Xiao-peng Cheng,Bo-wang Shu,Ya-jing Chang,Xin Li,Da-bin Yu

State Key Laboratory of Pulsed Power Laser Technology,College of Electronic Engineering,National University of Defense Technology,Hefei,230037,PR China

Keywords:Infrared Camouflage effectiveness Multi-fractal Similarity

ABSTRACT This paper reports an alternative approach to the evaluation of infrared camouflage effectiveness via a multi-fractal method.By calculating multi-fractal spectra of the target region and the background regions in an infrared image,the spectrum shape features and the discrete Fr′echet distances among these spectra were used to analyze the camouflage effectiveness of the target qualitatively and quantitatively,and the correlation coefficients of the spectra were further used as the index of camouflage effectiveness.It was found that the camouflaged target had better camouflage effectiveness than the target without camouflage in the same one background,and the same one camouflaged target had different camouflage effectiveness in different backgrounds.On the whole,the target matching well with its background had high camouflage effectiveness value.This approach can expand the application of multi-fractal theory in infrared camouflage technology,which should be useful for the research of infrared camouflage materials,the design of camouflage patterns as well as the deployment of military equipment in battlefield.? 2021 China Ordnance Society.Production and hosting by Elsevier B.V.on behalf of KeAi Communications Co.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

With the wide application of infrared(usually abbreviated as IR)camouflage materials such as coatings[1],dyes[2]and some others[3,4],the evaluation of the IR camouflage effectiveness of these materials has naturally caused increasing attention in many fields such as military applications[5],IR materials research and design[6],and IR image processing[7],etc.To realize IR camouflage,the IR materials were always designed into certain patterns to cover the surface of a target so as to conceal the target into the battlefield background thus protecting the target from risks of IR detecting systems.Thus,the design of the camouflage patterns on a target has a profound impact on its IR camouflage effectiveness[8].In order to get good camouflage effectiveness,the design of the camouflage patterns on a target should provide adequate mimic of the corresponding background[9].Generally,the more similar the patterns of a target are with those of its background,the better the patterns of this target match with those of the background,which means in this case we can assert that the IR camouflage effectiveness of this target is good.So it should be noted that measuring the similarity between a target and its background is the core point of the evaluation of the IR camouflage effectiveness of this target.However,compared to the target which is always designed with simple and regular patterns,the background is always‘chaotic’because there are always lots of vegetation,stones,trees,rivers and many other natural scenes in it.Therefore,it should be an interesting but challenging issue to measure the similarity between a target and its‘chaotic’background,when evaluating the IR camouflage effectiveness of this target.

Herein,we report an alternative approach to calculation of the similarity between a target and its background by using multifractal theory,aiming to evaluate the camouflage effectiveness of IR camouflaged targets.Developed by Mandelbot in the 1970s,fractal theory was widely used to describe and analyze the irregular,wiggly,and even self-similar structures in nature[10].In recent years,fractal theory has been developed to design IR camouflage patterns.For example,Sheng Bi et al.introduced the fractal geometry theory and the random midpoint displacement algorithm into the design of camouflage patterns,and this work provided a new tool for camouflage pattern generation and evaluation[11].Bossard et al.proposed a genetic algorithm to evolve optimal fractal random Cantor bars with multiple generators for mimicking the broadband filter functionality in the mid-IR and the near-IR broadband[12].Besides this,fractal theory has also been used to the detection of camouflaged targets[13].However,after many years of intensive investigation,it has been proved that one single fractal dimension is not sufficient for characterizing the structures in nature.Consequently,multi-fractal theory,which can also be used in camouflage field,has been developed[14].Plesa et al.presented a scientific method using multi-fractal theory for generating shapes and colors to mimic the environment[15].Yves Caron et al.had proposed a method for detecting objects using Legendre multi-fractal spectrum as early as 2002[16].Ying Li et al.utilized multi-fractal de-trended fluctuation analysis method to observe and detect targets in sea clutter[17].These works have proved that the multi-fractal theory has more advantages than the fractal theory in camouflage field because the former could reflect more characteristics of both the target and its background.Now that the fractal theory and the later multi-fractal theory can be used to design IR camouflage patterns and detect camouflaged targets,it can be,of course,used to evaluate the effectiveness of a camouflaged target because all of these are based on the similarity(or difference,on the contrary)between a target and its corresponding background.Specifically,from camouflage point of view,the more different a target is with its background,the easier this target is to be detected.And certainly the easier the target is to be detected,the worse its camouflage effectiveness is.So whether a camouflaged target could be detected or not is mainly determined by the camouflage effectiveness of it,which supports the potential for us to expand the camouflaged target detection algorithm to the evaluation of the camouflage effectiveness of a target.Inspired by the previous works,in this study we use multi-fractal theory to evaluate the camouflage effectiveness of IR targets.Up to now,there are still seldom reports concerning the use of multi-fractal theory in the evaluation of camouflage effectiveness of IR targets so far.

