Kluwer - Handbook of Biomedical Image Analysis Vol

Figure 7.4: A RGB representation of a color image.

one to exploit this interpretation of the reflected spectrum. A color image and its decomposition into the three channels, red, green, and blue, is shown in Fig. 7.4. Of course, the given interpretation holds only approximately: The red channel is not the spectral response to red illumination, but the red part of the spectral response to illumination with white light.

The RGB representation of a color fundus image is shown in Fig. 7.4. The red channel is relatively bright and the vascular structure of the choroid is visible. The retinal vessels are also visible but less contrasted than in the green channel (compare Fig. 7.4(b) with Fig. 7.4(c)). The blue channel is noisy and contains only few information. The vessels are hardly visible and the dynamic is very poor.

This phenomenon can be observed in all retinal images we have studied (about 200) with one difference: Sometimes, the blue channel contains information, sometimes, it does not. Indeed, the quality of the blue channel of the RGB

color space depends on the age of the patient and on the yellowing of the cornea; the cornea can be understood as a filter that screens out ultraviolet radiation. With the age, the cut-off frequency moves toward the blue, and even blue light can no longer pass.

This interpretation of the color content of fundus images privileges the use of the RGB color space, for the channels have a physical meaning. We have compared qualitatively the green channel of 30 fundus images with channels of other color spaces (HSV , HL S, Lab, Luv, principal components) and for all images, the green channel was better contrasted than any other channel (at least concerning all blood-containing features). Another advantage of the use of the green channel is that the choroidal vessels do not appear at all, whereas they do appear in the luminance channel for instance, for it is a combination of the three channels R, G, and B. This is why, we work mainly on the green channel.

7.3 Basic Morphological Operators

Mathematical morphology is a nonlinear image-processing technique. The interested reader may see [5] for an exhaustive discussion and [6] for a comprehensive introduction.

The basic operators presented in this chapter deal with two-dimensional discrete images (defined on E Z2). Binary images are defined as subsets of E and gray scale images as functions f : E → T , with T = {tmin, . . . , tmax} the set of gray levels.

7.3.1 Erosion and Dilation

Many operators in mathematical morphology are based on the use of a small “test-set” B called structuring element (SE). Its shape and size can be chosen in accordance with the segmentation or filtering task.

In order to calculate the morphological erosion of a binary image A, we test for each point x if the structuring element centered in x fits completely into A. If this is the case, x belongs to the eroded set ε A. The dilation can be seen as an erosion of the background.

Figure 7.5: Erosion and dilation with a circular SE of a retinal image (detail).

In Fig. 7.5, the effect of these operations are shown. We see that the erosion enlarges dark details and reduces bright ones and the dilation enlarges bright details and reduces dark ones.

7.3.2 Opening and Closing

Morphological openings γ B and closings φ B are the consecutive application of erosion and dilation:

˘ = {− }

with B b the transposed structuring element.

In order to understand the behavior of gray scale openings and closings, it is useful to consider the image f as a topographic surface: Pixels with low gray-levels correspond to valleys, pixels with high gray-levels correspond to mountains. A morphological opening removes all bright features that cannot contain the structuring element; “it razes the elevations.” A morphological closing removes all dark features that cannot contain the structuring element; “it fills the depressions in the surface.” The opening and closing of a retinal image are shown in Fig. 7.6: The opening removes the bright patterns (exudates) without enlarging the dark ones (Fig. 7.6(b)). The closing removes the dark patterns

7.4Contrast Enhancement and Correction of Nonuniform Illumination

There are three systematic problems that occur in nearly all segmentation tasks of color fundus images:

Nonuniform illumination

Poor contrast

Noise

For the attenuation of noise, we cannot propose filters that are applicable in general, because the exigency on such filters depends on the segmentation task. If big features are to be detected (e.g. the macula), strong filters can be used whereas algorithms dedicated to the detection of small details (e.g. microaneurysms) must rely on filters that preserve even small dark details.

In this section, we present an algorithm for contrast enhancement and shade correction. First, we propose a simple global contrast enhancement operator. Applying this operator locally enhances the contrast and corrects nonuniform illumination in one step.

7.4.1 Polynomial Contrast Enhancement

Let f : E → T be a gray-level image with T = {tmin, . . . , tmax} R a set of rational numbers. Let U = {umin, . . . , umax} R be a second set of rational numbers. An application

: T → U

u = (t)

is called gray-level transformation.

For convenience, the gray-level transformation is constructed in such a way that it assigns 12 (umin + umax) to the mean value µt of the original image f . Instead of t and u, we consider in the following the variables τ and ν defined by:

A polynomial gray-level transformation can then be defined as follows:

with the parameters r, a1, a2, b1, and b2. The parameter r can be chosen freely, the other parameters are determined in order to assure that the transformation

is continuous and that the resulting image covers the whole gray-level range (from umin to umax). These conditions can be expressed by

and with Eq. (7.6) we obtain for the parameters a1, a2, b1, and b2:

The corresponding graph is shown in the Fig. 7.9 for different µt . The resulting transformation is not symmetric to the point (µt , 12 (umax + umin)).

With r, we can control the strength of the contrast enhancement. For µt =

12 (tmin + tmax), we obtain a linear contrast stretching operator for r = 1. For r → ∞, we obtain a threshold operation with the thresh µt .

If this operator is applied to the whole image as a global contrast operator, the result is not satisfying due to the nonuniform illumination. In fact, the proposed gray-level transformation does not enhance the contrast for any subset of T , but only for subsets for which > 1. For instance, the contrast of a dark detail situated in a dark region may even be attenuated.