MF does not distinguish noisy pixels from regular pixels and processes every pixel in images, which results in increased computational load to process the signal pixels that do not need to be processed. Therefore, a good deal of computational load can be saved by performing noise reduction only on noise pixels. One of the ways to alleviate computing overload is to detect the noise pixels before performing any noise removal. The key point is how to detect noise pixels intelligently. To detect noise pixels in IR images, one must consider the imaging mechanism of IR images. The imaging mechanism is derived based on the heat radiation law and heat conduction law. The details have been described in [1].

According to the heat radiation law, the gray-level of pixels has a positive relationship with the temperature of the object surface. In addition, the heat conduction means that the temperature of the object varies monotonically on a surface. As a result, the crucial property of the IR imaging mechanism is that the gray-level of signal pixel of an object changes monotonically in IR images. On the contrary, the gray-level variations of noise pixels do not follow the IR imaging mechanism. FMBF is developed based on this crucial property of the IR imaging mechanism to detect noisy pixels.

A number of impulse detection methods [11–17] have been proposed that combind with MF to reduce the impulse noise. They include several different types of methods. All of these methods detect impulse noise in visual images. In this study, a noise detection algorithm has been developed based on IR imaging mechanism to detect the noise in IR images.

In order to explain how to perform the noise detection, some parameters of image pixels must be defined first. The terms x and y represent the horizontal and vertical coordinates of a pixel, respectively. p(x,y) is the pixel with coordinates x and y. g(x,y) is the gray-level of the pixel p(x,y). S_xy is the set that includes g(x,y) and its neighbor pixels. gm is the median gray-level of the S_xy, g(x,y) represents the gray-level of the central position pixel that will be processed, (s,t) denotes the coordinates of the pixels belonging to S_xy, and g(s,t) represents the gray-level of the pixels belonging to S_xy.

According to the imaging mechanism of IR images, the gray-level of an object varies monotonically, which results in an important property: if the gray-level g(x,y) is equal to the maximum gray-level g_max or the minimum gray-level g_min inside a filter window, the central position pixel p(x,y) will be a noisy pixel and should be replaced. Otherwise, it is considered as a signal pixel and should not be changed. Hence, this paper proposes the noise detection algorithm to find noisy pixels inside the filter window in IR images based on two steps. First, the noise detection method employs Equations (1a) and (1b) to find the maximum and minimum gray-level inside the filter window. Next, the algorithm checks the gray-level g(x,y) with g_min and g_max to consider whether the pixel p(x,y) is noise or signal. This noise detection can be performed in the following steps.

The proposed noise detection is based on the IR imaging mechanism. Therefore, FMBF can exactly identify the noisy pixels and replace them in IR images. The IR image’s edges, text information and details of objects are not damaged while being processed by the proposed FMBF.

According to the property of IR imaging mechanism, the pixel with median gray-level inside the window is adopted replacing the noisy pixels. In the proposed approach, the procedure to find out median gray-level is performed by the sort algorithm with a low computation complexity. In addition, it only processes the noisy pixels, but not the signal pixels.

Sorting is the main computation load of the noise removal, so in order to speed up the noise removal, reducing the computational load is critical. This study adopts a sort algorithm with low computational load. From an analysis of the properties of the versatile sort algorithms, a Radix sort with suitable parameters should run faster even than Quick sort or Heap sort [36]. The complexity of Radix sort is described as Equation (3): (3) Complexity of Radix sort   =   O ( n   log d   q )where n is the number of data, d is the number of digits, q is the range of the digit.

Thus in the proposed approach, a Radix sort algorithm with suitable d and q is selected to sort the pixel gray-level. In order to reduce the computational load of the Radix sort algorithm further, a Bit-plane processing technique is utilized to perform the Radix sort algorithm. The Bit-plane processing is easily implemented by using hardware in parallelization [37]. It can be achieved by the Bit-plane decomposition method [38], which decomposes the IR image into m bit-planes, where m is the number of bits in each pixel. Then the Radix sort algorithm is performed on these bit-planes to select the pixel with the median gray-level g_m inside a filter window. The selected pixel with gray-level g_m is utilized to replace the noisy pixel in the central position inside the filter window and the noise removal is accomplished.

