remote sensing image subpixel mapping based on adaptive ... · remote sensing image subpixel...

1306 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 42, NO. 5, OCTOBER 2012

Remote Sensing Image Subpixel Mapping Based onAdaptive Differential Evolution

Yanfei Zhong, Member, IEEE, and Liangpei Zhang, Senior Member, IEEE

Abstract—In this paper, a novel subpixel mapping algorithmbased on an adaptive differential evolution (DE) algorithm,namely, adaptive-DE subpixel mapping (ADESM), is developedto perform the subpixel mapping task for remote sensing images.Subpixel mapping may provide a fine-resolution map of classlabels from coarser spectral unmixing fraction images, with theassumption of spatial dependence. In ADESM, to utilize DE, thesubpixel mapping problem is transformed into an optimizationproblem by maximizing the spatial dependence index. The tradi-tional DE algorithm is an efficient and powerful population-basedstochastic global optimizer in continuous optimization problems,but it cannot be applied to the subpixel mapping problem in adiscrete search space. In addition, it is not an easy task to properlyset control parameters in DE. To avoid these problems, this paperutilizes an adaptive strategy without user-defined parameters, anda reversible-conversion strategy between continuous space anddiscrete space, to improve the classical DE algorithm. Duringthe process of evolution, they are further improved by enhancedevolution operators, e.g., mutation, crossover, repair, exchange,insertion, and an effective local search to generate new candidatesolutions. Experimental results using different types of remoteimages show that the ADESM algorithm consistently outperformsthe previous subpixel mapping algorithms in all the experiments.Based on sensitivity analysis, ADESM, with its self-adaptive con-trol parameter setting, is better than, or at least comparable to,the standard DE algorithm, when considering the accuracy ofsubpixel mapping, and hence provides an effective new approachto subpixel mapping for remote sensing imagery.

Index Terms—Differential evolution (DE), remote sensing, sub-pixel mapping, superresolution mapping.

I. INTRODUCTION

R EMOTE SENSING sensors with a spatial resolutionlarger than the extent of the classes on the ground yield

mixed pixels, i.e., pixels whose spectral signature is a com-posite of signatures of different classes [1]. These mixed pixelspose a difficult problem for land-cover mapping as their spectral

Manuscript received May 20, 2011; revised August 30, 2011 andDecember 1, 2011; accepted February 9, 2012. Date of publication April 11,2012; date of current version September 12, 2012. This work was supported inpart by the National Basic Research Program of China (973 Program) underGrant 2009CB723905, by the National Natural Science Foundation of Chinaunder Grants 40901213 and 40930532, by the Foundation for the Author ofNational Excellent Doctoral Dissertation of P. R. China under Grant 201052,by the Program for New Century Excellent Talents in University under GrantNECT-10-0624, and by the Fundamental Research Funds for the CentralUniversities under Grant 3103006. This paper was recommended by AssociateEditor H. Ishibuchi.

The authors are with the State Key Laboratory of Information Engineering inSurveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079,China (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSMCB.2012.2189561

characteristics are not representative of any single pure land-cover class [2]. Spectral unmixing techniques [3] and fuzzyclassifiers [4], as effective unmixing algorithms, can obtain thefraction images with different pure classes or “endmembers.”The fraction images indicate the percentage of each pixel that iscomposed of each class, but they do not provide any indicationabout the subpixel spatial distribution of the classes withinthe coarse pixel; for example, they cannot provide the exactsubpixel location of the interested target within the pixel.

Subpixel mapping techniques [5], [6], also termed superreso-lution mapping [1] or downscaling [7], can be used to obtain thesubpixel spatial distribution of any class, based on the fractionimages, by decomposing the pixel into smaller subpixels, basedon the spatial dependence phenomenon in which observationsclose together are more alike than those further apart [8].One of the earliest approaches to subpixel mapping is thatof Verhoeye and De Wulf [5], who proposed a deterministicsolution based on linear programming. Their algorithm mapsland cover within pixels from the proportions output by as-suming the maximum spatial dependence within and betweenpixels. This simple algorithm works efficiently on simulatedsatellite sensor imagery and Système Probatoire d’Observationde la Terre high-resolution visible imagery, recreating the initialscenes more accurately than traditional hard classification did.Mertens et al. [9] proposed a subpixel mapping algorithm basedon a genetic algorithm, using the same objective functions,as Verhoeye and De Wulf [5] did. A simple pixel-swappingstrategy, as used in spatial simulated annealing, was adopted byAtkinson [10] to construct subpixel or superresolution maps.This work applied the swapping algorithm to an initial purelyrandom subpixel map comprising the correct class fractionswithin each coarse pixel. Thornton et al. [8], [11] proposeda linearized pixel-swapping method for mapping rural lin-ear land-cover features from fine–spatial-resolution remotelysensed imagery. Artificial neural networks (ANNs), as pow-erful tools for nonlinear prediction, have also been appliedto subpixel mapping. Tatem et al. [12] trained a Hopfieldneural network to optimize an initial subpixel map used forfurther iterations, with the simultaneous objectives of coarsefraction reproduction and spatial autocorrelation maximization;this method was then successfully tested on an actual case study[13]. Their neural network approach was further extended to ac-count for indicator variogram models [14]. A back propagation(BP) neural network has also been used to improve subpixelmapping accuracy [15], [16]. Mertens et al. [17] combinedwavelets with neural network models to account for the reso-lution difference between fine class labels and coarse fractions.Geostatistics provides another solution for subpixel mapping.

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ZHONG AND ZHANG: REMOTE SENSING IMAGE SUBPIXEL MAPPING BASED ON ADAPTIVE DE 1307

Fig. 1. ADESM approach for remote sensing imagery.

Boucher et al. [1], [18] built up a sequential simulation frame-work based on a prior model of spatial structure or texture forgenerating alternative superresolution mapping of class labels.Atkinson et al. [7] proposed a superresolution mapping al-gorithm based on downscaling cokriging and demonstratedthe performance of the method using Landsat EnhancedThematic Mapper (TM) Plus (ETM+) images. In addition,Mertens et al. [19] proposed a direct-neighboring subpixel map-ping (DNSM) algorithm, which exploits spatial dependence in asimple manner, giving accurate results in a limited computationtime. The direct subpixel mapping algorithm was enhanced byusing the spatial-attraction model [spatial-attraction subpixelmapping (SASM)] [20] and showed increased accuracy whencompared with hardened soft classifications. More recently,a Markov random field model had been used to representthe spatial dependence within and between pixels, to give asubpixel mapping [21]. Ge et al. [22] proposed an advancedsubpixel mapping algorithm to provide detailed informationon the spatial distribution of land cover within a mixed pixel,but this algorithm cannot be applied to pixels falling into theboundary region of the degraded image [22], thereby losingthe image information. Although the current subpixel mappingalgorithms have obtained relatively satisfactory results, theycan be further improved by hybrid methods or by exploringnew strategies. For instance, to improve the subpixel mappingaccuracy, a hybrid method combining a spatial-attraction modeland a pixel-swapping algorithm [23] has been used to solve thesubpixel mapping problem.

In this paper, a new subpixel mapping algorithm based onthe differential evolution (DE) algorithm, namely, the adaptive-DE subpixel mapping (ADESM) algorithm, is proposed. InADESM, the subpixel mapping problem is considered as anoptimization problem, maximizing the spatial autocorrelationwithin the pixel and the image. DE, as an effective optimizationmethod, is used to solve the subpixel mapping optimization

problem in order to obtain the optimal subpixel mapping result.The DE algorithm, as proposed by Storn and Price [24], isa simple yet powerful population-based stochastic search andoptimization technique [25]. DE uses simple real mutation andcrossover operators to generate new candidate solutions in thecontinuous search space and applies a one-to-one competitionscheme to greedily decide whether the new candidate or itsparent will survive in the next generation, until finding the op-timal result [26]. Due to its simplicity, ease of implementation,fast convergence, and robustness, the DE algorithm has gainedmuch attention and a wide range of successful applications,such as numerical optimization [25], [27]–[29], mechanicalengineering [30], feedforward neural network training [31],digital filter design [32], image processing [33], [34], andpattern recognition [35], [36]. Although DE has been effectivein global optimization problems over continuous space, fewapplications of DE have been reported in the field of subpixelmapping. This is because the subpixel mapping problem needsto be solved in the discrete space as a permutation-based com-binatorial optimization problem, in which each subpixel withinthe pixel is allocated to different classes. The effectiveness ofDE for combinatorial discrete optimization problems is stillconsidered limited [37] as the major obstacle in applying DE tocombinatorial problems is caused by its working mechanism,which is based on real vectors. In addition, there are threecrucial control parameters involved in DE: the population sizeNP , scaling factor F , and crossover rate CR. These parametersare often kept fixed throughout the optimization process andmay significantly influence the optimization performance ofDE [38], [39]. If we want to obtain the appropriate valuesof the parameters for a specific optimization problem, DE isrun multiple times with different settings of the parameters[40]. This process requires high computational costs, and thetime for finding these parameters is often unacceptably longfor a remote sensing image process. Therefore, an adaptive


mechanism should be considered in the optimization process.To remedy these drawbacks and utilize the global optimalability of DE, this paper proposes an adaptive-DE approach tosolve the subpixel mapping problem for coarse remote sensingimages. The flowchart of the proposed algorithm is shown inFig. 1. Accordingly, several issues should be addressed.

1) Discrete-encoding and reversible-conversion strategies.It is necessary to encode the candidate solution for thesubpixel mapping in the population using a discrete valuebecause it consists of integer numbers, e.g., the classattribute of the subpixel. However, traditional DE cannotbe used in the discrete space. To solve this problem, areversible-conversion strategy is utilized to change thecandidate subpixel mapping solution from integer to realnumbers, as a real transformation, and then back to inte-ger numbers after crossover as an integer transformation.

2) Improved DE operators. After the conversion from in-teger numbers to real numbers, these discrete-encodingindividuals may be evolved to obtain a better solution bystandard mutation and crossover operators. However, theconversion process will also bring a high frequency ofinfeasible solutions, such as out of range. Three enhancedDE operators (repair, exchange, and insertion) were de-veloped to repair the infeasible solutions and improvethe subpixel mapping results. In addition, there is oftena possibility of stagnation in DE. To move away fromthe point of stagnation, a local search [41], [42] operatoris applied to find a more feasible solution in the localneighborhood after the selection process.

