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Ocular thermograms for both sets were obtained in a controlled environment where the temperature was kept at 25 ± 1◦C with a mean humidity of 78%. varioTHERM head II (Germany) (http://www.jenoptik-ir.com/) was the instrument used to capture the ocular IR thermal images in this study. The camera was placed 50 cm from the chin rest where subjects rested their chins for image taking. These thermal images were stored in the irbis format, which records temperature values after each thermogram shooting, and exported to jpeg format with a size of 256 × 256 pixels, by the built-in post-processing software for further processing.

5.3. Methods

The automated localization of the eye and cornea was achieved by using the snake algorithm and the target-tracing function minimized by the genetic algorithm.30,31 Snake,32 or active contour, is a series of points (dubbed snake points) moving under an external force field to lock onto nearby edges. We use it to delineate the edges of the eye under the external force field termed gradient vector flow (GVF).33 However, an accurate localization can be achieved only if the starting contour is closed to the feature of interest and is an appropriate shape. In this algorithm, the problem was overcome by proposing the target-tracing function and solved by using a genetic algorithm to search for the starting contour that is capable of correctly locating the eye. Afterwards, the corneal radius and its center are acquired using the snake that has localized the eye.

5.3.1. Snake and GVF

Traditionally, snake is a curve x(s) = [x(s) y(s)], where s [0, 1], defined on an image domain.32 It moves across the image spatial domain under a combination of external and internal forces to minimize the following energy functional:

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α andβ are parameters that control the length and rigidity of the snake33; x (s) and x (s) are the first and second derivatives with respect to the contour parameter, s. γ is a step-size, and A is a penta-diagonal banded matrix,32

where J = β, K = −α − 4β, L = 2α + 6β, M = −α − β, N = β.

Eext refers to the external energy, and Eext = [fx(x, y) fy (x, y)]. The external energy function opted for this work is the GVF field,33 Eext = v(x, y) = [u(x, y), v(x, y)]. The GVF is a dense vector field that minimizes

where µ is a parameter regularizing the first and second terms in the integrand and fe(x, y) is the edge map.30,31

fe = − | [Gσ1 (x, y) I(x, y)]| + re · Gσ2 (x, y) | sobI(x, y)| , (5.4)

where re is a control parameter and sob is the Sobel gradient operator. Gσ1 and Gσ2 are Gaussian blur functions with a standard deviation of σ1 and σ2, respectively.30,31 The proposed edge map combines two slightly different types of image derived forces to smoothen the ocular thermogram and to preserve the subtle pixel difference in the lower eyelid region.30,31

The solution to Eq. (5.3) is given by:

During the GVF snake algorithm run, the snake of interest often gets trapped in the border of the image due to the artifacts introduced during

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the calculation of Gaussian blur and GVF force field.30,31 This problem is resolved by expanding the image matrix, using the following formula and performing a number of iterative calculations. Let A be an image matrix, where the size is m × n. The expanded image after each iteration, labeled matrix B (with a size of [m + 2] × [n + 2]) is obtained by Eq. (5.5).30,31