FMBF does not act like the MF in sorting each pixel in a whole IR image; it only processes noisy pixels. In general, there are far fewer noisy pixels than signal pixels. FMBF has to expend extra computational load to determine the maximum and minimum gray-level in the noise detection procedure. The computation complexity of doing is O(n), which is still less than the computational complexity of Radix sort O(n log_d q). Thus, the proposed FMBF can theoretically save time in noise removal.

In order to automatically divide the histogram into hot objects and backgrounds for miscellaneous IR images, an adaptive threshold is needed. Abundant threshold selection methods have been described in [39] and have been categorized according to their information, such as entropy, histogram shape, spatial correlation, measurement space clustering, object attributes, and local gray-level surface. A fixed threshold is unsuitable to address the variations in different IR images. In order to conquer this real problem, this paper applies the iterative threshold selection method [40], based on the clustering-based thresholding method, to choose an adaptive threshold. The iterative threshold selection method has been effective in extracting adaptive thresholds from bimoded or non-bimoded histograms [1,41]. The procedures of the iterative threshold selection method are as follows:

Step 1. Initialize the threshold Th(1) (0 < Th(1) < 255) with a random value to segment the image into background and object.

The enhanced effects caused by HE will be less obvious on hot objects than on large area backgrounds. In order to remedy this shortcoming of HE, this paper proposes a HCHE, whose main property is the use of two different enhancement effects on hot objects and backgrounds: a large effect on hot objects and a small effect on backgrounds. The details for constructing a HCH follow.

In order to explain the generation of HCH, the cumulative histogram of HE, Equation (7), and its associated parameters are defined first: (7) T ( r k ) = ∑ j = 0 k p r ( r j ) = ∑ j = 0 k n j / n                 k = 0 ,     1 ,     2 , … ,   L − 1 (8) p r ( r k ) = n k / n           k = 0 ,     1 ,     2 , … ,   L − 1 (9) s k = int ( ( L − 1 ) × T ( r k ) )where r_k is the gray-level in the histogram of the source image, k is the gray-level, n is the total number of pixels in an image, n_k is the number of pixels with gray-level r_k, p_r(r_k) is the probability of the pixel with gray-level r_k, s_k is the pixel gray-level in the histogram of processed images, L is the number of gray-levels, L = 256 in our case, T(r_k) is the cumulative histogram of the HE generated by the source image, and int() is a function giving the integral part of a decimal number.

Based on the selected threshold Th, the histogram is divided into two groups: the pixels with gray-level smaller than Th, which belong to the background, and the pixels with gray-level equal to or larger than Th, which belong to the hot objects. The cumulative histogram to enhance the background is constructed by using the probability density function (PDF) of the background histogram, which is expressed in Equation (10a). The intensity density function (IDF) is defined as Equation (10c). The cumulative histogram to enhance hot objects is generated by using the IDF of the hot object histogram, which is expressed in Equation (10b). The HCH is constructed by combining Equations (10a) and (10b): (10a) T hb ( r k ) = { ∑ j = 0 k Arg   min { p r ( r j ) , p r ( r Th ) }                         0 ≤ k < Th (10b) T hb ( r k ) = { ∑ j = Th k p r ( r j ) × log 2 j                                                 Th ≤ k ≤ L − 1 (10c) IDF   =   p r ( r k ) × log 2 k 0 ≤ k ≤ L − 1where T_hb(r_k) is the HCH.

Equation (10a) accumulates the minimum of probability p_r(r_j) and p_r(r_Th) to generate the cumulative histogram for enhancing the background. Equation (10b) accumulates the IDF of probability p_r(r_j) to generate the cumulative histogram for enhancing hot objects. Combining Equations (10a) and (10b) generates the HCH of HCHE, which possesses two different enhancement effects on backgrounds and hot objects.

