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The performance of COE detection can be improved by weighting the CC matrix so that it emphasizes the region where the COE is most likely to be present and suppresses the regions of non-eye features within the eROI. The weighting function was derived by using prior knowledge about the distribution of annotated COEa with respect to the centre of the eROI.

Figure 5-28. Histogram showing the position of the annotated COE in reference to the corresponding centre of eROI. This histogram show a Gaussian distribution of COE where most of the COE are positioned close to the centre of eROI and only few COE are found at the edges of the eROI.

Figure 5-28 shows a histogram of the distribution of the annotated COE from the centre of the eROI of both eyes in each frame in the annotated reference database. This histogram shows that the distribution has an approximately a Gaussian shape and can be represented by a 2D circularly symmetric Gaussian function G(x, y) as shown in Figure 5-29.

Hence, to improve the detection of COE, the CC matrix of each eROI is multiplied with Gaussian matrix G(x, y) as shown Equation 5-9, to form weighted correlation matrix CCg.

( , )

g = × x y

CC CC G

Equation 5-9

Figure 5-30 shows an example where a falsely estimated COE in the left eROI of subject-1 was corrected by multiplying the CC of the eROI with a Gaussian matrix. In this eROI, the Gaussian matrix suppressed the false peak at the eyebrow region, which resulted in making the peak towards the true COE highest and correctly estimated as the COE position. In addition, the COE in the right eROI of the subject, which was already correctly estimated, was not affected by weighting the corresponding CC.

The respective left and right Gaussian functions shown in row two of Figure 5-30(b) and Figure 5-30(c) appear different because the parameters of the Gaussian matrix are dynamically derived for each eROI independently. For example, the size of the Gaussian matrix is set to be of same size as the CC of an eROI to be corrected.

Equation 5-10 gives the formula for deriving a Gaussian matrix. The centre (xc, yc) and standard deviations (σ_x, σ_{y) of the Gaussian matrix are derived from statistical information}

about the distribution of annotated COE (COEa) in the reference database. The centre coordinates (xc, yc) of Gaussian matrix is defined as the mean of respective x and y-coordinates of all COEa. Separate centres were derived for the left and right eROIs. Since the Gaussian matrix is chosen to be circularly symmetric the σ_{x and} σ_{y will be equal and is represented as}

single SD (σ_{). The} σ _{of the Gaussian matrix is derived by calculating the mean SDs of COEa}

distribution within both left and right eROI.

2 2 ( , ) c c x y x x y y u x y σ σ    ₋  ₋ =  +_ _     ( , ) ( , ) (1 ) exp 2 u x y x y = −p +_p× _− __     G Equation 5-10

(a)

(b) (c)

(d)

Figure 5-30. (a) Image showing the false estimation of eyebrow as the COE in the right eROI and correct estimation of COE in the left eROI. (b) Three rows sequentially showing the mesh plot of raw CC, Gaussian weighting function, and the weighted CC for the left eROI. (c) Same plot as (b) for the right eROI, where peak at the eyebrow region in CC is suppressed due to Gaussian function weighting making the peak close to the true COE towards the centre of the eROI highest peak in the CC. (d) Image showing correct COE estimation for the right eROI and unaffected COE estimation for the left eROI after the correction of COE estimation with Gaussian weighting function.

The Gaussian matrix is normalized so that its peak value is set to 1 and its minimum value is set to (1 – p). In Equation 5-10, p is a scaling factor, which defines the lower weighting limit of a Gaussian matrix. In other words, the p value of Gaussian matrix affects the weight put on edges of CC matrix. To determine an optimum p value for general population in the collected reference database, the CC matrix of each eROI in annotated frames was multiplied with different Gaussian matrixes derived with various p values ranging from 0 to 1 with increment of 0.1. Then, the p value for which the COE detection algorithm produced the most accurate (lowest error magnitude) result was selected as the optimum fixed p value used for every Gaussian matrix.

Figure 5-31 shows means of 90th percentile error magnitudes (plot on the left) and means of median error magnitudes (plot on the right) in estimating the COE’s y-coordinate of nine subjects based on various CCg derived with Gaussian matrixes with various p values. Each plot in Figure 5-31 shows the error magnitude in COE detection for three sub-groups of subjects, which are five subjects without glasses, four subjects with glasses, and combination of all nine subjects.

In the mean 90th percentile error magnitude plot of all nine subjects, the COE detection algorithm had the lowest error magnitude of 4.7 pixels in estimating COE when the p of the Gaussian matrix was set to 0.7. The mean medians error magnitude plot of the nine subjects indicated a lowest error magnitude of 2.1 pixels in estimating COE when the p was set to 0.6. The optimum p value for the Gaussian matrix was set to 0.7 as indicated by the 90th percentile plot because it represents the worst-case scenario. In addition, the mean median performance of COE detection is not compromised by setting the p value to 0.7 because there is very small difference between the average median error magnitudes when p = 0.6 and p = 0.7.