Active Contours Technique in Retinal Image Identification of the Optic Disk Boundary

Soufyane El-Allali, Stephen Brown

Department of Computer Science and Engineering

University of South Carolina, Columbia, SC. elallal@engr.sc.edu, brownst4@engr.sc.edu

Abstract. This paper describes the techniques used in detecting the optic disk boundary. Our approach follows progressive stages. These stages involve the following: Image Pre-processing, morphological techniques for noise filtering, windowing, as well as applying the derived gradient vector flow to the image. The concentrated segment of our project will involve reduction of the image size to a region that concentrates the optic disk using windowing. We further expose the optic disk boundary with an active contour derived by the Gradient Vector Flow (GVF). The main challenge that will be present is to incorporate both windowing and active contours, colloquially known as “snakes” [3,5], to the retinal images. The result obtained from these techniques is an isolation of the optic disk from a retinal image. This helps physicians to visually inspect any changes in size and shape of the optic disk due to possible disease development.

1 Introduction

The optic disk is the brightest hole in the back of the eye where axons of the receptor cells of the retina collect in a single spot. Blind spots exist because no receptor cells are located in this hole. Mapping topographic abnormalities of the optic disk shape and size is important for physicians’ diagnosis of disease developments such as degenerative myopia and glaucoma, which may lead to loss of vision.

Identifying the optic disk from retinal images is difficult due to several factors. The disk may be located in various positions of the retinal image, challenging the localization of the disk. Once the optic disk region is found, the intersecting blood vessels that converge in the middle of the disk create a heterogeneous section. Hence, running the active contours (snakes), which are an energy-minimizing splines guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges [3], results in incorrect boundary detection.

Our solution to these impediments entails multiple stages. Our application takes in a gray-scale retinal image as input, and passes through a statistical analysis of the image that defines a threshold and applies a windowing technique using this threshold parameter to size down the image. This leads to a localization of the optic disk region. After this

initial phase, we begin filtering using three consecutive morphological techniques. The first one is dilation of the area. Dilation is the addition of pixels to the boundaries of objects in an image. The second one is erosion. In contrast, erosion is used to remove pixels on object boundaries. The number of pixels added or removed from the objects in an image depends on the size and shape of the structuring element used to process the image. Lastly, we apply reconstruction by taking the maximum pixel value between the original image and the dilated/eroded image. Optimal results are achieved by applying our preprocessing as well as superposition of multiple gradient vector flow fields. The gradient vector flow field is defined to be a new external force for active contours, this force is computed as a diffusion of the gradient vectors of a gray level or binary edge map derived from the image [7].

2 Background

In image analysis and computer vision, segmentation is one of the most important steps in allowing successful completion of higher order tasks such as recognition and tracking. Many techniques have been proposed for segmentation, e.g., globally-based, edge-based, and region-based techniques [5]. An edge-based technique that has been developed in recent years is active contours, colloquially known as “snakes” [3,5]. Active contours simulate the fitting of an elastic curve to boundaries or objects of interest in an image. They have been applied to detection of roads and buildings in remote sensing images [2], feature extraction in faces [6], and 3D segmentation of MRI brain images [1], to name but a few examples.

2.1 Introduction of Active Contours

There have been numerous segmentation methods developed in image analysis and computer vision. Methods such as globally-based, edge-based, and region-based techniques [5,7]. A relatively new edge-based technique known as active contours was developed by Kass, Witkin, and Terzopoulos. These active contours are curves defined within an image domain that can move under the influence of internal forces coming from within the curve itself and external forces computed from the image data [7].

