An improved infrared image processing method based on adaptive threshold denoising

The infrared image characterizes the thermal properties (radiance) of the point in the scene with the brightness (grayscale) of each pixel. The wavelength of the infrared band is in the range of 0.75 to 1000 μm, and its short-wavelength boundary is connected to visible light [1]. In theory, the infrared radiation of an object depends on the surface temperature, apparent area, and emissivity ε of the object, but in fact, the infrared radiation of the surrounding environment (the sky, the ocean, other objects) must be measured, and the environmental factors have complex modulation effects on the infrared radiation of the target. The infrared radiation from the target will be selectively absorbed by some gases in the atmosphere before reaching the infrared sensor, and the airborne particles will also scatter the infrared rays, thereby causing the radiation flux to decay. When the infrared radiation from the target passes through the temperature change region, it will slightly deviate from the original direction. Since this area of heated air is unstable, the radiation deviation is irregular. This effect, called atmospheric flicker, causes the apparent direction and radiant flux of distant targets to change simultaneously.

For complex targets, their surface temperature or radiation flux will be inhomogeneous because of their strange shape and structure and many dispersed heat sources. The target operating speed also has a great influence on its convective cooling rate, which results in the change of target surface temperature. Target surface coating and thermal insulation measures, to a certain extent, can make the target infrared image blurred. Random noise generated by sensor elements is also an important obstacle to the transmission of infrared information.

To sum up, the infrared radiation of the target is very complex, which is because the factors affecting the infrared radiation of the target are multifaceted, including the radiation of the target surface, the absorption and reflectivity of the target, the climate conditions, the selective absorption and scattering of the atmosphere, the atmospheric scintillation, the velocity of the target, the surface coating and heat insulation, the noise of the circuit, and so on.

The diversity and complexity of the infrared target image information model determine the infinite dimension characteristics of the infrared target recognition observation sample space, especially the increasing infrared interference and the further development of infrared suppression technology, which makes great difficulties for the recognition of infrared targets. The effectiveness of the target feature strongly depends on the quality of the image preprocessing, which is equivalent to the ability to detect and segment the target effectively in the preprocessing phase of the primary vision processing image (including edge detection, noise filtering [2], image enhancement) under complex background conditions (low signal-to-noise ratio, low contrast, scene change, and so on) [3]. At this time, the processing effect of human vision of the traditional processing method is difficult to achieve, and the processing algorithm with human visual characteristics has a relative advantage. Wavelet transform is an ideal mathematical model for human primary visual information processing [4]. The decomposition of the image signal is not only in the spatial and frequency domains but also in the direction of the information expression local characteristics, which conforms to the basic characteristics of the human vision for the processing of image information. From the spatial domain, the size and direction of some image targets and other details are different, so the image needs to be processed at different resolution and direction. Here, wavelet transform can give full play to its advantages and can be analyzed from the frequency domain. Wavelet transform can decompose all kinds of mixed signals with different frequencies into block signals with different frequencies. Wavelet denoising has the advantages of multiresolution and diversity of wavelet basis selection, which makes it very suitable for image and other signal denoising [5]. Therefore, wavelet transform is a powerful mathematical tool for primary vision processing.

Simulation results show that the method proposed in this paper has a better suppression of noise and improves the image quality to a certain extent.

