IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 9 ...vsingh/brachy.pdfvolves implanting a group...

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 9, SEPTEMBER 2007 1291

Brachytherapy Seed Localization Using Geometricand Linear Programming Techniques

Vikas Singh*, Lopamudra Mukherjee, Jinhui Xu, Kenneth R. Hoffmann, Petru M. Dinu, and Matthew Podgorsak

Abstract—We propose an optimization algorithm to solvethe brachytherapy seed localization problem in prostatebrachytherapy. Our algorithm is based on novel geometricapproaches to exploit the special structure of the problem andrelies on a number of key observations which help us formulatethe optimization problem as a minimization integer program (IP).Our IP model precisely defines the feasibility polyhedron for thisproblem using a polynomial number of half-spaces; the solution toits corresponding linear program is rounded to yield an integralsolution to our task of determining correspondences betweenseeds in multiple projection images. The algorithm is efficientin theory as well as in practice and performs well on simulationdata ( 98% accuracy) and real X-ray images ( 95% accuracy).We present in detail the underlying ideas and an extensive set ofperformance evaluations based on our implementation.

Index Terms—Brachytherapy seed localization (BSL), integerprogramming, linear programming, geometric optimization,multiview point correspondence.

I. INTRODUCTION

PROSTATE cancer is one of the most common forms ofcancer in men and accounts for around 32 000 deaths each

year in the United States alone [1], [2]. It occurs when the cellsin the prostate (a gland in the male reproductive system) mu-tate and multiply uncontrollably. Fortunately, if diagnosed andtreated early, prostate cancer has a cure rate of over 90% [1].One of the most common methods of treatment is known as lowdose rate permanent seed brachytherapy [3]. This procedure in-volves implanting a group of radioactive seeds inside the softtissue (tumor). The specific seed configuration is based on the

Manuscript received February 8, 2007; revised May 4, 2007. Asterisk indi-cates corresponding author.

*V. Singh is with the Department of Computer Science and Engineering, TheState University of New York at Buffalo, Buffalo, NY 14260 USA and also withthe Toshiba Stroke Research Center at The State University of New York atBuffalo, Buffalo. NY 14214 USA.

L. Mukherjee and J. Xu are with the Department of Computer Science andEngineering, The State University of New York at Buffalo, Buffalo, NY 14260USA.

K. R. Hoffmann is with the Department of Neurosurgery, The State Universityof New York at Buffalo, Buffalo, NY 14209 USA and also with the ToshibaStroke Research Center at The State University of New York at Buffalo, Buffalo.NY 14214 USA.

P. M. Dinu is with the Department of Physics at The State University of NewYork at Buffalo, Buffalo, NY 14260 USA and also with the Toshiba Stroke Re-search Center at The State University of New York at Buffalo, Buffalo. NY14214 USA.

M. Podgorsak is with Roswell Park Cancer Institute, Buffalo, NY 14263USA.

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

Digital Object Identifier 10.1109/TMI.2007.900740

patient’s diagnosis and anatomy. An optimal spatial arrange-ment of the seeds will yield the best possible radiation dose dis-tribution for the tumor. Achieving an exact “implementation” ofthe prescribed seed configuration, however, is difficult becausethe procedure is done manually. Also, the prostate may undergocertain deformations (expansions or shrinkages) due to repeatedpiercings by the needles used to implant the seeds. Even minorinaccuracies in the implanted seed configuration lead to devi-ations from the optimal implant configuration and such dosi-metric deviations need to be carefully analyzed (see [4], [5] forrelated information). This is usually done during the monitoringand planning phase of the treatment. The analysis relies on firstdetermining the post-implant 3-D seed configuration which isthen “registered” with the optimal implant plan; patient specificdeviations can then be calculated. (Note: based on the practiceadopted at individual treatment centers, this may be done ei-ther during the implant process using a portable gantry or im-mediately after the implant). Usually, a computer tomography(CT) reconstruction of the region of interest (treatment volume)is performed in a subsequent patient visit and compared with theprescribed configuration. However, the initial treatment evalu-ation is done by acquisition of multiple fluoroscopic X-ray im-ages of the seed configuration. These 2-D projection images ofthe seed configuration are then analyzed to assess in a qualitativeway the appropriateness of the implant. Since this analysis re-lies on manual estimation of 3-D seed positions from 2-D X-rayprojections, several research efforts have focused on automatedreconstruction of the 3-D configuration of implanted seeds fromthese images [6]–[13]. It is expected that accurate reconstructionand seed localization techniques will mitigate the inaccuraciesdue to human estimation and help provide clinicians with ac-curate real-time feedback on the implanted seed configuration.This will give the clinician an improved ability in monitoringand controlling the treatment plan.

Reconstruction of an unknown 3-D object from two ormore 2-D projection views is a widely studied problem inmedical imaging. In general, information about the geometryrelating the coordinate systems of two or more views andknown corresponding points (the same 3-D point’s projec-tions on different imaging planes) are used to reconstructthe 3-D object. For our problem, we first need to determinethe correspondence of the 2-D seeds from the X-ray imagesin order to reconstruct the 3-D seed-configuration.1 Observethat if the relative geometry between the various coordinatesystems is known accurately, corresponding seed points in

1In the Biplane Geometry Determination problem, the user indicates corre-sponding points based on landmarks in the images, the geometry is then deter-mined based on known correspondences [14]–[16]. However, here correspon-dences are not known a priori.

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1292 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 9, SEPTEMBER 2007

the images can be determined—the projection lines or linesof sight (a 3-D line from the X-ray source of the respectivecoordinate system to the 2-D image point considered as a 3-Dpoint in its coordinate system) of two corresponding pointswill intersect, yielding the representative 3-D point [17]. Mostcommercial imaging systems, however, suffer from certainelectromechanical limitations that result in errors in the rela-tive geometries. In case of a configuration of brachytherapyseeds, the relative error introduced due to a small geometryerror could be significant. Second, the acquired images havesome distortion or segmentation errors (together referred to asimage error). Such errors make the task of the determinationof corresponding seeds in the images (seed matching) ratherchallenging. As can be expected, a reconstruction procedurewith incorrect correspondence information will be inaccurateand hence maybe unreliable for clinical purposes. In this paper,we propose efficient techniques for solving this brachytherapyseed localization (BSL) (or correspondence) problem.