3) Self-adaptive parameter control. A self-adaptation strat-egy for the parameters F and CR is designed to im-prove the intelligence and practicability of the proposedsubpixel mapping algorithm for different remote sensingimages. Here, the parameters to be adapted are encodedinto the individuals and undergo the evolution processusing genetic operators, such as crossover, mutation, andselection. The better individuals, with better values ofthese encoded parameters, will be more likely to surviveand produce offspring. This method reduces the time forfinding the appropriate parameters, the parameter controltechnique, and can produce a flexible DE for remotesensing subpixel mapping.

This paper proposes a novel adaptive-DE-based subpixelmapping approach for remote sensing imagery and addressessome important issues for this application, including discreteencoding, enhanced operators, and self-adaptive parameter con-trol. We provide comparisons with several subpixel mappingmethods for an artificial image and three remote sensing im-ages. The rest of this paper is organized as follows. Section IIprovides the mathematical formulation of the subpixel mappingalgorithm. Section III briefly introduces the basics of DE. InSection IV, we describe the proposed adaptive-DE approachfor solving the subpixel mapping problem. Experimental resultsand analysis are given in Section V. Section VI discusses themain properties of the ADESM, in theoretical and empiricalterms. Finally, the conclusions are provided in Section VII.

Fig. 2. Example of subpixel mapping (3 × 3 subpixels).

II. SUBPIXEL MAPPING PROBLEM

A. Basic Principles

Spectral unmixing or fuzzy classification techniques may beused to obtain the fraction images or membership images ofeach class [3], [4], but they do not give the subpixel attributionof each class in the coarse pixel. A subpixel mapping methodmay transform the fraction images to obtain the most suitablesubpixel locations for the different class fractions within a pixel.During this transformation, every pixel is divided into a numberof subpixels by defining a scale factor s. The scale factordescribes the difference of the resolution between the fractionimages and the subpixel mapping results. That is, each pixelwill consist of s2 subpixels.

The basic principle of subpixel mapping is shown in Fig. 2by a simple example with two classes (class 1 and class 2)in a raster grid of 3 × 3 coarse–spatial-resolution pixels. Thefraction image of land-cover class 1 is shown in Fig. 2(a), andthe proportion of another class, i.e., class 2, may be obtained bysubtracting one from the fraction of the corresponding class 1in the same pixel. If a coarse-resolution pixel zm,n is dividedinto nine subpixels, the scale s will be set to three, where mand n represent the mth lines and the nth samples in the image,respectively. Each subpixel corresponds to 11.11% coverage ofthe coarse-resolution pixel. As shown in Fig. 2(a), the fractionvalue of class 1 for zm,n, with 66.67%, represents six subpixelsbelonging to class 1 (black circle). Fig. 2(b) and (c) shows thetwo possible subpixel mapping solutions for Fig. 2(a).

The spatial dependence principle is utilized to comparethe two possible solutions, with the tendency for spatiallyproximate observations of a given property to be more alikethan is the case for more distant observations. Land cover isspatially dependent, both within and between pixels, on the con-dition that the intrinsic scale of variation is not less than thesampling scale imposed by the image pixels [6]. The higher thespatial dependence, the better the subpixel mapping solutions.In Fig. 2, the solution in Fig. 2(c) is better than that in Fig. 2(b).

B. Problem Formulation

According to spatial dependence principles, subpixel map-ping can be formulated as a maximum optimization problem,with a given appointed scale s. Suppose that the spectralunmixing model, or a soft classifier, yields fraction images forC land-cover classes and the coarse-resolution pixels are to bedivided into D subpixels, where D is equal to s2. The numberof subpixels for the land-cover class i is calculated by (1). The


number of subpixels that have to be assigned to land-cover classi is NCi and has been derived from the fraction images

NCi = round(D ∗ Fractioni) (1)

where D is the number of subpixels within a pixel, Fractioni

represents the fraction value of class i for the pixel, which de-scribes the percentage of the class, and round() is the operatorthat rounds its argument toward the closer integer.

A measure for the spatial dependence index (SDI) SDIij

will be calculated for land-cover class i and each subpixelj, considering the spatial correlation with neighboring pixels.Assuming that Nn neighboring pixels in a coarse-resolutionfraction image are considered, SDIij can be expressed as aweighted linear combination of Nn neighboring fractions ofland-cover class i using

SDIij =Nn∑k=1

wk · Fractioni,k (2)

where wk is the weight and Fractioni,k represents the fractionvalue of class i for the kth neighboring pixel around thesubpixel j. To satisfy the spatial dependence principle, wk isoften calculated as the inverse of the distance of the subpixel tothe coarse pixel center [6].

The subpixel mapping problem now becomes one of assign-ing land-cover classes to the subpixels while maximizing theSDI. The mathematical model of the subpixel mapping problemcan be denoted as follows:

Maximize f =C∑

i=1

D∑j=1

yij · SDIij (3)

where yij is an attribute value described whether the subpixel jis assigned to land-cover class i as follows:

yij ={

1, if subpixel j is assigned to land-cover class i0, otherwise

(4)

C∑i=1

yij = 1, j = 1, 2, . . . , D (5)

D∑j=1

yij = NCi, i = 1, 2, . . . , C (6)

where D is the number of subpixels within a pixel, C isthe number of land-cover classes, and NCi is the number ofsubpixels for the land-cover class i.

The example with the scale setting of three, shown in Fig. 2,is used to describe the subpixel mapping problem. Each pixelis divided into D subpixels, where D is equal to 9 (3 × 3).For a pixel Zm,n = {zt|t = 1, 2, . . . , 9}, its fractions are equalto 66.67% and 33.33% for classes 1 and 2, respectively. Thenumber of subpixels for classes 1 and 2 may be obtainedaccording to the following equations: NC1 = 9 × 66.67% = 6and NC2 = 9 × 33.33% = 3. It is worth noting that the sum ofNCi values should be equal to the number of subpixels D. Fol-lowing the aforementioned description, the subpixel mappingproblem can be described as the problem of how to locate this

Fig. 3. Permutation-based subpixel mapping problem.

land-cover class in D subpixels. Fig. 3 shows the representationof a possible subpixel mapping solution in Fig. 2(b), where thefirst, second, third, fourth, eighth, and ninth subpixels are allo-cated to class 1 and the other subpixels are allocated to class 2.For Fig. 2(c), the first, second, fourth, fifth, seventh, and eighthsubpixels are allocated to class 1, and other subpixels areallocated to class 2, as shown in Fig. 3. According to theaforementioned analysis, the subpixel mapping problem canbe formulated as a combinational optimization problem whichmaximizes the spatial dependence model f(y, SDI). In thispaper, we utilize DE as a powerful optimal algorithm to solvethe optimization problem of remote sensing subpixel mapping.

III. DE ALGORITHM

The DE algorithm, as invented by Storn and Price [24], is oneof the latest population-based stochastic global optimizers andis a simple yet powerful heuristic method for solving nonlinear,nondifferentiable, and multimodal optimization problems. Itadopts a floating-point encoding scheme and takes advantageof the differentiation information among the population to findthe global optimum in the continuous search space. The tech-nique combines simple arithmetic operators with the classicalevents of crossover, mutation, and selection, to evolve froma randomly generated initial population to the final individualsolution [26]. Successive populations are generated by addingthe weighted difference of two randomly selected vectors to athird randomly selected vector. There are several variants of DEbased on different mutation and crossover strategies [24], [43],[44]. In this paper, we use classical DE (DE/rand/1/bin) [24],[40] because this strategy is the most often used in practice. Itcan be described as follows.

To solve the minimal optimization problemmin f(x1, x2, . . . , xD), DE starts with an initial populationvector, which is randomly generated when no preliminaryknowledge about the solution space is available. LetXk,G = {x1

k,G, . . . , xDk,G} (k = 1, 2, . . . , NP for each

generation G) represent a candidate solution vector with Ddimensions, where NP represents the population size.

Step 1: Mutation. For each vector Xk,G in genera-tion G, its corresponding mutant vector Vk,G ={v1

k,G, v2k,G, . . . , vD

k,G} may be calculated by thefollowing equation:

Vk,G = Xr1,G + F · (Xr2,G − Xr3,G) (7)


where k = 1, 2, . . . , NP ; Xr1,G, Xr2,G, and Xr3,G

represent the randomly selected vectors; and theindices r1, r2, and r3 are mutually exclusive integersrandomly generated within the range [1, NP ], whichare also different, r1 �= r2 �= r3 �= k, so that NP ≥4 is required. The scaling factor F ∈ [0, 2] is apositive control parameter for scaling the differencevector. Larger values for F result in higher diversityin the generated population, and lower values causefaster convergence. According to the study, the valueof F is often set to 0.5 [24], [45].

Step 2: Crossover. After the mutation process, to increasethe potential diversity of the population, a crossoveroperation is applied to each pair of the targetvector Xk,G and its corresponding mutant vec-tor Vk,G to generate a trial vector Uk,G+1 ={u1

k,G, u2k,G, . . . , uD

k,G}. In the basic version, DEemploys the binomial (uniform) crossover, definedas follows:

utk,G+1 =

⎧⎨⎩

vtk,G, if (randt[0, 1] ≤ CR) or (t = trand),

t = 1, 2, . . . , Dxt

k,G, otherwise(8)

where randt[0, 1] is a uniform random number gen-erator and the crossover rate CR is a user-specifiedconstant within the range [0, 1], which controls thefraction of parameter values copied from the mutantvector. In [24] and [45], the experiential value ofCR is set to 0.9, and trand is a randomly choseninteger in the range [1,D] to ensure that at least oneparameter of Uk,G+1 is selected from the mutatedvector Vk,G.

Step 3: Selection. This approach must decide which vector(Uk,G+1 or Xk,G) should be a member of the next(new) generation G + 1. For a minimization prob-lem, the vector with the lower fitness value is chosenby a greedy selection scheme, as follows:

Xk,G+1 ={

Uk,G+1, if f(Uk,G+1) ≤ f(Xk,G)Xk,G, otherwise.

(9)

Step 4: Stopping condition. Once the new population isobtained, the process of mutation, crossover, andselection is repeated until the optimal solution islocated or a prespecified termination criterion issatisfied, e.g., the number of generations reaches apreset maximum Gmax.

