How To Define Fusion And Integration In Ategory Theory

Transcription

1 Fusion or Integration: What s the differene? Mark. Oxley Steven N. Thorsen Department of Mathematis and Statistis Department of Mathematis and Statistis Air Fore Institute of Tehnology Air Fore Institute of Tehnology 2950 Hobson Way 2950 Hobson Way Wright-Patterson AFB, Ohio Wright-Patterson AFB, Ohio U.S.A. U.S.A. mark.oxley@afit.edu steven.thorsen@afit.edu Abstrat The U. S. Air Fore 1 uses the term fusion in a very speifi manner. For example, the U.S. Air Fore Researh Laboratories have defined fusion on different objets, like sensors, data and lassifiers. Yet there is ambiguity in some instanes as to what is meant by it usage. Other Air Fore researh and aquisitions groups use the term integration to desribe the proess of ombining data, knowledge, ommand, ontrol, intelligene, surveillane, and reonnaissane. ven the U.S. Defense Advaned Researh Program Ageny (DARPA) has a program alled Integrated Sensing and Proessing (ISP) that aims to open the next paradigm for appliation of mathematis to the design and (o)operation of DoD sensor/exploitation systems and networks of suh systems. The program hopes to develop mathematial tools that enable the design and global optimization of systems that interatively ombine traditionally independent funtions of sensing, signal proessing, ommuniation, and exploitation. On the surfae it appears that integration is the same as fusion. In this paper, we define fusion and integration using the language of ategory theory. These definitions are in agreement with their usage in the Air Fore. Using ategory theory we show the differene (and similarities) between fusion and integration. Keywords: Fusion, integration, ategory theory, graph theory. 1 Introdution Several definitions have been given in the literature to apture the essene of fusion. Over the years these definitions have improved as we learned more about fusion proesses. Some definitions of fusion use integration to qualify the definition. Does this mean that integration and fusion are the same? Are fusion and integration synonyms? Integration, on the other hand, does not have several definitions floating around the literature. Yet the word is used profusely. Webster s Ditionary defines integrate as integrate v. tr. 1. To make into a whole by bringing all parts together; unify. integration n. 1. The at or proess of integrating. The use of the term integration in the ontext of fusion appears to be routine. That is, fusion is defined using integration. Why? The Journal of Information Sienes has a 1 The views expressed in this artile are those of the authors and do not reflet the offiial poliy or position of the United States Air Fore, Department of Defense, or the US Government. speial issue on web data integration [?] with a few papers disussing information integration[?]. The Defense Advaned Researh Program Ageny (DARPA) has a program alled Integrated Sensing and Proessing (ISP) where the goal is to open the next paradigm for appliation of mathematis to the design and (o)operation of DoD sensor/exploitation systems and networks of suh systems [?]. The program hopes to develop mathematial tools that enable the design and global optimization of systems that interatively ombine traditionally independent funtions of sensing, signal proessing, ommuniation, and exploitation. The U.S. DoD Command and Control (C2) nterprise Referene Arhiteture (1 Deember 2002) v defines: integration identifies funtional and data ommonalities among systems and eliminates redundany by aggregating these aspets into a redued number of modernized data systems in a shared environment. Bringing appliations into the shared environment an inlude some or all of the following; moving systems onto a ommon infrastruture; sharing data to provide a single logial authoritative soure, view and interpretation of data; sharing ode for ommon funtions that an redue ode maintenane osts aross separate appliations; and standardization of user interfaes to provide a ommon look and feel. This is a long and involved definition. The U.S. Air Fore Warfighter C2ISR Integration Page at has another definition. So, we ask the question does fusion and integration mean the same? If they do then we should NOT define fusion in terms of integration and vie versa. When writing a definition are must be taken so that the new term is defined using well understood and well-defined terms. 1.1 Motivation Our motivation for answering this question is two fold: (1) Based upon the similarities, researhers wishing to integrate an use existing fusion tehniques. Researhers may not be aware of the many fusion tehniques that would help