Because of its importance in clinical diagnosis and treatment,the BSL problem has been studied by a number of researchers inthe recent past. One of the first efforts to mathematically formu-late the problem was made by Siddon and Chin [6]. A cost-ma-trix data structure was constructed followed by an exhaustivebrute force search to find the lowest cost solution. Subsequently,researchers have proposed several heuristics to speed-up thesearch process, for example, a greedy choice using several dif-ferent cost metrics [7], performing a brute force search on asubset [8], [9], and using epipolar constraints to impose someform of partial ordering [10]. Other artificial intelligence heuris-tics like simulated annealing (see [11] and [12]) have also beenused with some success. Recently, Jain et al. [13] proposed anapproach which first constructs a multipartite graph and then ob-tains a solution by solving several instances of bipartite graphmatching. In fact, the connection of the correspondence deter-mination problem to the graph matching problem was formallyestablished earlier by Shafique and Shah [18] in the contextof point set tracking in an image frame sequence. They pro-posed a greedy algorithm by modeling correspondence matchesas cycles in digraphs and reported improvements over earlierwork on related problems [19]–[21]. The problems with suchapproaches is the fact that globally best choices for one bipar-tite graph are only locally good choices considering all viewsat once. In the presence of errors, the local choices in each bi-partite graph may no longer be even near-best choices whenconsidered globally. Similar issues also lead to difficulties ifthe digraph formulation in [18] (which is a reasonable strategyfor point set tracking from one frame to the next) is directlyapplied to the BSL problem. The tracking version of point setcorrespondence has been studied by other researchers to modelmany computer vision scenarios, see [22] and [23]. A more gen-eral version of the correspondence determination problem is theso-called object recognition problem in multiple views. How-ever, strong theoretical results are known only for very specialcases (e.g., planar objects). We refer the reader to [24] and ref-erences therein for a detailed discussion.

In short, significant progress in the last few years not with-standing, the problem remains far from solved. Part of thereason is that the BSL problem is closely related to several

notorious NP-hard problems (e.g., maximum clique problemand -partite matching problem), possibly indicating thatobtaining an optimal solution in polynomial time is unlikely.Heuristics may perform well on some data sets but may yieldpoor results on other instances of the problem. Purely proba-bilistic approaches may yield different solutions for each runwhile general purpose techniques such as simulated annealingand genetic programming may get trapped in local minima;additionally, such techniques provide no guarantees on eitherthe convergence speed or the quality of the solutions obtained.It seems that to come up with better solutions for this problem,a deeper theoretical analysis needs to be undertaken. However,a quest for stronger bounds could easily take the approachaway from the realm of practical feasibility. For instance, whileapproximation algorithms for the -partite matching problemyield theoretically good approximation ratios of[25], they are hardly suitable for most practical applications. Akey challenge lies in avoiding these pitfalls.

In this paper, we tackle the BSL problem in a two phasemanner assuming that the seeds have already been segmentedusing standard techniques yielding their centers of mass: 1)we first match 2-D points (correspondences) in the imagingplanes, , and then 2) reconstruct the 3-D pointsbased on the obtained matching information. Our paper focuseson a mathematical formulation for (1).

The rest of this paper2 is organized as follows. In Section II,we discuss some preliminary ideas and an intuition into theproblem formulation. In Sections III and IV, we cover an im-portant preprocessing step and then give a more formal descrip-tion of the problem and the key strategies towards a solution. InSections V and VI, we first introduce the integer program (IP)formulation of the problem and its linear program (LP) relax-ation and then discuss a natural extension of the algorithm toinclude ordering constraints in Section VII. We present the ex-perimental results on simulation, phantom, and clinical data inSection VIII and conclude in Section X.

II. PRELIMINARIES

A. Straightforward Formulation as a Graph Matching Problem

The most natural way to think of the 2-D point matching (i.e.,determining corresponding point-tuples) is to view it as a min-imum-weight -partite graph matching problem in a to-be-con-structed graph . A graph is multipartite if the setof vertices in the graph can be divided into nonempty subsets,which we call “parts,” such that no two vertices in the same parthave an edge connecting them. Furthermore, a complete multi-partite graph is a graph such that any two vertices that are notin the same part have an edge connecting them. The problemof minimum-weight matching in a multipartite graph is to findthe edge set of smallest total weight which “covers” each vertexfrom a part exactly once. At this stage, it is helpful to brieflydiscuss this formulation to get an intuitive understanding of theproblem. Consider each image plane as a “part” in the graph

; the vertices in that part are representative of the 2-D seedpoints on that imaging plane. Therefore, the vertex set is

2Our preliminary results on this problem appeared in [26].

SINGH et al.: BRACHYTHERAPY SEED LOCALIZATION USING GEOMETRIC AND LINEAR PROGRAMMING TECHNIQUES 1293

Fig. 1. Schematic construction of the image acquisition.

given by the union of all . Let us denote the th 2-D projec-tion point on the th imaging plane as a vertex, . In all,we have ‘parts’ denoting projection images in the graphwhere each part has vertices corresponding to projections of

3-D points. Since there may exist an edge in joining anypair of vertices not in the same part, can have no more than

edges where is the number of 3-D points. We desirea set of matchings, , such that each

is a -tuple of the form ,where each , and is one of the 2-D projection pointsin , and respectively. These -tuples should have novertex in common with each other, ensuring that each 2-D pointparticipates in one and only one matching. This constructiontransforms the BSL problem (see Fig. 1) into a -partite graphmatching problem in a graph with vertices [see Fig. 2(a) and(b)]. A random layout of the vertices on a 2-D plane is also avalid construction [see Fig. 2(c)].

Once all 2-D projection points on (the setsof imaging planes) have been represented as the vertex set in

, we assign weights to the edges joining the ver-tices. Intuitively, the weight of an edge should be a function ofthe likelihood that the two end vertices of the edge are a pos-sible match. That is, if two 2-D points are corresponding, theedge in joining the corresponding vertices should have a smallweight (where likelihood is given as the inverse of the weight).To this end, let us consider two projection lines [14], and ,for two known corresponding points on imaging planes, and

. If the situation were ideal (i.e., error free), the lines, and, would intersect, and the intersection point would yield the

actual 3-D point. Clearly, the minimum distance between andwould be 0. This leads us to believe that the distance between

the projection lines of the two points would be useful as theweight of the edge joining them.

