RESEARCH Open Access

An improved infrared image processingmethod based on adaptive thresholddenoisingYu Binbin1,2

Abstract

This paper combines the image adaptive threshold denoising algorithm and performs double threshold mappingprocessing to the infrared image, which effectively reduces the influence of these phenomena to the infrared image andimproves the quality of the image. In this paper, the infrared image denoising technology is studied, and aninfrared image denoising method based on the wavelet coefficient threshold processing is proposed. This method isbased on the noise distribution characteristics of infrared images, the multiplicative noise in the infrared image istransformed into an additive noise, and the wavelet transform coefficient of the transformed infrared image isprocessed to denoise the image. On this basis, the advantages and disadvantages of the soft and hard threshold functionsare deeply analyzed, and an adaptive threshold function with adjustable parameter is constructed. At the sametime, in order to suppress the Gibbs visual distortion caused by the absence of translation invariance of theorthogonal wavelet transform, the two-input wavelet transform with translation invariance is introduced, and a doublethreshold mapping infrared image processing method based on the adaptive threshold denoising algorithm based onthe two-input wavelet transform is formed. Simulation results show that the method proposed in this paperhas a better suppression of noise, maintains the integrity of image details, and improves the image quality toa certain extent.

Keywords: Infrared image, Binary wavelet transform, Image denoising, Threshold function, Double thresholdmapping

1 IntroductionThe infrared image characterizes the thermal properties(radiance) of the point in the scene with the brightness(grayscale) of each pixel. The wavelength of the infraredband is in the range of 0.75 to 1000 μm, and its short-wavelength boundary is connected to visible light [1]. Intheory, the infrared radiation of an object depends onthe surface temperature, apparent area, and emissivity εof the object, but in fact, the infrared radiation of thesurrounding environment (the sky, the ocean, other ob-jects) must be measured, and the environmental factorshave complex modulation effects on the infrared radi-ation of the target. The infrared radiation from the tar-get will be selectively absorbed by some gases in theatmosphere before reaching the infrared sensor, and the

airborne particles will also scatter the infrared rays,thereby causing the radiation flux to decay. When theinfrared radiation from the target passes through thetemperature change region, it will slightly deviate fromthe original direction. Since this area of heated air isunstable, the radiation deviation is irregular. This ef-fect, called atmospheric flicker, causes the apparentdirection and radiant flux of distant targets to changesimultaneously.For complex targets, their surface temperature or radi-

ation flux will be inhomogeneous because of their strangeshape and structure and many dispersed heat sources. Thetarget operating speed also has a great influence on itsconvective cooling rate, which results in the change of tar-get surface temperature. Target surface coating and ther-mal insulation measures, to a certain extent, can make thetarget infrared image blurred. Random noise generated byCorrespondence: yubb11@mails.jlu.edu.cn

1College of Computer Science and Technology, Jilin University, Changchun, China2Computer Science and Technology College, Beihua University, Jilin, China

EURASIP Journal on Imageand Video Processing

© The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 https://doi.org/10.1186/s13640-018-0401-8

sensor elements is also an important obstacle to the trans-mission of infrared information.To sum up, the infrared radiation of the target is very

complex, which is because the factors affecting the in-frared radiation of the target are multifaceted, includingthe radiation of the target surface, the absorption andreflectivity of the target, the climate conditions, the se-lective absorption and scattering of the atmosphere, theatmospheric scintillation, the velocity of the target, thesurface coating and heat insulation, the noise of the cir-cuit, and so on.The diversity and complexity of the infrared target

image information model determine the infinite dimen-sion characteristics of the infrared target recognitionobservation sample space, especially the increasing in-frared interference and the further development of in-frared suppression technology, which makes greatdifficulties for the recognition of infrared targets. Theeffectiveness of the target feature strongly depends onthe quality of the image preprocessing, which is equiva-lent to the ability to detect and segment the targeteffectively in the preprocessing phase of the primary vi-sion processing image (including edge detection, noisefiltering [2], image enhancement) under complex back-ground conditions (low signal-to-noise ratio, low con-trast, scene change, and so on) [3]. At this time, theprocessing effect of human vision of the traditional pro-cessing method is difficult to achieve, and the process-ing algorithm with human visual characteristics has arelative advantage. Wavelet transform is an ideal math-ematical model for human primary visual informationprocessing [4]. The decomposition of the image signalis not only in the spatial and frequency domains butalso in the direction of the information expression localcharacteristics, which conforms to the basic character-istics of the human vision for the processing of imageinformation. From the spatial domain, the size and dir-ection of some image targets and other details are dif-ferent, so the image needs to be processed at differentresolution and direction. Here, wavelet transform cangive full play to its advantages and can be analyzedfrom the frequency domain. Wavelet transform candecompose all kinds of mixed signals with differentfrequencies into block signals with different frequen-cies. Wavelet denoising has the advantages of multire-solution and diversity of wavelet basis selection, whichmakes it very suitable for image and other signal denoising[5]. Therefore, wavelet transform is a powerful mathemat-ical tool for primary vision processing.Strictly speaking, speckle noise of infrared image in-

