## Polynomial Contrast Enhancement

Let f : E ^ T be a gray-level image with T = {^¡n,..., imax) C R aset of rational numbers. Let U = {Wmin,..., umax} C R be a second set of rational numbers. An application r : T ^ U

is called gray-level transformation.

For convenience, the gray-level transformation is constructed in such a way that it assigns 1 (u^ + umax) to the mean value ¡xt of the original image f. Instead of t and u, we consider in the following the variables t and v defined by:

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

«1 ■ (t - Tmin)r + bi if T < 0 «2 ■ (T - Tmax)r + b2 if T > 0

with the parameters r, a1, a^ b1; and b2. The parameter r can be chosen freely, the other parameters are determined in order to assure that the transformation r is continuous and that the resulting image covers the whole gray-level range (from Unm to umax). These conditions can be expressed by f (Tmin) = vmin lim r+(T) = 0

and with Eq. (7.6) we obtain for the parameters ai, bi, and 62:

Vmin 2 (umax umin)

( Tmax)r (fit ^max)r b1 = vmin = 2(umin umax) b2 = vmax = (umax umin)

2(umax umin}

Cfii tmax)r

(t — tmin)r + umin if t < fit ' (t — tmax) + umax if t > fit

The corresponding graph is shown in the Fig. 7.9 for different ¡xt. The resulting transformation is not symmetric to the point , |(^max + umjn)).

With r, we can control the strength of the contrast enhancement. For ¡xt = 2 (tmin + ^max ), we obtain a linear contrast stretching operator for r = 1. For r ^to, we obtain a threshold operation with the thresh ¡xt.

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.

Umax

umin ax t

Figure 7.9: The graph of the gray level transformation for different ¡xt. 7.4.2 Contrast Enhancement and Shade Correction

In order to enhance the contrast all over the image independently from local illumination changes, we propose a shade correction operator based on the gray-level transformation shown in the preceding section.

Shade correction: A shade correction operator tries to remove the background information from an image. This is done by calculating a background approximation (for example with a low pass filter) and by subtracting it from the image. In order to avoid negative values, a constant is usually added:

In the corrected image, the gray-level values depend only on the difference between the original value and the background approximation.

The local contrast enhancement operator: In order to obtain a shade correction operator, which also enhances the contrast, we apply the gray-level transformation from Eq. (7.9) locally, i.e. we substitute the global mean ¡xt by a local background approximation.

One possibility is to calculate the mean value of f within a window W centered in the pixel x:

In this way, a contrast operator is obtained for which the transformation parameters depend on the mean value of the image in a window of a certain size. Hence, it is a shade correction and contrast-enhancement operator. Figure 7.9: The graph of the gray level transformation for different ¡xt. 7.4.2 Contrast Enhancement and Shade Correction

However, for pixels close to bright features, the background approximation may be biased by blurred bright objects. Indeed, we observe a "darkening" close to bright objects as the papilla or exudates (see Fig. 7.10b). This darkening is a real problem for segmentation algorithms, because these regions may then be confused with vessels, hemorrhages, or microaneurysms. Therefore, we propose to calculate the local mean value on a filtered image, where all these bright features have been removed. We have seen that the morphological opening removes bright features from an image (see section 7.3). However, we found it advantageous to apply an area opening y^  rather than a morphological one. Instead of using a SE, removes all bright objects if their area (number of pixels) is smaller than . The shade-correction operator can then be written as

The results obtained by the application of this operator are shown in Fig. 7.10 and Fig. 7.11. (a) Detail of the green channel of a fundus image containing hard ex-udates (b) The shade correction operator with the local mean value as background approximation (c) The final shade correction and contrast enhancement operator

Figure 7.10: The effect of filtering of the background approximation. Figure 7.11: A result for shade correction and contrast enhancement.

Figure 7.11: A result for shade correction and contrast enhancement.

The shade correction of retinal images is a prerequisite of several algorithms, for example, the detection of microaneurysms shown in section 7.6.1 and the detection of vessels shown in the next section.

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