Compared with most other evolution algorithms, DE is muchsimpler and significantly more straightforward to implement.The main body of the algorithm takes four to five lines to codein any programming language. Due to its simplicity of coding,DE has been applied to various engineering problems, includingnumerical optimization [25], [27]–[29] mechanical engineering[30], feedforward neural network training [31], digital filter de-sign [32], image processing [33], [34], and pattern recognition[35], [36].

IV. ADESM ALGORITHM

The subpixel mapping problem should be considered asa permutation-based combinational problem in the discretespace, which allocates a land-cover class to the position ofsubpixels (see also Section II-B and Fig. 3). However, thecanonical DE algorithm cannot be used to directly generatea discrete subpixel mapping problem since it was originallydesigned to solve continuous optimization problems. Therefore,in this section, we will present an enhanced DE approach,namely, an ADESM algorithm, to solve the subpixel mappingproblem. In ADESM, a reversible-conversion strategy is usedto transform the individual encoding of a subpixel mappingsolution from integer to real numbers. That is, the discretesubpixel mapping individuals are changed to real encoding.By this transformation, traditional DE operators for continuousspace may be retained and directly used, with the advantageof DE. In addition, a self-adapting parameter selection methodis proposed to adaptively choose the appropriate parametersduring the course of ADESM.

To describe ADESM, the following notations are used.

1) Let Z denote the input coarse fraction image, whereZ = {zmn : 1 ≤ m ≤ M, 1 ≤ n ≤ N} and M and Nrepresent the row number and column number of theimage, respectively.

2) Let zmn denote a pixel in the coarse fraction imageZ, where zmn = {zi

mn : i = 1, . . . , C}, zimn denotes the

fraction for the ith land-cover class, and C is the numberof the land-cover classes or endmembers.

3) Let s represent the scale of the subpixel mapping and Drepresent the number of the subpixels in a pixel. Here, Dis equal to s2.

ADESM consists of the following steps for each pixel zmn.

A. Calculating the Number of Subpixels for EachLand-Cover Class i

In the first step, the number of subpixels in land-cover classi, NCi, is derived according to the following equation:

NCi = round(D ∗

(zimn

)), i = 1, . . . , C

C∑i=1

zimn = 1

C∑i=1

NCi = D (10)

where round() denotes the operator that rounds its argumenttoward the closer integer. Based on (10), the sum of NCi valuesshould be equal to D.

By Step A, the number of subpixels for each land-cover classis calculated. The next process utilizes DE to obtain the optimalpositions of subpixels for each land-cover class within the pixel.


Fig. 4. Initial individual.

B. Initial Population and Discrete Encoding

After ADESM obtains the number of subpixels for eachland-cover class i, NCi, it needs to generate the initialpopulation PG = {X1,G, . . . ,Xk,G, . . . , XNP,G}, where Xk,G

represents the kth individual in the Gth generation andNP is the size of the population. Each individual Xk,G ={x1

k,G, . . . , xtk,G.., xD

k,G} denotes a candidate subpixel mappingsolution.

Each individual xtk,G in ADESM is discretely encoded to

describe the position of the subpixel with the correspondingclass attribute, where xt

k,G is unique and systematic, as shownin Fig. 4. For example, for Fig. 2(a), where NC1 = 6 andNC2 = 3, six and three subpixels belong to classes 1 and 2,respectively. According to the aforementioned encoding prin-ciple, for the candidate solution shown in Fig. 2(b), Xk,G isencoded to {1, 2, 4, 5, 7, 8, 3, 6, 9} (see also Fig. 3). Thatis, the first, second, fourth, fifth, seventh, and eighth subpixelsare allocated to class 1, and other subpixels are allocated toclass 2.

The initial population PG=0 is obtained according to theaforementioned discrete encoding using (11) and should coverthe entire search space as much as possible by uniformlyrandomizing individuals within the search space constrainedby the prescribed minimum and maximum parameterbounds Xmin = {x1

min, . . . , xtmin, . . . , xD

min} and Xmax ={x1

max, . . . , xtmax, . . . , x

Dmax}, where xt

min and xtmax represent

the bounds of the tth position. For the subpixel mappingproblem, xt

min and xtmax are equal to one and D, respectively

xtk,G=0 = (int)

(randt(0, 1) ·

(xt

max + 1 − xtmin

)+

(xt

min

)),

t = 1, . . . , D; xtk �∈

{x1

k, . . . , xt−1k

}(11)

where xtk,G=0 represents the tth position in the kth

individual Xk,G.

C. Calculation of the Objective Function Using an SDI

After Step B, the objective function of the new solution iscalculated. It should be noted that the purpose of solving thesubpixel mapping problem is to obtain the maximum spatialdependence using (3) while the purpose of DE is to obtain theminimum value of the objective function. Therefore, we mustchange the form of the objective function (3) using minimize Jin (12).

To construct the mathematical model, choice variables yij

are defined such that [5]

Minimize J = − f = −C∑

i=1

D∑j=1

yij · SDIij

yij =

{ 1, if subpixel j is assignedto land-cover class i

0, otherwise(12)

where D is the number of subpixels within a pixel, C is thenumber of land-cover classes, and SDIij is the measure forspatial dependence assigned to land-cover class i in the jthsubpixels.

The value of yij is obtained according to xtk,G. For ex-

ample, assuming that NC1 = 6 and NC2 = 3, a candidatesubpixel mapping solution {1, 2, 4, 5, 7, 8, 3, 6, 9} inFigs. 3 and 4 ensures that the first, second, fourth, fifth,seventh, and eighth subpixels are allocated to class 1 y1j ={11̄, 12̄, 0, 14̄, 15̄, 0, 17̄, 18̄, 0} and other subpixels are allocatedto class 2 y2j = {0, 0, 13̄, 0, 0, 16̄, 0, 0, 19̄}. The fitness of theindividual Xk,G is calculated as follows:

J(Xk,G)=− [(SDI1,1+SDI1,2+SDI1,4+SDI1,5+SDI1,7

+ SDI1,8) + (SDI2,3 + SDI2,6 + SDI2,9)] . (13)

The SDI can be calculated by (13). Unlike (2), the SDIij isthe average SDI, not the sum of the SDI values. The reason isthat each pixel has the nine neighboring pixels in the originalversion, while the number of neighboring pixels will changein the boundary pixel in the proposed algorithm. To bettercompare the SDI at the same scale, we use the average SDIvalue to replace the sum of the SDI values

SDIij =

Nn∑k=1

wk · Fractioni,k

Nn(14)

where wk is the weight, Nn is the number of the neighboringpixels, and Fractioni,k represents the fraction value of class ifor the kth neighboring pixel around the subpixel j. It is worthnoting that the value of the weight wk is inversely proportionalto the Euclidean distance (dis) between the subpixels and theneighboring pixels. In this paper, the weight is equal to thereciprocal of the distance by (15), and the distance is calculatedby (16). The shorter the distance, the higher the weight. In thisstep, it is necessary to project the coordinate of the subpixels(g, h) and the pixels (m,n) to the same coordinate systemsusing (17) and (18) and adaptively select the neighboring pixelsto avoid losing boundary information.

Coordinate Conversion: The aforementioned process isshown in Fig. 5 by a simple example with a scale of four. Theoriginal coordinate of the pixel (m,n) is (0.5, 0.5) in the XYcoordinate system, and the coordinate of the subpixel (g, h) is(1.5, 0.5) in the xy coordinate system. To calculate the distancebetween subpixels and pixels, the new Euclidean coordinatesystem, with vertical X-axis and horizontal Y -axis, is defined.In the new coordinate system, the pixel (m,n) with (0.5, 0.5) isconverted to (m′, n′), (2, 2), using (17), and the subpixel (g, h)with (1.5, 0.5) is converted to (g′, h′), (5.5, 4.5), using (18).


Fig. 5. Coordinate conversion and the distance calculation between the pixelsand subpixels.

According to (15) and (16), the weight of the neighboring pixelw in the upper left corner of the subpixels is equal to 0.233

w = 1/d (15)

d =√

(m′ − g′)2 + (n′ − h′)2 (16)(m′, n′) ← (s × m), s × n) (17)(g′, h′) ← (s + g, s + n). (18)

Adaptively Selecting Neighboring Pixels: The traditionalneighboring pixel zi,j for the pixel zm,n using the surroundingneighborhood [20] is selected as follows:

N [zm,n] = {zi,j |(i �= m ∨ j �= n) ∧ [(m − 1 ≤ i ≤ m + 1)∧(n − 1 ≤ j ≤ n + 1)] ∧ [(1 < m < M) ∧ (1 < n < N)]} .

(19)

In (19), the boundary pixels zm,n (m = 1, M or n = 1,N ) are not considered, to guarantee that each pixel has nineneighboring pixels. Thus, in some previous methods, suchas BP subpixel mapping (BPSM) and the subpixel mappingalgorithm proposed by Ge et al. [22], the subpixel mappingresults do not contain the boundary pixels and lose some sub-pixel information. To avoid this problem, the proposed methodimproves (19) to adaptively select valid neighboring pixels, asin the following equation:

N [zm,n] = {zi,j |(i �= m ∨ j �= n)∧ [(max(1,m − 1) ≤ i ≤ min(m + 1,M))

∧ (max(1, n − 1) ≤ j ≤ min(n + 1, N) ]∧ [(1 ≤ m ≤ M) ∧ (1 ≤ n ≤ N)]} (20)

where M and N represent the row number and column number,respectively. The functions max() and min() return the largestand smallest values from the numbers provided, respectively.

Fig. 6. Self-adapting encoding strategy.

D. Real Transformation

To apply DE to solve the subpixel mapping problem,ADESM transforms the subpixel mapping solution Xk,G ={x1

k,G, . . . , xtk,G, . . . , xD

k,G} in the discrete space to the contin-uous real space. In this paper, if the precision of the value to begenerated is set to a defined decimal place a, an integer numberxt

k,G is transformed by the following equation:

xtk,G =

xtk,G

10−a(21)

where the range of the variable xtk,G is between 10a and D∗10a.

For example, when a is equal to two, then the value of theindividuals in Fig. 2 is within [100, 900].