2.Experiments and methods

2.1.Materials and design of simulative targets

Materials:Ordinary green coatings,IR camouflage coatings and a kind of square steel plates(0.5 m×0.5 m)were used to fabricate the simulative targets in our laboratory.The IR emissivity of the ordinary green coating was 0.94,and the IR emissivity of three kinds of IR camouflage coatings was controlled to be 0.94,0.79,and 0.64,respectively.

Design of simulative targets:The target without IR camouflage was fabricated through using the square steel plate coated by the ordinary green coating(Target 1,abbreviated as T1 hereafter).The IR camouflaged target was fabricated by using the square steel plate and the IR camouflage coatings,and it was designed with six different disruptive patches by the three kinds of IR camouflage coatings(Target 2,T2).

2.2.Backgrounds and weather conditions

Backgrounds:A sparse grass background covered by ca.60%grass(abbreviated as B1),a thick grass background covered by ca.90%grass(B2)and a bush background(B3)were used as the typical backgrounds.

Weather conditions:It was sunny,with ca.23°C room temperature and ca.60% relative humidity.The wind was southeast with the speed of 6-16 mph.

2.3.Experimental process

The IR images were obtained by a ThermoPro TP9 thermal imager(Wuhan Guide IR Co.Ltd,China),and the corresponding visual images were captured using a D7200 camera(Nikon Corporation,Japan).To acquire the texture features of both the target and its surrounding backgrounds in IR images as clearly as possible,the simulative targets were placed in the middle of the images,and the ratio of the target area to the background area in each image was ca.1:16.

As shown in Fig.1,T1 and T2 were placed in the same background B1,respectively,to analyze the camouflage effectiveness of different targets in the same one background.To compare the camouflage quality of the same one target in different backgrounds,the target T2 was placed in B1,B2 and B3,respectively.

3.Evaluation of infrared camouflage effectiveness

3.1.Overview

Benefiting from multi-fractal theory,it is easy for us to investigate unordered and irregular structures,as well as to quantify the self-similarity between local segments and the whole,which is fundamental for camouflage effectiveness evaluation.However,most of patterns and textures in natural backgrounds are not so uniform and strictly self-similar.Comparing with fractal theory,multi-fractal theory can overcome this problem and reveal the selfsimilarity between a target and its background,and hence the physical fractal characteristics of the patterns in an IR image can be characterized into various dimensions,normally in the form of multi-fractal spectrum.By this method,the fractal features of the self-similar patterns in the image can be extracted completely and thus we can multi-dimensionally compare the difference between local segments(the target region)and the whole(the background region).Apparently,it can pave a new way for us to quantify the camouflage effectiveness of a target based on the similarity between this target and its background.

3.2.Calculation of the multi-fractal spectrum

The calculation of the multi-fractal spectrum is based on the differential box-counting method[18]and the Legendre transform theory[19].

In general,a grayscale image of sizeM×Mcan be considered as the representation of a surface in a three-dimensional space(x，y，z).Here,(x，y)marks the position of the pixels in the image andzrepresents the grayscale level of the corresponding pixel.Then,assume the image space is partitioned in boxes of sizes×s×s,namely the scaling ratior=s/M,thus we can say that the measurePij(s)has multi-fractal behavior if

whereαis called Holder exponent which characterizes the average strength of singularity in the measurePij(s).

Obviously,it is difficult to calculate the multi-fractal spectrum according to its definition.In practice,letkbe the index of the box containing the maximum gray level in the column(i，j)andlbe the index of the box containing the minimum gray level in the column(i，j),thus the number of boxes which can cover the image local segment region is

Hence,the number of the boxes which can cover the whole image is

Fig.1.The real IR images of the simulative targets and their corresponding backgrounds:(a)T1 in B1;(b)T2 in B1;(c)T2 in B2;(d)T2 in B3.In each IR image,the bottom-left inset was the corresponding visual image of the target.

So the fractal dimension could be estimated:

Afterwards,the measurePij(s)could be constructed into a concrete variableμr(i，j)in practical calculation:

If the textures in the IR image are multi-fractals,this measure μr(i，j)can perform multi-fractal behavior.So we can calculate the spectrum based on the Legendre transform:

Here,qis any real integer in a non-empty interval ofR,and it acts as a scanning tool scrutinizing the denser and rarer regions of the measureμr(i，j)but in practice we usually limit it to a finite intervalQ.