The derivative of a function is defined as the ratio of Δy and Δx. The terms Δy and Δx represent the argument increment of a function in Y-axis and X-axis, respectively. In general, the argument in Y-axis is the output and the argument in X-axis is the input of a function. Thus, the derivative of cumulative histogram is large which reveals the enhancement effect of cumulative histogram is large. Similarly, the derivative of cumulative histogram is small which reveals the enhancement effect of cumulative histogram is small. Therefore, the derivative of cumulative histogram can represent the enhancement effect of cumulative histogram.

Comparing Equations (7) with (10a), it is clear that the derivative of Equation (10a) is less than or equal to the derivative of the cumulative histogram of the HE when the gray-level r_k belongs to the background. Then Equation (10b) constructs the cumulative histogram of the hot objects. Comparing Equations (7) with (10b), it is easy to observe that the derivative of Equation (10b) must be larger than or equal to the derivative of Equation (7) when the gray-level r_k represents the hot objects. Thus, the hot objects’ enhanced effect caused by the HCH is larger, and the backgrounds’ is smaller. This result can be proven based on the backward difference approximation of the first derivative (BDAFD) [42]. According to the BDAFD, the first derivative of a cumulative histogram of the HE can be expressed as: (11) T ' ( r k ) = ( T ( r k ) − T ( r k − 1 ) ) / ( r k − r k − 1 )                           = ( ∑ j = 0 k p r ( r j ) − ∑ j = 1 k − 1 p r ( r j ) ) / 1                           = p r ( r k )where T’(r_k) is a first derivative of the cumulative histogram T(r_k).

In the same manner, the first derivative of the HCH can be expressed as: (12) T hb ' ( r k ) = ( T hb ( r k ) − T hb ( r k − 1 ) ) / ( r k − r k − 1 )           = { Arg   min   { p r ( r k ) , p r ( r Th ) } 0 ≤ k < Th p r ( r k ) × log 2 r k Th ≤ k < L − 1where T_hb’(r_k) is a first derivative of the HCH T_hb(r_k). If gray-level r_k belongs to the background, i.e., 0 ≤ k < Th, it yields: Arg   min { p r ( r k ) , p r ( r Th ) } ≤ p r ( r k )

So: (13) T hb ' ( r k ) ≤ T ' ( r k )                                                         0 ≤ k < Th

Thus, the background’s enhancement effect caused by the HCH Equation (10a) is smaller than or equal to that caused by the cumulative histogram of the HE. When the gray-level r_k belongs to the object, i.e., Th ≤ k < L − 1, it yields: p r ( r k ) × log 2 r k ≥ p r ( r k )

Thus: (14) T hb ' ( r k ) ≥ T ' ( r k )                                                         Th ≤ k < L − 1meaning that the enhancement effect on objects caused by the HCH, Equation (10b) is larger than or equal to that caused by the cumulative histogram of the HE.

As a result, the enhancement effect on hot objects caused by HCHE is larger than or equal to that caused by the HE. On the other hand, the enhancement effect on backgrounds caused by HCHE is smaller than or equal to that caused by the HE. This characteristic is especially significant when the hot objects are much smaller than the backgrounds. Consequently, HCHE proposed in this study can alleviate the limitation of HE.

In addition, in order to avoid saturation of the equalized gray-level of transformed pixels, the maximum gray-level of transformed pixels must remain less than or equal to the maximum scale of the gray-level. A normalized version of HCHE is needed and is expressed as: (15) T n ( r k ) = T hb ( r k ) / T hb ( r L − 1 )                                                         0 ≦ k ≦ L − 1where T_n(r_k) is the normalization of T_hb(r_k).

In order to verify the validity of the proposed approach, four life-time IR images of palm-dorsa are collected and used as the test samples for our study. Each IR image has 640 H × 480 V pixels and each pixel is represented by 256 gray-levels. As shown in Figure 2, the IR images all display low contrast and brightness. The vein-patterns are hard to observe in these images. They have to be preprocessed to improve the image quality for the future postprocessing.