cludes both multiplicative noise and additive noise.Aiming at this noise, this paper firstly derives the loga-rithms of the infrared image and transforms the multi-plicative noise into the additive noise. At the same

time, an adaptive threshold function with adjustable pa-rameters is constructed. Then, the binary wavelet trans-form with translation invariance is used for denoising.The specific contributions of this paper can be expressedas follows:

1. To solve the problem of infrared image denoising, aninfrared image denoising method based on waveletcoefficient threshold processing is proposed. In thismethod, the speckle noise of the infrared image istransformed into additive noise by logarithmictransformation, and the threshold of the waveletcoefficients of the transformed infrared image isobtained to realize image denoising.

2. On the basis of infrared image denoising methodbased on wavelet coefficient threshold processing,the advantages and disadvantages of soft and hardthreshold functions are deeply analyzed, and anadaptive threshold function with adjustableparameters is constructed.

3. In order to suppress the Gibbs visual distortioncaused by the lack of translation invariance in theorthogonal wavelet transform, the binary wavelettransform with translation invariance is introduced.

4. In the process of infrared image denoising, doublethreshold mapping is adopted, and finally, a dualthreshold mapping infrared image processingmethod based on adaptive wavelet denoisingalgorithm based on binary wavelet transform isformed.

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

2 Related workNowadays, the denoising method based on wavelet trans-form has become an important branch of image denoisingand restoration. Among the many image denoisingmethods based on wavelet transform, the wavelet thresh-old denoising method proposed by Donoho [6] and othershas been widely used because of its simple principle, easyimplementation, and remarkable denoising effect. But itsinherent shortcomings, such as the discontinuity of hardthreshold function, the constant deviation caused by softthreshold function [7–10], and the lack of scale adaptabil-ity, also limit its further development. Later, many scholarsand researchers have studied it deeply from the point ofview of constructing new threshold function and search-ing for optimal threshold and proposed many adaptiveimage denoising methods. Hong and Yang [11] have com-pared the signal-to-noise ratio (SNR) and root meansquare error (RMSE) according to the shortcomings ofsoft threshold function and hard threshold function in the

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 2 of 12

traditional wavelet threshold denoising algorithm andconstructed a new threshold function to denoise thesignal. To a large extent, the shortcomings of poor con-tinuity and inherent deviation of these two functionswere overcome in practical processing, which made thenoise processing more effective and superior. Junchengand Qiang [12] applied the translation invariant to thesecond-generation wavelet transform based on theexisting threshold denoising method of the second-gen-eration wavelet transform and realized the more rapidand effective denoising of seismic signal and haveachieved good denoising effect in the trial calculationof analog data and actual data. Hong [13] improved theclassical wavelet threshold denoising algorithm andproposed a new threshold denoising method of minevideo monitoring image in the wavelet transform do-main. The image was decomposed by three-level wavelettransform, the low-frequency decomposition coefficientswere filtered by Wiener filter, and the high-frequencydecomposition coefficients were denoised by improvedwavelet I value denoising algorithm. Experiments showedthat it can effectively reduce noise and get an effective andcomplete image. Xiaofei and Xiaohui [14] combined theadvantages of typical wavelet threshold function, inte-grated some improved methods, and proposed an im-proved threshold function for its shortcomings. Thisfunction is not only continuous at the threshold; the coef-ficient is a symptotic original coefficient of the waveletestimation, but also has differentiability, and it is easy toimplement adaptive learning of the gradient algorithm.Fengbo, Changgeng, and Hongqiu [15] proposed an im-proved threshold algorithm based on lifting wavelet trans-form. First of all, the advantages of the wavelet transform,multiresolution, and diversity of wavelet bases wereexploited. On the other hand, a new threshold functionwas proposed to improve the signal-to-noise ratio and re-duce the RMSE. Denoising with the improved thresholdfunction is superior to the commonly used threshold func-tion in both visual effects and mean square error and peaksignal-to-noise ratio performance analysis. Xiaoyan andAbdukirimturki [16] combined the advantages of typicalwavelet threshold function with some improved methodsand proposed an improved new threshold function. Theexperimental results show that the denoised image isbetter than the traditional soft and hard thresholding andthe existing thresholding in terms of visual effect, meansquare deviation, and peak signal-to-noise ratio. Thesemethods not only open up a broad prospect for the fulladvantage of the wavelet threshold denoising method butalso provide a basis for further exploration of adaptivedenoising methods. However, in general, these studiesare almost all based on the orthogonal wavelet trans-form, using the Mallat algorithm. Mallat algorithm [17]can achieve satisfactory speed, but due to the lack of