Nowadays, the denoising method based on wavelet transform has become an important branch of image denoising and restoration. Among the many image denoising methods based on wavelet transform, the wavelet threshold denoising method proposed by Donoho [6] and others has been widely used because of its simple principle, easy implementation, and remarkable denoising effect. But its inherent shortcomings, such as the discontinuity of hard threshold function, the constant deviation caused by soft threshold function [7–10], and the lack of scale adaptability, also limit its further development. Later, many scholars and researchers have studied it deeply from the point of view of constructing new threshold function and searching for optimal threshold and proposed many adaptive image denoising methods. Hong and Yang [11] have compared the signal-to-noise ratio (SNR) and root mean square error (RMSE) according to the shortcomings of soft threshold function and hard threshold function in the traditional wavelet threshold denoising algorithm and constructed a new threshold function to denoise the signal. To a large extent, the shortcomings of poor continuity and inherent deviation of these two functions were overcome in practical processing, which made the noise processing more effective and superior. Juncheng and Qiang [12] applied the translation invariant to the second-generation wavelet transform based on the existing threshold denoising method of the second-generation wavelet transform and realized the more rapid and effective denoising of seismic signal and have achieved good denoising effect in the trial calculation of analog data and actual data. Hong [13] improved the classical wavelet threshold denoising algorithm and proposed a new threshold denoising method of mine video monitoring image in the wavelet transform domain. The image was decomposed by three-level wavelet transform, the low-frequency decomposition coefficients were filtered by Wiener filter, and the high-frequency decomposition coefficients were denoised by improved wavelet I value denoising algorithm. Experiments showed that it can effectively reduce noise and get an effective and complete image. Xiaofei and Xiaohui [14] combined the advantages of typical wavelet threshold function, integrated some improved methods, and proposed an improved threshold function for its shortcomings. This function is not only continuous at the threshold; the coefficient is a symptotic original coefficient of the wavelet estimation, but also has differentiability, and it is easy to implement adaptive learning of the gradient algorithm. Fengbo, Changgeng, and Hongqiu [15] proposed an improved threshold algorithm based on lifting wavelet transform. First of all, the advantages of the wavelet transform, multiresolution, and diversity of wavelet bases were exploited. On the other hand, a new threshold function was proposed to improve the signal-to-noise ratio and reduce the RMSE. Denoising with the improved threshold function is superior to the commonly used threshold function in both visual effects and mean square error and peak signal-to-noise ratio performance analysis. Xiaoyan and Abdukirimturki [16] combined the advantages of typical wavelet threshold function with some improved methods and proposed an improved new threshold function. The experimental results show that the denoised image is better than the traditional soft and hard thresholding and the existing thresholding in terms of visual effect, mean square deviation, and peak signal-to-noise ratio. These methods not only open up a broad prospect for the full advantage of the wavelet threshold denoising method but also provide a basis for further exploration of adaptive denoising methods. However, in general, these studies are almost all based on the orthogonal wavelet transform, using the Mallat algorithm. Mallat algorithm [17] can achieve satisfactory speed, but due to the lack of translation invariance, the reconstruction accuracy after image denoising is not high, and the Gibbs phenomenon is easy to appear in the denoised image and cannot meet the requirements of human vision. The fast algorithm based on the binary wavelet transform [18]—à trous algorithm [19]—not only has the invariance of translation but also makes the representation of the image in the domain of binary wavelet transform very redundant, and the disturbance of partial coefficients will not lead to the serious distortion of the reconstructed image. In addition, for the Gibbs visual distortion in a denoised image caused by the wavelet threshold denoising method using orthogonal wavelet transform, many researchers have made improvements such as literature [20–23]. The results show that translation invariance is an important property of effectively suppressing Gibbs phenomenon and improving denoising effect. These conclusions provide a basis for the research of adaptive threshold image denoising based on binary wavelet transform.

2.2 Binary wavelet theory

In the Mallat algorithm, due to the downsampling operation, the singular points of the image edges are easily lost, resulting in Gibbs phenomenon at the edges of the reconstructed images after denoising, resulting in image edge distortion or even blur. In order to overcome this drawback, wavelet transform with translation invariance needs to be considered. The binary wavelet transform only discretizes the scale factor in the continuous wavelet transform and keeps the translation factor continuously changing, thus effectively maintaining the translation invariance. This feature of the binary wavelet transform, coupled with the fast decomposition and reconstruction algorithm, à trous algorithm, has certain advantages in the fields of image denoising and image feature extraction.

It should be noted that the à trous algorithm mentioned in this paper is a modified version of the à trous algorithm form described in the Mallat book, and the algorithm is different from the proof in the Mallat book.

Assuming that the sample value a_{0, k} of the input discrete signal is the local average of f in the domain of t = k, it can be written as:

$$ {a}_{0,k}=<f(t),\varphi \left(t-k\right)> $$

(8)

where φ and \( \tilde{\varphi} \) are the two scale functions generated by the infinite cascade calculation of filter (h, g) and its dual filter \( \left(\tilde{h},\tilde{g}\right) \), which can be expressed as:

$$ {\displaystyle \begin{array}{l}\varphi (t)=\sqrt{2}\sum \limits_k{h}_k\varphi \left(2t-k\right)\\ {}\tilde{\varphi}(t)=\sqrt{2}\sum \limits_k{\tilde{h}}_k\tilde{\varphi}\left(2t-k\right)\end{array}} $$

(9)

For any j ≥ 0, remember:

$$ {a}_{j,k}=<f(t),{\varphi}_{2^j}\left(t-k\right)> $$

(10)

where \( {\varphi}_{2^j}(t)=\frac{1}{\sqrt{2^j}}\varphi \left(\frac{t}{2^j}\right) \).