E. Adaptive Mutation and Crossover

By Step D, the population changes from integer to real num-ber. These values may be used in the traditional DE mutationand crossover operators. There are two control parameters thatneed to be adjusted by the user: the scaling factor F and thecrossover rate CR, according to (7) and (8), respectively. Suit-able control parameters are different for different real problems.In some cases, the time for finding these appropriate parameterscan be unacceptably long, particularly for large-volume remotesensing images.

To solve the problem, a self-adaptive strategy for the controlparameters is used. The control parameters F and CR areencoded to each individual, as shown in Fig. 6. That is, eachindividual has its corresponding F and CR and can be adjustedduring the process of evolution [40]. The subpixel mappingsolution (Fig. 6) is represented by the D-dimensional vectorXk,G and two control parameters Fk,G and CRk,G in the Gthgeneration, k = 1, . . . , NP .

For each transformed vector Xk,G at the generation G, itsassociated mutant vector Vk,G = {v1

k,G, v2k,G, . . . , vD

k,G} canbe generated via the strategy DE/rand/1/bin, which is the strat-egy most often used in practice [24], [40], [44]. The mutationoperators are as follows:

Vk,G =Xr1,G+F · (Xr2,G−Xr3,G), k=1,. . ., NP(22)

where the indices r1, r2, and r3 are mutually exclusive inte-gers randomly generated within the range [1, NP ], r1 �= r2 �=r3 �= k.

The better objective values in (12) of these encoded controlparameters are more likely to survive and produce offspring,which leads to better individuals and improves the probabilityof finding the optimal solution. To adaptively determine the


mutation rate pm according to the objective values of eachindividual, the process is as follows:

pm =J(Xk,G) − min (J(Xk,G))

max (J(Xk,G)) − min (J(Xk,G)). (23)

New control parameters in the G + 1th generation Fk,G+1

and CRk,G+1 are updated as follows, with probability pm:

Fk,G+1 =

{1 − rand

(1− GGmax )b

1 , if rand2 < pm

Fk,G, otherwise(24)

where randt, t ∈ {1, 2}, denotes the uniform random valueswithin the range [0, 1], G is the iteration number, Gmax isthe maximal iteration number, b is a parameter to decide thenonconforming degree, and the experiential value is set to three[46]. In the initial generations, ADESM tends to search thesolution space uniformly, and in the later generations, it tendsto search the space locally, i.e., closer to its descendants, toguarantee convergence.

After the mutation phase, a crossover operation is applied togenerate a trial vector Uk,G = {u1

k,G, u2k,G, . . . , uD

k,G} for themutant vector Vk,G as follows:

utk,G =

{vt

k,G, if (randt[0, 1] ≤ CR) or (t = trand)xt

k,G, otherwise,

t = 1, 2, . . . , D; k = 1, . . . , NP. (25)

CRk,G+1 is updated using the following [40]:

CRk,G+1 ={

rand4, if rand3 < pm

CRk,G, otherwise (26)

where randt, t ∈ {3, 4}, denotes the uniform random valueswithin the range [0, 1]. Fk,G+1 and CRk,G+1 are obtainedbefore the mutation is performed. Therefore, they influence themutation, crossover, and selection operations of the new vectorXk,G+1.

F. Integer Transformation

After the mutation and crossover processes, ADESM obtainsthe set of the solution Uk,G = {u1

k,G, u2k,G, . . . , uD

k,G} [see also(25)] in continuous space. The data type of these solutions isfloating. To find the possible subpixel mapping solution in thediscrete space, integer transformation is used to convert the realvalue back to the integer, as given in (27), assuming ut

k,G to bethe real value obtained from (25)

int[ut

k,G

]=

utk,G

10a. (27)

The value utk,G is rounded to the nearest integer using

an inverse process of (21). Real transformation and integertransformation effectively allow DE to optimize permutativesolutions for discrete subpixel mapping problems.

Once the set of the solution Uk,G = {u1k,G, u2

k,G, . . . , uDk,G}

is obtained after the transformation, it is then checked for fea-sibility. Feasibility, measured using (28), only refers to whether

the solutions are within the bounds. If a solution is detected tohave violated a bound, it is dragged to the offending boundary

utk,G =

{xt

min, if utk,G < xt

min

xtmax, if ut

k,G > xtmax

(28)

where xtmin and xt

max represent the bounds of the tth position.For the subpixel mapping problem, xt

min and xtmax are equal to

one and D, respectively.

G. Repair

Although the value utk,G in Uk,G ranges from xt

min to xtmax,

the discrete individual of unique values cannot be guaranteed.To solve this problem, a repair operator is added. To repair theindividual, we need to record the positions of the repeatingvalues and the missing values, denoted by RV = {rvi} andMV = {mvi}, respectively. The position of the repeating val-ues rvi is selected randomly, and the rvith value is replaced bythe missing value mvi. Since each value is randomly selected,the value has to be removed from the array after selection, toavoid duplication.

H. Enhanced Operators

The traditional DE algorithm should be able to obtain a sat-isfactory result by mutation and crossover operators because itoperates in real continuous space. However, to solve the discretesubpixel mapping problem using the aforementioned process,ADESM may still be unable to find an appropriate solution, dueto some replicated solutions. In this section, enhanced operators(exchange and insertion) are added to extend the search spaceand improve the quality of the individuals.

Exchange Operator: The exchange operator, as an improve-ment technique, may explore random regions in the searchingspace to find a better subpixel mapping solution by simplyexchanging two values in an individual [41]. Two uniquerandom values are selected: r1 and r2 ∈ [1,D], where r1 �=r2. The values indexed by these values are exchanged as

ur1k,G

exchange↔ ur2k,G, and the individual is evaluated. If the ob-

jective value improves, then the new solution is accepted in thepopulation.

Insertion Operator: Insertion may provide greater diversityto the solution than the standard mutation in discrete space,which can be regarded as a more complicated mutation method[41]. In this step, two unique random numbers are selected: r1

and r2 ∈ [1,D]. The r1th value ur1k,G is removed to behind the

other value ur2k,G.

To explain the process from Step G to Step H, a simple ex-ample is shown in Fig. 7. For example, assuming an infeasiblesubpixel mapping solution obtained by integer transformation,Uk,G = {1, 3, 8,1, 5, 7, 2,9,9}, D = 9.

1) Repair: As shown, the values 1 and 9 are repeated inthe solution, so the positions of the repeated valuesare recorded: rv1 = 4, and rv2 = 8. We can find thatthe values 4 and 6 are missing: mv1 = 4, and mv2 =6. ADESM randomly selects the missing value fromMV to the located rvith position in Uk,G. It assumes


Fig. 7. Simple example using repair and enhanced operators.

that the missing values 4 and 6 are arranged to thefourth and eighth positions, respectively, and Uk,G ={1, 3, 8,4, 5, 7, 2,6, 9}.

2) Exchange: Two random numbers are generated withinthe bounds: Rnd = {2, 7}. u2

k,G and u7k,G in Uk,G

are exchanged to obtain the new solution Uk,G ={1,2, 8, 4, 5, 7,3, 6, 9}.

3) Insertion: Two random numbers are generated within thebounds: Rnd = {3, 6}. Then, u3

k,G is removed to behindu6

k,G, Uk,G = {1, 2, 4, 5, 7,8, 3, 6, 9}.

I. Selection

After the calculation of the objective function using (12), aselection operation is performed. The objective function valueof each trial vector J(Uk,G) is compared with that of itscorresponding target vector J(Xk,G) in the current population.If the trial vector has a lower or an equal objective functionvalue, compared with the corresponding target vector, the trialvector will replace the target vector and enter the population ofthe next generation. Otherwise, the target vector will remain inthe population for the next generation. The selection operationcan be expressed as follows:

Xk,G+1 ={

Uk,G, if J(Uk,G)≤J(Xk,G), k=1, 2,. . ., NPXk,G, otherwise.

(29)

J. Local Search

There is always a possibility of stagnation in DE [41]. Toavoid stagnation, a local search operator is used to find a morefeasible solution in the local neighborhood. The basic processof a local search is shown in Fig. 8. The set of the selectednumber A is set to empty in the initialization. On each iteration,a random number I is chosen, with its range within [1,D].Another random number j is chosen in another set B. Thevalues in the individual indexed by i and j are then swapped.The objective function of the new individual is calculated, andonly if there is an improvement is the new solution accepted.

Fig. 8. Local search process.

K. Stopping Condition

If the generation G does not meet the maximum generationnumber Gmax, go to Step C. Otherwise, output the best indi-viduals as the subpixels of the pixel zmn. The aforementionedprocess is repeated from Step A until the subpixels of all pixelsin the image Z are located. Finally, the proposed algorithmoutputs the subpixel mapping image. The flowchart for theADESM framework is shown in Fig. 1.

V. EXPERIMENTS AND ANALYSIS

The aforementioned ADESM was coded in Visual C++ 6.0and tested on different images. To demonstrate the effectivenessof the proposed approach, three different types of data sets wereused to test its performance. Consistent comparisons betweenADESM and the previous subpixel mapping algorithms—theDNSM algorithm, the subpixel mapping algorithm based onspatial-attraction models (SASM), and the BP neural networksubpixel mapping algorithm (BPSM)—were carried out. Tocompare ADESM with other evolutionary algorithms (EAs),the subpixel mapping algorithm based on a genetic algorithm[genetic-algorithm subpixel mapping (GASM)] was also usedin all the experiments. The surrounding neighboring type wasused for all subpixel mapping algorithms, and the estimationsof the subpixel mapping accuracy for the several subpixelmapping algorithms are provided.

A. Experiment 1: Synthetic Artificial Images

In experiments 1 and 2, synthetic imagery was used. Syn-thetic images are images that have been degraded to a coarserscale using an averaging filter. Degradation of a hard classifi-cation yields fraction images for each class in the classifica-tion. These images can be interpreted as a soft classification.Synthetic imagery has the advantage of lacking coregistrationerrors between the lower and higher resolution images. The re-sulting soft classification does not contain any uncertainty as itoriginates from degradation instead of a classification process.Consequently, the subpixel mapping error solely reflects theperformance of the proposed method. Validation is facilitated,as the original hard classification can be used as referencematerial [14].