translation invariance, the reconstruction accuracy afterimage denoising is not high, and the Gibbs phenomenonis easy to appear in the denoised image and cannot meetthe requirements of human vision. The fast algorithmbased on the binary wavelet transform [18]—à trous algo-rithm [19]—not only has the invariance of translation butalso makes the representation of the image in the domainof binary wavelet transform very redundant, and the dis-turbance of partial coefficients will not lead to the seriousdistortion of the reconstructed image. In addition, for theGibbs visual distortion in a denoised image caused by thewavelet threshold denoising method using orthogonalwavelet transform, many researchers have made improve-ments such as literature [20–23]. The results show thattranslation invariance is an important property of effect-ively suppressing Gibbs phenomenon and improvingdenoising effect. These conclusions provide a basis for theresearch of adaptive threshold image denoising based onbinary wavelet transform.

2.1 Infrared image noise analysis2.1.1 The meaning and classification of image noiseImage noise is generally defined from two aspects of hu-man perception and mathematics. From the mathematicalpoint of view, the image information can be regarded as aspatial function f, and image noise is the factor thatdegrades the information expressed by this function, thatis, under the influence of noise, the image is degraded tofn. Image noise can be divided into different categories ac-cording to different methods. From the mathematicalpoint of view, the image can be divided into additive noiseand multiplicative noise according to the way in which theimage is degraded. These two relations can be expressedby the following formula:

f n ¼ f þ n ð1Þf n ¼ f � n ð2Þ

where n represents noise, formula (1) represents additivenoise, and (2) represents multiplicative noise.If it is divided according to the physical factors of noise

generation, it can be classified into electronic noise, photo-electron noise, photosensitive particle noise, speckle noise,and so on.

2.1.2 Measurement of image denoising effectBecause of the randomness of noise generation itself,the noise contained in an image can only be describedby a statistical method, that is to say, the noise gener-ation is regarded as a random process. The probabilitydensity function of this random process is used to de-scribe the overall behavior of noise. But in many cases,the probability density function of noise distribution isvery difficult to obtain, or it is difficult to express by

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 3 of 12

some mathematical function. In this case, we usuallystudy some digital characteristics of noise distribution,such as the mean m, variance σ, correlation coefficientρ, and so on. This paper describes the following indica-tors for measuring the quality of image denoising:

1. Normalized mean square error (NMSE)

NMSE ¼

Xm

Xn

f m; nð Þ− f 0 ðm; nÞh i2

Xm

Xn

f m; nð Þ2 ð3Þ

where f(m, n) is the original image, and f'(m, n) repre-sents the restored image after processing.

2. Standard deviation s

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXMi¼1

XNj¼1

f i; jð Þ− f� �2MN

vuuuutð4Þ

where f(i, j) is an image, f ¼

XMi¼1

XNj¼1

f ði; jÞMN , and M

and N are the total number of rows and columns ofthe image matrix.

3. The ratio between signal and mean squared error(F/MSE)

F=MSE ¼ 10 lg

XMm¼1

XNn¼1

f m; nð Þ½ �2

XMm¼1

XNn¼1

f0m; nð Þ− f ðm; nÞ

h i2 dBð Þ

ð5Þ

4. Image relative signal to noise ratio

5. Smoothness: In most cases, the image after denoisingshould be at least as smooth as the original image.

6. The variance estimation of image and originalimage after similarity denoising should be the leastvariance in the worst case (minmax estimator).

2.1.3 Noise characteristic analysis of infrared imageSpeckle noise [24] exists extensively in the imaging processof infrared images. The formation of speckle noise is mainlydue to the interaction of infrared waves in the imagingprocess. Besides, it is also closely related to the roughnessof the imaging tissue surface. From the visual point of view,the noise in the image presents a speckle distribution andaccording to the density of speckles can be divided intothree types of noise.From the nature of noise analysis, strictly speaking,

speckle noise contains both the components of multi-plicative noise and the components of additive noise,which can be expressed by Eq. (7):

f n ¼ f � nm þ na ð7ÞHowever, in most cases, the influence of additive noise

on the image is much smaller than that of the multiplicativenoise, so the additive noise can be ignored. Finally, thespeckle noise is considered to be a multiplicative noise.

