On approximately fair cost allocation in Euclidean TSP games Ulrich Faigle University of Twente Twente , Netherlands S?andor P. Fekete ? Mathematisches Institut Universitat zu Koln 50923 Koln , Germany Winfried Hochstattler y Mathematisches Institut Universitat zu Koln 50923 Koln , Germany Walter Kern University of Twente Twente , Netherlands ? Parts of this work were done while the author was staying at the Department of Applied Mathematics and Statistics , SUNY Stony Brook , NY 11794 -LCB- 3600 , during a visit at RUTCOR , Rutgers Univeristy , and during a stay at Tel Aviv University ; support by NSF Grants ECSE-8857642 and CCR-9204585 . yParts of this work were done while the author was visiting at RUTCOR , Rutgers University , supported by SFB 303 -LRB- Deutsche Forschungsgemeinschaft -RRB- . Euclidean TSP Games 2 Abstract : We consider the problem of fair cost allocation for Travelling Salesman games for which the triangle inequality holds . We give examples showing that the core of such games may be empty , even for the case of Euclidean distances . On the positive , we develop an LP-based allocation rule guaranteeing that no coalition pays more than times its own cost , where is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation , which is also known as the solution over the \ subtour polytope '' . A well known conjecture states that < = 43 . We also exhibit examples showing that this ratio can not be improved below 43 . Euclidean TSP Games 3 1 Introduction Travelling Salesman games are well-studied examples of so-called \ cooperative games '' . Formally , such games are defined as follows : Definition 1.1 Let N be a finite set , the set of players . A cooperative game is then defined by a function c : 2N ?! IR ; i.e. a function that associates a cost c -LRB- S -RRB- to every coalition S ? N . The coalition N is also called the grand coalition . Several interesting games arise from combinatorial optimization problems . Example 1.2 Let N = f1 ; : : : ; ng , N0 : = N -LSB- f0g and consider the complete undirected graph with vertex set N0 and edges weighted with real numbers dij , i ; j 2 N0 . For every S ? N , let c -LRB- S -RRB- = minimum cost of a tree spanning all vertices in S -LSB- f0g : This game is known as the Minimum Spanning Tree game , see -LSB- 9 -RSB- . Example 1.3 Same as in Example 1.2 , except for c -LRB- S -RRB- = minimum length of a travelling salesman tour visiting all vertices in S -LSB- f0g : This game is known as the Travelling Salesman game , see -LSB- 20 -RSB- . Example 1.2 can be interpreted as follows : Consider vertex as a supply node , e.g. an electricity supply . The cost c = c -LRB- N -RRB- of the grand coalition is the total cost of supplying electricity to all vertices or \ players '' . The problem is to find a fair way to allocate this amount to all the players . More precisely , let xi denote the amount of money player i is asked to pay , thus P i2N xi = c . A coalition S ? N which is asked to pay more than its own cost , i.e. P i2N xi > c -LRB- S -RRB- , will refuse to pay and prefer to split off the grand coalition by connecting to the supply on its own . Thus a fair allocation is one for which P i2S xi < = c -LRB- S -RRB- holds for every coalition S ? N . Example 1.3 can be interpreted in a similar way . Consider vertex as the home city of a speaker who has to give talks at the universities located in vertices 1 ; : : : ; n . The total travel cost equals c = c -LRB- N -RRB- . Again the problem is to find a fair cost allocation , such that no coalition S will split off and invite the speaker to visit only the universities i 2 S. Definition 1.4 Consider a cooperative game defined by c : 2N ?! IR . An allocation is a vector x 2 IRN such that P i2N xi = c -LRB- N -RRB- . The core of the game is defined as the set of all allocation vectors x 2 IRn satisfying X i2S xi < = c -LRB- S -RRB- for all S ? N : Any vector x in the core is called a core allocation vector , or simply a fair allocation . In Example 1.2 above , it is straightforward to verify that a fair allocation can be obtained as follows : Construct a minimum cost spanning tree T in KN0 , and direct the edges of T such that node becomes the root . Then for each i 2 N , let xi be the length of the -LRB- unique -RRB- edge of T entering node i. Euclidean TSP Games 4 As for the case of the Travelling Salesman games , it has been observed that the core of such games may be empty , see Tamir -LSB- 24 -RSB- . -LRB- Note that in that paper , a slightly more general setting is used , allowing non-hamiltonian graphs . -RRB- In the present paper , we will consider TSP games for Euclidean distances , which are among the most natural classes of TSP games , but have not received any special attention so far . The rest of this paper is organized as follows : Section 2 presents a minimal example of a Euclidean TSP game with empty core . This motivates the search for \ reasonably fair '' allocations . One possible definition was given in -LSB- 5 -RSB- , where the notion of \ '' - approximately fair allocations '' is introduced . Definition 1.5 Consider a game defined by c : 2N ?! R . An '' - approximately fair allocation is an allocation vector x 2 IRN , such that x -LRB- N -RRB- > = c -LRB- N -RRB- and x -LRB- S -RRB- < = -LRB- 1 + '' -RRB- c -LRB- S -RRB- for all S ? N : The idea is that for small '' > 0 , an '' - approximately fair allocation may still be acceptable , since the additional overhead costs for a coalition S after splitting off the grand coalition is likely to exceed a fraction of '' of its cost . Conversely , a limited deficit for the supplier may be acceptable . Other ways of defining '' - approximate core allocations have been proposed in the literature , see -LSB- 5 -RSB- for more references . In Section 3 we prove that TSP games with triangle inequality always have '' - approximately fair allocations for '' = 12 . If a well-known conjecture concerning the Held-Karp relaxation of the Euclidean TSP is true , our allocation rule presented in Section 3 would even achieve '' = 13 . Moreover , these allocations can be computed in polynomial time , even if the optimal tour is not known . In Section 4 we show that no allocation rule can guarantee '' < 13 . We end with some open problems and concluding remarks in Section 5 . 