Fig. 9. Subpixel mapping results in experiment 1 (scale = 4). (a) Original artificial image (400 × 400). (b) Original hard classification result (400 × 400).(c) Fraction images (100 × 100). (d) DNSM. (e) SASM. (f) BPSM. (g) GASM. (h) ADESM. (i) Zoom images for all the subpixel mapping results.

In this experiment, synthetic artificial imagery was used toenable the refinement of the technique. The synthetic imagewas created by the authors to aid in the design and develop-ment of the algorithm. A visual checking of the algorithm’sperformance is easy with the synthetic image as it representssimple geometric figures. To better simulate the real remotesensing scene in the artificial image, we designed four land-cover classes—water, tree, agricultural field, and imperviouslayer (e.g., building and road)—to illustrate the functionalityof the technique, as shown in Fig. 9(a). A maximum-likelihoodhard classification algorithm was applied to the synthetic artifi-cial images to provide real imagery to be used for degradationand accuracy testing, which is shown in Fig. 9(b). Fig. 9(c)shows the degraded fraction image with a scale of four from ahard classification image. Subpixel mapping using the DNSM,SASM, BPSM, GASM, and ADESM algorithms is shown inFig. 9(d)–(h). For a clear comparison of ADESM with the otheralgorithms, two small images of the S1 and S2 areas are zoomedand are shown in Fig. 9(i).

In this experiment, ADESM could adaptively choosethe most user-defined running parameters, as described inSection IV-E. The number of iterations was equal to 100.The configuration for BPSM (using one hidden layer) was asfollows: The number of hidden layers, the learning rate, themomentum rate, and the number of iterations were 1, 0.25, 0.9,and 1000, respectively. The crossover rate and mutation rate forGASM were set to 0.5 and 0.05.

From the comparison between Fig. 9(b) and Fig. 9(d)–(h), itcan be observed that ADESM shows better subpixel mappingresults than DNSM, BPSM, SASM, and GASM do. Serratededges exist between different classes in the subpixel mappingimages obtained by DNSM and BPSM, with many subpixels ofthe image allocated to incorrect positions, as shown in Fig. 9(i),S1. It is worth noting that BPSM obtains better subpixel map-ping for two ridges of the field on the middle of the results thanthe other three subpixel mapping algorithms from Fig. 9(i), S2.However, BPSM does not obtain the subpixel result of the edge,which is noted by the black pixel in Fig. 9(f). In general, SASM,


TABLE IADJUSTED CONFUSION MATRICES OF THE SUBPIXEL MAPPING RESULTS DERIVED FROM THE SYNTHETIC ARTIFICIAL IMAGE IN EXPERIMENT 1

TABLE IICOMPARISON OF FIVE SUBPIXEL MAPPING ALGORITHM PERFORMANCES FOR THE SYNTHETIC ARTIFICIAL IMAGE IN EXPERIMENT 1

GASM, and ADESM can obtain smoother visual results for allclasses.

For a more detailed verification of the results, we comparedhard classification with the subpixel mapping results and as-sessed the accuracy of each algorithm quantitatively, using theoverall accuracy (OA) measure and Kappa coefficient basedon the confusion matrices [47], as shown in Tables I and II.In addition, an adjusted confusion matrix was utilized. Theadjusted confusion matrix is identical to the original confusionmatrix except that it is calculated only for mixed pixels, whichmeans that it ignores the subpixels that have a pure pixel as

parent. These subpixels all belong to the same class and mayraise the Kappa coefficient without providing information aboutthe algorithm’s prediction abilities [9]. Based on the adjustedconfusion matrix, we calculated the adjusted OA (OA∗), theadjusted Kappa coefficient (Kappa∗), the average of the pro-ducer’s accuracy, the average of the user’s accuracy, and theaverage of Short’s mapping accuracy index (Short’s index)[48]. Among them, the average of the producer’s accuracy, theaverage of the user’s accuracy, OA, and Kappa value are widelyused in the validation of land-use/land-cover classification [4],[48]. The average of Short’s mapping accuracy index is the


arithmetic mean of this index [49], [50] (a monotonic functionof the harmonic mean of the user’s and producer’s accuracies)that explicitly combines both user’s and producer’s accuraciesin one measure [4]. This measure is supposed to be supplemen-tary to the average of the user’s accuracy and the average of theproducer’s accuracy.

Table I lists the confusion matrices of comparisons betweenthe hard reference classification image and the subpixel map-ping results produced by five subpixel mapping algorithms:DNSM, SASM, BPSM, GASM, and ADESM. It is worthnoting that the number of total subpixels for BPSM is lessthan that for the other subpixel mapping algorithms becauseBPSM did not place, on the subpixel map, the edge of the low-resolution image, as shown in Fig. 9(f). For a more detailedverification of the results, we assessed the accuracy of eachmethod using the producer’s and user’s accuracy measures (seeTable I). As an example, for C1 simulated water class, theproducer’s accuracies of DNSM, SASM, BPSM, GASM, andADESM are 73.38%, 96.69%, 82.62%, 94.88%, and 96.91%,respectively. The producer’s accuracies of these methods showthat some of the C1 pixels were wrongly identified as C2class. A similar situation occurred for C2 and C4 classes. Acareful evaluation of the confusion matrix reveals that there isconfusion when discriminating among C1, C2, and C4 classesin DNSM, SASM, GASM, and BPSM. It can be observedthat there is also significant confusion when discriminatingthe C4 class from the other classes, particularly when usingDNSM, SASM, GASM, and BPSM. ADESM recognizes theC4 class better than the other methods. It also exhibits thehighest producer’s accuracy for C1, C2, and C4 classes amongthe five methods considered. It is worth noting that BPSMhas obtained the highest producer’s and user’s accuracies forC3 class, i.e., 89.45%. However, it is difficult for BPSM torecognize the other three classes. Although the producer’s anduser’s accuracies of SASM are higher than those of DNSMand BPSM, the producer’s and user’s accuracies provided byADESM are further increased, to some degree, than SASM.

Table II shows the global performance in terms of the sub-pixel mapping accuracy yielded by DNSM, SASM, BPSM,GASM, and the proposed ADESM. From the table, we canobserve that ADESM exhibits the highest OA, Kappa value,adjusted OA, adjusted Kappa, and average of Short’s map-ping accuracy index. DNSM has the lowest subpixel map-ping accuracy because it did not consider the distance of theneighboring pixels. BPSM did not obtain a satisfactory result,as there is a sawtooth on the edges. Moreover, BPSM stillneeds an additional training data set. SASM utilized the spatial-attraction models, improving the results of DNSM by consider-ing neighboring information. However, employing SASM didnot guarantee obtaining the optimal subpixel mapping result. Toobtain the optimal result in GASM and ADESM, the subpixelmapping problem is transformed into an optimization problemof the SDI. ADESM, as a new subpixel mapping algorithmbased on a DE algorithm, like GASM, belongs to the family ofevolutionary computation and is a population-based stochasticglobal optimizer. However, there are some major philosophicaldifferences between GASM and ADESM. Although ADESMalso uses selection, crossover, and mutation operators, the op-

erations need to be redefined. In ADESM, candidate subpixelmapping solutions are represented by chromosomes based ondiscrete numbers. In the mutation process of an ADESM algo-rithm, the weighted difference between two randomly selectedpopulation members is added to a third member to generate amutated solution. Then, a crossover operator follows to com-bine the mutated solution with the target subpixel mappingsolution, to generate a trial solution. Thereafter, a selectionoperator is applied to compare the fitness function values ofboth competing solutions, namely, target and trial solutions, todetermine which can survive for the next generation. Followingthe aforementioned process, ADESM obtains the optimal sub-pixel mapping results for the synthetic artificial image in theexperiment.

B. Experiment 2: Synthetic Images—TM and HJ-1A Images

Synthetic imagery is also used for real remote sensing im-ages. This helps in avoiding the uncertainty inherent in realimagery that is caused by the sensor point spread function,atmospheric and geometric effects, and spectral unmixing orclassification errors. Work in progress is aimed at applying thetechnique to real imagery and reducing such uncertainty. Twodifferent types of remote sensing images, namely, Landsat TMimages and Chinese HJ-1A images, were chosen to demonstratethe same spatial dependence present among features in bothimages at different resolutions. For ADESM algorithms, wecarried out the experiments with the number of iterations equalto 100, and the other three parameters, i.e., the number ofindividuals in the population NP , the scaling factor F , andthe crossover rate CR, were adaptively obtained. The numberof hidden layers, the learning rate, the momentum rate, andthe number of iterations were fixed to 1, 0.25, 0.9, and 1000,respectively, in the BPSM method. The crossover rate andmutation rate for GASM were set to 0.5 and 0.05.

First, we tested the subpixel mapping algorithm proposedin experiment 2 using the 30-m resolution multispectral Land-sat TM image shown in Fig. 10(a) [51]. The image (400 ×400 pixels) was acquired in Wuhan City, Hubei Province,China, on October 26, 1998. The survey area is part of thecity, and the primary objective image was expected to fall intofour classes: water, vegetation, road/bare land, and building.The classification image shown in Fig. 10(b) was obtained byan unsupervised artificial immune classifier [51]. Fig. 10(c)shows the degraded fraction images with a scale of four froma hard classification image. The subpixel mapping results,using DNSM, SASM, BPSM, GASM, and ADESM algorithms,are shown in Fig. 10(d)–(h). To ensure clarity in comparingADESM with the other algorithms, two small images of S1 andS2 areas are zoomed and are shown in Fig. 10(i).

To evaluate the subpixel mapping, the original hard classi-fication result shown in Fig. 10(b) is considered as the realsubpixel mapping result. As shown in Fig. 10 [specifically,Fig. 10(i) zoom images for all subpixel mapping results],SASM, GASM, and ADESM obtained satisfactory visual sub-pixel mapping results, which means that their results weresmoother and more continuous than those of DNSM andBPSM. When comparing ADESM with SASM and GASM in


Fig. 10. Subpixel mapping results for the Wuhan TM image in experiment 2 (scale = 4). (a) Wuhan TM image red-green-blue (RGB) (4, 3, 2) (400 × 400).(b) Original hard classification result (400 × 400). (c) Fraction images (100 × 100). (d) DNSM. (e) SASM. (f) BPSM. (g) GASM. (h) ADESM. (i) Zoom imagesfor all subpixel mapping results.

Fig. 10(h), S2, one can see that ADESM can retain the linearfeatures from a coarse image, e.g., a road (white) on the rightof the S2 images.