III. PREPROCESSING STEP

In this section, we discuss our preprocessing technique whichexploits the implicit geometric information in the spatial ori-

entation of the set of projection lines. Notice that in the pre-vious graph construction step, we may add an edge betweenany pair of vertices that are not in the same part. However,this construction loses a great deal of geometric information in-herent in the 2-D images. For example, consider three imagingplanes , where projection lines are oriented as shown inFig. 3(a). The lines are pair-wise close (the associated edges ingraph may have a near-zero weight) but are well-separated inspace and enclose a region of a relatively large area; it is unlikelythat the three projection lines correspond to the same 3-D point.Though stronger examples can be constructed, even this simpleillustration [Fig. 3(a)] suggests that the two “close” projectionlines are not a sufficient indication of a correspondence. Rather,a correspondence is likely only when at least projection linescome together within a close neighborhood. While this infor-mation is obvious in the figures, retrieving the same from theconstructed graph is slightly more complicated. For this reason,it becomes important to a) remove the edges which are unlikelyto be part of a match and b) somehow encode this informationfor use by our algorithm. In the preprocessing phase, we willaddress item a) that has an additional advantage of speeding upthe implementation ( is smaller). Later, using item a), we willenforce item b) by a mathematically precise condition.

A. Upper Bound on the Maximal Error

Let the total error introduced in the imaging system be upperbounded by a constant (which can be determined from theprecision of the imaging system). The “maximal” error of thereconstructed 3-D points, in the presence of an error , can beupper bounded by . Here, is a function that canbe easily calculated by introducing the maximum errors in thecomponents of the rotation matrix, , and translation vector,

relating the imaging systems3. Also for two projection linescorresponding to the same 3-D point, their shortest 3-D distancewould be upper bounded by . In other words, in the presence ofan error , the 3-D projection lines, that may have intersected inthe absence of error, move apart by a distance no larger than .This observation forms the basis for our preprocessing step—aball of radius 2 , sliding (sweeping) along each projection linestarting from the origin or X-ray source and stopping at its cor-responding 2-D point, would hit (intersect) each projection linethat is a likely candidate for a match. It may also hit other pro-jection lines which are not matches but such redundancy can betackled later in an update step. For the ball sliding along a pro-jection line, say , the set of all other projection lines it hits,

, will be no larger than . The correct (unknown)matches for will be included in . For our preprocessing,we first compute a sequence of s for each projection line, asthe ball sweeps. This process is similar to a standard computa-tional geometry technique called plane-sweep [27]; we sweep aball from the X-ray source to its corresponding 2-D image point.When some projection line enters (intersects) the ball along itssweep-path, we treat it as an event. The “ball-sweep-status” isupdated to reflect this event. Similarly, when any projection linecurrently on the ball-sweep-status leaves the ball (ball moves

3In clinical imaging systems, R is determined by the gantry angles right/leftanterior oblique (i.e., RAO-LAO) and cranial-caudal (i.e., CRA-CAU), and t isthe translation between the isocenters.


Fig. 2. (a) Representation of a fully connected k-partite graph, each part i corresponds to the imaging plane, P ; (b) k-partite graph with fewer edges (some edgesremoved via preprocessing, see Section III). Each vertex v 2 P represents the projection lines of the jth image point on P . The weight of the edge denotes theshortest distance between the respective projection lines. (c) Random layout of the k-partite graph.

Fig. 3. (a) Three lines are pair-wise close but enclose a region of some area. (b)Placement of balls such that at least three lines (one from each plane) intersectevery ball.

away far enough), the ball-sweep-status is again appropriatelyupdated. Once the sweep process for a particular projection lineis completed (ball has reached the 2-D image point), the sweep isevaluated. At any “event” in the sweep, if the ball-sweep-statushad at least projection lines, a ball of radius 2 is placed inthat spatial event position. In other words, when unique pro-jection lines intersect the ball at an event, it denotes a possiblepositioning for a 3-D seed point. The ball-sweep process is re-peated for each projection line and redundant ball positions areremoved. The process yields a set of all possible ball positions, aset of less than s for each ; the unknown 3-D pointsare located inside of these balls (see Fig. 3). Since planesweeptakes time, we have the following simple results.

Lemma 1: The ball-sweep for each projection line takes nomore than time. The preprocessing step requiresat most time.

Lemma 2: If denotes the set of all ball positions deter-mined, is bounded above by .

We create a comprehensive set of balls,where is given by . Each

ball , defines a possible location of 3-D seed points. Ifa projection line hits a ball, , the corresponding vertex “be-longs” to the ball . If two vertices belong to the same ball,

, their edge also belongs to . Next, we reduce the numberof edges in the constructed graph by using the preprocessinginformation. For a pair of vertices not belonging to any singleball, their edge is removed from [compare Fig. 2(a) with (b)].We assume that following the removal of unnecessary edges,each vertex will be connected to at most vertices in every otherplane (for a suitable constant, ). Therefore, .We also create a binary matrix, . is 1 if theedge is in ball , and 0 otherwise.

IV. MAIN IDEAS

A. Finding Complete Mutually Disjoint Subgraphs (CMDS)

The objective now is to find complete mutually disjointsubgraphs (CMDS) of (where and are thevertex set and the edge set respectively) such that

• each CMDS has exactly nodes, one from each imagingplane.

• each CMDS “belongs” entirely to one of the balls.In other words, each CMDS should form a clique structure withone and only one vertex from each imaging plane. Ideally, it isalso desired that the sum of weights of all CMDS, given by sum-ming over all its constituent edges, is minimized globally. EachCMDS can then be used to reconstruct the corresponding 3-Dpoint. In addition, the cliques are required to be vertex disjoint

because each 3-D seedpoint has a unique projection. Since each CMDS (clique) willtogether correspond to one such 3-D point, vertices are allowedto participate in only one CMDS, justifying the disjointness con-dition, see Fig. 4.