2 A minimal example with empty core Kuipers -LSB- 14 -RSB- showed that under quite general assumption -LRB- including the Euclidean case -RRB- , every TSP-game with up to 5 players has a nonempty core . The following example shows that for n = 6 players , there exist Euclidean instances with empty core . Example 2.1 See Figure 1 . Consider an isosceles triangle of side length l = p3 , centered at 0 . Label the vertices by 1 , 2 , 3 . Place three more nodes 4 , 5 , 6 at equal distance d from the center , such that node i lies on the line segment ; i ? 3 . The distance d will be chosen appropriately . Let f denote the distance d56 = d46 = d45 ; clearly , f = p3d . Let h denote the distance d15 = d16 = d24 = d26 = d34 = d35 . Applying Pythagoras ' Theorem to the triangle ? -LRB- 5P 3 -RRB- , we get h = v u u t p3 2 + p3 2 d ! 2 + ? -LRB- 1 ? d -RRB- 2 ? 2 ; we conclude that h = p1 + d + d2 . Finally , let g : = d36 = d25 = d14 , i.e. g = 1 ? d . As a consequence , the two obvious candidates for an optimal TSP tour of length L have the following lengths L1 and L2 -LRB- see Figure 2 -RRB- : Euclidean TSP Games 5 lgdfh0123456P Figure 1 : A minimum Euclidean example with empty core 12345601234560 Figure 2 : The two candidates for optimal TSP tours Euclidean TSP Games 6 L1 = l + 3g + 2d + h = 3 ? d + p3 + p 1 + d + d2 ; L2 = 2l + 2g + 2d + f = 2 + 2p3 + p3d : Now suppose x 2 IR6 were a core allocation . In particular , we must have 1 . P6i = 1 xi = L ; 2 . xi + xi +1 + xi +3 + xi +4 < = 2d + 2g + l for i = 1 ; 2 and x3 + x1 + x6 + x4 < = 2d + 2g + l : Adding up the three inequalities , we get L = 6X i = 1 xi < = 3 2 -LRB- 2d + 2g + l -RRB- = 3 + 3 2 p3 : If the optimal tour has length L1 , we get L1 = 3 ? d + p3 + p 1 + d + d2 < = 3 + 3 2 p3 ; which is equivalent to d > = 1 4 -LRB- p3 ? 1 -RRB- > 1 3 : If the optimal tour has length L2 , we get L2 = 2 + 2p3 + p3d < = 3 + 3 2 p3 ; which is equivalent to d < = 1p3 ? 1 2 < 1 10 : We conclude that there is no fair cost allocation if neither of these two conditions is satisfied , e.g. for d = 14 . 3 Approximately fair allocations for Euclidean instances We have seen in the previous section that not all TSP instances allow a fair cost allocation . This makes it desirable to examine approximately fair cost allocations , where the customers can be overcharged by a certain percentage , or the supplier is allowed to run a certain deficit . -LRB- See Definition 1.5 . -RRB- Interesting allocation rules -LRB- different from core allocations -RRB- have been studied in the context of TSP games , see -LSB- 20 -RSB- . In this section , we will approach the question from a geometric point of view and make use of linear programming duality . There is another good reason for considering approximately fair cost allocations : Since computing the length L of an optimal TSP tour is NP-hard , we can not expect to find an efficient way of computing cost allocations , whether they are fair or not . These computational difficulties make it desirable to consider performance bounds on cost allocations which can be computed in polynomial time . In section 3.2 below , we will introduce a modified game with cost function c -LRB- HK -RRB- < = c , for which we can efficiently compute core allocation vectors z . It is known that c -LRB- HK -RRB- > = 32 cChr , Euclidean TSP Games 7 Figure 3 : The fare zones of Hamburg , reproduced with kind permission by Hamburger Verkehrsverbund HVV where cChr denotes the cost function corresponding to the well-known Christofides heuristic . Hence , by scaling the vector z by a factor = cChr -LRB- N -RRB- cHK -LRB- N -RRB- < = 32 , we get an '' - approximately fair allocation for the original TSP game with '' < = 12 -LCB- see below for details . In case an optimal global tour of length L is known , we may of course scale the vector z by a factor of = L cHK -LRB- N -RRB- < = to obtain an '' < = '' < = 12 . 3.1 Geometric cost allocation We encounter methods of allocating the cost of transportation in many instances in everyday life . Two of the easiest allocation rules are also the most common ones : A taxi charges by the distance that is traveled by an individual customer . On the other hand , public mass transportation typically charges a flat fee for anyone who uses it , regardless of the distance . The practicality of these two allocation rules relies on the fact that the number of players in a game is either extremely small , i.e. 1 , or arbitrarily large . -LRB- Insufficiencies of the latter assumption are reflected by the deficits of most public transportation . -RRB- In many cities , there are attempts to refine the fares by using a \ zone structure '' : The region is subdivided into traffic zones , and a customer is charged a certain amount for crossing from one zone into the other . -LRB- See Figure 3 for a practical example . -RRB- For our purposes , a subdivision of the plane into fixed zones -LRB- e.g. , concentric circles around the depot -RRB- is much too crude . Instead , we have to take into account the relative position of the customers . In the following , we will describe a geometric cost allocation method that follows this idea . Euclidean TSP Games 8 Definition 3.1 For a given set of vertices in the plane , a moat is a simply closed strip of constant width that separates two nonempty complementary subsets of the vertices . The inside of the moat is the region containing the depot , the other region we call the outside . A moat packing is a collection of moats with pairwise disjoint interior . The cost of a moat packing is twice the sum over all widths . Note that any tour has to cross every moat twice , hence the cost of a TSP tour is greater than or equal to the cost of a moat packing . Figure 4 shows a moat packing for an instance of cities in the American midwest , distances are taken from Nemhauser and Wolsey , -LSB- 15 -RSB- , p. 530 . Chicago Omaha St. Louis Kansas City Springfield Dallas Oklahoma City Wichita Minneapolis Denver Figure 4 : A moat packing for an instance of 10 cities in the midwestern United States 3.2 Moats and the subtour polytope The cost of any moat packing can be allocated as follows : Distribute twice the width of any moat among the vertices on the outside -LRB- in an arbitrary way -RRB- . It is intuitively clear that if the total cost of the moat packing is distributed this way , the resulting distribution is such that no coalition pays more than its TSP cost . -LRB- We will prove that formally later on . -RRB- Note , however that by distributing the cost of a moat packing we will in general not get an allocation vector of the TSP game , since the cost of a moat packing is in general strictly less than the cost of an Euclidean TSP Games 9 optimal TSP tour . The situation where the two costs -LRB- maximum moat packing and minimum TSP tour -RRB- coincide occurs precisely if there is a moat packing and a tour such that : 1 . no part of the given tour is left uncovered , and 2 . each moat is crossed by the tour exactly twice . In the general situation , i.e. when the maximum moat packing has a cost strictly less than the optimum TSP tour , we can still get an allocation by simply scaling the cost distribution above appropriately . This yields an approximately fair allocation for the TSP game , which will be discussed in detail below . The allocated cost for any moat packing is twice the sum of the moat widths . Moat widths are required to be nonnegative and the sum of the widths of moats separating two vertices must not exceed their distance . This motivates the following linear program : max @ 2 X S ; S2M wfSSg 1 A -LRB- M -RRB- subject to the constraints wfSSg > = for all fS ; Sg 2 M ; X -LRB- i ; j -RRB- 2ffi -LRB- S -RRB- wfSSg < = d -LRB- i ; j -RRB- for all i ; j 2 V : Here M denotes the set of all nontrivial partitions -LRB- S ; S -RRB- of V -LRB- assuming that the depot is contained in S -RRB- and ffi -LRB- S -RRB- is the set of all edges that join a vertex from S to a vertex from S. Note that there are exponentially many variables . Furthermore , there may be solutions to the linear program which do not correspond to a moat packing , since there may be positive wfS1S1g , wfS2S2g with S1 \ S2 < > ; and S1 6 ? S2 and S2 6 ? S1 , in which case the two corresponding moats are forced to intersect . However , for any instance of -LRB- M -RRB- , there is an optimal solution for which the sets with wfSSg > have the special structure of a nested family : Definition 3.2 A family of partitions -LRB- S1 ; S1 -RRB- ; : : : ; -LRB- Sk ; Sk -RRB- is called nested , if for any two partitions -LRB- Si ; Si -RRB- and -LRB- Sj ; Sj -RRB- , we have Si \ Sj = ; or Si ? Sj or Sj ? Si . For details on nested families , see Pulleyblank -LSB- 21 -RSB- ; a proof for the above claim can be found in Cornu?ejols , Fonlupt and Naddef -LSB- 3 -RSB- . It is not hard to see that a solution with the structure of a nested family allows a moat packing . The details of using a moat packing for allocating the optimal value of -LRB- M -RRB- will be discussed in the following section . We will assume that triangle inequality and -LRB- for the sake of simplicity -RRB- symmetry hold for the distance function . We assume the distance function to be defined for all -LRB- i ; j -RRB- ; i ; j 2 V . How can we solve -LRB- M -RRB- in polynomial time ? Consider its dual , i.e. the following linear problem : Euclidean TSP Games 10 min X -LRB- i ; j -RRB- 2V 2 d -LRB- i ; j -RRB- xij -LRB- D -RRB- subject to the constraints xij > = for all i ; j ; X -LRB- i ; j -RRB- 2ffi -LRB- S -RRB- xij > = 2 for all S ; S 2 M ; Consider an optimal solution to -LRB- D -RRB- and suppose there was a vertex i with P j < > i xij > 2 . If there was only one vertex k < > i with xki > 0 , we would have xik > 2 and we could lower xik and stay feasible . Since d -LRB- i ; k -RRB- > = 0 , this would not increase the objective value . So assume there are two vertices k1 and k2 with xik1 > and xik2 > 0 . In that case we can lower both xik1 and xik2 and raise xk1k2 by the same value and maintain feasibility . Since by triangle inequality d -LRB- k1 ; i -RRB- + d -LRB- i ; k2 -RRB- > = d -LRB- k1 ; k2 -RRB- , this does not increase the objective value . So we may assume that for any vertex i , we have P j < > i xij = 2 . Furthermore , there can be no edge -LRB- k1 ; k2 -RRB- with xk1k2 > 1 , since otherwise we would get the violated constraint X -LRB- i ; j -RRB- 2ffi -LRB- fk1 ; k2g -RRB- xij = X i6 = k1 xik1 + X i6 = k2 xik2 ? 2xk1k2 = 4 ? 2xk1k2 < 2 : This means we may consider the following linear program -LRB- HK -RRB- instead of -LRB- D -RRB- : min X -LRB- i ; j -RRB- 2V 2 d -LRB- i ; j -RRB- xij -LRB- HK -RRB- subject to the constraints xij > = for all i ; j ; xij < = 1 for all i ; j ; X -LRB- i ; j -RRB- 2ffi -LRB- S -RRB- xij > = 2 for all S ; S 2 M ; X j < > i xij = 2 for all i 2 V : The feasible region for this second linear program is known as the subtour polytope Sn , the program itself as Held-Karp relaxation : Any of the constraints corresponding to a moat variable is a so-called subtour elimination constraint . These subtour elimination constraints were first introduced by Dantzig , Fulkerson and Johnson -LSB- 4 -RSB- . Grotschel and Padberg showed that they are facet-inducing for n > = 4 . Grotschel , Lov?asz and Schrijver -LSB- 10 -RSB- , and Karp and Papadimitriou -LSB- 13 -RSB- , showed that a polynomial method for solving the separation problem for a polytope yields a polynomial method for optimization by means of the ellipsoid method . Padberg and Hong -LSB- 17 -RSB- -LRB- see also Padberg and Wolsey -LSB- 19 -RSB- , and Padberg and Rao -LSB- 18 -RSB- -RRB- demonstrated how to solve separation for the subtour polytope in polynomial time by using the method of Gomory and Hu -LSB- 9 -RSB- for finding the minimum cost cut in a graph . Thus we know that optimization over the subtour polytope is possible in polynomial time by means of the ellipsoid algorithm . For a comprehensive study of optimizing over the subtour polytope , see Boyd -LSB- 1 -RSB- and also Boyd and Pulleyblank -LSB- 2 -RSB- . Summarizing , we state : Euclidean TSP Games 11 Theorem 3.3 We can determine an optimal moat packing in polynomial time . Since the feasible region Qn of the Travelling Salesman Polytope is contained in Sn , any optimal solution to minimizing over Sn is a lower bound for the optimal value of TSP . It was proved by Wolsey -LSB- 26 -RSB- and by Shmoys and Williamson -LSB- 22 -RSB- that for any distance function d satisfying the triangle inequality , this bound can be at worst 2/3 of the optimum : Theorem 3.4 -LRB- -LSB- 26 -RSB- , -LSB- 22 -RSB- -RRB- If the distances satisfy the triangle inequality then the optimum value of -LRB- HK -RRB- is at least 23 of the length of a shortest tour . It is a well-known open conjecture that the factor of 2/3 can be replaced by 3/4 . It should be noted that Shmoys and Williamson have shown an even stronger version of Theorem 3.4 : Theorem 3.5 -LRB- -LSB- 22 -RSB- -RRB- If the distances satisfy the triangle inequality then the optimum value of -LRB- HK -RRB- is at least 23 of the length of a tour obtained by the method of Christofides . This means we can guarantee allocation of 2/3 of the cost of an approximate tour where both the allocation and the tour have been computed in polynomial time . 