The adjusted confusion matrices and global performanceindices obtained by applying DNSM, SASM, BPSM, GASM,and ADESM to this data set are shown in Tables III and IV. Asan example, one can see from Table III that, for the road/bare-land class, the producer’s accuracies of DNSM and BPSM are49.42% and 54.28%, respectively. They are characterized bya significant amount of confusion between the road/bare-landclass and the other classes. Many road/bare-land subpixels aremisclassified as building or vegetation classes, as can be seenin Fig. 10. In addition, the subpixel mapping results obtainedby DNSM and BPSM are fuzzy and not smooth, as a whole.By contrast, SASM and ADESM have better visual subpixelmapping results, as shown in Fig. 10(e) and (h), comparedwith Fig. 10(b). According to Table III, SASM and ADESM

also exhibited a higher accuracy (i.e., 76.76% and 77.38%for the water class, respectively) than the other subpixel map-ping techniques. On the other hand, there is confusion whendiscriminating vegetation from building, and many vegetationsubpixels are incorrectly allocated to the building class bySASM and GASM. It can also be observed that SASM andGASM misclassified many building subpixels to the vegetationclass.

In Table IV, we can observe that the proposed ADESMobtained the highest OA and Kappa value, i.e., 77.54% and0.6911, respectively. ADESM also has the highest adjusted OAand Kappa value, compared with the other subpixel mappingresults. These results are also confirmed by the other qualityindices considered, including the average of Short’s mappingaccuracy index. The reason for this may be attributed to thefact that ADESM can obtain the maximum of the SDI byusing evolution operators, mutation, crossover, repair, etc. In


TABLE IIIADJUSTED CONFUSION MATRICES OF THE SUBPIXEL MAPPING RESULTS DERIVED FROM THE WUHAN TM IMAGE IN EXPERIMENT 2

TABLE IVCOMPARISON OF FIVE SUBPIXEL MAPPING ALGORITHM PERFORMANCES FOR THE WUHAN TM IMAGE IN EXPERIMENT 2

addition, ADESM uses the local search strategy to improve theconnectivity property. It may also add the probability to find theoptimal subpixel mapping solution. Hence, with this data set,the ADESM technique is superior to the other three algorithmsconsidered.

Another synthetic image, based on the HJ-1A image, wasused to test the performance of the proposed subpixel mappingalgorithm by comparing it with the DNSM, BPSM, SASM,and GASM algorithms. The HJ-1A satellite is a small Chineseenvironmental satellite. It was launched on September 6, 2008,

from the Taiyuan Satellite Launch Center of Shanxi Province[52]. It has a sun-synchronous circular orbit with an orbitalaltitude of 649 km. The HJ-1A satellite has a hyperspectralsensor with 115 bands, including blue, green, red, and near-infrared spectra. It has a spatial resolution of 100 m and aspectral range of 0.45–0.95 μm. At present, the satellite imag-ing area can cover large parts of Asia, specifically, more thanten countries, including China, India, Pakistan, Kazakhstan,Mongolia, South Korea, North Korea, Japan, the Philippines,and Thailand [53]. The HJ-1A image (256 × 256), shown in


Fig. 11. Subpixel mapping results for the Hanchuan HJ-1A image in experiment 2 (scale = 4). (a) Hanchuan HJ-1A image RGB (95, 73, 43) (256 × 256). (b)Original hard classification result (256 × 256). (c) Fraction images (64 × 64). (d) DNSM. (e) SASM. (f) BPSM. (g) GASM (h) ADESM. (i) Zoom images for allsubpixel mapping results.

Fig. 11(a), was acquired on August 19, 2009 (path: 01; raw: 79),and was used as the original image for the experimental results.The study site is located in Hanchuan City, Hubei Province,central China, and its surrounding area. Four land-cover classes,i.e., city, agricultural land, water, and vegetation, characterizethis image. Fig. 11(b) shows the hard classification result for theHanchuan HJ-1A image obtained by a support vector machine(SVM) in ENVI software [54]. The fraction images of the fourclasses were obtained, as shown in Fig. 11(c), by resamplingwith a scale of four. Fig. 11(d)–(h) shows the subpixel mappingresults obtained by using DNSM, SASM, BPSM, GASM, andthe proposed algorithm ADESM (100 iterations). To quantita-tively evaluate the subpixel mapping accuracy, the original hardclassification result, as the real subpixel mapping result, wasused to test the performance of the five algorithms. To allow usto evaluate all the subpixel mapping algorithms from the visual

results, two small images of the S1 and S2 areas are zoomedand are shown in Fig. 11(i).

As shown in Fig. 11(d)–(i), DNSM and BPSM did notprovide satisfactory visual results, as there is blurring andmany structural features are lost. In addition, the sawtoothphenomenon can be observed in the BPSM result. By contrast,SASM, GASM, and ADESM obtained better visual results,being smoother and with most classes’ structural informationpreserved. Quantitative comparisons of the five algorithms,based on the adjusted confusion matrices and global perfor-mance indices obtained by applying DNSM, SASM, BPSM,GASM, and ADESM to this data set, are shown in Tables Vand VI. As with the visual results, the subpixel mappingaccuracies of DNSM and BPSM are lower than those of theother two algorithms. For instance, one can see from Table Vthat the average producer’s accuracies of DNSM and BPSM


are equal to 61.34% and 68.72%, respectively. SASM andGASM improve the subpixel mapping accuracy and locate thesubpixels’ position better than the aforementioned algorithms,resulting in producer’s accuracies of 72.89% and 72.61%. Itwas ADESM, however, that obtained the highest subpixel map-ping result, having a producer’s accuracy of 74.12%. This isbecause the proposed algorithm directly takes into account thespatial dependence as the optimization problem, to guaranteethe connectivity property of subpixels and the smoothness ofthe result. In Table V, it can be seen that ADESM obtainsthe highest OA, Kappa coefficient, adjusted OA, and adjustedKappa coefficient, which are 79.83%, 0.7263, 74.04%, and0.6479, respectively. These results are also confirmed by theother quality indices considered, e.g., the average of Short’smapping accuracy index being 0.59, thus pointing out thesuperiority of ADESM over the other four algorithms for thisdata set.

C. Experiment 3: Real Image

The synthetic imagery used in experiments 1 and 2 lackscoregistration errors, which have been shown to be an importantsource of errors in classification or spectral unmixing. Neitherdo the fraction images contain any uncertainty, as they originatefrom degradation instead of a classification or spectral unmix-ing process. A comprehensive discussion on the production ofsynthetic imagery is provided by Mertens et al. [12]. Althoughsynthetic images may avoid the reflection of other errors (i.e.,a coregistration error), the subpixel mapping methods were notdirectly applied to subpixel mapping for real imagery. In thethird experiment, we used the subpixel mapping methods toobtain a higher resolution classification image from a real low-resolution remote sensing image (i.e., 30-m ETM image). Toevaluate the result, a real high-resolution classification imagebased on a 4-m IKONOS image of the same research area wasused to test the performance of these subpixel mapping algo-rithms. Although there are other errors, such as coregistrationerrors and spectral unmixing errors, these errors are the samefor all the subpixel mapping methods, and relative comparisonsare fair.

The Landsat ETM standard product (98 × 30) over theShenzhen City zone of China, shown in Fig. 12(a), was acquiredon December 31, 2001. This product was georeferenced with aspatial resolution of 30 m. The IKONOS data over the sameregion, with a 4-m spatial resolution, shown in Fig. 12(b), wereacquired on December 20, 2001 [55], and were used to obtaina high-resolution classification image as the evaluated standardof the subpixel mapping results. Although an 11-day differenceexists between the two images, both of them were acquired inthe same season; hence, their reflectance spectra are assumedto be similar. In such a case, the spectral signature may beobtained through inverse spectral mixture analysis to deriveendmember signatures or training pixels, given the fractionalcovers that can be obtained from the IKONOS image. As aresult, we first performed a geometric correction on the imageso that each corrected pixel of the output image has the sameUniversal Transverse Mercator coordinate as the ETM data.The ETM spatial resolution was resampled to 32 m, which

is eight times the IKONOS data (4 m). Three steps wereincluded to calculate the fractional land cover for each ETMpixel. First, the IKONOS image was classified into four classes(city, vegetation, bare soil, and water) using an SVM with fieldsurveying, assisted by interpretation, as the shade reflectionintertwines with water. The classification image is shown inFig. 12(c). Recent studies have found that the typical scaleof urban reflectance is between 10 and 20 m [56]. Therefore,most 4 × 4 m IKONOS pixels are spectrally homogeneousand can be reasonably assumed to be pure materials. Second,the classified IKONOS image was registered with the ETMimages using the ground control points with a total root meansquare error of less than 0.03. Third, the fraction images ofthe ETM image were obtained by the least squares linearspectral mixture analysis method [3], using ENVI remotesensing image processing software [54]. The fraction imagesare shown in Fig. 12(d). Fourth, the subpixel mapping re-sults using different methods—DNSM, SASM, BPSM, GASM,and ADESM—were acquired, based on the fraction imagewith scale = 8. The subpixel mapping results are shown inFig. 12(e)–(i). Finally, the classification map of the IKONOSimage was overlaid on the subpixel mapping results to estimatethe performance of the different subpixel mapping algorithmsat the IKONOS data scale. To ensure a clear comparison ofADESM with the other algorithms, two small images of S1 andS2 areas are zoomed and are shown in Fig. 12(j). In the subpixelmapping process, the number of iterations for ADESM andGASM was set to 100. The main parameters of BPSM were setas follows: number of hidden layers = 1, learning rate =0.25, momentum rate = 0.9, and number of iterations =1000. In addition, BPSM needs training samples to train the BPneural network. In this experiment, approximately 50 samplesfor each class were selected randomly as the training samplesfor BPSM.

