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A Framework for Decision Fusion in Image Forensics Based on Dempster-Shafer Theory of Evidence

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  IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 8, NO. 4, APRIL 2013 593 A Framework for Decision Fusion in Image ForensicsBased on Dempster–Shafer Theory of Evidence Marco Fontani  , Student Member, IEEE  , Tiziano Bianchi  , Member, IEEE  , Alessia De Rosa,Alessandro Piva  , Senior Member, IEEE  , and Mauro Barni  , Fellow, IEEE   Abstract—  In this work, we present a decision fusion strategy forimage forensics. We de fi ne a framework that exploits informationprovided by available forensic tools to yield a global judgmentabout the authenticity of an image. Sources of information aremodeled and fused using Dempster–Shafer Theory of Evidence,since this theory allows us to handle uncertain answers from toolsand lack of knowledge about prior probabilities better than theclassical Bayesian approach. The proposed framework permitsus to exploit any available information about tools reliability andabout the compatibility between the traces the forensic tools look for. The framework is easily extendable: new tools can be addedincrementally with a little effort. Comparison with logical disjunc-tion- and SVM-based fusion approaches shows an improvementin classi fi cation accuracy, particularly when strong generalizationcapabilities are needed.  Index Terms—  Decision fusion, Dempster–Shafer, forgery detec-tion, image forensics, image integrity, image tampering. I. I  NTRODUCTION I MAGES have always played a key role in the transmissionof infor mation, mainly because of their presumed objec-tivity. However, in the last years the advent of digital imaginghas given a great impulse to image manipulation, and nowadaysimages are facing a thrust crisis. Image Forensics, whose goalis to investigate the history of an image using passive ( blind  )approaches, has emerged as a possible way to solve the abovecrisis.The basic idea underlying Image Forensics is that most, if not all, image processing tools leave some (usually impercep-tible) traces into the processed image, and hence the presence of these traces can be investigated in order to understand whether  Manuscript received June 04, 2012; revised January 04, 2013; acceptedFebruary 11, 2013. Date of publication February 25, 2013; date of currentversion March 07, 2013. This work was supported in part by the REWINDProject, funded by the Future and Emerging Technologies (FET) programmewithin the 7FP of the EC under Grant 268478, and by the European Of  fi ceof Aerospace Research and Development under Grant FA8655-12-1-2138:AMULET-A multi-clue approach to image forensics. The associate editor coordinating the review of this manuscript and approving it for publication wasDr. Siome Klein Goldenstein.M.FontaniandM.BarniarewiththeDepartmentofInformationEngineeringand Mathematical Sciences, University of Siena, 53100, Siena, Italy (e-mail:marco.fontani@unisi.it; barni@dii.unisi.it).T. Bianchi is with the Department of Electronics and Telecommunications,Politecnico di Torino, I-10129, Torino, Italy (e-mail: tiziano.bianchi@polito.it).A. De Rosa is with the National Inter-University Consortium for Telecom-munications, University of Florence, 50139, Florence, Italy (e-mail: alessia.derosa@uni fi .it).A. Piva is with the Department of Information Engineering, University of Florence, 50139, Florence, Italy (e-mail: alessandro.piva@uni fi .it).Color versions of one or more of the  fi gures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identi fi er 10.1109/TIFS.2013.2248727 the image has undergone some kind of processing or not. In thelastyearsmanyalgorithmsfordetectingdifferentkindsoftraceshave been proposed (see [1] for an overview) which usually ex-tract a set of features from the image and use them to classifythe content as exposing the trace or not. Very often, the creationof a forgery involves the application of more than a single pro-cessing tool, thus leaving a number of traces that can be usedto detect the presence of tampering; this consideration suggeststo analyze the authenticity of images by using more than onetamper detection tool. Furthermore, existing forensic tools arefar from ideal and often give uncertain or even wrong answers,so, whenever possible, it may be wise to employ more than onetool searching for the same trace. On top of that, it may also bethe case that the presence of one trace inherently implies the ab-sence of another, because the traces are mutually exclusive byde fi nition. For these reasons, taking a  fi nal decision about theauthenticityofanimagerelyingontheoutputofasetofforensictools is not a trivial