2.2 Binary wavelet theoryIn the Mallat algorithm, due to the downsampling oper-ation, the singular points of the image edges are easily lost,resulting in Gibbs phenomenon at the edges of the recon-structed images after denoising, resulting in image edgedistortion or even blur. In order to overcome this draw-back, wavelet transform with translation invariance needsto be considered. The binary wavelet transform only dis-cretizes the scale factor in the continuous wavelet trans-form and keeps the translation factor continuouslychanging, thus effectively maintaining the translation in-variance. This feature of the binary wavelet transform,coupled with the fast decomposition and reconstructionalgorithm, à trous algorithm, has certain advantages in thefields of image denoising and image feature extraction.It should be noted that the à trous algorithm men-

tioned in this paper is a modified version of the à trousalgorithm form described in the Mallat book, and the al-gorithm is different from the proof in the Mallat book.Assuming that the sample value a0, k of the input

discrete signal is the local average of f in the domain oft = k, it can be written as:

a0;k ¼< f tð Þ;φ t−kð Þ > ð8Þwhere φ and ~φ are the two scale functions generated bythe infinite cascade calculation of filter (h, g) and its dual

filter ð~h; ~gÞ, which can be expressed as:SNR ¼ f

sð6Þ

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 4 of 12

φ tð Þ ¼ffiffiffi2

p Xk

hkφ 2t−kð Þ~φ tð Þ ¼

ffiffiffi2

p Xk

~hk~φ 2t−kð Þ ð9Þ

For any j ≥ 0, remember:

aj;k ¼< f tð Þ; φ2 j t−kð Þ > ð10Þ

where φ2 jðtÞ ¼ 1ffiffiffiffi2 j

p φð t2 jÞ.

The binary wavelet coefficients are calculated as follows:

d j;k ¼ Wf 2 j; k� � ¼< f tð Þ;ψ2 j t−kð Þ > ð11Þ

where ψ2 jðtÞ ¼ 1ffiffiffiffi2 j

p ψð t2 jÞ and ψ and ~ψ are the wavelet

functions generated by the infinite cascade calculation of

the filter (h, g) and its dual filter ð~h; ~gÞ , which can beexpressed as:

ψ tð Þ ¼ffiffiffi2

p Xk

gkφ 2t−kð Þ~ψ tð Þ ¼

ffiffiffi2

p Xk

~gk~φ 2t−kð Þ ð12Þ

For the à trous algorithm, the binary wavelet decom-position algorithm is expressed as:

ajþ1;k ¼Xn

hna j;kþ2 jn; j ¼ 0; 1;⋯;

d jþ1;k ¼Xn

gna j;kþ2 jn; j ¼ 0; 1;⋯:ð13Þ

The binary wavelet reconstruction algorithm is expressedas:

aj;k ¼ 12

Xn

~hna jþ1;k−2 jn þXn

~gnd jþ1;k−2 jn

!; j

¼ 0; 1;⋯:

ð14Þ

The proofs of formulas (13) and (14) are givenbelow. For formula (13), the following can be obtainedfrom the definition of aj, k and the relationship be-tween two scales.

ajþ1;k ¼< f tð Þ;φ2 jþ1 t−kð Þ >¼Z

Rf tð Þ 1ffiffiffiffiffiffiffiffiffi

2 jþ1p φ�

t−k2 jþ1

� �dt

¼Xn

hnaj;kþ2 jn; j ¼ 0; 1;⋯:

ð15Þ

In the same way, there are:

d jþ1;k ¼< f tð Þ;ψ2 jþ1 t−kð Þ >¼Z

Rf tð Þ 1ffiffiffiffiffiffiffiffiffi

2 jþ1p ψ� t−k

2 jþ1

� �dt

¼Xn

gna j;kþ2 jn; j ¼ 0; 1;⋯:

ð16Þ

For formula (15), from the dual-scale relationshipb~φð2wÞ ¼ 1ffiffi2

p b~hðwÞb~φðwÞ , b~ψð2wÞ ¼ 1ffiffi2

p b~gðwÞb~φðwÞ and the

binary condition ψ�ðwÞb~ψðwÞ þ φ�ðwÞb~φðwÞ ¼ φ�ðw=2Þb~φðw=2Þ, we get:ffiffiffi

2p

φ w�2

� � ¼ b~h� w�2

� �φ wð Þ þ b~g� w�

2� �

ψ wð Þ ð17Þ

Time domain is expressed as:

2ffiffiffi2

pφ 2tð Þ ¼

Xk

~h2kφ t þ kð Þ þXk

~g2kψ t þ kð Þ ð18Þ

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

aj;k ¼< f tð Þ;φ2 j t−kð Þ >¼Z

Rf tð Þ 1ffiffiffiffi

2 jp φ�

t−k2 j

� �dt

¼ 12

Xk

~hkajþ1;k−2 jn þXk

~gkd jþ1;k−2 jn

!; j ¼ 0; 1;⋯:

ð19Þ

The above theorem describes the à trous algorithm ofthe one-dimensional signal. The structure of this algo-rithm is shown in Fig. 1. For the two-dimensional imagesignal, the one-dimensional à trous algorithm is appliedalong the rows and columns of the image to realizeimage decomposition and reconstruction.Although the à trous algorithm has the same algorithm

structure as the Mallat algorithm, there is a substantialdifference: the à trous algorithm does not perform thedownsampling operation, which not only preserves thetranslation invariance well but also the data length undereach scale is the same as the original data length. Com-pared with the data of the Mallat algorithm, there is agreat redundancy, which is convenient for spectrum ana-lysis of details and profiles at each scale.

3 Proposed method3.1 Adaptive threshold denoising analysis of wavelettransform3.1.1 The basic principle of image denoising based onwavelet transformThe idea of image denoising based on wavelet trans-form can be roughly divided into two kinds: one is that

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 5 of 12

the image signal and noise signal show different laws indifferent resolutions after wavelet transform. Thethreshold can be set at different resolution levels, andthe wavelet coefficients can be adjusted to achieve thepurpose of denoising. The other is to combine wavelettransform with traditional image denoising algorithm toimprove the performance of the original image denois-ing algorithm by utilizing the characteristics of wavelettransform multiresolution and time-frequencylocalization. The general steps of wavelet transform forimage denoising are as follows:

1. Two-dimensional discrete wavelet transform is usedto get the wavelet coefficients at different resolutions.

2. The wavelet coefficients are analyzed andadjusted accordingly or processed according tosome traditional methods.

3. Wavelet transform is used to get the processed image.

From the above description, the key of image denois-ing using wavelet transform is the selection of waveletbasis function and the analysis and correction of wave-let coefficients. Based on the threshold processingmethod of wavelet coefficients, the main idea of thismethod is to divide wavelet coefficients into noise-con-taminated points and information points and then filterwavelet coefficients representing noise points. Theremaining wavelet coefficients are used to reconstructthe image to get the denoising results.

3.1.2 Research on threshold processing of wavelet coefficientFrom the energy point of view, the energy of the noiseon the wavelet domain is distributed over all waveletcoefficients, while the signal energy is concentrated ononly a small fraction of the wavelet coefficients. There-fore, the wavelet coefficients can be divided into twotypes. The first type of wavelet coefficients is only ob-tained by noise transformation. The wavelet coeffi-cients have small amplitude and a large number. Thesecond type of wavelet coefficients is derived from thesignal and contains the influence of noise. These wave-let coefficients have large amplitudes and a small num-ber. In this way, the threshold can be set by thedifference in the amplitude of the wavelet coefficients,

and a denoising method can be constructed. A waveletcoefficient larger than this threshold is considered tobelong to the second type of coefficient, that is, a trans-formation result containing both signals and noise,which may retain a simple reservation or a subsequentoperation. And a wavelet coefficient smaller than thethreshold is considered to be the first type of waveletcoefficient, that is, it is completely transformed bynoise, and the removal of these coefficients achievesthe purpose of denoising. At the same time, the imagedetails can be better preserved by retaining most of thewavelet coefficients containing the signal. The waveletthreshold denoising method can be described by thefollowing three-step strategy:

1. Calculate the orthogonal wavelet transform ofnoisy images

Select the appropriate wavelet base and wavelet decom-position layer number J, and use the Mallat decompositionalgorithm to perform the J-layer wavelet decomposition ofthe noisy image to obtain the corresponding wavelet de-composition coefficient.

2. Threshold quantization of the decomposed high-frequency coefficients

For each layer from 1 to J, an appropriate thresholdand an appropriate threshold function are selected tothreshold quantize the decomposed high-frequency coef-ficients to obtain an estimated wavelet coefficient.

3. Perform inverse wavelet transform

According to the low-frequency coefficient (scale coeffi-cient) of the Jth layer after image wavelet decompositionand the high-frequency coefficients (wavelet coefficients)of each layer after threshold quantization, the Mallat re-construction algorithm is used for wavelet reconstructionto obtain the denoised image.It can be seen from the above description that the se-

lection of the threshold value, the threshold function,the selection of the wavelet basis, and the determinationof the decomposition layer are the key factors in thewavelet threshold denoising method.