The hard classification result of the IKONOS image, asshown in Fig. 12(c), as the true result, was used to test theperformance of the different subpixel mapping algorithms. Asshown in Fig. 12, when comparing the subpixel mapping resultsof DNSM, SASM, BPSM, GASM, ADESM, and the high-resolution IKONOS classification image, DNSM and BPSMdid not provide satisfactory subpixel mapping results. For ex-ample, the structural information of the water class has been loston the right of the images, and the structure of the playground isvery fuzzy, as shown in Fig. 12(j), S1. The whole visual resultof GASM is fuzzy, particularly for the vegetation and baresoil classes. Although SASM showed better visual results thanDNSM, GASM, and BPSM and provided a more integratedsubpixel mapping result, the continuity of the vegetation classwas inadequate on the left of the results, as shown in Fig. 12(j),S2, in which many vegetation subpixels were cut. By contrast,ADESM had better visual results and provided more integratedand continuous subpixel mapping results, e.g., for the vegeta-tion and water classes.

To quantitatively evaluate the subpixel mapping accuracy,the adjusted confusion matrices and global performance indiceswere obtained by applying DNSM, SASM, BPSM, GASM,and ADESM to this data set (see Tables VII and VIII). It isworth noting that BPSM did not map the boundary of the image


TABLE VADJUSTED CONFUSION MATRICES OF THE SUBPIXEL MAPPING RESULTS DERIVED FROM THE HANCHUAN HJ-1A IMAGE IN EXPERIMENT 2

TABLE VICOMPARISON OF FIVE SUBPIXEL MAPPING ALGORITHM PERFORMANCES FOR THE HANCHUAN HJ-1A IMAGE IN EXPERIMENT 2

in the subpixel layer, for example, the black lines shown inFig. 12(g). The subpixels in the boundary were not considered,to test the performance of the BPSM subpixel mapping result.The number of real subpixels for different classes in the BPSMimage is less than that in the subpixel mapping images of theother subpixel mapping algorithms. BPSM obtained the highestsubpixel mapping producer’s accuracies of the vegetation andwater classes, i.e., 22.28% and 21.57%, respectively. However,it was difficult for BPSM to differentiate the city class fromthe other classes. In addition, one can see from Table VII that,for the vegetation class, the producer’s accuracies of DNSM

and SASM are equal to 21.66% and 21.85%, respectively, andbetter than that of ADESM. However, this class is significantlyconfused with city and water, i.e., only 26.99% and 27.17%,respectively, of the vegetation subpixels are subpixel mappedcorrectly. By contrast, ADESM recognized the vegetation classbetter than the aforementioned algorithms, resulting in theuser’s accuracy of 33.83%. Moreover, ADESM had the highestsubpixel accuracy for the city class, with the producer’s anduser’s accuracies of 76.71% and 64.19%, respectively.

In Table VIII, we can observe that, in general, ADESMobtained the highest adjusted OA (OA∗) and adjusted Kappa


Fig. 12. Subpixel mapping results for the Shenzhen TM image in experiment 3 (scale = 8). (a) Shenzhen ETM image RGB (4, 3, 2) (98 × 30). (b) ShenzhenIKONOS image (784 × 240). (c) Shenzhen IKONOS classification image. (d) Fraction images of ETM image (98 × 30). (e) DNSM. (f) SASM. (g) BPSM.(h)GASM. (i) ADESM. (j) Zoom images for all the subpixel mapping results.

(Kappa∗) value, with gains of 2.97%, 2.81%, and 2.08% overDNSM, SASM, and BPSM for OA∗, respectively. It is worthnoting that BPSM provides a higher subpixel mapping resultthan DNSM and SASM do for difficult real remote sensingimages. Comparing ADESM with GASM, the two algorithmshave similar adjusted OA, i.e., 53.89% and 54.21%, but thevisual result of GASM is poorer than ADESM and the subpixelmapping result of GASM is fuzzy, as shown in Fig. 12(h) and(j). In addition, we can find that the subpixel accuracies usingthe real data are lower than those for the synthetic data forall algorithms. The possible reasons for this are as follows:1) Georeferenced and geometric corrected errors may exist;2) the spectral unmixing error of the ETM image and theclassification error of the IKONOS image may influence the

evaluated accuracies; 3) there are radiometric correction andreflectance retrieval between IKONOS and ETM; and 4) thedifference among homogeneous spectra is large. It can thereforebe stated that the quality of the input soft classification orspectral unmixing result is a key factor to the success of thesubpixel mapping. These results are also confirmed by theother quality indices considered, thus pointing to the superiorityof ADESM over the other four algorithms for this data set,particularly the data set of the real image.

VI. SENSITIVITY ANALYSIS OF ADESM

The scale of subpixel mapping is an important parameter forADESM. In Section V, we only analyzed the subpixel mapping


TABLE VIIADJUSTED CONFUSION MATRICES OF THE SUBPIXEL MAPPING RESULTS FOR THE SHENZHEN ETM IMAGE IN EXPERIMENT 3

TABLE VIIICOMPARISON OF FIVE SUBPIXEL MAPPING ALGORITHM PERFORMANCES FOR THE SHENZHEN ETM IMAGE IN EXPERIMENT 3

result when the scale was equal to four. In this section, thesensitivity analysis between scale and five subpixel mappingalgorithms (DNSM, SASM, BPSM, GASM, and ADESM) wascarried out to evaluate the performance of these algorithms.In ADESM, we added exchange, insertion, and local searchoperators to improve the subpixel mapping algorithm. To in-vestigate the effectiveness of these operators, the sensitivityanalysis between the improved operators and ADESM wasundertaken. In addition, we analyzed the effectiveness of theself-adaptive strategy for ADESM. The proof of convergence ofEAs with self-adaptation is difficult because control parameters

are changed randomly and the selection does not affect theirevolution directly [40], [57], [58]. Since DE is a particular in-stance of EA, it is interesting to investigate how self-adaptivitycan be applied to it [59]. In this section, to test the validity of theself-adaptive strategy in ADESM, we analyzed the sensitivityof three control parameters for the original DE subpixel map-ping (DESM) algorithm: amplification factor of the differencevector F , crossover control parameter CR, and population sizeNP . In this section, the synthetic artificial image and twosynthetic real images were tested using different parametervalues.


Fig. 13. Sensitivity of subpixel mapping algorithms in relation to s. (a) Artificial image. (b) Wuhan TM image. (c) Hanchuan HJ-1A image.

Fig. 14. Sensitivity of DESM in relation to improved operators. (a) Artificial image. (b) Wuhan TM image. (c) Hanchuan HJ-1A image.

A. Sensitivity in Relation to Parameter s

To study the ADESM sensitivity in relation to scale s, theother parameters were kept the same as those in experiments 1and 2. The values of the scale were assumed as follows: s ={2, 4, 8, 16}.

As shown in Fig. 13, the higher the s value, the lower theadjusted OA (OA∗). Overall, the values of OA∗, for all thesubpixel mapping algorithms proposed in this paper, take onthe tendency of descension. This is because the distribution ofland-cover classes is more complex when the scale s increases,resulting in an increase in the difficulty of subpixel mapping.Although the OA∗ of the artificial image and two synthetic im-ages decreases as the scale increases, those of the subpixel map-ping images, overall, are superior than those of the degradedimages and provide more subpixel information. Compared withthe other subpixel mapping algorithms, ADESM is a morestable subpixel mapping algorithm, which can obtain bettersubpixel mapping results than DNSM, GASM, and BPSM.When the scale s is equal to two and four, ADESM has a higheraccuracy than SASM. For scales 8 and 16, ADESM and SASMhave similar subpixel mapping accuracy.

B. Sensitivity in Relation to Improved Operators

To investigate the effectiveness of the improved operators,including exchange, insertion, and local search operators, weremoved these operators of ADESM one by one. The otherparameters were kept the same as those in experiments 1 and 2.

To denote these algorithms, the following notations are used.ADESM-Exchange represents the ADESM algorithm withoutexchange operator, ADESM-Insertion is the ADESM algorithmwithout insertion operator, ADESM-Exchange + Insertion rep-resents the ADESM without exchange and insertion operators,and ADESM-Localsearch describes the ADESM algorithmwithout local search operators. In addition, to better evaluatethe performance of ADESM, the subpixel mapping algorithmsbased on other adaptive-DE algorithms, for example, JADE (anadaptive differential evolution proposed by J. Zhang and A. C.Sanderson [38]), namely, the subpixel mapping algorithm basedon JADE (JADESM), were used to compare with ADESM.

As shown in Fig. 14, JADESM obtained the best OA∗ values,i.e., 90.72% and 67.23%, for experiments 1 and 2, respectively,and ADESM-Localsearch obtained the best OA∗, i.e., 74.14%,for experiment 3. It was noted that the original JADE also can-not be used in the discrete space. Thus, the discrete-encodingand reversible-conversion strategies of ADESM were utilized inJADE for subpixel mapping. Although JADESM and ADESM-Localsearch slightly increased the subpixel mapping accuracy,their computational costs are too much, being approximatelyfive to six times greater than ADESM. That is, the localsearch operator of ADESM can decrease the computationalcost and find a better feasible solution in the local neighbor-hood. Compared with ADESM-Exchange, ADESM-Insertion,and ADESM-Exchange + Insertion, ADESM is a more stablesubpixel mapping algorithm, which gave appropriate subpixelmapping results, with satisfactory computation times of 80.32,386.85, and 163.85 s for the three images.


Fig. 15. Sensitivity of DESM in relation to NP . (a) Artificial image. (b) Wuhan TM image. (c) Hanchuan HJ-1A image.

Fig. 16. Sensitivity of DESM in relation to CR. (a) Artificial image. (b) Wuhan TM image. (c) Hanchuan HJ-1A image.

C. Sensitivity in Relation to Parameter NP

To compare our self-adaptive version of the DESM algo-rithm, i.e., ADESM, with the original DESM algorithm, thebest control parameter setting for DESM may be needed. TheDESM algorithm does not change the control parameter valuesduring the subpixel mapping process, except for the analyzedparameter. The experiential parameters of the original DESMalgorithm were set as follows: CR = 0.9 and F = 0.5, in [24]and [45].

The number of the initial population NP is very importantin maintaining the diversity of the population and extendingthe search range in the feature space. To analyze the sensi-tivity in relation to parameter NP , the other parameters didnot change, and NP assumed the following values for threeexperimental images: NP = {10, 20, 30, 40, 50, 60, 70, 80, 90,100, 110, 120, 130, 140, 150, 160}. Fig. 15 shows the sensitiv-ity of DESM in relation to the parameter NP by comparing itwith the adjusted OA provided by the DESM algorithms.