2.1.1 Mathematical Analysis of Active Contours : Continuous Approach

A gray scale image is a two dimensional image I(x, y) in the Cartesian system. Each pixel I(x0, y0) has an intensity value that ranges 0 to 255. The active contour is defined as a parametric time-varying curve v(s, t) = [x(s, t) ; y(s, t)]T in the plane, with s Є [0, 1], and t

as time [7]. This contour moves through the spatial domain in order to minimize the following energy functional:

dstsvEtsvEE extsnake 1

0int ,, (1)

This energy functional is constructed from the summation of two types of energy, an internal energy Eint and an external energy Eext. The internal energy is due to the kinetic and potential energy terms, and the external energy represents energy due to an external force field that is generated by the image in which the contour is embedded [7]. A mathematical representation of the internal energy can be shown as follows [3,5,7]:

2

2

222

2

2

int 21

sv

sv

tv

vE , (2)

where is the snakes’ elasticity, i.e. the snakes’ resistance to stretching. is the snakes’ rigidity, i.e. the snakes’ bending resistance. The description for the terms is listed in the following:

2

2

2

tv

is a kinetic energy term due to the motion of the snakes at every point.

2

sv

is the potential energy due to stretching of the active contour.

2

2

2

sv

is the potential energy due to bending of the active contour.

Subsequently, the external energy Eext depends on the image that embodies the contour of interest. Two formulations of the external energy are given in formula (3):

2)2(

2)1(

,,

,,,

yxIGyxE

yxIyxEyxE

ext

extext

, (3)

where denotes the gradient operator, and denotes the standard deviation. The )1(extE

uses a simple image gradient based on the intensity that results in minimal values where

the gradients have higher values. Whereas, )2(extE uses the gradient of the convolution of

the Gaussian function with the image intensity. This will generate minima where the filtered Gaussian function has maxima [7]. It is important to note that larger ’s result in blurry boundaries, yet such large ’s are often necessary to increase the capture range of the contour [7].

By applying the calculus of variations to formula (4), we can result in the Euler Lagrange equation (5). A snake that minimizes snakeE must satisfy the Euler equation [7]:

1

02

2

,,,s

dssv

sv

vsFvE (4)

04

4

2

2

extEsv

sv

(5)

This can also be defined using a force balance equation (6) [7]. 0int extFF , where

4

4

2

2

int sv

sv

F

is an internal force that reduces stretching and bending. And

extext EF is an external force that pulls the snake toward the desired image edges [7].

As snakes approach the edge, a damping opposite force must be applied in order to bring the snakes to an equilibrium state. This can be achieved by adding two terms with positive coefficients, a damping term and an inertial term as in equation (6):

04

4

2

2

2

2

extEsv

sv

tv

tv

(6)

In this paper, the value of is set to zero, since we are neglecting the snake’s mass.

2.1.2 Discrete Active Contours

The active contour is defined as a set of N nodes:

nni

nini ihy

ihxy

xv

,

,, , (7)

where h = 1/N and the initial condition is it kv 0, with ik defining random fixed

positions. For a closed contour, the first and last nodes must be neighbors. Using finite difference methods we can approximate equation (6) as in (8):

nininininininini vvvvvvvv ,,1,2,1,,11,, 22

0

1,

1,222 ,2,1,,1,,1

nif

nifvvvvvv

y

xnininininini , (8)

where

1,,1,1,,1,

1,1,

niyniniyni X

exty

X

extx y

Enif

xE

nif

Let TnNnnon xxxx ,1,1, .,..,, and TnNnnon yyyy ,1,1, .,..,, . The

following matrix expression describes the evolution of the contour nodes in time:

11

11

1

11

1

,

,0

0 nny

nn

n

n

n

n

yxf

yxfx

y

x

BAIBAI

yx

, (9)

where A and B are tridiagonal and pentadiagonal matrices incorporating the elasticity and stiffness parameters, respectively [7].

2.2 Gradient Vector Flow

Due to the effect that there exist many configurations for the snake which are local energy minima, but do not provide the boundary of interest to us [7]. Also, traditional snakes primitively developed by Kass and his colleagues do not converge in the case of concave shapes. In response to this, Xu and Prince proposed an improved method called the gradient vector flow field h(x, y) (10):

yxqyxp

yxhF GVFext ,

,, (10)

The h(x, y) field minimizes the following energy function:

dydxghgyq

xq

yp

xp

Eyx

GVF

22

2222

, (11)

where g is the edge map of the image as defined in (3), is a regularization parameter that controls trade-off between the first and second terms. This has the effect of increasing the effective range of edges at locations distant to edges [7]. The GVF field is then found by solving the Euler system where p and q are treated as functions of time allowing for the system to converge to a solution:

00

222

222

yxy

yxx

gggqqgggpp

(12)

In our experiment, is set to 0.2 and the number of iterations for the calculation of the GVF field is set to 80.