task, thus justifying the design of proper de-cisionfusionmethodsexplicitlythoughtforthisscenario.Giventhat new forensic tools are developed continuously, we wouldlike our decision fusion method to be easily extendable, so thatnewtoolscanbeincludedassoonastheybecomeavailable.An-other key issue regards the creation of training datasets for thefusionstage:whileproducingdatasetsfortrainingsingletoolsisarather simpletask,creating datasetsrepresenting thevarietyof  possible combinations of traces that could be introduced duringthe creation of a realistic forgery is extremely challenging.As an answer to the above needs, we propose a decisionfusion framework for the image forensics scenario based onDempster–Shafer Theory of Evidence (DST); the proposedmodel is easily extendable and, as a key contribution, allowsincremental addition of knowledge when new tools becomeavailable. With respect to more classical approaches to in-ference reasoning, the use of DST avoids the necessity of assigning prior probabilities (that would be extremely dif  fi -cult to estimate) and also provides more intuitive tools for managing the uncertain knowledge provided by the forensictools. This paper extends a previous work by Fontani  et al. [2] both from a theoretical and an experimental point of view.The most signi fi cant novelty is that tools and searched tracesare modeled in a more  fl exible way, speci fi cally, a mechanismfor hierarchical fusion of traces is introduced, leading to akey improvement of framework extendability. Moreover, thenumber of implemented tools has been raised to  fi ve and testshave been performed also over a realistic (handmade) forgerydataset. Differences with respect to the previous work will behighlighted when necessary. 1556-6013/$31.00 © 2013 IEEE  594 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 8, NO. 4, APRIL 2013 The rest of the paper is organized as follows. In the nextSection I-A,we brie fl y introduce the problem of decision fusionin an image forensics scenario, then we give some basic notionof DST (Section II) and describe in detail the proposed frame-work (Section III). In Section IV, we present experimentalresults regarding a scenario in which the outputs of   fi ve imageforensic tools ([3]–[7]) are fused to give a global judgementabout image authenticity. The results show a clear performanceimprovement with respect to more classical decision fusionstrategies when realistic forgeries are examined.  A. Decision Fusion in the Image Forensics Scenario The problem of taking a  fi nal decision about an hypothesis by looking at the output of several different tools is an impor-tant task in decision fusion; there are basically three kinds of approaches to tackle with it. The  fi rst is to perform fusion at the  feature  level: a subset of the features extracted by the tools isselected and used to train a global classi fi er. The second is toconsider the (usually scalar) output provided by the tools andfuse them (fusion at the  measurement  , or   score , level). The lastapproach consists in fusing the binary answers of the tools, usu-ally obtained by binarizing their soft outputs (fusion at the  ab- stract   level). An effective example of how these three strate-gies can be applied to a problem similar to the one addressedin this paper is illustrated in [8], where fusion is used cast in asteganalysis framework. In fact, both in steganalysis and imageforensics, tools usually extract some features from the image, perform measurements/classi fi cation on them and  fi nally pro-duce an output, often probabilistic, which can be thresholded toyield a binary classi fi cation.Although being promising in terms of performance, fusionat the feature level has some serious drawbacks, most impor-tantly the dif  fi culty of handling cases involving a large number of features (commonly addressed as “curse of dimensionality”)and the dif  fi culty to de fi ne a general approach to feature selec-tion, since ad-hoc solutions are needed for different cases. Fur-thermore, feature selection in most cases is followed by somemachine learning, that by de fi nition is effective only when atraining dataset can be prepared that is representative of a large part of the global population of samples. If this can be done for training a single detector, creating a representative dataset of all possible image forgeries is practically unfeasible, especially inthe case of photorealistic ones.Working at the other extreme, the abstract level, suffers fromthe complementary problem: lots of information is discardedwhen outputs are thresholded, so the discrimination power of the various tools is not fully exploited. In image forensics, mostoftheexistingworksarebasedonthe fi rstapproach[9]–[11];anhybrid