3.3.Camouflage effectiveness evaluation

Based on the IR image,we can utilize the difference of multifractal characteristics between the target region and the background region to evaluate the camouflage effectiveness of a camouflaged target.

Intuitively,the shape features of multi-fractal spectrum can be used to superficially analyze the difference between a target and its background.For example,the spectrum widthΔα=αmax-αmincan reflect the heterogeneity of the measurement.Specifically,for the image it represents the uniformity of the grayscales distribution and the smaller the value ofΔαis,the more concentrated the grayscale level in the image is.The value off(αmin)andf(αmax)represents the fractal dimensions of small probability subset and large probability subset,respectively.Therefore,the value of Δf=f(αmin)-f(αmax)actually can be used to analyze whether small probability subset or large probability subset is dominant in this multi-fractal.

Certainly,several discrete values of the multi-fractal characteristics are not enough to discriminate the target and its background from an image completely and accurately.Therefore,for evaluating the camouflage effectiveness of a target comprehensively,it is necessary to consider the overall difference between the multifractal spectra of both the target and its background.It has been proved that discrete Fr′echet distance can be used to quantify the similarity of curves[20].Thus we can quantify the similarity of the multi-fractal spectra of both the target and its background based on the Fr′echet distance theory,and further evaluate the camouflage effectiveness of the target.The discrete Fr′echet distance of two spectrafA(α(q))andfB(α(q))are obtained through:

Here,dis the Euclid distance between the two points(αA，fA)and(αB，fB).Thus,we can evaluate the camouflage effectiveness of a target according to the similarity between the multi-fractal spectra of both this target and its corresponding background.

Moreover,although we can already judge the camouflage effect of the target based on the discrete Fr′echet distance,it is still necessary to seek a kind of criterion which can further normalize the camouflage effectiveness value ranging between 0 and 1 for the sake of according with the habit of human cognition and making the result more intuitive.The correlation coefficients between the multi-fractal spectra naturally range between 0 and 1,so it is reasonable to regard the correlation coefficient as a judgment criterion for the camouflage effectiveness.Practically,the correlation coefficient can be calculated through:

Here,Nis the samples count.Obviously,if the multi-fractal spectra of a target and its background are very similar,the value ofρwould be greater and correspondingly the camouflage effectivenessCEof the target would be better.

4.Results and discussions

4.1.Camouflage effectiveness of different targets in the same one background

Based on the IR images shown in Fig.1(a)and(b),we can compare the IR camouflage effectiveness of different targets in the same one background.As shown in Fig.2,the target region(128×128 pixels)was selected,and the surrounding neighborhood of the target was divided into eight regions(128×28 pixels)which were clockwise marked fromN1toN8.Afterwards,we can calculate the multi-fractal spectra of the target region and its neighborhood regions.Moreover,because the multi-fractal patterns can satisfy scale invariance,we also take the comparison between the whole background(512×512 pixels)and the target region into consideration.According to the multi-fractal characteristics of these image regions,we can analyze the difference between a target and its background and further quantify the IR camouflage effectiveness of this target.

According to the mathematical model above,we can calculate the multi-fractal spectra of those image regions.Fig.3 shows the multi-fractal spectrum of T1,the multi-fractal spectra of T1 and its neighborhood regions,the multi-fractal spectrum of the corresponding whole background,and the multi-fractal spectra of both T1 and its whole background.

Similarly,Fig.4 showed the multi-fractal spectrum of T2,the multi-fractal spectra of T2 and its neighborhood regions,the multifractal spectrum of the corresponding whole background,and the multi-fractal spectra of both T2 and its whole background.

So far,we can briefly analyze some important multi-fractal characteristics of the target and the corresponding backgrounds.The value of spectrum widthΔα,f(αmin),f(αmax)andΔfof each image regions are displayed in Table 1.

Obviously,from the multi-fractal spectra shown in Figs.3 and 4,we can find that the spectrum shape feature of T2 is more similar with that of the background.Moreover,drawing a comparison between T1 and T2,we can find that the multi-fractal characteristic values of T2 are relatively closer to those of the background.It demonstrated that T2 fused into its background to some extent.Therefore,we can qualitatively analyze that the camouflage effectiveness of T2 is better than that of T1.Certainly,for either T1 or T2,its neighborhood regions are more similar with the whole background than the target itself because these neighborhood regions are naturally a part of the background.It proves again that artificial targets should provide adequate mimic of the natural background for good camouflage.