Notice that imposing the clique structure requirement on eachCMDS and globally minimizing the CMDS (clique) weights areboth necessary for a good solution. It makes intuitive sense thatany set of 2-D projection points constituting a -tuple shouldbe “close” to each other. The main difficulty of this approachlies in how to find the disjoint cliques efficiently. It is known


Fig. 4. (a) Weighted graph constructed after the preprocessing phase. (b) Min-imum weighted cliques of size 3 (in bold).

that even the problem of finding one minimum weight max-imum clique4 in a graph (i.e., the maximum-clique problem) isstrongly NP-hard. For our problem, several others factors likethe geometric information, etc., further complicate the problem.To overcome these difficulties, our strategy is to exploit our ob-servations about the special structure of the problem (see Sec-tion III) and design a practical combinatorial approach. Morespecifically, we use the constructed graph [see Fig. 2(c)] toformulate an IP. Each of our observations is transformed intoequivalent linear constraints in the IP model. The constraints arehalf-spaces (in a high dimensional space) whose intersection de-fines the feasibility polyhedron for the BSL problem. The ratio-nale is that this feasibility polyhedron and the objective functioncan be used effectively to find a solution to this problem. In thesubsequent sections, we discuss our IP model, followed by therelaxed LP version, and finally, a method for rounding the LPsolution to derive an integral solution for our problem.

V. INTEGER PROGRAMMING FORMULATION

To facilitate the presentation, we will start with a brief de-scription of the variables. We will then introduce the IP modeland discuss each constraint in detail.

is the graph discussed in Section II-A; anddenote the number of vertices and edges in . Each edge

has an associated weight represented by or .As discussed earlier, the number of imaging planes is denotedby and the input, , denotes the total number of 3-D pointsin the seed configuration casting a unique projection on eachimaging plane. The notation refers toa vertex, where is the index of the plane to which the vertexbelongs (e.g., ), and is the index of the 2-D point in . Notethat the set of all incident on is represented aswhere . The 2-D arrayis maintained to enforce geometric consistency as discussed inSection III.

We now discuss the IP variables. As in literature, we representthe indicator variable as . Eachedge has a corresponding which isa linear array of binary values. If a row, is , itimplies that edge is not “chosen” in the solution. On the otherhand, if the th entry of is 1, is chosen as part of the th

4A clique is a graph, such that it has an edge between every pair of vertices.

Fig. 5. Schematic illustration of a few indicator variables and their effect onthe components of the final solution.

CMDS in the IP solution. The indicator variable, , imposesthe clique condition on the vertices chosen as part of a CMDS.If the th vertex (point) in image plane hasa chosen edge (is mapped) to vertex in image plane

, then the value of is 1. can therefore be thoughtof as a variable that keeps track of connectivity information ofvertices (on a CMDS-by-CMDS basis), see Fig. 5. The indicatorvariable, , is used to impose the geometric constraints. is1 if CMDS belongs entirely to a ball, , otherwise it is 0.

We now present our IP formulation. The constraints are indi-vidually explained at the end of the formulation

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)


(9)

...

(10)

(11)

For any triplet of vertices , and such that

(12)

(13)

The term in (1) is the objective that we would like to minimize,(2)–(13) represent constraints that must be satisfied for a goodsolution to our problem and denote the half-spaces that definethe feasibility polyhedron. The first constraint that we imposeis that an edge in the final solution can belong to at mostone CMDS [see (2)]. Next, we define the relationship betweenthe variable corresponding to an edge and the indicatorvariables that correponds to the end vertices of . We knowthat if an edge is chosen (i.e., ), the respective

variable corresponding to the end vertices will be set to one,otherwise it will be zero. In either case, the difference betweenthem must be zero [see (3)]. The next constraint in (4) simplyensures that each subgraph or CMDS will have exactlyedges (as each CMDS is a complete graph with vertices).

Recall that in Section III, we discussed the requirement thatthe correspondences chosen should be mutually close to ruleout cases such as those in Fig. 3(a). We use (5)–(6) to ensurethat a CMDS belongs to one ball “completely.” The notion of“completely” will become clear in the following discussion.First, we want that there should be at least one ball for eachCMDS, such that all constituent edges of that CMDS belongto that ball. The array, , initialized in the preprocessingphase, is useful here. The binary indicator variable is 1 or0 depending on whether or not the CMDS belongs to theball . We have already imposed the requirement in (4) thateach CMDS has edges. Now, if all edges belonging toCMDS belong to some , then whenever is 1,will be 1. Therefore, (keeping and fixed) must be1, implying that if ,and otherwise. While these obser-vations are helpful, they are useless if not incorporated correctlywithin the constraint set. We approach this by a simple trick ofimposing a lower bound of 0 and an upper bound ofin (5). Whenever CMDS must be 0 for the inequalityto be satisfied. If , the lower bound is violated. Similarly,if when CMDS , the upper bound is violatedas . Therefore, must be 1 whenCMDS , thereby enforcing the desired requirement. In(6), we specify that each CMDS must belong completely to atleast one ball by imposing a lower bound of 1 on the sum ofover all . These constraints together ensure our geometricrequirements of the solution discussed in Section IV-A.

In (7), we impose the constraint that there should be exactlyedges incident on any vertex in the final solution. We also

need to verify that these edges belong to the same CMDS. Thisrestriction is imposed using the sequence of conditional con-straints in (8)–(10). The final constraint in this group requiresthat the vertex should be connected to one (and only one) vertexfrom each of the other planes [see (11)]. We use (12) for im-posing the clique condition. Observe that if two vertices on dif-ferent planes are individually connected to a common vertex,these two vertices should be connected among themselves (12).In other words, for any “triplet” of vertices, the sum of thevariables for that triplet should never be equal to 2. To addressthe “ ” sign in (12), we interpret it as follows:

(14)

The “connectivity” constraint in (12) on a triple-by-triple basisalso reinforces the closeness criteria for an entire clique, it isuseful if very high input errors are expected. If closeness is in-terpreted as the existence of a 3-D point for a CMDS that is“close” to every participating projection line , then themutual closeness for the entire CMDS can indeed be expressedas closeness for each triple in that CMDS as given by the fol-lowing observation.

