Enhancement function and its properties
The aim for enhancing images is to improve the quality of the images so that image perceptions required by further processes can be obtained. So a welldesigned enhancement algorithm can amplify the details of image, can adjust the image contrast, and meanwhile, can remove the noise of signals. It means that the coefficients of the image should be processed separately. So it is a good choice to use the nonlinear function as the enhancement model.
The enhancement model used in our enhancement algorithm is proposed by A.F. Laine in 1996 [14]. Since then, many scholars have applied this function to image enhancement processing and have achieved good results.
$$ f(x)= a\left[\mathrm{sigm}\left( c\left( x b\right)\right)\mathrm{sigm}\left( c\left( x+ b\right)\right)\right] $$
(1)
Where \( a=\frac{1}{\mathrm{sigm}\left( c\left(1 b\right)\right)\mathrm{sigm}\left( c\left(1+ b\right)\right)} \), sigm is defined as \( \mathrm{sigm}(x)=\frac{1}{1+{e}^{ x}} \)。
There are two parameters named c and b in this function, where b is used to control the enhancement range, and its value range is (0, 1); while c is used to control the enhancement intensity, which usually takes the fixed value between 20 and 50. In the existing researches, many enhancement algorithms took the selection of the two parameters of b and c as key factors to improve the nonlinear enhancement effect. Figure 1 demonstrates the curves with different b and c values in different colors. As shown in Fig. 1, the shape of the enhancement function varies greatly with different b and c values. At the same time, it also can be found that (take the positive direction of X axis as an example) two special points play important roles in the form of the function curve. One point is the first intersection between the curve and the line f(x) = x. It determines whether the image coefficient should be suppressed by denoising or enhanced by amplifying. It means that the horizontal coordinate of the point is the boundary of denoising and enhancement. The other important point is the intersection between the function curve and line y = 1, it determines the extent of the enhancement. The coefficients with greater value than the horizontal coordinate of the point will be increased to the maximum value of the coefficients, while the smaller coefficients will be amplified nonlinearly according to the function curve.
The two important points on the function curve, in this paper, are proposed as two thresholds to determine the form of the enhancement function. The thresholds will be selected adaptively according to the image multiresolution decomposition coefficients of different scales and directions.
Selection of the thresholds
We set two thresholds of \( {T_k^l}_{\mathrm{low}} \) and \( {T_k^l}_{\mathrm{high}} \) in the enhancement function. \( {T_k^l}_{\mathrm{low}} \) is the threshold to distinguish the coefficients between the enhancement set and the suppression set, \( {T_k^l}_{\mathrm{high}} \) is used to control the contrast stretching degree of the enhanced coefficients. The two thresholds correspond to the two special points in the enhancement curves analyzed in 2.1.
Shearlet transform can decompose images into different scales and directions. And the coefficients of different scale have different features. The highfrequency coefficients often correspond to the image edges and details. This kind of coefficient needs obvious enhancement, but noise signals also mostly exist in the highfrequency coefficients. So it is required to suppress the highfrequency coefficients which have smaller absolute values. The lowfrequency coefficients correspond to the image background or smooth parts, such coefficients should not be sharply enhanced but should try to maintain its contrast. Since there is less noise signals in lowfrequency coefficients, their demand for denoising is not obvious. Therefore, we will set the thresholds for different decomposition scales and directions adaptively.
\( {T_k^l}_{\mathrm{low}} \) corresponds to the horizontal coordinate of the enhancement function curve at the point of x = f(x). It is supposed that \( {T_k^l}_{\mathrm{low}} \) is proportional to the standard deviation of the decomposition coefficients, and the standard deviation calculation function satisfies formula (2).
$$ \sigma =\sqrt{\frac{1}{\mathrm{MN}}{\displaystyle \sum_{m=1}^M{\displaystyle \sum_{n=1}^N{\left({c}_k^l\left( m, n\right)\mathrm{mea}{\mathrm{n}}_c\right)}^2}}} $$
(2)
Where σ is the standard deviation of the coefficients; \( {c}_k^l\left( m, n\right) \) is the value of the coefficient in the position of (m, n); mean_{c} is the mean value of the coefficients of k scale l direction.
The frequency of decomposition coefficient rises with the increase of the decomposition scales. Noise signals are more likely to be gathered in the highscale coefficients. So the calculation of \( {T_k^l}_{\mathrm{low}} \) needs to consider the scale factors. The calculation function of \( {T_k^l}_{\mathrm{low}} \) is shown as formula (3).
$$ {T_k}_{\mathrm{low}}^l=\frac{1}{8}\sigma *\mathrm{level} $$
(3)
Where \( {T_k^l}_{\mathrm{low}} \) represents the first threshold of the coefficients in k scale l direction, σ is the standard deviation of the decomposition coefficients, level is the decomposition scale. Formula (3) means that the value of \( {T_k^l}_{\mathrm{low}} \) is propositional to the decomposition scale. That is, the greater the value of the decomposition scale, the more stringent the coefficients denoising.
\( {T_k}_{\mathrm{high}}^l \) corresponds to the value of \( \frac{9}{c}+ b \) because from this point the derivative of the enhancement function begins to approach zero [15]. Its calculation function satisfies formula (4)
$$ {T_k}_{\mathrm{high}}^l={T}_{\mathsf{low}}^l* n/\mathrm{leve}{\mathrm{l}}^2 $$
(4)
Where \( {T_k}_{\mathrm{high}}^l \) represents the second threshold of the coefficients in k scale l direction; n is a constant, it is assigned to 150 in our experiments; \( {T_k}_{\mathrm{high}}^l \) is inversely proportional to the decomposition scale. That is, the higher the frequency of the decomposition coefficient, the stronger the enhancement degree of the coefficient. While for the lowfrequency coefficient, the enhancement curve tends to be gentle.
The overall framework of the algorithm
A flow chart of the proposed enhancement algorithm is illustrated in Fig. 2, which is composed of seven steps:

(1)
Decompose the original image by NSST to K levels and L orientations.

(2)
Normalize the two highest frequency scales of coefficients in order to prepare them to be enhanced.

(3)
Calculate the standard variance of the normalized coefficients in each layer and direction.

(4)
Calculate the two enhancement thresholds for each layer and direction and use the thresholds to determine the form of the enhancement function.

(5)
Enhance the normalized coefficients by the enhancement function.

(6)
Antinormalize the enhanced coefficients.

(7)
Recompose the image by the enhanced highfrequency coefficients and the original lowfrequency coefficients.
Our algorithm is suitable to enhance grayscale images. Since it can suppress noise signals by the first threshold, it can also be used to enhance noisy images.