3.3 Tours and Moats We now proceed to describe how to obtain allocation vectors from moat packings . The idea is to distribute twice the width of every moat in an arbitrary way -LRB- e.g. uniformly -RRB- among the vertices on its outside . Let w ? be an optimal solution of the linear program -LRB- M -RRB- and define xi : = 2 ? X i2S 2 ? S w?S ; ? S jSj : By linear programming duality this vector is in the core of the game associated with the linear program -LRB- HK -RRB- in a natural way : Define the cost of a coalition S ? N by c -LRB- HK -RRB- -LRB- S -RRB- : = min X i ; j d -LRB- i ; j -RRB- zij subject to the constraints zij > = for all i ; j ; P -LRB- i ; j -RRB- 2ffiS -LRB- T -RRB- zij > = 2 for all 2 T ae S -LSB- f0g : -LRB- Here ffiS -LRB- T -RRB- denotes the set of edges joining T to -LRB- S -LSB- f0g n T :-RRB- -RRB- The general idea to apply linear programming duality to combinatorial games goes back to -LSB- 16 -RSB- . The vector x 2 IRN is easily seen to be a core vector of the game defined by the cost function c -LRB- HK -RRB- : Indeed , by linear programming duality we have X i2N xi = 2 ? X i2N X i2S 2 ? S w?S ; ? S jSj = 2 ? X fS ; Sg2M X i2S 1 jSjw?S ; ? S = 2 ? X fS ; Sg2M w?S ; ? S = c -LRB- HK -RRB- -LRB- N -RRB- : To see that this is in fact a fair allocation vector , recall that for a coalition S ? N , its cost c -LRB- HK -RRB- -LRB- S -RRB- is defined as the optimum of an LP . Its dual is given by Euclidean TSP Games 12 c -LRB- HK -RRB- -LRB- S -RRB- = max 2 ? X 62T aeS wT ; -LRB- S -LSB- f0g -RRB- nT subject to the constraints X 62T aeS ffiS -LRB- T -RRB- 3 -LRB- i ; j -RRB- wT ; -LRB- S -LSB- f0g -RRB- nT < = d -LRB- i ; j -RRB- 8i ; j wT ; -LRB- S -LSB- f0g -RRB- nT > = : Now an optimum solution w ? of -LRB- M -RRB- induces a feasible solution of the LP above by ~ wT : = X S \ U = T w?U . Hence we get c -LRB- HK -RRB- -LRB- S -RRB- > = 2 ? X 62T aeS ~ wT = 2 ? X -LRB- i ; T -RRB- i2T?S 1 jT j ~ wT = X i2S 2 ? X i2U w?U jU \ Sj > = X i2S 2 ? X i2U w?U jU j = X i2S xi : Figure 5 : A moat packing Note that we can choose any distribution of the cost of a moat among the outside vertices , i.e. instead of taking xi : = 2 ? X i2S 2 ? S wS ; ? S jSj ; Euclidean TSP Games 13 we could choose any xi : = 2 ? X i2S 2 ? S > = SiwS ; ? S with P i2S > = Si = 1 ; < = > = Si < = 1 : without changing the validity of any of the above statements . If the optimal tour is not an optimal solution of -LRB- HK -RRB- then for any moat packing either at least one moat is traversed more than two times -LRB- see Figure 5 -RRB- or the optimal tour runs through territory which is not covered by any moat -LRB- see Figure 6 below -RRB- . Proposition 3.6 We would like to note that even if c -LRB- N -RRB- > c -LRB- HK -RRB- -LRB- N -RRB- , the core may be nonempty . The easiest example for this situation arises from Example 2.1 , see Figures 1 and 5 : It can be shown that an optimal moat packing has cost 3 -LRB- f2 + g + l2 -RRB- +2 -LRB- d ? f2 -RRB- = -LRB- 3 ? d -RRB- + -LRB- 3 + d -RRB- p32 = : L3 . Now consider the case where d satisfies one of the bounds given in Example 2.1 with equality ; in that case , we have a tour of length L = 3 + 3 p32 > L3 , meaning that we can not distribute the cost of the tour by the Held-Karp allocation rule . However , one can check that allocating p3 to each of the players on the outside vertices and 1 ? p32 to each of the players on the inside vertices is a fair allocation . 3.4 Approximately fair allocations Summarizing the results of the previous two sections , we state : Theorem 3.7 For TSP games with triangle inequality , there is a vector x 2 IRN , which can be computed in polynomial time and satisfies the following conditions : -LRB- i -RRB- x is an '' - approximative core allocation for '' = 12 . -LRB- ii -RRB- If the \ 43-conjecture '' on the Held-Karp bound is true , x is an '' - approximative core allocation for '' = 13 . -LRB- iii -RRB- x '' - approximately allocates the cost of an approximative TSP tour obtained by the Christofides heuristic for '' = 12 . Proof . Let x as above denote the fair allocation of the associated LP-game . Let c ? -LRB- HK -RRB- : = c -LRB- HK -RRB- -LRB- N -RRB- denote the optimum value of -LRB- HK -RRB- and let L be the length of a shortest tour . Obviously , ^ x : = L c ? x is a vector in IRN satisfying 1 . ^ x -LRB- N -RRB- = L and 2 . ^ x -LRB- S -RRB- = L c ? x -LRB- S -RRB- < = L c ? L -LRB- S -RRB- < = 3 2L -LRB- S -RRB- : We will see in the following Section 4 that it is impossible to achieve a better general bound than '' = 13 , even in the case of Euclidean distances . Remark 3.8 Potter , Curiel and Tijs -LSB- 20 -RSB- have shown that in the case of a distance function induced by the Euclidean metric for a planar arrangement of points , any convex arrangement of players guarantees a fair cost allocation . As it was shown by Fekete and Pulleyblank -LSB- 6 -RSB- , any such arrangement has a moat packing with cost equal to the length of the optimal tour , implying that this special case is covered by our above approach . Euclidean TSP Games 14 Figure 6 : A generic example with empty core Remark 3.9 We would like to point out that the above worst case estimate for the ratio L c ? -LRB- HK -RRB- is far from the average case ratio . It can be shown -LRB- cf. -LSB- 23 -RSB- , -LSB- 8 -RSB- -RRB- -RRB- that if p1 ; : : : ; pn ; : : : are independently uniformly distributed in the unit square , and L -LRB- p1 ; : : : ; pn -RRB- denotes the length of a shortest tour through p1 ; : : : ; pn and c ? -LRB- HK -RRB- -LRB- p1 ; : : : ; pn -RRB- denotes the optimum value of the Held-Karp relaxation , then there exist constants TSP and -LRB- HK -RRB- , such that L -LRB- p1 ; : : : ; pn -RRB- pn n ! 1 ?! TSP a.s. and c ? -LRB- HK -RRB- -LRB- p1 ; : : : ; pn -RRB- pn n ! 