Furthermore,as shown in Table 2,to evaluate the camouflage effectiveness quantitatively,we calculate the discrete Fr′echet distances of these multi-fractal spectra.The closer the distance between the multi-fractal spectra of a target and that of its background is,the more similar the target is with the background.Thus,if the discrete Fr′echet distance value is small,we can alert that the camouflage effect of the target is good.In addition,we also calculated the correlation coefficients between the multi-fractal spectra of a target and that of its background,and further obtained the camouflage effectiveness value of this target because the correlation coefficient can be regarded as the direct judgment criterion of the camouflage effectiveness(mentioned in Section 3.3).

Apparently,a quantitative camouflage effectiveness value makes the camouflage quality of the targets more intuitive.From Table 2,we can see that T2(with camouflage)has higher camouflage effectiveness value than T1(without camouflage)in the same one background,which means the camouflage quality of T2 is better than that of T1.The result is also in accordance with human observations.

4.2.Camouflage effectiveness of the same one target in different background

Based on the IR images showed in Fig.1(b),(c)and(d),we can compare the IR camouflage quality of the same one target in different backgrounds.As shown in Fig.5,we calculated the multifractal spectra of T2 in the backgrounds B2 and B3,respectively.

Similarly,according to the correlation coefficients listed in Table 3,we can also evaluate the camouflage effectiveness of T2 in the backgrounds B2 and B3,respectively.In addition,to compare the camouflage quality of T2 in different backgrounds intuitively,the correlation coefficient of T2 in B1(already listed in Table 2)was illustrated in Table 3 again.

Obviously,from Table 3,we can see that T2 has different camouflage effectiveness in different backgrounds.Directly observing the IR images by the naked eye,we can clearly see that the patterns in the background B1 are disruptive,whereas those in B2 are relatively even.The size of patches in T2 is closer to that in B1,thus the patterns in T2 match better with the patterns in B1 than those in B2.According to the human observation,T2(with disruptive patterns)can fuse into B1(also with disruptive patterns)to some extent,whereas T2 cannot fuse into B2(with relative even patterns).So it can be said that the camouflage effectiveness of T2 in B1 is better than that in B2.Moreover,the patterns in B3 are very complicated(with trees,stones and bushes etc.),and the size of the patches near the target region is very small while the size of the patches in T2 is relative big.Therefore,the shape features of the patterns in T2 and B3 are not similar,thus resulting in the low camouflage effectiveness of T2.

Fig.4.(a)The multi-fractal spectrum of T2,(b)the multi-fractal spectra of T2 and its neighborhood regions,(c)the multi-fractal spectrum of the corresponding whole background,(d)the multi-fractal spectra of both T2 and its whole background.

Table 1Some important multi-fractal characteristic values of the image regions.

Table 2The discrete Fr′echet distance,the correlation coefficient,and the evaluated camouflage effectiveness.

4.3.Comparison with other existing method

The human observation is the most commonly simple method for camouflage effectiveness evaluation.As mentioned,the camouflage effectiveness evaluation results are in accordance with human observations.Despite this,we also evaluated the camouflage effectiveness of the targets based on image saliency method proposed before[21].Calculated by image saliency method,the camouflage effectiveness evaluation results of T1 and T2 in the corresponding background are 0.2011(T1-B1),0.6754(T2-B1),0.4758(T2-B2),and 0.4405(T2-B3),respectively,which are in good consistence with these in this work.

Fig.5.(a)The multi-fractal spectra of both T2 and its background in B2,(b)the multi-fractal spectra of both T2 and its background in B3.

Table 3The discrete Fr′echet distance,the correlation coefficient,and the evaluated camouflage effectiveness of T2 in three kinds of different backgrounds.

5.Conclusion

In conclusion,based on a kind of multi-fractal method,the quantitative IR camouflage effectiveness value of a target was successfully evaluated.With the calculated multi-fractal spectra of the targets and the corresponding backgrounds,the discrete Fr′echet distances among these spectra were used to quantitative the similarity between a target and its corresponding background.Based on the correlation of the multi-fractal characteristics of a target and its background,the camouflage effectiveness of the target was evaluated.On the one hand,this method can be used to differentiate the camouflage effectiveness of different targets in the same one background.On the other hand,it can also quantitatively reflect the IR camouflage effectiveness of a target in different backgrounds.Although more detailed investigations are still needed,this approach paves a new way to the evaluation of IR camouflage effectiveness and is beneficial for the commander to optimize the deployment of military equipment in battlefield for seeking better camouflage effectiveness.

Declaration of competing interest

The authors declare no conflicts of interest.

Acknowledgements

This work was supported by the State Key Laboratory of Pulsed Power Laser Technology,College of Electronic Engineering,National University of Defense Technology,Hefei,China.