(a) (b)

Fig. 1 a à trous decomposition algorithm. b à trous reconstruction algorithm

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 6 of 12

3.1.3 Adaptive threshold function selectionIn the wavelet threshold technique, the choice of thresh-old is a key factor in determining the performance of thealgorithm. On the one hand, it needs to be large enoughto remove as much noise as possible, and on the otherhand, it needs to be small enough to retain as much sig-nal energy as possible. In many practical applications, itis difficult to meet both of these requirements by relyingon a single global uniform threshold. Since the wavelettransform is a multiresolution analysis, the selection ofthe threshold can be considered on each layer. That is,the local feature of the image is used as a parameter forthreshold selection so that the threshold can be changedas the specific characteristics of the image change. Inthis paper, an adaptive threshold function is designed.Compared with the hard threshold and soft thresholdfunction, it has better superiority and flexibility. It canrealize different processing of wavelet coefficients andcomplete adaptive denoising of images.Although the hard threshold and soft threshold methods

have been widely used in practice, they have achieved goodresults, but the method itself has some potential defects:

1. The hard threshold function uses the threshold asthe boundary between the signal and the noise. Thecoefficient smaller than the threshold is set to zero,and the coefficient larger than the threshold isdirectly retained. This method preserves most ofthe details of the image well, which brings greatconvenience to the processing of the image. But itdoes not match the interference of the noise signalin the wavelet coefficient larger than the thresholdin the actual situation, and the lack of continuity atthe threshold makes the estimated signal produceadditional oscillations, resulting in visual distortionsuch as pseudo-Gibbs phenomenon in the recon-structed image.

2. The continuity of the soft threshold function atthe threshold can effectively overcome thepseudo-Gibbs phenomenon caused by the hardthreshold function, but the constant deviation be-tween the estimated wavelet coefficients and thewavelet coefficients of the soft threshold functionhas a direct impact on the approximation be-tween the reconstructed image and the realimage. The result of denoising by soft thresholdmethod is relatively smooth, and it is easy tocause blurred image edges.

In this paper, based on the advantages and disadvantagesof the soft and hard threshold functions, an adaptivethreshold function is constructed in combination with theliterature [25–28]. This function can be regarded as ageneralization of soft and hard threshold functions. It is

based on the nonlinear hyperbolic tangent function of BPneural network; the translation of the function is adjustedby introducing the threshold δ. And based on parameterm, the difference between the estimated wavelet coeffi-cient and the wavelet coefficient can be adjusted. The ex-pression is as follows:

W i; j ¼ sgn wi; j� � jwi; jj−mδ

1−e− jwi; jj−δð Þ1þ e− jwi; jj−δð Þ

!; j wi; j j ≥δ

0; j wi; j j< δ

8><>: ;m≥0

ð20Þ

Looking at the above adaptive threshold function, itcan be seen that when parameter m = 0, this function isa hard threshold function; when m = 1, and wi, j→∞,the function is a soft threshold function; when ∣wi,

j ∣ →∞, j wi; j−wi; j j →mδ , that is, as ∣wi, j∣ increases,the absolute value of the deviation of the estimatedwavelet coefficient wi; j and the wavelet coefficient wi, j

gradually approaches mδ. Since m is a parameter, theabsolute value of the deviation can be controlled byadjusting the size of m, which effectively improves theconstant deviation between the estimated wavelet coeffi-cient and the wavelet coefficient generated in the softthreshold function. Compared with the soft and hardthresholding function, the adaptive thresholding func-tion proposed in this paper is a more superior and flex-ible choice. It can realize the different processing ofwavelet coefficients and complete the adaptive denoisingof images only by properly determining the size of m ac-cording to different images.

3.2 Infrared image processing based on binary waveletadaptive threshold denoisingIt can be seen from Section 2.1 that the infrared imagenoise is mainly due to the coherence of infrared lightwaves, including both multiplicative noise and additivenoise. But in most cases, the effect of additive noise onthe image is much less than that of multiplicative noise.However, most of the current wavelet denoising methodsare designed for additive noise interference, which obvi-ously cannot meet the needs of infrared image process-ing in this subject. Therefore, a binary wavelet adaptivethreshold denoising scheme is designed for speckle noisein the infrared image.Considering that speckle noise is dominated by multi-

plicative noise, the multiplicative noise is transformed intoadditive noise by taking the logarithm of infrared image,then the binary wavelet adaptive threshold denoising algo-rithm is used to process the image, and then the exponen-tial restoration of the processing results can be achieved.According to the calculation, the pixel values of the infra-red image used in the experiment are all positive, so thelogarithm scheme can be adopted.

