Well-exposedness Measure

Well-exposedness measure evaluates how well a pixel is exposed. Mertens et al.

[22] utilized a Gaussian distribution to model the exposedness of a pixel depending on how close its luminance is to the target luminance values 128 (the middle value of luminance range [0, 255]). That is, the pixels with luminance value closer to gray level 128 should have a larger weight while the pixels with luminance far away from 128 should have a smaller weight when computing the well-exposedness measure.

Generally, the well-exposedness measure is defined as follows [22]:

2 .

where E_k(x, y) is the well-exposedness value of the pixel located at (x, y), and σ is the standard deviation of the Gaussian distribution which is set as 0.2×255 (the luminance range). From Eq. (21), the well-exposedness value is bounded to be the range [0, 1].

Further, the pixel with luminance value closer to gray level 128 will be assigned a original luminance values. Then, distinct target luminance values are defined for

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different classes. Intuitively, the target luminance values Y_i^t (i = 1, 2, …, m) can be determined by finding the center of each equally-spaced interval (see Fig. 12 for an example with cluster number m = 3).

Fig. 12 The equally-distributed target luminance values Y_i^t with m = 3.

This method assumes that the whole luminance range (256) is divided into m equally-spaced intervals having width R = 256/m. Further assuming that the luminance values in each class are Gaussian distributed with width equals R = 6σ_e where σ_e is the standard deviation of the Gaussian distribution. Thus, the target luminance value associated with class Ω_i is

. 2) (i 1 R

Y_i^t    (22)

However, the equally-spaced target luminance values, without considering property of the input image, should be adjusted to appropriate values. For example, if the input image is a dark one consisting of many dark pixels in Ω₁, the target luminance value Y₁t should be adjusted as well. Similarly, if the input image is a bright one consisting of many bright pixels in Ωm, the target luminance value Y_m^t should be adjusted in according with the number of pixels belonging to Ω_m. Another approach, considering

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the number of pixels in each class, will take into account the image property to find

target luminance values. Let pi denote the probability of the pixels belonging to class Ωi in the input image. Then, the cumulative probability Cum(i) for each class Ωi is

According to the probability of each class, the target luminance value is defined as the middle value in each target luminance range:

m

Further, the mean luminance value of the pixels belonging to the largest class is used to determine whether an input image is a dark one or a bright one. Let N_i denote the number of pixels belonging to class Ω_i. The index of the largest class is defined as

follows:

Then the mean luminance value of the largest class _n^o

max is computed:

29 However, if the luminance value of a pixel is near the boundary of two classes, it is hard to determine which class this pixel really belongs to. That is, it is hard to determine its target luminance value and thus it is impossible to correctly evaluate the appropriate exposedness value. Therefore, we exploit the concept of fuzzy clustering to determine the probability that a pixel belongs to a class. Let _i^o denote in the input image the average luminance of those pixels belonging to cluster Ω_i:

m

Then, the probability, computed as the likelihood that a pixel value is from each class, is modeled as a Gaussian function:

2 ,

where Y is the input image, σ_i is the standard deviation computed from the luminance values of those pixels having luminance values in the range [_{i 1}^o_, _{i 1}^o_ ]. Note that

o

0 is set to 0 and _{m 1}^o_ is set to 255. Since the range [_i^o₁, _{i 1}^o ] is larger for those i

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with 1im and thus the corresponding standard deviation is multiplied by 0.75 such that every σi is computed from similar range. Fig. 14 illustrates the above concept.

Fig. 14 The luminance range for determining σ_i with m = 3.

To utilize the fuzzy clustering concept, the well-exposedness value of the pixel in Yk

associated with class Ω_i is defined as follows:

m

where E_k(x, y) denotes the well-exposedness value associated with the pixel located at (x, y) in virtual exposure image Y_k. By applying different target luminance values to different classes, pixels will be adjusted toward different luminance value and thus the global contrast can be reserved. Fig. 15 and Fig. 16 show the well-exposedness maps produced by using the exposedness measure computed by using a single target luminance value 128 proposed by Mertens el al. [22] and the proposed classified

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exposedness measure. From Fig. 15, by observing the sky region, we can see that those lower exposure dark images have larger weight values than those in the brighter images. As a result, the luminance of the sky region will decrease in the fused image.

From Fig. 16, however, by using the proposed method, E-2 has larger weight values in the sky region and can preserve the luminance much better than Mertens’s method.

Fig. 17 and Fig. 18 show another example of computed well-exposedness maps by using the exposedness measure proposed by Mertens el al. [22] and the proposed classified exposedness measure.

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig. 15 Exposedness maps generated by using the exposedness measure proposed by Mertens et al. [22] (a) E-9 (b) E-8 (c) E-7 (d) E-6 (e) E-5 (f) E-4 (g) E-3 (h) E-2 (i) E-1 (j) E0 (k) E1 (l) E2 (m) E3 (n) E4 (o) E5

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig. 16 Classified well-exposedness maps (a) E-9 (b) E-8 (c) E-7 (d) E-6 (e) E-5 (f) E-4 (g) E-3 (h) E-2 (i) E-1 (j) E0 (k) E1 (l) E2 (m) E3 (n) E4 (o) E5

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig. 17 Exposedness maps generated by using the exposedness measure proposed by Mertens et al. [22] (a) E-14 (b) E-13 (c) E-12 (d) E-11 (e) E-10 (f) E-9 (g) E-8 (h) E-7 (i) E-6

(j) E-5 (k) E-4 (l) E-3 (m) E-2 (n) E-1 (o) E0

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig. 18 Classified well-exposedness maps (a) E-14 (b) E-13 (c) E-12 (d) E-11 (e) E-10 (f) E-9 (g) E-8 (h) E-7 (i) E-6 (j) E-5 (k) E-4 (l) E-3 (m) E-2 (n) E-1 (o) E0

36 of weights among these 2M+1 weight maps equals 1:

.

Fig. 19 and Fig. 20 show two example of final weight maps produced by multiplying the proposed contrast measure and the classified exposedness measure.

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig 19 Weight maps, Wk,of each exposure image Yk generated by using the proposed classified exposedness measure. (a) W-9 (b) W-8 (c) W-7 (d) W-6 (e) W-5 (f) W-4 (g) W-3

(h) W-2 (i) W-1 (j) W0 (k) W1 (l) W2 (m) W3 (n) W4 (o) W5

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig 20 Weight maps, Wk,of each exposure image Yk generated by using the proposed classified exposedness measure. (a) W-14 (b) W-13 (c) W-12 (d) W-11 (e) W-10 (f) W-9 (g) W-8 (h) W-7 (i) W-6 (j) W-5 (k) W-4 (l) W-3 (m) W-2 (n) W-1 (o) W0

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