1 ?! -LRB- HK -RRB- a.s. Numerical experiments are reported -LSB- 8 -RSB- saying that TSP ss :709 and -LRB- HK -RRB- ss :7 . This means that in the average one can expect '' - approximately fair allocations for '' ss :013 . 4 More examples with empty core and a lower bound on '' In this section we present an infinite family of examples with empty core . We also show that the bound '' < = 13 which arises from the conjecture on the Held-Karp bound is tight by displaying a geometric instance where this bound is asymptotically met . First , consider again an isoceles triangle with center 0 . On each \ spoke '' , we place a large number of points at small distance , see Figure 6 . Suppose that the length of a spoke equals 1 , i.e. the triangle has side length p3 . Then the length of an optimum tour comes arbitrarily close to L = 4 + p3 . On the other hand , a simple argument , similar to that of Section 2 , yields that every potential core vector x 2 IRN must satisfy x -LRB- N -RRB- < = 32 -LRB- 2 + p3 -RRB- . This shows that such instances of TSP-games can not have '' - approximate core allocations for '' < = 2 -LRB- 4 + p3 -RRB- 3 -LRB- 2 + p3 -RRB- ? 1 = :0239 : : : This observation can be generalized to wheels with an arbitrary odd number 2k + 1 of spokes of length 1 , see Figure 7 for the case k = 3 . If s is the shortest distance between the endpoints of two adjacent spokes , the shortest tour Euclidean TSP Games 15 Figure 7 : Odd stars have an empty core has length close to L = ks + 2k + 2 , while any potential core vector must satisfy x -LRB- N -RRB- < = -LRB- 2k + 1 -RRB- -LRB- 2 + s -RRB- 2 = ks + 2k + 1 + s 2 : Since s2 < 1 , the core must be empty . Furthermore , there are no '' - approximately fair allocations for '' < = ks + 2k + 2 ks + 2k + 1 + s2 ? 1 = 1 ? s2 ks + 2k + 1 + s2 : A natural question arising in this context is to determine the worst case bound for '' . We have seen in the previous section that -LRB- subject to the conjecture on the Held-Karp bound -RRB- there is a way to guarantee '' < = 13 . In the following , we present an example with '' arbitrarily close to this bound of 13 . Theorem 4.1 The bound '' < = 13 resulting from the conjecture on the Held-Karp bound is best possible . Proof . Consider a set of n points , equally distributed on 3 columns , as shown in Figure 8 . Let h denote the height of the columns . Since the distance d between two columns can be made arbitrarily small -LRB- provided we place a sufficient number of points on each column -RRB- , a shortest tour has length approximately equal to 4h . On the other hand , any coalition consisting of two of the three columns is willing to pay at most 2h + 4d . Hence any potential core vector must satisfy x -LRB- N -RRB- < = 32 -LRB- 2h + 4d -RRB- = 3h + 6d . This shows that there are no '' - approximately fair allocations for '' < = 4h 3h + 6d ? 1 . If we let d tend to 0 , we get the desired result . Combined with our results in Section 3 , this bound implies a previous result due to Goemans -LSB- 8 -RSB- , stating that there are TSP instances for which the ratio between the length of a shortest tour and the optimal value of the Held-Karp relaxation is arbitrarily close to 43 . Euclidean TSP Games 16 dh0 Figure 8 : An example with '' ?! 13 5 Concluding remarks A proof of the \ 34-conjecture '' on the Held-Karp relaxation would completely settle the question about the worst case analysis of the '' - approximability problem . On the other hand , it may be possible to find completely different methods of constructing approximately fair allocations with guaranteed worst case approximation error '' = 13 . For example , Faigle and Kern -LSB- 5 -RSB- propose some allocation rules based on the minimum spanning tree allocation and the distance from the supply node . An empirical study of randomly generated small instances -LRB- n < = 10 -RRB- seems to indicate that the Held-Karp allocation rule is by far preferable to other heuristic rules based on minimum spanning trees and relative distance functions , cf. Hunting -LSB- 11 -RSB- . Another interesting question concerns the average case behavior . We conjecture that if the points are independently and uniformly generated , say , in the unit square , then the probability of the core being empty tends to zero as n ?! 1 , where n is the number of players . One way of approaching this problem may be using the so-called zero-or-one law from probability theory , see Feller -LSB- 7 -RSB- , Theorem 3 , Chapter IV .7 , volume II . This is easily seen to imply the following : Let X0 , X1 , X2 ; : : : be independent random variables that are uniformly distributed in the unit square . Let A ? ? -LSB- 0 ; 1 -RSB- 2 ? 1 be defined as A : = f -LRB- a0 ; a1 ; a2 ; : : :-RRB- j the TSP game has nonempty core for almost all begin sequencesg : Then either P -LRB- A -RRB- = or P -LRB- A -RRB- = 1 , where P is the infinite product measure on ? -LSB- 0 ; 1 -RSB- 2 ? 1 . Acknowledgements : The second author would like to thank Bill Pulleyblank for many helpful discussions about moat packings . We appreciate the friendly permission by the Hamburger Verkehrsverbund HVV to use Figure 3 and the technical assistance by KarMa Hochstattler in its digital reproduction . References -LSB- 1 -RSB- S.C.Boyd , The subtour polytope of the travelling salesman polytope , Ph.D. . Thesis , University of Waterloo , Waterloo , Ontario , 1986 . -LSB- 2 -RSB- S.C.Boyd , W.R.Pulleyblank , Optimization over the subtour polytope of the travelling salesman polytope , Mathematical Programming , 49 -LRB- 1990 -RRB- , 163 -LCB- 187 . 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O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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High-Performance Image Analysis and Visualization for Three-dimensional Light Microscopy Jie Chen , John Aarsvold , and Chin-Tu Chen , Frank Center for Image Analysis Melvin Griem , Department of Radiation and Cellular Oncology Peter Davies , Department of Pathology and Medicine University of Chicago , Chicago , IL 60637 Terry Disz , Ian Foster , Randy Hudson ? , Man K. Kwong , Biquan Lin , and Peter Tangy Mathematics and Computer Science Division Argonne National Laboratory , Argonne , IL 60439 disz , foster , hudson , kwong , blin , tang@mcs.anl.gov Abstract Three-dimensional microscopy imaging is important for understanding complex biological assemblies . Computers and digital image acquisition systems have made possible the three-dimensional reconstruction of images obtained from optical microscopes . Since processing such images requires tremendous CPU , storage , and I/O capacities , a high-performance computing facility is necessary in order to achieve real-time performance . However , it is not practical for many medical centers or biological research institutions to operate their own large computer system . One solution is to provide access to remote high-performance computers via highspeed networks . In this paper , we present such a prototype system that integrates microscopy image acquisition , parallel deblurring , 3D image visualization , and high-speed networks . 