The binary wavelet coefficients are calculated as follows:

$$ {d}_{j,k}= Wf\left({2}^j,k\right)=<f(t),{\psi}_{2^j}\left(t-k\right)> $$

(11)

where \( {\psi}_{2^j}(t)=\frac{1}{\sqrt{2^j}}\psi \left(\frac{t}{2^j}\right) \) and ψ and \( \tilde{\psi} \) are the wavelet functions generated by the infinite cascade calculation of the filter (h, g) and its dual filter \( \left(\tilde{h},\tilde{g}\right) \), which can be expressed as:

$$ {\displaystyle \begin{array}{l}\psi (t)=\sqrt{2}\sum \limits_k{g}_k\varphi \left(2t-k\right)\\ {}\tilde{\psi}(t)=\sqrt{2}\sum \limits_k{\tilde{g}}_k\tilde{\varphi}\left(2t-k\right)\end{array}} $$

(12)

For the à trous algorithm, the binary wavelet decomposition algorithm is expressed as:

$$ {\displaystyle \begin{array}{l}{a}_{j+1,k}=\sum \limits_n{h}_n{a}_{j,k+{2}^jn},j=0,1,\cdots, \\ {}{d}_{j+1,k}=\sum \limits_n{g}_n{a}_{j,k+{2}^jn},j=0,1,\cdots .\end{array}} $$

(13)

The binary wavelet reconstruction algorithm is expressed as:

$$ {a}_{j,k}=\frac{1}{2}\left(\sum \limits_n{\tilde{h}}_n{a}_{j+1,k-{2}^jn}+\sum \limits_n{\tilde{g}}_n{d}_{j+1,k-{2}^jn}\right),j=0,1,\cdots . $$

(14)

The proofs of formulas (13) and (14) are given below. For formula (13), the following can be obtained from the definition of a_{j, k} and the relationship between two scales.

$$ {\displaystyle \begin{array}{l}{a}_{j+1,k}=<f(t),{\varphi}_{2^{j+1}}\left(t-k\right)>={\int}_Rf(t)\frac{1}{\sqrt{2^{j+1}}}{\varphi}^{\ast}\left(\frac{t-k}{2^{j+1}}\right) dt\\ {}\kern1.00em =\sum \limits_n{h}_n{a}_{j,k+{2}^jn},\kern0.5em j=0,1,\cdots .\end{array}} $$

(15)

In the same way, there are:

$$ {\displaystyle \begin{array}{l}{d}_{j+1,k}=<f(t),{\psi}_{2^{j+1}}\left(t-k\right)>={\int}_Rf(t)\frac{1}{\sqrt{2^{j+1}}}{\psi}^{\ast}\left(\frac{t-k}{2^{j+1}}\right) dt\\ {}\kern1.00em =\sum \limits_n{g}_n{a}_{j,k+{2}^jn},\begin{array}{cc}& j=0,1,\cdots .\end{array}\end{array}} $$

(16)

For formula (15), from the dual-scale relationship \( \widehat{\tilde{\varphi}}(2w)=\frac{1}{\sqrt{2}}\widehat{\tilde{h}}(w)\widehat{\tilde{\varphi}}(w) \), \( \widehat{\tilde{\psi}}(2w)=\frac{1}{\sqrt{2}}\widehat{\tilde{g}}(w)\widehat{\tilde{\varphi}}(w) \) and the binary condition \( {\widehat{\psi}}^{\ast }(w)\widehat{\tilde{\psi}}(w)+{\widehat{\varphi}}^{\ast }(w)\widehat{\tilde{\varphi}}(w)={\widehat{\varphi}}^{\ast}\left(\raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\widehat{\tilde{\varphi}}\left(\raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \), we get:

$$ \sqrt{2}\widehat{\varphi}\left(\raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)={\widehat{\tilde{h}}}^{\ast}\left(\raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\widehat{\varphi}(w)+{\widehat{\tilde{g}}}^{\ast}\left(\raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\widehat{\psi}(w) $$

(17)

Time domain is expressed as:

$$ 2\sqrt{2}\varphi (2t)=\sum \limits_k{\tilde{h}}_{2k}\varphi \left(t+k\right)+\sum \limits_k{\tilde{g}}_{2k}\psi \left(t+k\right) $$

(18)

At this time, for formula (14), there is:

$$ {\displaystyle \begin{array}{l}{a}_{j,k}=<f(t),{\varphi}_{2^j}\left(t-k\right)>={\int}_Rf(t)\frac{1}{\sqrt{2^j}}{\varphi}^{\ast}\left(\frac{t-k}{2^j}\right) dt\\ {}\kern1.00em =\frac{1}{2}\left(\sum \limits_k{\tilde{h}}_k{a}_{j+1,k-{2}^jn}+\sum \limits_k{\tilde{g}}_k{d}_{j+1,k-{2}^jn}\right),j=0,1,\cdots .\end{array}} $$

(19)

The above theorem describes the à trous algorithm of the one-dimensional signal. The structure of this algorithm is shown in Fig. 1. For the two-dimensional image signal, the one-dimensional à trous algorithm is applied along the rows and columns of the image to realize image decomposition and reconstruction.

Although the à trous algorithm has the same algorithm structure as the Mallat algorithm, there is a substantial difference: the à trous algorithm does not perform the downsampling operation, which not only preserves the translation invariance well but also the data length under each scale is the same as the original data length. Compared with the data of the Mallat algorithm, there is a great redundancy, which is convenient for spectrum analysis of details and profiles at each scale.

