, θ, a) = exp − 2 σx2 σy xθ = xcosθ + ysinθ yθ = −xsinθ + ycosθ where j = (−1), θ denotes the orientation of Gabor filter, a is the frequency of complex exponential, and σx and σy are the standard deviations of the Gaussian envelope along x and y axes, respectively. We use the real part of Gabor expression. It is also be called as even-symmetric Gabor filter, 2 1 xθ yθ2 he (x, y, θ, a) = exp − + 2 ·cos(axθ ) 2 σx2 σy Fig. 2 shows an even-symmetric Gabor filter and its modulation transfer function (MTF). To apply Gabor filter to an image, the following parameters must be specified: σx , σy , θ and a. The selection of the values of σx and σy involves a trade-off. The larger the values, the more robust to noise the filters are but the more likely the filters will create spurious vessels. On the other hand, the smaller the values of σx and σy , the less likely the filters will create spurious vessels; consequently they will be less effective in removing the noise. The values of σx and σy were set to 4.0 and 4.0, respectively based on empirical data. The orientation θ is completely determined by the local vascular orientation. The frequency a is determined by the width of vessels. Let I be the enhanced image, θ be the orientation image, the filtered image If is obtained as follows:
he (u, v, Iθ (x, y), a)I(x−u, y −v). If (x, y) = (u,v)∈W
where the size of the Garbor filter are 11×11 pixels. To detect the vessels with various widths, we apply Gabor filter to a retinal image twice with two different frequencies (a=2 and a=4). The results of the Gabor filter with different frequencies are shown in Fig. 4.
3. RESULTS AND CONCLUSIONS We applied our method on 20 images chosen from DRIVE database which is publicly available. The results of enhancement of two cases are indicated in Figure 4 (b). Figure 4 (a) shows the original images. In order to make it clear that the proposed method are of good performance and to eliminate the noise brought by enhancing, a Gabor filter was applied to the enhanced images. The results of filtering by Gabor are presented in Figure 4 (c). From Figure 4 (b), it can be seen that the background has been normalized and the vascular network are more 6 Proc. of SPIE Vol. 6914 691422-6
at) Figure 4. Results of enhancement and Gabor filter. (a) Enhanced image. (b)-(c) Results of Gabor filter with different frequency a = 2 and 4. (d) the Ground Truth.
distinct than the original images. After filtering by Gabor filter, the vascular network are further highlighted and most noise in enhanced images are eliminated. Figure 4 (d) demonstrates the Ground Truth. To the best of our knowledge, this is the first study to apply anisotropic Gaussian kernels with various directions to weight the enhancement. We take into account the mean and variance values of a local neighborhood to estimate the desired mean and variance, and fulfill the normalization and enhancement in a single step. The proposed method can normalize the background and enhance the vessels in a single step. The experimental results demonstrate a distinct vascular network of retinal image. Vessels which even cannot be recognized by a human observer in original image became obvious after enhancement. However, this method will bring some noise when the weighting function is not set appropriately. Fortunately, most of the noise can be eliminated by Gabor filter. By applying our method, many micro-vessels are displayed clearly. Figure 4(c) shows a very distinct microvessel in the place where we cannot find any micro-vessel by using many other algorithms. However, the enhancement technique will also bring a lot of noise when the weighting function (see section II) is not set appropriately. Fortunately, the most of the noise can be eliminated by using Gabor operator and local threshold processing algorithm. According to the property of Gabor operator, we can infer that we will get the maximum value of response when the direction of the Gabor operator is consistent with the vascular’s. In this paper, we use Gabor operator that is in the same direction as directional field at each pixel rather than use Gabor operator with 18 directions proposed by Soares.15 So using our algorithm, the amount of calculation is one eighteenth comparing to the Soares’s method theoretically. But the speed cannot reach 18 times because of the FFT adopted by Soares and the time consumed on calculating directional field. Our algorithm has another advantage that the computational complexity will not increase when we increase the directional resolution of Gabor filter. Comparatively, the computational complexity of Soares’s algorithm will grow linearly when increasing directions.
ACKNOWLEDGMENTS This paper is supported by the Project for the National Key Basic Research and Development Program (973) under Grant No.2006CB705700, National Science Fund for Distinguished Young Scholars of China under Grant No. 60225008, the Joint Research Fund for Overseas Chinese Young Scholars under Grant No.30528027, the National Natural Science Foundation of China under Grant No. 30370418, 90209008, 60302016, 60532050, 30500131, Beijing Natural Science Fund under Grant No. 4051002, 4042024. 7 Proc. of SPIE Vol. 6914 691422-7
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