The value of NP is of importance to the DESM algorithmin maintaining the diversity of the population, because NPdetermines the population size used to extend the search spaceand gives the mutation a good chance to proceed toward thesolution. As shown in Fig. 15, there was an upward trend inthe adjusted OA (OA∗) of the subpixel mapping algorithmbased on the original DE algorithm (DESM) when the valueof NP was changed from 10 to 160. When NP was equalto 120, 140, and 160, for the three experimental images,the highest OA∗ values of DESM are 90.73%, 66.93%, and73.74%, respectively. It was noted that, the higher the NPvalue, the greater the computational cost of DESM, because

the population size is increased. When the NP is equal to 160,DESM has computational costs of 92.37, 444.88, and 188.42 sfor the artificial image, Wuhan TM image, and Hanchuan HJ-1A image, respectively. Compared with DESM, ADESM, withNP = 160, has computational costs of 80.32, 386.85, and163.85 s for the three experiments. In a comparison of the OA∗

value with ADESM, ADESM obtained higher OA∗ values, i.e.,67.06%, and 74.04%, for the two synthetic real images.

D. Sensitivity in Relation to Parameter CR

For all three experimental images, the DESM algorithm wasperformed with CR taken from [0.1, 1.0] by steps of 0.1, whilethe other parameters were set as follows: NP = 100, F = 0.5,and s = 4. First, we set the control parameter CR = 0.1 andkept it fixed during 30 independent runs. Then, we set CR =0.2 for the next 30 runs, etc. The results were averaged over 30independent runs. The experimental results for different imagesare shown in Fig. 16.

The best adjusted OA (OA∗) values of DESM for the arti-ficial image and Wuhan TM image, i.e., 90.74% and 67.06%,respectively, were obtained by CR = 0.4 (NP = 100, F =0.5, and s = 4). The values are slightly higher than or equal tothose of ADESM, i.e., 90.60% and 67.06%. For the HanchuanHJ-1A image, ADESM obtained a better OA∗ value, i.e.,74.04%, than the best OA∗ value of DESM, i.e., 73.97%, whenCR was equal to 0.5. Although DESM can obtain satisfactoryresults by adjusting the value of parameter CR, ADESM mayadaptively provide similar or better subpixel mapping results,without prior knowledge or experience.


Fig. 17. Sensitivity of DESM in relation to F . (a) Artificial image. (b) Wuhan TM image. (c) Hanchuan HJ-1A image.

E. Sensitivity in Relation to Parameter F

According to (7), F is a key parameter for the DESMalgorithm. Hence, we further investigated the effect of F onsolution quality. In accordance with the conclusion from Stornand Price’s results, where F is in the range of 0.4–0.9 andCR is in the range of 0.5–0.9 [24], we set F from 0.1 to 1.0with steps equal to 0.1 and fixed the other parameters withNP = 100 and CR = 0.9. Similar computational experimentswere then conducted. The experimental results are presented forthree different images in Fig. 17.

From Fig. 17, it follows that, as F increases, the adjusted OA(OA∗) produced by the DESM algorithm decreases. The bestadjusted overall accuracies of DESM, i.e., 90.72%, 66.88%,and 73.93%, for the three different images, were obtained whenF was equal to 0.1. As shown in Fig. 17; although DESM mayobtain higher OA∗ than ADESM for a synthetic artificial image,ADESM obtained better results than the DESM algorithm anddid not need any other prior knowledge for the two syntheticreal images.

Based on the aforementioned sensitivity analysis, there aretwo disadvantages in the original DESM. First, parametertuning requires multiple runs, and it is usually not a feasiblesolution for problems that are very time consuming. Second,the best control parameter settings of DESM are problemdependent. The proposed ADESM overcomes these disadvan-tages, so there is no need for multiple runs to adjust the con-trol parameters; moreover, self-adaptive DESM is much moreproblem independent than the original DESM is. Therefore,we conclude that ADESM is a robust and effective subpixelmapping algorithm.

VII. CONCLUSION

Based on the DE theory, this paper has proposed a newmapping strategy for the subpixel mapping of remote sensingimages. In line with this strategy, the subpixel mapping prob-lem is transformed to an optimization problem in the discretesubpixels’ space by maximizing the SDI. The traditional DEalgorithm only obtains the optimal solution in the continuousspace, it does not process the subpixel mapping problem inthe discrete space. In addition, DE needs to choose the propercontrol parameters, employing prior experience of the users,population size NP , crossover rate CR, and scaling factor F .

This is quite a difficult task because the best settings for the con-trol parameters are not easy to determine for complex problems.In this paper, to apply DE to the subpixel mapping problemeffectively, an improved DE approach, the ADESM algorithm,has been proposed. The proposed method is a combination ofa discrete binary DE approach with an adaptive strategy, whichis enhanced by advanced evolution operators and local search.It utilizes real transformation and integer transformation toencode candidate subpixel mapping solutions from the discretespace. During the subpixel mapping process, the enhancedDE operators—repair, exchange, and insertion—are added toimprove the subpixel mapping results, along with the traditionalevolution operators—mutation and crossover. The local searchstrategy, as an improvement strategy, is also used to find morefeasible solutions in the local neighborhood. In addition, theproposed self-adaptive method is an attempt to determine theoptimum values of control parameters F and CR.

Our self-adaptive DESM algorithm (ADESM) has been im-plemented and tested on subpixel mapping problems usingdifferent remote sensing images taken from the literature. Theexperimental results in this paper consistently show that theADESM algorithm, with its self-adaptive control parametersettings, provides better results than the traditional DNSM andSASM algorithms, as well as the latest BP neural networksubpixel mapping algorithm from the literature. In the threeexperiments using different images (a synthetic artificial im-age, two synthetic remote sensing images, and a real remotesensing image), ADESM was able to achieve better results,with a higher subpixel mapping accuracy. This evinces thatADESM is appropriate for the subpixel mapping of remotesensing images and is an improved alternative subpixel map-ping algorithm. In the comparison of the subpixel mappingaccuracies between synthetic images and real images, ADESMaccomplishes satisfactory subpixel mapping results for threesynthetic images, with an accuracy of more than 67%. Withthe real data of ETM and IKONOS, the proposed method alsoachieves a higher accuracy than the other algorithms, but theaccuracy is lower than that for the synthetic images becausethere are geometric correction, classification, or spectral unmix-ing errors. The sensitivity analysis of the scale demonstratesthat the subpixel mapping accuracy decreases when the valueof the scale is increased from 2 to 16. To evaluate the va-lidity of the self-adaptive strategy in ADESM, we analyzedthe sensitivity in relation to three parameters, namely, NP ,


CR, and F , by using different user-defined values. The resultsshow that our algorithm, with its self-adaptive control para-meter settings, is better than, or at least comparable to, thestandard DESM algorithms. However, our self-adaptive methoddecreases the user-defined process and can be simply used tosolve the subpixel mapping problem with a satisfactory level ofaccuracy.

Although we tried to apply ADESM to a real remote sensingimage, and obtained better subpixel mapping results than tradi-tional algorithms do, it is still difficult to process a real remotesensing image with many errors, e.g., spectral unmixing errors.Our future work will explore further methodologies and aimsto find a significantly better data set to improve the subpixelmapping accuracy for real applications, thereby investigatingthe adaptability of the SDI for a complex real image (e.g.,hyperspectral image [60]).

ACKNOWLEDGMENT

The authors would like to thank the Editor-in-Chief, theAssociate Editor, and the three anonymous reviewers for thehelpful comments and suggestions that improved this pa-per. Dr. Y. Zhong would also like to thank Prof. K. Tangof the University of Science and Technology of China andDr. A. M. Zhou of East China of Normal University for theirhelpful discussions and constructive suggestions.

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Yanfei Zhong (M’11) received the B.S. degree ininformation engineering and the Ph.D. degree inphotogrammetry and remote sensing from WuhanUniversity, Wuhan, China, in 2002 and 2007,respectively.

Since 2007, he has been with the State Key Lab-oratory of Information Engineering in Surveying,Mapping and Remote Sensing, Wuhan University,where he is currently a Professor. He was a Refereeof Pattern Recognition. He has published more thanten peer-reviewed articles in international journals,

such as IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING andInternational Journal of Remote Sensing. His research interests include multi-and hyperspectral remote sensing image processing, artificial intelligence, andpattern recognition.

Dr. Zhong was the recipient of the National Excellent Doctoral DissertationAward of China (in 2009) and the New Century Excellent Talents in Universityof China (in 2009). He was a Referee of IEEE TRANSACTIONS ON SYSTEMS,MAN AND CYBERNETICS—PART B and IEEE JOURNAL OF SELECTED

TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING.

Liangpei Zhang (M’06–SM’08) received the B.S.degree in physics from Hunan Normal University,Changsha, China, in 1982, the M.S. degree in opticsfrom Xi’an Institute of Optics and Precision Mechan-ics, Chinese Academy of Sciences, Xi’an, China,in 1988, and the Ph.D. degree in photogrammetryand remote sensing from Wuhan University, Wuhan,China, in 1998.

He is currently with the State Key Laboratoryof Information Engineering in Surveying, Mappingand Remote Sensing, Wuhan University, as the Head

of the Remote Sensing Division. He is also a “Chang-Jiang Scholar” ChairProfessor appointed by the Ministry of Education, China. He is currently thePrincipal Scientist for the China State Key Basic Research Project (2011–2016)appointed by the Ministry of National Science and Technology of Chinato lead the remote sensing program in China. He is an Executive Member(Board of Governor) of the Chinese National Committee of InternationalGeosphere–Biosphere Programme. He also serves as an Associate Editor ofInternational Journal of Ambient Computing and Intelligence, InternationalJournal of Image and Graphics, International Journal of Digital MultimediaBroadcasting, Journal of Geo-spatial Information Science, and Journal ofRemote Sensing. He has more than 230 research papers and is the holder of5 patents. His research interests include hyperspectral remote sensing, high-resolution remote sensing, image processing, and artificial intelligence.

Dr. Zhang is a fellow of the Institution of Electrical Engineers, an executivemember of China Society of Image and Graphics, and others. He regularlyserves as a Cochair of the series SPIE Conferences on Multispectral ImageProcessing and Pattern Recognition, Conference on Asia Remote Sensing, andmany other conferences. He edits several conference proceedings, issues, andthe Geoinformatics Symposiums.

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