Observation 1: In 3-D Euclidean space, there exists nostraight lines in which every three lines are close but some

choice of lines (where ) are not close.Conditional constraints like “or” such as the ones in (8)–(10)

and (14) above cannot be directly represented in the form of alinear inequality. Fortunately, with some manipulation, they canbe converted to ordinary constraints by introducing additionalauxiliary binary variables, foreach pair in (8)–(10) as follows. For each vertex

(15)

(16)

is satisfied when , andis satisfied when . Observe that these two are con-

ditionally satisfied even though the two constraints in (16) arealways satisfied. This type of conversion is important to preventthe constraint set from becoming ill-posed.

Using a similar approach, the conditional constraint in (14)is converted to an ordinary constraint by introducing anothervariable (one for each triplet in ).

(17)

(18)

The number of constraints in the model is given by the fol-lowing lemma.


Lemma 3: The number of constraints in the formulation ispolynomial.

Proof: Let be given as . From Section III-A,we know that for a constant . The numberof constraints in the constraint set have the following orders:(2) and (3) contribute at most constraints each; (4)contributes at most constraints; (5) contributes at most

constraints because ; (6) contributes at mostconstraints; (7) contributes at most constraints.

Each individual constraint in (8)–(10) contributes at mostconstraints, therefore, together they contribute no more

than constraints. The constraints in (11) contributesat most constraints; (12) contributes at mostconstraints. Since the number of constraints above are additive,the lemma follows.

VI. LINEAR PROGRAM

At this stage, the IP proposed above can be solved directlyusing one of the many standard techniques like branch andbound [28] implemented in any free or commercially availableIP solver such as CPLEX and DASH. The problem is that,in general, integer programming problems are very difficultto solve efficiently.5 In most practical situations, where thevariables are bounded, these problems fall in the class ofNP-hard problems. 0-1 integer programming, of the form wehave here, is a special case of IP where variables are requiredto be binary (0 or 1). This special case is also NP-hard as isthe decision version of the problem (also see [29]). To avoidthese pitfalls, we employ a methodology standard in the fieldof combinatorial optimization. We convert the integer programto its corresponding linear program by relaxing the integralityconstraints and try solving the relaxation optimally instead.This means that variables are no longer strictly binary, they canbe fractional values in . The solution to the linear programwill be an assignment of values in for the variables, whichminimizes the objective function optimally while satisfying theconstraints. For a minimization program such as ours, it is astandard strategy to use the optimal solution obtained from theLP as a lower bound for the IP solution.

The formulation of our linear program is quite identical to theIP. The objective function is represented as (1). The constraintset for the LP is similar to (2)–(7), (11) and the pair in (16)and (17) and is omitted. The primary difference is that now theindicator variables can have values between 0 and 1

(19)

Theorem 1: The LP formulation in Section VI can be solvedoptimally in polynomial time.

Proof: The theorem follows from Lemma 3 and the well-known fact that linear programs can be solved optimally in poly-nomial time using interior point method [30], [31].

5Most mixed integer program (MIP) solvers employ heuristics to speed upthe search and may perform reasonably fast for moderate values of n. This iscertainly a viable option for real-world applications if sufficient computationalresources are available.

A. Rounding Approach

The linear program discussed in the previous section is solvedusing the interior point method, a standard tool for solving linearprograms. We then perform rounding on the fractional LP solu-tion to obtain an integral (0–1) solution for our problem. In gen-eral, rounding can be done deterministically or using randomized(probabilistic) rounding techniques (see [32] for details) with anemphasis on not moving too far away from the optimal LP solu-tion while keeping the solution feasible. Our rounding techniquealso tries to achieve a similar objective. More precisely, we mustnot only determine a low cost solution but must also ensure theformation of disjoint CMDS in the final solution, i.e., the so-lution obtained by rounding should be a feasible solution for thecorrespondence determination problem. To do this, we employa rounding conceptually similar to randomized rounding, but ateach step we perform additional checks to ensure that the require-ment of the formation of CMDS is not violated.

Randomized rounding works under the principle that higherthe fractional value of an indicator LP variable, the more likelyit is to be rounded up to 1. We have two variables, and ,that are directly associated with an edge being chosen. For eachedge with end vertices and , we compute its value as

. The edges are then sorted in non-ascendingorder with respect to their values. Each edge in the sorted list isthen evaluated and rounded to 1 provided that the formation ofcliques is maintained. Since the edges are added incrementally,thefollowingstepsneedtobetakentoavoidconflictwiththeedgesalready selected (or rounded to 1). For any edge .

1) If is already connected to some via roundingto 1 in a previous step, it cannot be

connected to another member in . Hence, the is dis-carded (rounded to 0).

2) If and are connected to different vertices andin the same plane , then is discarded (rounded to 0).

3) If and have a common “connectee,” i.e., they bothhave edges to the same vertex in some plane , thenis chosen (rounded to 1).

4) If the projection line corresponding to the vertex doesnot belong to at least one ball to which and all otherverticesalreadyconnected to belong, then isdiscarded(rounded to 0).

These checks can be easily implemented using appropriate datastructures. This strategy directly yields a good solution in al-most all cases. However, this approach may not give us distinctCMDS directly in extremely degenerate cases; for example, ifa wrong edge is chosen to be a part of a “wrong” CMDS earlyon, it prevents the end vertices of that edge from belonging totheir respective “correct” cliques, at a later step. Here, we mayhave cliques of size less than . However, the sum of vertices thatare a part of such incomplete cliques are always a multiple of .Hence they can be split and merged between themselves to createthe required number of sized cliques using local modifications,if required.