2 Report Doumentation Page Form Approved OMB No Publi reporting burden for the olletion of information is estimated to average 1 hour per response, inluding the time for reviewing instrutions, searhing existing data soures, gathering and maintaining the data needed, and ompleting and reviewing the olletion of information. Send omments regarding this burden estimate or any other aspet of this olletion of information, inluding suggestions for reduing this burden, to Washington Headquarters Servies, Diretorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subjet to a penalty for failing to omply with a olletion of information if it does not display a urrently valid OMB ontrol number. 1. RPORT DAT RPORT TYP 3. DATS COVRD to TITL AND SUBTITL Fusion or Integration: What?s the differene? 5a. CONTRACT NUMBR 5b. GRANT NUMBR 5. PROGRAM LMNT NUMBR 6. AUTHOR(S) 5d. PROJCT NUMBR 5e. TASK NUMBR 5f. WORK UNIT NUMBR 7. PRFORMING ORGANIZATION NAM(S) AND ADDRSS(S) Department of Mathematis and Statistis,Air Fore Institute of Tehnology,2950 Hobson Way,Wright-Patterson AFB,OH, PRFORMING ORGANIZATION RPORT NUMBR 9. SPONSORING/MONITORING AGNCY NAM(S) AND ADDRSS(S) 10. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATMNT Approved for publi release; distribution unlimited 13. SUPPLMNTARY NOTS 14. ABSTRACT 11. SPONSOR/MONITOR S RPORT NUMBR(S) 15. SUBJCT TRMS 16. SCURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. RPORT unlassified b. ABSTRACT unlassified. THIS PAG unlassified Same as Report (SAR) 18. NUMBR OF PAGS 6 19a. NAM OF RSPONSIBL PRSON Standard Form 298 (Rev. 8-98) Presribed by ANSI Std Z39-18

3 them. (2) Based upon the differenes, new fusion tehniques may have to be reated to handle integration problems. But, to do this we need to know what is integration? 1.2 Problem Statement Can the fusion and integration proesses be desribed in a unifying mathematial manner? We believe the branh of mathematis known as ategory theory [1, 2, 3], provides a way to desribe the proesses. One a proess desription is attained, the relationships that exist in fusion proesses may be explored using the algebrai theorems and properties of ategory theory. In this paper we will define fusion and integration using well-defined mathematial terminology. We will show the differenes and similarities of fusion and integration. 2 Bakground In this setion we give a short literature review of fusion and integration. Then a subsetion overing ategory theory follows. We do not wish the reader to think that we are being harsh in the following ritique of the definitions. We wish to onvene the sense that the definitions have improved over the years as the ommunity has learned more and understood more about fusion proesses. ven our definition of integration may need to hange as we understand more about integration proesses. In mathematis, one writes a definition using welldefined terms to avoid irularity. Although following these terms to their origins will result in terms that are defined using undefined terms, like set, element and is an element of. These are undefined, but assumed to be understood. Writing a definition one may use these terms (and terms derives from these, like a mapping) and irularity will be avoided. 2.1 Fusion The literature on fusion is vast in the ommunity of information fusion and sensor fusion. The definition of data fusion as defined by the Department of Defense of the United States of Ameria, Data Fusion Subpanel of the Tehnology Panel for C3 (ommand, ontrol, ommuniations) of the Joint Diretors of Laboratories (JDL) [?]as a multilevel, multifaeted proess dealing with the automati detetion, assoiation, orrelation, estimation, and ombination of data and information from single and multiple soures. This definition is more general than the previous one [?] with respet to the types of information than an be ombined. Other use of fusion has ourred in omputer siene ommunity where they speifi horizontal fusion and vertial fusion as well. 2.2 Category Theory It is apparent that using direted graphs will be very useful in developing a model of fusion. The branh of mathematis known as Category Theory quite naturally takes advantage of this. In fat, the basi definition of a ategory inludes a definition of a direted graph as well. Other useful elements will beome apparent later, but exploring the full power of ategory theory in order to produe a theory of fusion is part of the researh. In this setion, we have drawn upon various authors presentations to explain the basis of ategory theory [4, 5, 2, 3]. Definition 1 (Category) A ategory C onsists of the following: A1. A olletion of objets denoted Ob(C). A2. A olletion of maps denoted Ar(C). A3. Two mappings, alled Domain (dom) and Codomain (od), whih assign to an arrow f Ar(C) a domain and odomain from the elements of Ob(C). Thus, for arrow f, given by O 1 od(f) = O 2. f O 2, dom(f) = O 1 and A4. A mapping assigning eah objet O Ob(C) an unique arrow 1 O alled the identity arrow, suh that O 1O O and suh that for any existing element, x, of O, we have that x 1 O x. A5. A map,, alled omposition, A A A. Thus, given f, g A with od(f) = dom(g) there exists an unique h A suh that h = g f. Notie that Axioms A1 - A3 define a direted graph, where the objets are the nodes and the arrows are the