solution has been investigated in [12], but still focusingon feature fusion.In order to get around the above problems, we choose to per-form fusion at the measurement level. This choice delegatesthe responsibility of selecting features and training classi fi ers(or other decision methods) to each single tool, thus keepingthe fusion framework more general and easy to extend, whileavoiding the loss of important information about tool responsecon fi dences. Speci fi cally, we present a fusion framework basedonDempster–Shafer’s“Theoryofevidence”(DST)[13]thatfo-cuses exclusively on fusion at the measurement level. The pro- posed framework exploits knowledge about reliability of toolsand about compatibility between different traces of tampering,and can be easily extended when new tools become available.It allows both a “soft” and a binary (tampered/nontampered) in-terpretation of the fusion result, and can help in analyzing im-ages for which taking a decision is critical due to con fl ictingdata. Note that a fusion approach involving DS Theory has al-ready been proposed in [14], but such a scheme applies fusionat the feature level hence inheriting the general drawbacks of feature-level fusion, noticeably the lack of scalability and theneed to retrain the whole system each time a new tool is added.Also, in [15] the authors exploit Dempster’s combination rule,which provides only a limited part of the expressive capabilityof the DST framework, to devise an image steganalysis schemethat combines three algorithms to improve detection accuracy;however, our goal is deeply different from that pursued in [15],since we do not aim at providing a speci fi c multiclue forgerydetection tool, but at de fi ning a theoretical model that allowsfusing a generic set of tools targeting splicing detection. As wewill see later in the paper, the combination rule by itself is notsuf  fi cient to address our problem, since we must deal with het-erogeneous and evolving sources of information.II. D EMPSTER   –S HAFER  ’ S  T HEORY OF  E VIDENCE Dempster–Shafer’s theory of evidence was  fi rstly introduced by A. Dempster [16] and further developed by G. Shafer [13]. Itcan be regarded as an extension of the classical Bayesian theorythat allows representation of ignorance and of available infor-mation in a more fl exible way. When using classical probabilitytheory for de fi ning the probability of a certain event , the ad-ditivity rule must be satis fi ed; so by saying thatone implicitly says that , thus committing the probability of an event to that of its complementary . Mostimportantly, the additivity rule in fl uences also the representa-tion of ignorance: complete ignorance about a dichotomic eventin Bayesian theory is best represented by setting(according to the maximum entropy principle), but this probability distribution also models perfect knowledgeabout the probability of each event being 0.5 (as for a cointossing), thus making it dif  fi cult to distinguish between igno-ranceandperfectlyknownequiprobableevents.Sincereasoningin a Bayesian framework makes an extensive use of prior prob-abilities, which are often unknown, a wide usage of maximumentropy assignments is often unavoidable, leading to the intro-duction of extraneous assumptions. To avoid that, DS theoryabandons the classical probability frame and allows to reasonwithout  a priori  probabilities through a new formalism.  A. Shafer’s Formalism Let the frame de fi ne a  fi nite set of  possible values of a variable ; a proposition about variableis any subset of . We are interested in quantifying how muchwe are con fi dent in propositions of the form “the true value of is in ”, where (notice that the set of all possible propositions is the power set of , ). To give an example,  FONTANI  et al. : FRAMEWORK FOR DECISION FUSION IN IMAGE FORENSICS BASED ON DST 595 let us think of a patient that can either be affected by cancer or not: we can model this scenario de fi ning a variable withframe where is the proposition “patient isaffectedbycancer”, istheproposition“patientisnotaffected by cancer”, and is the doubtful proposition “patient isor is not affected by cancer”. The link between propositions andsubsets of allows to map logical operations on propositionsinto operations among sets. Each proposition is mapped onto asingle subset and is assigned a basic belief   mass  through a BasicBelief Assignment, de fi ned over the frame of the variable.  