A complete numerical algorithm implementation has been fully illustrated in the Xu and Prince paper.

3 Method

3.1 Image Preprocessing

Image preprocessing is an essential step in preparing retinal images for running gradient vector fields. Without filtering prior to running the GVF, incorrect results occur as GVF fields converge along edges created by blood vessels. These blood vessels make up a distinctive edge-map leading the GVF field forces toward scattered regions. Figure 1 illustrates the original image without any applied pre-processing, the applied GVF field, and the resulting active contours.

Fig. 1. Boundary detection of the optic disk on the original image using snakes. (a) Original image. (b) Gradient vector flow. (c) Result of active contours. A morphological technique approach proved to be very effective in filtering out unwanted blood vessels. This process uses the combination of dilation, erosion, and reconstruction operations respectively yielding a healthier image to work with.

3.1.1 Thresholding and Windowing

For larger images, the calculation of the GVF fields is expected to be time consuming. However, in this case of running the GVF to find the optic disk in retinal images, we have prior knowledge of the image intensity of the optic disk. It is known that the optic disk corresponds to the brightest 2% region within gradient images. To reduce the size of these images, we can set a threshold that filters the unwanted sections. This threshold is found by setting a gradient marker level obtaining only the region of interest. A histogram is generated to provide a view of the data intensity repartition of the original retinal image in figure 1.a. Figure 2 shows this result.

Fig. 2. Histogram of the data intensity from original retinal image.

Given this, the image is cropped to a more concise size that still includes the area of interest. Figure 3 shows the cropped image.

Fig. 3. Cropped image as a result of thresholding and windowing.

Furthermore, applying this process minimizes the GVF vector set. This gives a more concentrated area to analyze.

3.1.2 Dilation, Erosion, and Reconstruction

Dilation and erosion are two fundamental morphological operations. Dilation (13) causes objects to dilate or grow in size by adding pixels to the boundaries of objects in an image, while erosion (14) causes objects to shrink by removing pixels on object boundaries. The amount and the way that they grow or shrink depend upon the choice of the structuring element. This structure element is a vital component of 1s and 0s used to probe the input image during dilation and erosion. Typically, the choice of the structure element depends on the object needing to be processed in the image. In our case, we processed the image using a disk shaped structured element that defines the neighborhood region around its origin with a radius of 12 (the number of 1s counted in a radial direction from the origin). This 12 corresponds to the width of the blood vessels needed to be filtered out, which is in our case 8 pixels on average.

yxyxyxIzyzxGzz

zzGzyzxIzyzxIyxGIyxyx

,.,,, max,,,

(13)

yxyxyx

IzyzxGzz

zzGzyzxIzyzxIyxGIyxyx

,.,,, min,,,

, (14)

where denotes the dilation operation and denotes the erosion operation. The overbar represents averaging, and the dot indicates element by element multiplication.

After dilation and erosion, a reconstruction (15) of the image is necessary in order to restore the distinct edges of the original image. Figure 4, shows the progress stages in dilation, erosion, and reconstruction on the cropped retinal image.

GGIyxII rec ,,max (15)

Fig. 4. Pre-processed images after windowing. (a) Dilated image. (b) Eroded and reconstructed image.

3.1.3 Significance of Image Preprocessing

This preprocessing is a critical step in preparing the image for a more efficient GVF execution. Otherwise, from figure 3 there are two main blood vessels that cross through the center of the optic disk creating a low intensity across the image; therefore, distorting the homogeneous structure of the optic disk. It is clear from figure 4 that a preprocessed image will help reduce the distraction of GVF field vectors. Therefore, converging to a more accurate boundary.