Figure 3 shows the Figure 2(a–d) with impulse noise of 10%, 20%, 30% and 40%, and exhibit the results of performing FMBF to iteratively filter the noisy images. Considering Figure 3(a1–a4) as examples, Figure 3(a1) is filtered once, Figure 3(a2) filtered once, Figure 3(a3) is iteratively filtered two times, Figure 3(a4) is iteratively filtered three times, respectively. Figure 3(b1,b2) demonstrate how the impulse noise is filtered out and the edges, textures and detail information preserved simultaneously. Figure 3(b3,b4) demonstrate how the impulse noise is almost filtered out except for a few noise with large area. The other filtered images as shown in Figure 3 are quite consistent with the Figure 3(b1–b4).

The PSNR is applied to assess the noise reduction performance of FMBF. The PSNRs of Figure 3(a–d) caused by FMBF are addressed in Table 1. Using Figure 3(a1) as an example, the PSNR caused by FMBF is 57.90 dB, meaning that the degree of noise reduced by FMBF is 20.82 dB. It is easy to observe, the noise in Figure 3(b1) is less than in Figure 3(a1), which shows the consistence with the PSNR. Table I also shows the PSNRs caused by FMBF of Figure 3(b2–b4) are 51.32 dB, 53.28 dB and 52.60 dB, respectively. This shows that FMBF reduces noise by 17.26 dB, 20.98 dB and 21.52 dB for Figure 3(b2–b4), respectively. Compared to the MF, FMBF improves the PSNRs 2.67 dB, 1.03 dB, 1.44 dB and 2.03dB for Figure 3(a1–a4), respectively.

Table 1 also illustrates the sort time performance of applying FMBF to noisy images shown as Figure 3(a1–a4), (c1–c4), (e1–e4) and (g1–g4). FMBF performs a sorting algorithm for noisy pixels only, performing the sorting algorithm 204,500 times for Figure 3(a1). Hence, the computation load is reduced by the proposed FMBF by 33.4%. FMBF reduces the computation load 44.1%, 32.1% and 25.2% for Figure 3(a2–a4), respectively. These experimental results are quite consistent with the aforementioned theoretical analysis.

In the same manner, Table 1 also illustrates the results of Figure 3(a1–a4) filtered by FMBF. All the Figure 3 and Table 1 show that the performances of the proposed FMBF can reduce impulse noise and computer load more effectively than MF. In addition, FMBF performs the Radix sort algorithm with bit-plane, so it is easily implemented by hardware in parallelization. Thus, FMBF can save further computational time when performed in a parallel hardware architecture.

Figure 4 demonstrates the experimental results of Figure 3(b2) enhanced by the HE and HCHE with a threshold of 51. The image enhanced by HE is shown in Figure 4(e), and the histogram of Figure 4(e) is shown as Figure 4(g). Figure 4(e,g) reveal that the intensities of the backgrounds are increased, and the contrast is spread. Intensities of the hot objects, vein-patterns, are also increased to near-maximum gray scale, but their contrast is compressed to a narrow gray-level range. On the other hand, when one observes the derivative of the HCH of the HCHE, as shown in Figure 4(d), one can see that the derivative in the interval from gray-level 37 to 51 is less than that in the interval from gray-level 51 to 74. The former represents gray-levels belonging to backgrounds, and the latter represents gray-levels belonging to hot objects. Thus, the enhancement effect on backgrounds induced by HCHE is less than that on hot objects, and the contrast of backgrounds is decreased, whereas the contrast of hot objects is increased. The enhanced image by the HCHE is addressed in Figure 4(f), and the histogram of Figure 4(f) is shown as Figure 4(h). The gray-levels of backgrounds in Figure 4(h) are decreased and the contrast is compressed, while the gray-levels of hot objects are increased and the contrast is spread. These figures show that the experimental results in the cumulative histogram and the HCH match well with Equations (8,9) and the proposed HCHE can remedy the limitation of the HE.

Figure 5 demonstrates the performance of HCHE. The results of Figure 3(b1–b4) enhanced by HCHE are shown as Figure 5(a1–a4). Figure 5(b1–b4) show the histograms of Figure 5(a1–a4). Compared to Figure 3(b1–b4), the contrast of images enhanced by HCHE is better than original images and the vein-patterns are much clearer and easier to observe. Figure 5 exhibits the high performance of HCHE to enhance vein-patterns IR images.