Computational Methods for Feature Detection in Optical Images
enough to represent the local distribution, but small enough so that gradual spatial changes from illumination effects will not cause a perturbation:
f(i, j) |
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g(i, j) |
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255 |
[ W(f) − W(fmin)] |
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(2.8) |
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[ W(fmax) − W(fmin)] |
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where the sigmoidal function is:
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exp |
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f W − f |
−1 |
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(2.9) |
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σW |
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W |
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= |
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while fmin and fmax are the minimum and maximum values of intensity within the image I.
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f W(i,j) = |
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f(k, l) |
(2.10) |
M2 |
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(k,l) W(i,j) |
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σW2 (f) = |
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(k,l) W(i,j) |
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Prior to running the adaptive local contrast enhancement algorithm, retinal images are first converted from RGB to HIS using methods described in Sec. 2.2.1.1, then a Gaussian smoothing kernel (32 × 32, σ = 1) is passed over the intensity (I) channel to reduce noise. The results of M = 49 window can be seen in Fig. 2.6.
2.2.2.4. Histogram transformations
Histograms can be used to describe image color or intensity density distribution by discretely counting pixels in the bins of a specified width (range). The user-defined width of bins is the range in which a pixel must fall for inclusion, with more bins reducing the width of each bin. The histogram is a useful image statistic, providing insight into necessary further contrast enhancement by fitting a histogram into a desired shape, ultimately transforming image pixels from the original image. We will discuss several types of histogram transformations that have been used as retinal image preprocessing steps in the literature, histogram equalization, histogram specification, and multi-level histogram equalization.
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Michael Dessauer and Sumeet Dua
Fig. 2.7. (a) Original image, (b) RGB images and histograms, (c) equalized RGB images and histograms, and (d) equalized image.
An image made up of pixels that occupy the entire range of intensity values with an approximately uniform distribution can be considered to exhibit high contrast.8 High contrast with minimal noise, as stated earlier, is a goal of image processing methods. Most digital retinal images do not initially display a highly distributed histogram (Fig. 2.7). Histogram equalization can be applied to transform an image with compact pixel intensity distributions. For discrete pixel values, we will represent the probability of the occurrence of intensity level rk in an image as approximated by:
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k = 0, 1, 2, . . . , L − 1. |
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pr (rk ) = n |
(2.12) |
Here, n is the total pixel count of the entire image, nk is the number of pixels that have intensity rk, and L is the range of intensity values (0– 255 for eight-bit images). The histogram-equalization transform function is then:
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sk = T(rk ) = |
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pr (rj ), |
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j=0 |
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nj |
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= j=0 |
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k = 0, 1, 2, . . . , L − 1. |
(2.14) |
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Computational Methods for Feature Detection in Optical Images
Each image pixel rk is mapped to sk through transform T , resulting in an image with a histogram that is much more evenly spread across the range [0 · · · L − 1] of pixel intensities. Histogram equalization has been used as a preprocessing step for retinal images, giving retinal anatomical constituents stronger contrast (Fig. 2.7). Although enhanced contrast is typically valuable for image segmentation, analysis issues in using histogram-equalized images arise with inter-image variations, due to the uniqueness of intraimage transformations.
A second method of enhancing contrast through histogram manipulation is histogram specification. This method uses a reference model histogram (usually from a retinal image considered optimal in contrast for algorithm success) to transform an image’s histogram to closely match that of the model. This method provides a baseline histogram shape that decreases inter-image variations that are prevalent in using histogram equalization.
The method begins similarly to histogram equalization where we first determine sk from the PDF of pr (r). Next, we specify a probability density function pz(z) that we wish the PDF of pr (r) to closely approximate. This discrete formulation has the form8:
k |
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vk = G(zk) = pz(zi) = sk k = 0, 1, 2, . . . , L − 1. |
(2.15) |
i=0
In Fig. 2.8, we display the output of histogram specification between two retinal images. Histogram specification has been shown to achieve superior results in increasing separation between lesion type clusters when used for color normalization.12
A third method of histogram transformation, multi-level histogram equalization (MLE), first equalizes the entire image (global), then performs local window equalization. This method will enhance local areas larger than features that are being detected (drusen, cotton wool spots, etc …), due to sometimes close similarity to the background. This method has been applied to lesion detection using sequential, non-overlapping windows, which decrease in size until a lesion is detected.13 The calculation is performed using the same equations as typical histogram equalization, with the addition of the iterative windows of equalized histograms (Fig. 2.9). Because the window size is dependent on detection results, varying levels of equalization occur for inter-image analysis.
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Michael Dessauer and Sumeet Dua
Fig. 2.8. Histogram specification of green intensity channel performed on Image 1 (left) with model histogram (top) to produce an image with similar intensity distribution (right).
Fig. 2.9. Example of MLE performing two levels of enhancement, using a 3 × 3 non-overlapping window.
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