VII. EXTENSION FOR ‘ORDERING’ CONSTRAINTS

In this section, we discuss an extension of the optimizationframework to handle ordering constraints. As an illustration,consider the schematic construction in Fig. 6 that is given from


Fig. 6. (a) Schematic construction of the acquisition setup given from the estimate of the pose parameters. The 3-D seed configuration in the center is forillustration alone and is unknown. (b) Graph representation of the construction in (a). Notice that the correspondence information in (a) indicated by red arrowsis inconsistent with the geometric setup.

the estimate of the geometries that relate the coordinate systemof the multiple views. The construction shows that views, and

are related by a 120 rotation about the -axis. In isvertically above . Similarly, in view 3, is directly above .Hence, we cannot possibly have a correspondence set that maps

to and to because of the inconsistency due to orderswap given the orientation geometry. This suggests that evenif the available orientation information is only approximatelyaccurate, some correspondences are geometrically impossible.If vertex indeed maps to must map to some vertexvertically lower than on to maintain geometric consis-tency and that the two correspondences indicated by bidirectedarrows in Fig. 6(a) and (b) are inconsistent and cannot be truesimultaneously. In other words, an exclusive-or-like (XOR) re-lationship exists between the two correspondences (edges). Ifone is chosen, the other must be discarded. This implies that inFig. 6(b), the variable cannot be 1 simultaneously for edges

and . Such “conflict” information canbe retrieved by the algorithm using some simple preprocessingbased on epipolar geometry. Our objective now is to somehowincorporate this information within our IP and LP models so thatthe optimization proceeds making sure that these requirementsare maintained automatically.

We will first convey the intuition using convex constraints.We will then discuss how the requirement can still be imposedin linear form by some manipulations. Let us assume that suchconflict information is available via preprocessing in the formof a conflict matrix defined as follows.

Let us introduce an indicator variable defined as

(20)

Our objective of imposing the XOR constraint can be accom-plished using the following convex constraint:

(21)

For (21) to be satisfied, if is 1 ( and must not chosensimultaneously), either or should be 0 (discarded). Onthe other hand, if is 0, (21) will be satisfied irrespectiveof the values of and . The constraint, however, involvesthe product of two variables. To represent this in linear form, weemploy some additional manipulations. We achieve our objec-tive in the following matrix argument form:

(22)

In (22), is a row vector and is acolumn vector. If two edges and are chosen, we have

. Therefore, will exceed 1 if . For (22)to be satisfied, either (no conflict between edgesand ) or one among and is 0.

VIII. EVALUATION AND RESULTS

The evaluation of the proposed technique was performed byimplementing the algorithm in C++ using CGAL [33], LEDA[34], and GLPK [35] on a machine running GNU/Linux.


A. Simulation Data

To artificially simulate a seed implant, input configurationsof seed points were generated using a normal dis-tribution inside a 3-D box with sides 5 cm. Hereafter, we callthis the simulated distribution. To perform evaluations on clin-ical distributions, we used the reconstruction results based oncorrespondence solutions calculated for clinical 2-D datasets(clinical evaluations are discussed later in Section VIII-C). Notethat while the correspondences from the algorithm may not be“perfect,” the 2-D reprojection of the reconstructed 3-D distri-bution (50–60 seeds) was consistent (qualititative and quantita-tive) with the input 2-D views, hence, it serves as a good 3-Dclinical distribution. Hereafter, we call this the clinical distribu-tion. The 3-D points were then projected onto multiple imagingplanes, i.e., , yielding sets of 2-D points6. Thesource-to-image distance (SID) was 120 cm (for simulated dis-tribution) and 144 cm (for clinical distribution). For the simu-lated distribution, the magnification (SID/source-to-object dis-tance) was assumed to be 1.2, and the pixel size was 0.02 cm.The input image error introduced in the 2-D image points wasselected from normal distributions having values of 0 cm,0.01 cm, 0.02 cm, 0.03 cm, 0.04 cm, 0.05 cm (up to 2 pixels).The input geometry error introduced in the angles of the relativegeometries was selected from normal distributions with values

. The initial values forthe two primary angles (RAO-LAO and CRA-CAU) of the ge-ometry acquired from the imaging system were independentlychosen in the range, and , with at least15 relative separation from another geometry in the -set. Foreach level of input error, 50 realizations were performed, andthe mean of the results was analyzed.

In Fig. 7, we plot our evaluation results for simulated andclinical distributions. In Figs. 7(a) and (b), we plot the per-centage of correct matchings (cliques) determined as a functionof introduced image error and geometry error, respectively. Forlow introduced image error [in Fig. 7(a)], the technique deter-mines an almost perfect set of mappings. For high image error(up to 0.05 cm or pixels), the algorithm consistently de-termines about 94%–96% of the matches correctly for simu-lated and clinical distributions. These results are encouragingbecause the image error introduced in the 2-D points was notcorrelated—for instance, in our simulations, two neighboring3-D points in the seed-configuration will cast close 2-D projec-tions on the imaging planes. If the introduced image error isgreater than the distance between those two 2-D point pro-jections, it is likely that in some cases, their mutual ordering willbe reversed (after error introduction). A small percentage of theincorrect matches reported in the plots above can be attributedto such degeneracies. In Fig. 7(b), we illustrate the performanceof the algorithm as the introduced geometry error is increased.We notice that the algorithm performs extremely well, even forintroduced geometry error of up to 1.5 , and determines thecorrect mapping in about 98% of the cases. Once the correctmappings were determined by the algorithm, the corresponding

6With reasonably accurate geometries (errors� 1 ), root mean square (rms)errors in reconstruction tend to reduce significantly for k � 5 views; this hasbeen observed independently [13].

point -tuples (cliques of size ) were used to reconstruct theset of 3-D points by a simple stereo triangulation procedure.The reconstructed points were then compared to original points,and the 3-D rms errors were calculated [see Figs. 7(c) and (d)].We notice that the 3-D rms errors increase almost linearly forsimulated and clinical distributions with introduced image error[Fig. 7(c)], as expected. In Fig. 7(d), we plot the errors as afunction of introduced geometry error. Additionally, one mayalso want to calibrate the geometry using determined correspon-dences (e.g., via trifocal tensors) for improved accuracy in termsof 3-D rms errors.

B. Phantom Data

To evaluate the effects of gantry and other real imagingsystem errors on the performance of the algorithm, testing wasperformed on data acquired from a phantom experiment. Weimplanted 24 seeds (ball-bearings) inside a phantom (sides ofabout 5 cm). The diameter of the seeds used was about 2.2mm. Using a rotational angiography protocol, 210 images wereacquired at 1 intervals using a C-arm gantry (Toshiba Infinix).The acquisition parameters were as follows: resolution was512 512, exposure time was 9 ms, field-of-view was in,SID was 110 cm, source to object distance (SOD) was 82 cm,pixel size was 0.039 cm, and acquisition rate was 40 frames persecond.