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In summary, the algorithm is described as follows:

1. Assuming that the additive noise in speckle noise isignored, formula (8) can be rewritten as:

f n ¼ f � nm ð21Þ

2. Take the logarithm of 4 as the base on both sides offormula (24), we can get:

log4 f n¼ log4 f þ log4nm ð22Þ

In fact, in order to avoid taking the logarithm of 0 asthe base when implementing the algorithm, special orderfn = fn + 0.0000001

3. The infrared image log4fn performs three-layer bin-ary wavelet decomposition: select appropriate wave-let base for log4fn, and apply à trous decomposition

algorithm for J-layer decomposition to obtain alow-frequency (scale) coefficient aj, m, n and a set ofhigh-frequency coefficients di

j;m;n (i = 1, 2, 3), wherei = 1, 2, 3 respectively indicate horizontal, vertical,and diagonal directions.

4. Threshold quantification of high-frequency coefficients:First, the high-frequency coefficient di

j;m;n (i= 1, 2, 3) isrewritten as a one-dimensional sequence di

j;k(i= 1, 2,

3), and then the estimation coefficient dij;k (i= 1, 2, 3) is

obtained by double threshold processing of dij;k (i= 1,

2, 3) with the adaptive threshold function given in thispaper. The expression of the adaptive threshold functionis rewritten as follows:

d∧ i

j;k ¼

sgn dij;k

� jdi

j;k j−mλ j1−e− jdi

j;k j−λ jð Þ1þ e− jdi

j;k j−λ jð Þ

!; j di

j;k j ≥λ j

dij;k ;

λ j

2≤ j di

j;k j ≤λ j

0; j dij;k j<

λ j

2

8>>>>>><>>>>>>:;m≥0

ð23ÞThe upper sgn(⋅) is a symbolic function, λ j ¼ σffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 lnðMNÞp

= lnð jþ 1Þ is a scale adaptive threshold,

a b

c dFig. 2 Results graph of denoising for different threshold functions of males. a The original image. b Soft threshold function denoising. c Hardthreshold function denoising. d Adaptive function denoising

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 8 of 12

M and N are the rows and columns of image pixels, jis the decomposition scale, and σ is the noise stand-ard deviation, expressed as: σ2 ¼ medianfdi

j;m;ng=0:6745, i = 1, 2, 3, 1 ≤ j ≤ J.

5. Reconstruction of infrared image log4fn: Writethe estimated coefficient d

ij;k (i = 1, 2, 3) into a

two-dimensional form dij;m;n (i = 1, 2, 3);

combined with the low-frequency coefficient aj,m, n, the image is reconstructed by the à trousreconstruction algorithm.

6. The image obtained in the previous step is subjectedto a power of 4 to obtain a denoised image.

4 Results and discussionThe simulated image is uniformly imaged by an infraredimage of 182 × 208 pixels, which is taken by a medicalinfrared camera of Chongqing Weilian Technology Lim-ited. The selected wavelet functions are sym4 waveletswith good symmetry and suitable for image denoising.Firstly, the performance of the adaptive threshold func-

tion selected in this paper is analyzed, and the traditionalsoft and hard threshold functions are used as a compari-son. The results of infrared image processing are shown in

Fig. 2. Figure 2a is the original image, Fig. 2b is the resultof soft threshold function denoising, Fig. 2c is the result ofhard threshold function denoising, and Fig. 2d is the resultof adaptive threshold function denoising designed in thispaper. As can be seen in Fig. 2, the resulting image ob-tained by soft and hard threshold function denoisingstill contains a lot of noise compared with the originalinfrared image, and the visual effect is not much im-proved compared with the original infrared image.The overall effect of the image obtained by adaptivethreshold denoising in this paper is relatively smoothand clear, so it can be said that the adaptive thresholdfunction proposed in this paper is better for infraredimage denoising.In order to better compare the performance of adap-

tive threshold function processing proposed in thispaper, formula (6) is used to calculate the signal-to-noise ratio of the original image, and formula (5) isused to calculate the signal-to-mean square error ratioof image denoising. The comparison results are shownin Table 1.In order to better show the denoising effect of soft and

hard thresholding functions and adaptive thresholdingfunctions of this algorithm, the standard deviation ofdenoising of different thresholding functions for men

Table 1 Comparison of different threshold function denoising data

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 /

Soft threshold denoising image 78.25 13.03 9.1367e−005 77.53 12.96 9.0304e−005

Hard threshold denoising image 77.57 12.91 8.9812e−005 77.11 13.23 8.8911e−005

The denoising image of algorithm in this paper 21.07 15.65 6.5287e−005 20.99 15.47 6.7044e−005