direted edges of the graph. Axioms A3-A5 lead to the assoiative and identity rules: Assoiative Rule. Given appropriately defined arrows f, g, and h we have that (f g) h = f (g h). Identity Rule. Given arrows A f B and B g A, then there exists 1 A suh that 1 A g = g and f 1 A = f. Definition 2 (Subategory) A subategory B of A is a ategory whose objets are some of the objets of A and whose arrows are some of the arrows of A, suh that for eah arrow f in B, dom(f) and od(f) are in Ob(B), along with eah omposition of arrows, and an identity arrow for eah element of Ob(B). A ategory of interest is the ategory Set, whih has as objets sets and arrows all total funtions, with omposition of funtions as the omposition. Clearly this onstrut has identity arrows and the assoiative rule applies, so it is indeed a ategory. The subategories of interest to us are subategories of partiular types of data sets, denoted D, with objets similar types of data sets and arrows only

4 the identity arrows, and subategories of partiular types of feature sets, denoted F, with objets similar types of feature sets, and arrows only the identity arrows. The objets and arrows of these ategories shall orrespond to a partiular sensor system, so will represent all of the possible data (or feature) sets that an be generated by the sensorproessor system. For example, the data generated by a partiular sensor system may be 2x2 real-valued matries. In this ase, D = (R 2x2, id D, id D, ) represents the ategory with only the identities as arrows, and being the usual omposition of funtions. Another useful ategorial onstrut is a funtor. Definition 3 (Funtor) A funtor F between two ategories A and B is a pair of mappings F = (F Ob, F Ar ) suh that Ob(A) F Ob Ob(B) Ar(A) F Ar Ar(B) while preserving the assoiative property of the omposition map and preserving identity maps. Thus, given ategories A, B and funtor F : A B, if A Ob(A) and f, g, h, 1 A Ar(A) suh that f g = h is defined, then there exists B Ob(B) and f, g, h, 1 B Ar(B) suh that (i) F Ob (A) = B. (ii) F Ar (f) = f, F Ar (g) = g. (iii) h = F Ar (h) = F Ar (f g) = F Ar (f) F Ar (g) = f g. (iv) F Ar (1 A ) = 1 Ob(A) = 1 B. In general, if a funtor between two ategories of fusion an be developed or disovered, it ould possibly demonstrate an isomorphism between the two. In fat, if a fusion rule in the first ategory were optimal, and there existed an isomorphi funtor to the seond, then there might exist an optimal fusion rule in the seond fusion ategory as well (though not neessarily the image of the optimal rule under the funtor). Thus, if one fusion ategory were well understood and was isomorphi to a seond, the seond would be well understood. Finally, we need the definition of a natural transformation between funtors. Definition 4 (Natural Transformation) Given ategories A and B and funtors F and G with A F B and A G B, then a Natural Transformation is a family of arrows ν = {ν A A A} suh that for eah f Ar(A), A f A, A A, the square ommutes. We then say the arrows ν A, ν A are the omponents of ν : F G, and all ν the natural transformation of F to G. Definition 5 (Funtor Category A B ) Given ategories A and B, the notation A B denotes the ategory of all funtors F, B F A. This ategory has all suh funtors as objets and the natural transformations between them as arrows. Definition 6 (Produt Category) Let {C i } n i=1 be a finite olletion of ategories. Then n C i = C 1 C 2 C n i=1 is the orresponding produt ategory. 3 Main Results In this setion we give mathematial definitions of fusion and integration and demonstrate the similarities and differenes between the two. Let C be a subategory of the Set ategory throughout this setion. For n N we define the artesian produt C n to be C n = C C C }{{} n times whih an be shown to be another ategory. (See Adamek [2] for details.) In partiular, sine C = (Ob (C), Ar (C), Id(C), ) then C n = (Ob (C) n, Ar (C) n, Id(C) n, ). We wish to investigate artesian produts of C, but do not need to be speifi about the number of produts. Thus, we define the olletion of all artesian produts of C to be Cart(C) = {C n : n N}. Therefore, given ategory A Cart(C) there exists an n N suh that A = C n. 3.1 Fusion Definition 7 (Fusion Rule) Let {C i } N i=0 be subategories of a ategory A. Let n i=1 C i be a produt ategory. Then n i=1 a fusion rule is a funtor R C Ci 0. Thus, a fusion rule is a pair of mappings (R Ob, R Ar ). A fusor is a fusion rule of an event-deision proess whih performs by means of a funtional on its orresponding ROC urve better than any branh of the graph of the original proesses before applying a fusion rule. Definition 8 (Fusor, Fusion) Let C be a ategory. Let S be a olletion of fusion rules in N n i=1 n=1{c Ci 0 }. Let ϕ be a nonnegative real-valued funtional defined on S. Assume the optimization problem F(A) ν A G(A) max{ϕ(r) : R S} F(f) F(A ) ν A G(f) G(A ) has a maximizer R S. We say R is a fusor with respet to ϕ on S. The proess of determining a fusor is alled fusion.