After acquisition, the image sequence was corrected for dis-tortion (note: this was the only form of postprocessing employedfor error correction). However, as is common with all acquisi-tions, other forms of errors in the system still propagate into theimage data. For instance, we know from experience that the er-rors in the angles (precision) reported by the gantry used for theexperiments are up to 1 and the image errors are up to 1 pixel.Additional errors come from gantry instability (wobbling andsagging). From past calibration experiments, we know that theerrors in the center of rotation (isocenter) and focal spot posi-tioning are each about 1 cm. The evaluations of the algorithmreported in this section correspond to performance in the pres-ence of such errors.

A simple edge detection and segmentation (with morpholog-ical operators) using Matlab was performed on the output fromthe distortion correction method and fed to the algorithm asinput (see Fig. 8). Note that the correspondence determinationalgorithm is transparent to the image processing performed inthe first stage; the input to the algorithm is sets of 2-D points[center of mass denoted as ], hence, segmentation in thepresence of image noise must be addressed using more sophisti-cated image processing routines. In case of low contrast clinicalimages, a user indication to identify the center of mass of theseeds must be adopted (as shown later in Fig. 12).

To establish a ground truth model for the evaluations, the 2-Dseeds were tracked through the image sequence to determinecorrect seed correspondences. The complete set of 210 projec-tion images were also used for a CT reconstruction using a fil-tered back projection procedure for use as “truth” (see Fig. 9).A reconstruction of the seed configuration subsequent to cor-respondence determination was registered with the truth anddistance measures were calculated between the two reconstruc-tions.


Fig. 7. Evaluations for simulated distributions (in solid black) and clinical distributions (in dashed red): (a) percentage of correct matches as a function of imageerror; (b) percentage of correct matches as a function of geometry error; (c) 3-D rms errors in reconstructed points as a function of image error; (d) 3-D rms errorsin reconstructed points as a function of geometry error.

In all, 200 runs of the program were performed on thephantom data. For each run, views with at least 20relative separation were chosen from the image sequence andused as input to the algorithm. Since the correct correspondenceinformation was known, we calculated the percentage of incor-rect matches determined by the algorithm for all runs. This wasdone as follows. We labeled a CMDS to be incorrect if it hada single incorrect vertex as a member. Therefore, a CMDS (or

-clique) qualified to be correct only if all its members werepicked correctly7. We observed that the algorithm correctlydetermined % cliques perfectly on average. Among theremaining 7%, about 5% of the matches had just one incorrectvertex. The other 2% were due to erroneous choices made inthe rounding procedure.

In Fig. 10(a), we plot the histograms of 3-D rms errors. Thiswas done by first reconstructing the unknown seed configura-tion based on calculated CMDS. This configuration was thentransformed to the CT coordinate system and one-to-one map-pings were established between the 3-D points in the segmented

7The incorrect k-clique counting procedure is rather strict because even if onevertex in a k-clique is incorrect (ideally should have been assigned to anotherk-clique), we consider the entire k-clique to be incorrect.

CT volume and the 3-D points in the calculated reconstruction(using bipartite graph matching [36]). We then quantified thedistance variations and plotted the histograms. As illustrated inFig. 10(a), the 3-D rms errors were about 2 mm on average.We also noticed that the rms errors were lower in cases wherea fourth projection view was used because of the additional in-formation given by the additional view.

Our final evaluation of the algorithm was aimed towardsquantifying its performance in terms of global optimality. To dothis, we determine the performance ratio of our solution withrespect to the optimum integral solution. The idea is briefly asfollows. If is a solution determined by our algorithm andis the unknown optimal, if the relationship, holds(where ) for all solutions, then is a -approximationsolution to the problem (see [32] for an in-depth discussion).The optimal solution, , is obviously unknown. Luckily, afew standard ideas still allow us to estimate the performanceby using the guaranteed optimal linear program solution as alower bound on the best possible integral solution (i.e., ). It isstraightforward to see that the performance ratio for the lowerbound extends to . Since (1) is a minimization program, wetake the ratio of the linear program solution and the solution


Fig. 8. Images (a)–(c) show three views of the seed configuration in the image sequence. Images (d)–(f) show the 2-D points (center of mass) for use as input tothe algorithm. We used a simple segmentation to obtain (d)–(f), but for low contrast clinical images more sophisticated segmentation or user indications may berequired.

Fig. 9. CT reconstruction of the seed configuration on a Vitrea workstation. CT data was used for evaluations.

after rounding with respect to the corresponding values ofthe objective function. This value directly indicates the lossin optimality due to rounding for each instance. The plots inFig. 10(b) show that the empirical performance of the algorithm(in terms of quality of solutions) is very good, we are alwayswithin a factor of 1.25 of the lower bound.8

8This is a slight underestimation of the performance because the best possibleintegral solution will be higher than the LP lower bound.

C. Clinical Data

The algorithm was evaluated on clinical data (four patientcases) acquired from the Radiation Oncology division atRoswell Park Cancer Institute, Buffalo, NY. Several views ofan implant are shown in Fig. 11. Fig. 12 shows an enlargedversion of the images showing the region of interest (the userindication of the center of mass of the seeds is in red for bettervisibility). The centers of the circular regions (user indications


Fig. 10. (a) Three-dimensional rms errors in reconstruction. (b) Performancein terms of approximation ratios obtained in phantom evaluations.

of seeds) in the images were extracted for use by our algorithm(see Fig. 12). The acquisition parameters were: cmand cm.