1 Introduction There are two major approaches to performing 3D optical sectioning microscopy . One is to use a confocal mi - ? Is also of the Networking Services Department of the University of Chicago . yThis work was supported by the Mathematical , Information , and Computational Sciences Division subprogram of the Office of Computational and Technology Research , U.S. Department of Energy , under Contract W-31-109-Eng-38 . croscope , which uses optics to reject directly the light that comes from the out-of-focus planes and optically dissect the specimen -LSB- 1 -RSB- -LSB- 2 -RSB- . The other approach is to use a conventional light microscope , acquire stacks of images at different Z positions , and then use computational methods to remove the out-of-focus information from the acquired images -LSB- 3 -RSB- -LSB- 4 -RSB- -LSB- 5 -RSB- . The confocal approach is easy to implement and can get clean images directly . However , it has several disadvantages : much higher cost than that of a conventional light microscope ; low light efficiency because of the rejection of more than 95 % of the fluorescent light ; limited choice of fluorescent dyes available for confocal imaging , which results in a limited selection of cell components that can be imaged ; and severe photo-bleaching . On the other hand , with the development of high-resolution , low-noise CCD -LRB- Charge-Coupled Device -RRB- cameras and high-speed , general-purpose computers , the computational approach to 3D optical sectioning microscopy is becoming more and more appealing . By adding a CCD camera , a Z positioner , and the necessary software , a conventional light microscope can easily be converted into a 3D microscope . Because the light microscope can image the cell in its natural aqueous environment and because a variety of fluorescent dyes are available , it is possible to image specific cell components and -LRB- possibly -RRB- the functions of these components . Once out-of-focus information is removed , the whole real cell structure can be viewed in 3D , providing much richer information for biologists to study various biological phenomena . This paper presents a prototype system that integrates digital image acqui - sition , high-speed restoration algorithms , volume rendering , and high-speed network connections to provide cost-effective and near real-time 3D optical microscopy imaging . 2 System Overview Our 3D microscopy imaging system consists of the following modules : 3D Microscopy Image Acquisition System . An image acquisition system modified from an ordinary light microscope . A cooled CCD camera is used to obtain a high-quality digitized image . We have also designed software to control the piezoelectric crystal positioner , the CCD camera exposure timing , the image acquisition , and the shutter of the microscope light path . Restoration Algorithms . Computational algorithms to remove out-of-focus information from the acquired image by using the information in the 3D point spread function -LRB- PSF -RRB- . Currently , we have implemented two restoration methods , the nearest-neighbor deconvolution algorithm and the Wiener filtering algorithm . The algorithms have been parallelized on an IBM SP2 supercomputer , located at Argonne National Laboratory , to achieve fast deconvolution and can be easily ported to other parallel and distributed computers . Volume Rendering . Three-dimensional visualization software to visualize the processed 3D data sets . The volume visualization has been implemented on both local SGI workstations and the CAVETM -LSB- 6 -RSB- -LSB- 7 -RSB- , a virtual reality environment developed by the Electronic Visualization Laboratory at the University of Illinois at Chicago . Client/Server Computing . A client/server suite to transfer images between the medical center of the University of Chicago and Argonne National Laboratory -LRB- 20 miles distant -RRB- . The software is based on a high-speed ATM link between Argonne and the university . We have implemented it by using Nexus -LSB- 8 -RSB- , a runtime task-parallel tool that supports heterogeneous architectures and protocols . We are also implementing it by using CC + + , a small set of extensions to C++ for parallel computing . The CC + + language is based on Nexus , but is easier to use for application developers . 3 Description of System Com - ponents In this section , we describe each module of the system in more detail . 3.1 3D Microscopy Image Acquisition System The image acquisition system we developed is based on a conventional light microscope . The following components are integrated into the system : Cooled CCD Camera . A cooled CCD camera is attached to the microscope to obtaining high quality digital images directly . The CCD chip is capable of acquiring images of 1317 ? 