5 If the olletion ontains only two fusion rules, say S = {R 1, R 2 }, and if ϕ(r 1 ) ϕ(r 2 ) then we an define a partial ordering on C n i=1 Ci, (with respet to ϕ) to be R 1 R 2 if and only if ϕ(r 1 ) ϕ(r 2 ). Definition 9 (Homogenous Fusion) Let C be a ategory and define HR (C n, C) = {r : A C A = C n for some C Ob(C)}. Let ϕ be a nonnegative-valued funtional defined on HR (C n, C). The proess of find a maximizer r HR (C n, C) is alled homogeneous fusion. Definition 10 (Heterogenous Fusion) Fusion that is not homogeneous is said to be heterogeneous fusion. 3.2 Integration Let A and B be two ategories. Let objet A Ob(A) and objet B Ob(B). Assume there is a mapping A t B that we an think of as the truth mapping. We wish to approximate this mapping with a olletion of mappings that ompose together. For example, suppose there are mappings A f C and C g B suh that the omposition g f is defined and approximates t. Let ψ be a real-valued funtional defined on B A, then ψ(g f) ψ(t) quantifies the approximation of g f to t. We define the olletion of mappings that are omposable {A f C, C g B} to be a system. Assume the mapping A f C existed but no mapping from C to B existed. We define integration to be the proess of determining a mapping C g B suh that the approximation error is as small as possible. That is, min ψ(g f) ψ(t) = ψ(g f) ψ(t). g C B We all g an integrator. Now we write definitions using rigorous mathematis. Assume a mapping t B A is given as above, as well as a ψ funtional. we need artesian produts of ategories that may be different. Let C 1, C 2,..., C n be n ategories (possibly distint). Form the artesian produt C 1 C 2 C n. A funtor that maps this produt into another ategory C (possibly distint also) is an integration rule. suh that g (f 1, f 2,..., f n } B A for some n N, for some C 1 C 2, C 2 C 2,..., C 1 C n for ategories C 1, C 2,..., C n. Observe that the graph for a series system is a hain. The graph for a parallel system is fan out- fan in graph. See the figures. Note that the ategories C 1, C 2,..., C n might not be distint. Definition 13 (System of systems) A PS-system for A t B is a finite olletion of P-system of S-sytems. A PPsystem (P 2 -system) for A t B is a finite olletion of P- system of P-sytems. Observe that an S-system of S-systems is another S- system. A S-system of P-systems will be onsidered later. This proess an be ontinued any number of times to get a P n S-system or P n -system. Definition 14 (Integration Rule) Let S 1, S 2 be two systems that are not onneted. An integration rule is a mapping that onnetes S 1 and S 2 into a single system S 3. Definition 15 (Integrator) Let T be a olletion of integration rules for systems S 1, S 2. Let ψ be a nonnegative real-valued funtional defined on T. Assume the optimization problem max{ψ(i) : I T} has a maximizer g T. We say g is an integrator with respet to ψ on T. The proess of determining a integrator is alled integration. 4 xamples xample 1. Consider the simple problem of onneting two S-systems. The integration rule i is D i F suh s D Fig. 1: System 1. F L Fig. 2: System 2. Definition 11 (Series System) A series system (S-system) for A t B is a finite olletion of mappings {A f1 f 2 f n f n+1 C 1, C 1 C2,, C n 1 Cn, C n B} suh that f n+1 f n f 1 B A for some n N, for some C 1 C 2, C 2 C 2,..., C 1 C n for ategories C 1, C 2,..., C n. that we get the system in quation 1. Hene, the integration rule is a proessor, and the final system is an event-deision system. xample 2. s D i F L (1) Integration rule i is a data interfaer i so that given two systems suh as Definition 12 (Parallel System) A parallel system (Psystem) for A t B is a finite olletion of mappings {A f1 C 1, A f2 C 2,, A fn C n, C 1 C 2 C n g B} and p s F L