De   fi nition 1:  Let be a frame. A functionis called a Basic Belief Assignment (BBA) over the frame if:(1)where the summation is taken over every possible subset of .Continuing the previous example, a doctor after examiningthe patient could provide information that lead us to write thefollowing basic belief assignment:(2)Each set such that is called a  focal element   for . In the following, we will omit the frame when it is clear from the context, writing instead of ; furthermore, whenwriting mass assignments only focal elements will be listed (sothelastrowof(2)wouldnotappear).BBAsaretheatomicinfor-mation in DST, much like probability of single events in proba- bility theory. By de fi nition, is the part of belief that sup- ports exactly but, due to lack of knowledge, does not supportany strict subset of , otherwise the mass would “move” intothe subsets. In the previous example, if we had assigned mass0.85 to proposition and 0.15 to it would havemeant that there is some evidence for the patient being affected by cancer but, basing on current knowledge, a great part of our con fi dencecannotbeassignedtononeofthetwospeci fi cpropo-sitions. Whenever we have enough information to assign all of the mass to singletons, 1 DST collapses to probability theory.Intuitively, if we want to obtain the total belief for a set , wemust add the mass of all proper subsets of plus the mass of itself, thus obtaining the  Belief    for the proposition .  De   fi nition 2:  Given the BBA in (1), the Belief functionis de fi ned as follows:summarizes all our reasons to believe in with theavailable knowledge. There are many relationships between, and other functions derived from these; herewe just highlight that andis the lack of information (or theamount of doubt) about . 1 A singleton is a set with exactly one element.  B. Combination Rule If we have two BBAs de fi ned over the same frame, whichhave been obtained from two independent sources of informa-tion, we can use Dempster’s  combination rule  to merge theminto a single one. Notice that the concept of independence be-tween sources in DST is not rigorously de fi ned (as it is, for ex-ample, in Bayesian theory): the intuition is that we require thatthe different pieces of evidence have been determined by dif-ferent ( independent  ) means [17].  De   fi nition 3:  Let and be belief functions over thesame frame with BBAs and . Let us also assume that, de fi ned below, is positive. Then for all nonemptythe function de fi ned as:(3)where , is a BBA functionde fi ned over and is called the  orthogonal sum  of and, denoted by .isa measureofthe  con   fl  ict   between and :thehigher the , the higher the con fl ict. The meaning of can be under-stood from its de fi nition, since is obtained by accumulatingtheproductofmassesassignedtosetshavingemptyintersection(which means incompatible propositions). Furthermore, we seethat Dempster’s combination rule treats con fl ict as a normaliza-tion factor, so its presence is no longer visible after fusion.Recall the example in Section II-A, and suppose that we ob-tain evidence coming from another doctor, who is not a cancer specialist, about the variable . Let us call the BBA in (2)and the new assignment; so we have:for for for for  Note that since the second doctor is not a specialist the informa-tion he provides is quite limited: most of the mass is assigned todoubt.FusingthetwopiecesofinformationaccordingtoDemp-ster’s rule results in:for for We see that after fusion values are not far from those alreadyassigned by : this is perfectly intuitive, since the seconddoctor did not bring a clear contribution to the diagnosis. No-tice also that for the same reason, and for the low con fi dence of  fi rst doctor about absence of cancer, little con fl ict is observed.Dempster’srulehasmanyproperties[18],inthisworkwearemainly interested in its associativity and commutativity, that is:(4)(5)Despite its many desirable properties, Dempster’s rule is notidempotent; this means that observing twice the same evidenceresults in stronger beliefs. This is the reason why we need to  596 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 8, NO. 4, APRIL 2013 introduce the hypothesis of independent sources in Dempster’scombination rule. In practice, before letting a new source of in-formation enter the system, we must always look at how thenew information is collected, to ensure that we are not countingtwice the same evidence. In our example, we must be sure thatdoctors did not talk with each other, did not use the same tech-nology when performing measurements, and so on.The combination rule expressed in (3) is applicable if the twoBBAs, and , are de fi ned over the same frame, whichmeans that they refer to the same propositions. Whenever weneed to combine BBAs de fi ned over different frames, we haveto rede fi ne them on the same target frame before the combina-tion. This can be done by using  marginalization  and  vacuousextension .  