1035 pixels with 12 bits/pixel depth . Image acquisition parameters such as image size and exposure time can be changed by software . Z Position Stage . To obtain optical slices of a 3D object , we developed a piezoelectric crystal precision positioning stage . This stage has a piezoelectric crystal inside a cylinder at the center and has six nylon beams attached to the center cylinder to support the whole stage . The tension on each beam is carefully calibrated so that stage can move up and down without lateral shift . Linear Amplifier . A high-voltage linear amplifier is used that provides voltages of 0V to 1000V to the piezoelectric crystal . It is controlled by the acquisition software through a serial line connection . Light Shutters . There are two computer-controlled shutters . One controls the light from a mercury lamp to the sample . This shutter minimizes the exposure of the samples to fluorescent light . Another shutter , which is part of the CCD camera , is in front of the CCD chip ; it controls the exposure of the CCD array by the specimen . Vibration Isolation System . An air-table supports the whole system -LRB- except the power supply for the mercury lamp -RRB- . It is used to isolate the system from the vibrations of the building and the power supply . Computer and Software . An SGI Indigo workstation is used to run the software package that controls all of the system components and to perform image analysis . A client/server suite allows the use of a remote supercomputer to perform the computationally intensive deconvolution and volume rendering procedures . The image acquisition module provides an X/Motif-based graphic user interface -LRB- GUI -RRB- to the user . It accepts the imaging parameters the user specifies , translates them into low-level commands , and sends them to the CCD camera . It monitors the status of the CCD camera and transfers acquired images from the CCD camera to disk files . The image acquisition module has several features that help automate the image acquisition . 3.2 Restoration Algorithms In this subsection , we discuss several technical issues concerning our restoration algorithms and parallel implementations . Measuring Point Spread Function -LRB- PSF -RRB- . The first step in image restoration is to characterize the optical system . The characterization of the optical system can be obtained by measuring its PSF . The PSF represents information describing how the optical system translates physical objects into images . To measure the PSF , we use a small bead as a point source . A light source can be considered a point source when it is sufficiently small compared with the spatial resolution of the optical system . In our experiments , we used fluorescent latex beads with 0.100 ? 0.007 ? m diameter and found that the size is within the requirements for a point source for our system . Five bead images are acquired and averaged as the PSF of the system in our restoration algorithm . The Nearest-Neighbor Deblurring Method . This deconvolution algorithm is a widely used microscopy image deconvolution method -LSB- 3 -RSB- . The algorithm assumes that the most blurring within a slice is caused by the light scattering from its neighboring slices , one above and one below , and that the intra-plane PSF is much smaller compared with the inter-plane PSF . By these assumptions , the algorithm can be easily implemented in the following manner . Let ^ S -LRB- ? ; ? ; z -RRB- and G -LRB- ? ; ? ; z -RRB- be the 2D Fourier transforms of the estimated image and the acquired image of the zth slice in Z direction , respectively . Then , ^ S -LRB- ? ; ? ; z -RRB- = G -LRB- ? ; ? ; z -RRB- ? c ? -LSB- H1 -LRB- ? ; ? -RRB- ? G -LRB- ? ; ? ; z ? 1 -RRB- + H ? 1 -LRB- ? ; ? -RRB- ? G -LRB- ? ; ? ; z + 1 -RRB- -RSB- ; where H1 -LRB- ? ; ? -RRB- and H ? 1 -LRB- ? ; ? -RRB- are the Fourier transforms of the inter-plane PSF for the two neighboring planes below and above the central plane . The constant c in the preceding formula is a weighting factor that adjusts the contribution of the two neighboring planes to the central plane ; a value of 0.49 gives a good deconvolution image . Simple Parallel Algorithm . In the simple parallel algorithm , each processor independently reads three images -LRB- its current working image , the image one step above , and the image one step below -RRB- . Each node also has available the three central slices of the PSF data . Each node deblurs its working image and outputs its deblurred image . The advantage of this model is that the original deconvolution algorithm can be ported to parallel computers without any modifications . On 39 IBM parallel nodes , the time required for deconvolution of a 39 slice data set -LRB- 39 ? 1024 ? 1024 -RRB- is reduced from 40 minutes to about five minutes . Efficiency is limited because each 2MB images is read three times . Message-passing Parallel Algorithm . In the messagepassing algorithm , each processor reads only its own working image , and communicates with other nodes to receive the images one step above and one step below . Using the PSF data , each node processes its current working image and outputs its deblurred image . Because internal data exchange is much faster than data exchange between memory and disk -LRB- for example , the difference in exchange speed is a factor of 9 in an IBM SP2 system -RRB- , the message-passing model significantly reduces the time for image input . Currently , it takes less than three minutes to finish deconvolution of one of our data sets . Further improvements can be obtained by exploitation of parallel I/O features and by code optimization . 3.3 Volume Visualization The volume visualization has been implemented on both local SGI workstations and the CAVE virtual reality environment . In this subsection , we focus on CAVE - based volume visualization . The CAVE is a fully immersive , projector-based virtual reality -LRB- VR -RRB- system -LSB- 6 -RSB- -LSB- 7 -RSB- . The volume visualization module encompasses a set of techniques for displaying all or part of a 3D field in the CAVE . One such technique is the modeling and display of isosurfaces in the field ; another is direct volume rendering . The data is volume rendered as a translucent substance of varying opacity , the opacity representing the density of the imaged material . Volume rendering allows the entire 3D data field to be viewed at once . Until recently , little volume rendering has been done in virtual environments because the time required to render a single animation frame has defeated the need in VR for real-time animation . However , on the SGI ONYX , 3D texture mapping , done in hardware , can be used for volume rendering in real time . The Software . Our volume visualization tool uses the 3D texturing hardware the same way as does SGI 's volren application -LSB- 9 -RSB- , and as described in -LSB- 10 -RSB- -LSB- 11 -RSB- . In brief , the data is defined as a 3d texture which is then sam - pled by a stack of equally-spaced slicing planes . There are many ways to use 3D texture mapping in direct volume rendering . We currently let the input values act as indices into a texture lookup table , which in turn , contains the