6 we end up with s i p whih is the final event-deision system. xample 3. In this example, we show that integration is fusion. Given a first system and a seond system D 3 s 1 p s 2 F then integration rule i is a fusion rule i D 3 so that F L s 1 i p D 3 s 2 is the final event-deision system. Thus, fusion is a speial ase of integration. xample 4. Given system 1 and system 2 F s D p1 F 1 1 L D p2 F 2 2 L we see that our integration rule needs label (deision) fusion and is 1 F 1 L p 1 s D i L p 2 2 F 2 L L L xample 5. Here is a nontrivial integrated system that ontains feature fusion and label fusion. 1 F 1 L 1 p 1 s 1 p 2 F f 2 2 L 3 s 2 p 3 F f 1 3 L 2 xample 6. [Mammography]. Suppose sensor s is an x- ray mammography mahine. A woman will have 4 x-rays taken (2 different views on eah breast). The woman is an outome e from a population set. The x-ray mammogram provides 4 x-ray images, alled a ase. This is the data d; 4 x-ray images. Let D denote the olletion of all possible 4-tuples of images. Therefore, d = s(e) and s : D. A radiologist (a person trained to find anerous regions in a mammogram) looks at the ase to determine if any region appears to be anerous. (They look for ertain features and harateristis) This region is alled a region of interest (ROI) in the medial ommunity. A biopsy is performed to determine if the tissue in the ROI is anerous or not, thus, the biopsy proedure labels the region of interest. Radiologists desire to able to look at a ase and determine for ertain if aner is present. But, radiologists are not perfet, not onsistent and not tireless. They may miss a aner (a type II error alled a false negative). A false negative error is very ostly for the woman, whereas for a false positive the woman goes through a painful biopsy and mental stress to be relieved that there is no aner. (Of ourse, she has to pay for this proedure after all is done, but the ost is less than her life) Some health are related ompanies are working on devies that aid the radiologist to minimize these errors. Some devies try to emulate the radiologists. Radiologists will look for regions where miroalifiations our and also regions where breast tissue is dense. Some aner will form spiulated massy tissue, thus radiologists look for these regions as well. s D 5 Conlusion 1 F 1 L 1 p 1 p2 2 F 2 L f 2 3 L 3 L 4 Integration an be defined without having to speifi the systems elements. Fusion is the proess of mapping several objets to a single objet in an optimal fashion. Integration is the proess of onneting systems into a larger system. Thses systems may have fusion in them.

7 Aknowledgements We wish to aknowledge the finanial support of the U.S. Air Fore Researh Laboratory, Sensor Diretorate, Automati Target Reognition Branh (AFRL/SNAT) at Wright- Patterson AFB, Ohio and the Defense Advaned Researh Program Administration (DARPA), Mathematial Sienes Division, Arlington,Virginia. Referenes [1] S. MaLane and G. Birkhoff. Algebra. The MaMillan Company, Toronto, [2] J. Adámek, H. Herrlih, and G. Streker. Abstrat and Conrete Categories. John Wiley and Sons, In, New York, [3] F. William Lawvere and Stephen H. Shanuel. Coneptual Mathematis, A First Introdution to Categories. Cambridge University Press, Cambridge, [4] Saunders Ma Lane. Categories for the Working Mathematiian, Seond dition. Springer, New York, [5] Colin MClarty. lementary Categories, lementary Toposes. Oxford University Press, New York, 1992.