3.2 Experiments and Discussion

3.2.1 Snake Initialization Model

Initialization is a crucial step in the active contour model proposed by Kass, Witken and Terzopoulos. Without correct initialization, the active contours are dictated by incorrect vector sub fields that converge along boundaries relating to other unwanted objects as in figure 5.

Fig. 5. Improper snakes initialization results. (a) Snake result when initialized in a null GVF field. (b) Snake result when initialized too large and influenced by outside sub GVF fields.

A solution to this problem is derived using the results of the pre-processing stage. After pre-processing the image, the optic disk is known to be located at the center of the cropped image. Therefore, setting an initial circular snake within the center of the image allows correct initialization.

3.2.2 Superposition of the Gradient Vector Flow Fields

The Xu and Prince implementation of the GVF provided some initial non-accurate results as depicted from figure 1, because Xu and Prince’s implementation needs a simple curve-like object that has little or no noise. After pre-processing a better result is obtained as in figure 4. As stated in the theoretical section a regularization parameter is set to 0.2 and the number of iterations for the calculation of the GVF field is set to 80. Also the number of snake iterations is set to 25, which allow enough time for the active contours to stabilize.

Fig. 6. Results of the GVF field and the snakes after pre-processing on the cropped image. (a) GVF field. (b) Active contours converging towards the lower concavity of the veins.

As observed from figure 6.b, the snakes do not converge along the circular boundary, it converges toward the blood vessels’ delta shape. This inaccuracy is a result of the GVF

field converging inside the optic disk provided that greater external energy forces than the internal energy forces are pushing inwards (figure 6.b).

Our proposed solution to this issue is to generate multiple gradient vector flow fields with different properties converging to different areas of interest. The summation of these different GVF fields will converge to the optimal optic disk boundary. The reasoning behind this proposed method is that creating more external forces pushing inward will balance the GVF field summation at the optic disk. The next addressed task is defining the correct forces to ensure correct convergence. The solution to this problem is purely statistical. A small range for the Gaussian functions’ standard deviation is set to [1, 5] in order to find the correct values, which make up the final GVF field. The best resulting values used in this experiment are = 2 for the first GVF field and = 4 for the second one. Lower values do not destroy the edge boundary, thus keeping the features that make the GVF detect the edge, yet still not capturing the entire region of interest. Larger values capture the entire region of interest, i.e. the optic disk, yet making the edge more blurry. Figure 7 shows the final result of our experiment. It is clear that, using the proposed method, the active contours settled along the boundary of the optic disk.

Fig. 7. Final results on the cropped image applying superposed GVF fields. (a) Superposed GVF field. (b) Active contours result using superposition of the GVF fields.

For the purposes of clinical analysis, it is important to relate the optic disk to other objects of the retinal image. Therefore, it was important to put the cropped image back into its exact location on the original image. Thus, resulting in our final work as illustrated in figure 8.

Fig. 8. Final result of the identification of the optic disk on the original image.

4 Conclusions

Snakes and the gradient vector flow models developed by Kass, Witkin, and Terzopoulos along with Xu and Prince have proven effective without the incorporation of noise in images. However, images in the medical field have a great deal of noise, and need to be treated differently. This project dealt with finding the optic disk in the gray-scale retinal images using an improved method of the gradient vector flow. The pre-processing step readies the image for our superposition GVF fields’ technique. The pre-processing step includes: thresholding, windowing, dilation, erosion, and reconstruction. Then correct initialization of the active contours is crucial for accurate deformation of the snakes. This deformation process is strongly influenced by our proposed superposition of the GVF fields. Given the correct initialization and by setting different standard deviation for GVF field’s Gaussian functions results in precise boundary conversion.

Based on this, future work using the superposition technique of GVF fields may be explored further for different types of images. Our technique should be further tested to determine the error percentage level. This can be achieved by running the developed algorithms on a large number of images, and analyzing the results. We would also like to use our improved method to obtain the perimeter of the optic disk, as well as investigate the effect of the superposition technique to help determine the abnormalities in optic disks. This will have a great potential in ophthalmology.

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