De   fi nition4:  Let beaBBAfunctionde fi nedoveraframe, and let be another frame. The vacuous extension of tothe product space , denoted with , is de fi ned as:if otherwiseThis allows to extend the frame of a BBA without introducingextraneous assumptions (no new information is provided about propositions that are not in ). That said, vacuous extension isnot the only possible way to extend a BBA to a larger frame: it just provides the “least informative” extension.The inver se operation of vacuous extension is marginaliza-tion.  De   fi nition 5:  Let beaBBA functionde fi ned on adomain, its marginalization to the frame , denoted with ,is de fi ned aswheretheindexofthesummationdenotesallsets whose projection on is .To outline the projection operator, let us introduce two product frames and , that are obtained as the cartesian product of the frames of some variables. Formally, we haveand , whereis the frame of the -th variable and is a subset of theindices in . Each element of will be a vector whose-th component is a value in . 2 The projection operator mapseach element into an element of by removingfrom all the components whose indices are not in . 3 The importance of extension and marginalization is that theyallow to combine over a common frame BBAs srcinally refer-ring to different frames, hence enabling us to fuse them withDempster’s rule.III. DST-B ASED  D ECISION  F USION IN  I MAGE  F ORENSICS By using the basic instruments of DST introduced in the pre-vious section, we developed a framework for combining evi-dence coming from two or more forgery detection algorithms.In particular we focus on the splicing detection problem, which 2 For instance, if one possible element of is ,where , and . 3 For example, if we project the set to theelement reduces to . consists in determining if a region of an image has been pastedfrom another. As already stated, during this process some tracesare left into the image, depending on the modality used to createthe forgery. The presence of each of these traces can be re-vealed by using one (or more) image forensic tools, each of which provides information about the presence of the trace it islooking for. Note that, in splicing  detection  tasks, most forensictools assume knowledge of the suspect region. That said, if no information is available, we could still run all tools in a block-wise fashion, and fuse their outputs at the block level. Aswe willhighlightinSectionV,forgery localization  isadifferent problem, that we will consider from the decision fusion point of view in future works.  A. Assumptions Our framework for decision fusion relies on the basic as-sumptions listed below:1) Each tool outputs a number in [0,1], where higher valuesindicate higher con fi dence about the analyzed region con-taining the searched trace;2) Compatibility relations among some or all of the consid-ered traces are known, at least theoretically (for instance,we may know that two tools search for mutually-exclusivetraces).3) Information about tools reliability, possibly image depen-dent, is available (for instance such an information couldderive from published results or from experimental evi-dence);4) Each tool gathers information independently of other tools(i.e. a tool is never employed as a subroutine of another,and no information is exchanged between tools), and bydifferent means (each tool relies on a different principle or effect);These assumptions are very reasonable in the current imageforensics scenario; nevertheless, some of them can be relaxedwith a limited impact on our framework. For example, inSection III-D we discuss how to handle the case where somerelationships between tracesarenot known,and in Section IV-Bwe show that errors in estimating tool reliabilities do not affectoverall performance signi fi cantly. Notice that assumption 4 is needed to ensure that we can fusetoolresponsesusingDempster’srule.Intuitively,itmeansthatif we observe two different tools supporting the same proposition,we are more con fi dent than observing only one. On the other hand, if two tools that search for the same trace exploiting thesame model are available, it makes sense to discard the lessreliable one, since its contribution will be limited or null. Thatsaid, and also considering that the concept of independence inDST is not equivalent to statistical independence, we believethat possible limited dependencies between algorithms wouldnot undermine the developed framework.  B. Formalization for the Single-Tool Case For sake of clarity, we start by formalizing the DST frame-work whenonly one toolis available,let uscallit ,whichreturns a value and has a reliability . Whilein our previous work [2] we directly modeled the output of thetool with a variable, here we propose a different point of view,
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