brightness values and opacities that are actually displayed . Widgets are available for separate control of brightness and opacity . This arrangement allows the user to segment the data by making certain ranges of values visible or invisible . The opacity can be adjusted to vary the display from translucent to opaque , the latter being equivalent to an isosurface display . Brightness can be adjusted for a pseudo-lighting effect . The data may be rotated in texture space to be viewed from any orientation . Results are best if the slicing planes are kept perpendicular to the user 's line of sight . In a stereographic virtual environment , where the user can move in world space , the lines of sight can have any orientation , and the planes must be rotated to remain perpendicular to them . Since the planes can slice the texture volume at any orientation , the slicing polygon vertices must in general be recomputed each animation frame . The Data . The ONYX at Argonne is currently equipped with RM4 raster manager boards . These contain 4 Mbytes of texture memory , 2MB of which are usable for the texture itself . Three-dimensional texture mapping on the ONYX currently requires the texture pixel resolution in each of the three dimensions to be a power of two . This means that if we subsample each input image to 2562 , and use only 32 of them , they will just fit into texture memory . If we use MIP-mapping -LRB- which builds a resolution pyramid of filtered images to be used when zooming in and out -RRB- , part of the 2MB is needed for these other images . 3.4 Client/Server Computing A client/server suite enables one to use highperformance computers located at remote supercomputer centers . The local workstation acts as a client , and the remote supercomputer can be configured as a server . During image acquisition , one can select a server . Once a server is selected , the acquired images are not saved into a local hard disk but instead are sent with the processing commands and parameters through the network to the remote server . The remote server receives the data , the processing commands , and the associated parameters . It then starts the corresponding subroutines needed to do the deconvolution calculations and the volume rendering . Results can be displayed on the local workstation where the images are acquired , or in the CAVE . A high-speed ATM -LRB- Asynchronous Transfer Mode -RRB- network link between the University of Chicago and Argonne National Laboratory is used to transfer the images . The current peak speed of the network is 34 Mbits/sec -LRB- about 20 seconds per 39 slice data set -RRB- , but but will be upgraded to 155 Mbits/sec in the near future . With this high-speed link , it will be possible to send the acquired images to the Argonne supercomputer during the interval of normal image acquisition . Because of the nature of the deconvolution algorithm , the server does not need to obtain all the 2D slice images before starting the deconvolution process . It can start as soon as some 2D slice images have been received . In the case of the nearest-neighbor method , the deconvolution can start once the images of the current slice and its above and below slices have been received . When the entire 3D image acquisition is completed , the server is already half done with its calculation . Therefore , image restoration can be done soon after the entire 3D image acquisition is completed . 4 Conclusions We have built a prototype system that exploits supercomputing facilities , high-speed networks , and 3D visualization systems for research and applications in microscopy imaging . The system also provides a client/server model for parallel computer applications in large-scale image processing . In future work , we plan to optimize further the system just described , in order to support real-time 3D display of microscopic images . This will make it possible to conduct interactive experiments in studying the response of living specimens to external stimuli . References -LSB- 1 -RSB- G.J. Barkenhoff , E.A. Van Spronsen , H.T.M.Van Der Voort , and N.Nanninga , \ Three-dimensional confocal fluorescent microscopy , '' Methods in Cell Biology , vol . 30 , pp. 379 -LCB- 398 , 1989 . -LSB- 2 -RSB- J.B. Pawley , Handbook of Biological Confocal Microscopy , Plenum Press , 1990 . -LSB- 3 -RSB- D.A. Agard , \ Optical sectioning microscopy : cellular architecture in three dimensions , '' Annual Reviews in Biophysics and Bioengineering , vol . 13 , pp. 191 -LCB- 219 , 1984 . -LSB- 4 -RSB- S. Joshi and M.I. Miller , \ Maximum a posteriori estimate with Good 's roughness for threedimensional optical-sectioning microscopy , '' Journal of Optical Society of America , vol . 10 , no. 5 , pp. 1078 -LCB- 1085 , 1993 . -LSB- 5 -RSB- W.A. Carrington , R.M. Lynch , E.D. Moore , G. Isenberg , K.E. Fogarty , and F.S. Fay , \ Superresolution three-dimensional images of fluorescence in cells with minimal light exposure , '' , Science , vol . 268 , pp. 1483 -LCB- 1487 , June 9 , 1995 . -LSB- 6 -RSB- C. Cruz-Neira , D.J. Sandin , T.A. DeFanti , R.V. Kenyon , and J.C. Hart , \ The CAVE : audio visual experience automatic virtual environment , '' Communications of the Association for Computing Machinery , vol . 35 , no. 6 , pp. 65 -LCB- 72 , June , 1992 . -LSB- 7 -RSB- C. Cruz-Neira , D.J. Sandin , and T.A. DeFanti , \ Surround-screen projection-based virtual reality : the design and implementation of the CAVE , '' Computer Graphics -LRB- Proceedings of SIGGRAPH ' 93 -RRB- , pp. 135 -LCB- 142 , August , 1993 . -LSB- 8 -RSB- I. Foster , Designing and Building Parallel Programs , Addison-Wesley , 1995 . -LSB- 9 -RSB- R. Fraser , \ Interactive volume rendering using advanced graphics architectures , '' Online Tech . Center white paper -LCB- http://www.sgi.com/Technology/volume/ VolumeRendering.html , Silicon Graphics , Inc. . -LSB- 10 -RSB- B. Cabral , N. Cam , and J. Foran , \ Accelerated volume rendering and tomographic reconstruction using texture mapping hardware , '' Proceedings of the 1994 Symposium on Volume Visualization , pp. 91 -LCB- 94 , Washington , D.C. , October , 1994 . -LSB- 11 -RSB- M. Teschner , \ Texture Mapping , '' IRIS Universe , no. 29 , pp. 8 -LCB- 11 , SGI , Fall 1994 .

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deblurring
optical sectioning microscopy
parallel computing
volume rendering