The 2-D projection points were used as input to our algorithmtogether with available geometry information; consistent withpractice, all clinical data corresponds to anterior–posterior (AP),lateral, and two oblique views. Once the corresponding CMDSwere determined, they were used for a 3-D reconstruction of theimplanted seed configuration. The reconstructed 3-D configura-tion was then reprojected onto the imaging planes. To calculatedistance measures, a bipartite graph matching was employedbetween the reprojected seeds and the input 2-D seeds. Then,2-D distance deviations were calculated. We found that the av-erage errors were about 1.2 mm ( distance) and are listed inFig. 13(a). A representative histogram of reprojection errors forone of the clinical datasets is shown in Fig. 13(b). These smallerrors and consistency observed between reprojected and input2-D points indicate that the algorithm is robust and performswell on clinical data. It should also be noted that while a frac-tion of the 2-D reprojection errors can be attributed to a smallnumber of incorrect matches determined by the algorithm [rightend of the histogram in Fig. 13(b)], a fair percentage of the errorcomes from errors in the input data (converting the radiographslides into images, segmentation, user indications, etc.). WhileFig. 13(a) summarizes the performance of the algorithm on alldatasets, the distribution of errors can be better inferred fromFig. 13(b)—we can observe that over 95% of the errors are con-centrated around the region from 0.75 mm to 1.5 mm. Even withperfect correspondence determination based on expert indica-tion, reprojections will still have error usually as a function oferrors in the input image data. If higher accuracy is desired, onepossibility is to use a pose calibration technique [16] after cor-respondences have been determined.

D. Note on Experimental Evaluations

Shape of Seeds: Our phantom experiments used ball-bear-ings instead of real elliptical seeds, therefore, the diameter of the“seeds” in the phantom was larger than those employed in clin-ical practice. In general, the algorithm is immune to the shapeand size of the seeds because it uses the center of mass of theseeds as input— sets of 2-D points (e.g., where each point isa tuple ). However, in practice, segmentationof elliptical seeds (and calculating the center of mass) may notbe perfect which may introduce additional errors in the input2-D points. These may propagate as errors in the reconstruc-

tion of the configuration. Therefore, while the shape of the seedsdoes not matter because we are concerned only with its center ofmass (as discussed above), one may expect the 3-D rms errorswith noncircular shapes to vary slightly especially if an imper-fect segmentation algorithm is employed. To avoid such addi-tional image processing errors, we used ball-bearings that alloweasy morphological segmentation.

Angles between the views: Our simulation and phantom ex-periments used both limited angle views ( separation)and views with sufficient separation , for which wediscussed various error measures. The clinical data we had ac-cess to corresponds to AP, lateral, and two oblique views. Forthis data, we discussed reprojection accuracy because postoper-ative CT reconstruction was unavailable. From our simulationand phantom experiments, we did not observe a significant re-lation between reconstruction errors and the angular separationbetween the views. However, we noticed that bringing in an ad-ditional view with a good separation does help in reducing the3-D rms errors.

IX. DISCUSSION

A. Implementation Issues

Hidden Seeds: Like other existing approaches [8], [13] forthis problem, our algorithm also does not attempt to tackle thehidden seeds problem (where a seed projection on one view may“correspond” to two seed projections in one or more of the otherviews), see Section II-A. Since it is standard clinical practice toacquire views that yield significantly more mutual information,typically only a few seeds are hidden. Hence, this issue can beaddressed in a heuristic postprocessing phase with nominal userintervention by first executing the algorithm to find the CMDS(cliques) in and then reprojecting the reconstructed configu-ration onto the views to determine seeds that are likely to be“hidden.” This can be done in a manner similar to the two-viewcase by “assigning a group of pixels in one image to a singlepixel in the other image” (see [37]) and evaluating the improve-ment in reprojection consistency. One may also automate thisprocess by using ideas in [38] to determine the hidden seeds viaprobabilistic evaluation based on marginal densities.

Exploiting the Sparse Matrix Structure: The input modelto most LP solvers is supplied in matrix form. Notice that thematrix for the set of linear inequalities in our model (2)–(17)has only a small fraction of nonzero entries. If all relevant geo-metric information has already been determined, the LP can bedirectly modeled in AMPL [39]. The analysis of the zero (andnonzero entries) and the use of such information efficiently ishandled by the solver internally; GLPK [35] allows the user toprovide this information via the function call,other libraries also have related functions after which the solvertakes only 30–40 s to obtain a solution. If the algorithm is em-ployed for the general correspondence determination problem,the sparse structure of the matrix can be exploited via such op-tions.

B. Improvements and Future Works

We believe that an interesting direction of future research is toexplore if the constraint set can be further refined. For instance,


Fig. 11. Digitized images of three views of a prostate seed implant.

Fig. 12. Enlarged views of the region of interest in the three views of seed implant. The user indication of the center of mass of the seeds represents it as a point(x; y), which is then used by the optimization algorithm.

Fig. 13. (a) Reprojection error evaluations for clinical cases. (b) Representativehistogram of 2-D errors between input 2-D points and reprojected points fromreconstructed 3-D seed configuration on clinical data (plot corresponds to case#1).

we observed that the linearization of (12) given by (14) can bestrengthened by using the following “rectangular” form:

Additionally, (4) can be discarded, the other constraints (collec-tively) are sufficient to enforce this requirement.

X. CONCLUSION

We have proposed a novel algorithm for solving an impor-tant problem in prostate brachytherapy. The formalization usesparadigms from computational geometry and mathematical pro-gramming and enables transforming the BSL problem into aninteresting optimization problem. To perform the optimizationefficiently, we formulate it as an IP and then solve the equiva-lent linear program. We propose techniques for rounding the LPsolution to yield an integral solution to our problem. We focuson the theoretical and practical aspects of the formulation andpresent evaluations based on our implementation. We believethat the algorithm can be implemented as is to tackle the gen-eral problem of determining point correspondences in multipleviews and for providing seed visualization and implant analysisin the clinic. In many applications, if imaging system errors arenot atypically pathological, some constraints (that are used to


address worst case situations) may be discarded based on ex-perimental analysis. The algorithm works very well in practiceand yields % accuracy in matching corresponding 2-D seedpoints. In addition to the BSL problem, we believe that the ideaswill be useful in designing efficient strategies to solve other re-lated problems as well.

ACKNOWLEDGMENT

This work was supported in part by NIH Grant HL 52567,NSF CAREER award CCF-0546509 and Toshiba Medical Sys-tems Corporation. We thank Barbara Cunningham for acquiringthe patient images.

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