Fig. 3 Results graph of denoising for different threshold functions of females

Binbin EURASIP Journal on Image and Video Processing (2019) 2019:5 Page 9 of 12

and women in Table 1 and SNR or F/MSE is averaged,and the histogram is drawn as shown in Fig. 3. Combin-ing Fig. 3 and Table 1, we can see that the soft and hardthreshold functions can denoise the image and raise thestandard deviation by about 5 and the SNR or F/MSE byabout 1 dB. The adaptive threshold function proposed inthis paper can denoise the image and raise the standarddeviation by about 60 and the SNR or F/MSE by about3 dB.In order to validate the performance of the infrared

image denoising algorithm designed in this paper, thetraditional wavelet threshold denoising algorithm andthe algorithm designed in this paper are used to denoisethe infrared image. The results are shown in Fig. 4, inwhich Fig. 4a is the original image, Fig. 4b is the denois-ing result of the traditional wavelet threshold denoisingalgorithm, and Fig. 4c is the denoising result of the algo-rithm designed in this paper. In Fig. 4, we can see thatthe traditional wavelet threshold denoising algorithmcan effectively improve the performance of infraredimage, but the image denoised by this algorithm retainsthe edge features of the image and effectively improvesthe Gibbs phenomenon near the edge and the image de-tails are clearer and the visual effect is obviously betterthan the traditional wavelet denoising method.Similarly, Table 2 gives the performance data of the trad-

itional wavelet threshold method compared with the algo-rithm in this paper. In order to better show the denoisingeffect of the traditional wavelet threshold algorithm and

the algorithm in this paper, the standard deviation of dif-ferent threshold functions for men and women in Table 2and SNR or F/MSE is averaged, and the histogram isdrawn as shown in Fig. 5. Combining Table 2 and Fig. 5,we can see that the traditional wavelet threshold algorithmcan improve the standard deviation by about 5 and theSNR or F/MSE by about 1 dB, while the image denoisingby this algorithm can improve the standard deviation byabout 67 and the SNR or F/MSE by about 6 dB.In summary, experiments show that the proposed

method for infrared image denoising has better perform-ance than the traditional wavelet threshold denoising al-gorithm and can effectively suppress the Gibbs visualdistortion caused by the lack of translation invariance oforthogonal wavelet transform.

5 ConclusionsWith the implementation of the Digital Earth Project andthe development of infrared technology, infrared imageprocessing has become the main factor restricting the de-velopment of infrared technology. Therefore, the key tothe breakthrough development of infrared image process-ing is to adopt new and effective mathematical analysismethods instead of traditional analysis methods. Based onthe traditional wavelet threshold infrared image denoisingmethod, this paper fuses the image adaptive thresholddenoising algorithm and carries on the double thresholdmapping processing to the infrared image. Firstly, on thebasis of traditional wavelet threshold processing of

(a) (b) (c)

Fig. 4 Simulation results of males. a Original image. b Traditional wavelength threshold denoising results. c Denoising result of this algorithm

Table 2 Data comparison of image denoising

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

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infrared images, the advantages and disadvantages ofsoft and hard threshold functions are analyzed in depth,and an adaptive threshold function with adjustableparameters is constructed, and a double threshold map-ping method is used to process infrared images. Sec-ondly, in order to suppress the Gibbs visual distortioncaused by the lack of translation invariance of orthog-onal wavelet transform, a binary wavelet transform withtranslation invariance is introduced. Finally, a dualthreshold mapping infrared image processing methodbased on the adaptive threshold denoising algorithm ofbinary wavelet transform is formed. Simulation resultsdemonstrate the effectiveness and superiority of theproposed method.

AbbreviationsF/MSE: The ratio between signal and mean squared error; NMSE: Normalizedmean square error

AcknowledgementsThe author thanks the editor and anonymous reviewers for their helpful commentsand valuable suggestions.

About the authorsJilin University at Changchun, No.2699 Qianjin Rd. Chaoyang District, Changchun,Jilin.Binbin Yu was born in Jilin City, Jilin Province. P.R. China, in 1984. He receivedthe Master degree from Northeast Normal University, P.R. China. Now, he is aPh.D. candidate, College of Computer Science and Technology, Jilin University.He works in Computer Science and Technology College, Beihua University. Hisresearch interests include information fusion of Internet of things and so on.

FundingNo funding.

Availability of data and materialsPlease contact author for data requests.

Authors’ contributionsBY did everything in this paper. The author read and approved the finalmanuscript.

Competing interestsThe author declares that he has no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in publishedmaps and institutional affiliations.

Received: 2 September 2018 Accepted: 21 December 2018

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