The simulated image is uniformly imaged by an infrared image of 182 × 208 pixels, which is taken by a medical infrared camera of Chongqing Weilian Technology Limited. The selected wavelet functions are sym4 wavelets with good symmetry and suitable for image denoising.

Firstly, the performance of the adaptive threshold function selected in this paper is analyzed, and the traditional soft and hard threshold functions are used as a comparison. The results of infrared image processing are shown in Fig. 2. Figure 2a is the original image, Fig. 2b is the result of soft threshold function denoising, Fig. 2c is the result of hard threshold function denoising, and Fig. 2d is the result of adaptive threshold function denoising designed in this paper. As can be seen in Fig. 2, the resulting image obtained by soft and hard threshold function denoising still contains a lot of noise compared with the original infrared image, and the visual effect is not much improved compared with the original infrared image. The overall effect of the image obtained by adaptive threshold denoising in this paper is relatively smooth and clear, so it can be said that the adaptive threshold function proposed in this paper is better for infrared image denoising.

In order to better show the denoising effect of soft and hard thresholding functions and adaptive thresholding functions of this algorithm, the standard deviation of denoising of different thresholding functions for men and women in Table 1 and SNR or F/MSE is averaged, and the histogram is drawn as shown in Fig. 3. Combining Fig. 3 and Table 1, we can see that the soft and hard threshold functions can denoise the image and raise the standard deviation by about 5 and the SNR or F/MSE by about 1 dB. The adaptive threshold function proposed in this paper can denoise the image and raise the standard deviation by about 60 and the SNR or F/MSE by about 3 dB.

In order to validate the performance of the infrared image denoising algorithm designed in this paper, the traditional wavelet threshold denoising algorithm and the algorithm designed in this paper are used to denoise the infrared image. The results are shown in Fig. 4, in which Fig. 4a is the original image, Fig. 4b is the denoising result of the traditional wavelet threshold denoising algorithm, and Fig. 4c is the denoising result of the algorithm designed in this paper. In Fig. 4, we can see that the traditional wavelet threshold denoising algorithm can effectively improve the performance of infrared image, but the image denoised by this algorithm retains the edge features of the image and effectively improves the Gibbs phenomenon near the edge and the image details are clearer and the visual effect is obviously better than the traditional wavelet denoising method.

Similarly, Table 2 gives the performance data of the traditional wavelet threshold method compared with the algorithm in this paper. In order to better show the denoising effect of the traditional wavelet threshold algorithm and the algorithm in this paper, the standard deviation of different threshold functions for men and women in Table 2 and SNR or F/MSE is averaged, and the histogram is drawn as shown in Fig. 5. Combining Table 2 and Fig. 5, we can see that the traditional wavelet threshold algorithm can improve the standard deviation by about 5 and the SNR or F/MSE by about 1 dB, while the image denoising by this algorithm can improve the standard deviation by about 67 and the SNR or F/MSE by about 6 dB.

Gender	Male			Female
Index	Standard deviation s	SNR or F/MSE(dB)	NMSE	Standard deviation s	SNR or F/MSE(dB)	NMSE
Original infrared image	82.12	12.01	/	82.25	12.13	/
Traditional wavelet threshold denoising image	78.25	13.03	9.1367e−005	77.53	12.96	9.0304e−005
Image denoising based on this algorithm	15.22	18.53	4.1217e−005	15.78	18.62	4.2077e−005

In summary, experiments show that the proposed method for infrared image denoising has better performance than the traditional wavelet threshold denoising algorithm and can effectively suppress the Gibbs visual distortion caused by the lack of translation invariance of orthogonal wavelet transform.

With the implementation of the Digital Earth Project and the development of infrared technology, infrared image processing has become the main factor restricting the development of infrared technology. Therefore, the key to the breakthrough development of infrared image processing is to adopt new and effective mathematical analysis methods instead of traditional analysis methods. Based on the traditional wavelet threshold infrared image denoising method, this paper fuses the image adaptive threshold denoising algorithm and carries on the double threshold mapping processing to the infrared image. Firstly, on the basis of traditional wavelet threshold processing of infrared images, the advantages and disadvantages of soft and hard threshold functions are analyzed in depth, and an adaptive threshold function with adjustable parameters is constructed, and a double threshold mapping method is used to process infrared images. Secondly, in order to suppress the Gibbs visual distortion caused by the lack of translation invariance of orthogonal wavelet transform, a binary wavelet transform with translation invariance is introduced. Finally, a dual threshold mapping infrared image processing method based on the adaptive threshold denoising algorithm of binary wavelet transform is formed. Simulation results demonstrate the effectiveness and superiority of the proposed method.