Finite-Element Level-Set Curve Particles

Tingting Jiang and Carlo Tomasi∗

Department of Computer Science, Duke UniversityDurham, NC 27708

{ruxu, tomasi}@cs.duke.edu

Abstract

Particle filters encode a time-evolving probability den-sity by maintaining a random sample from it. Level setsrepresent closed curves as zero crossings of functions oftwo variables. The combination of level sets and particlefilters presents many conceptual advantages when trackinguncertain, evolving boundaries over time, but the cost ofcombining these two ideas seems prima facie prohibitive. Aprevious publication showed that a large number of virtuallevel set particles can be tracked with a logarithmic amountof work for propagation and update. We now make level-set curve particles more efficient by borrowing ideas fromthe Finite Element Method (FEM). This improves level-setcurve particles in both running time (by a constant factor)and accuracy of the results.

1. Introduction

Tracking an animal in video, a wildfire on the ground, ora drifting cloud in the sky involves following boundaries onthe plane that move and deform over time: the silhouette ofthe animal, the fire front, a vapor iso-density contour in thecloud pattern.

When the region within the boundary splits and recom-bines, and as it develops and refills holes, it becomes the-oretically more elegant and practically simpler to representthe region’s boundary as the zero-crossing of a scalar func-tion of the two coordinates on the plane. This is the wellknown level set representation of curves. Although encod-ing a function of two variables is more expensive than en-coding a set of curves, changes in geometry and topologybecome trivial to handle. Computation is now much moreuniform and leads to practical efficiency and simple code.

In many applications, knowledge about the curve’s posi-tion and shape (the curve’s “state” for short) comes from asparse set of uncertain measurements. Particle filters havebeen used successfully in many contexts to track and re-

∗This research was partially funded under NSF grant IIS-0534897.

fine the state of evolving objects from sparse measurements.These filters maintain random samples (a set of particles)out of the probability density that represents the currentknowledge about the object’s state. Particles are propagatedas the state changes, and are updated when new measure-ments become available.

Tempting as the combination of level sets and particlefilters may be, it immediately raises the specter of compu-tational complexity: the space of curves is infinite, and adensity in such a space needs many samples indeed for afaithful representation. If each particle is a set of curvesencoded by a level set, the complexity of the naıve com-bination of these two representational ideas multiplies thenumber of particles by the cost of handling a level set func-tion. This product is invariably a large number, and this ex-plains why the merge of these ideas has not been attempteduntil recently.

The internal structure of individual particles as sets ofcurves was exploited in [10]. It was shown that it makesmathematical sense to “add” two sets of curves by addingthe level set functions that represent them. Using this in-sight, one can in effect track C particles while reaping thecombinatorial benefit of implicitly considering the superpo-sition of all the 2C combinations of the C sets of curvesthat are explicitly tracked. This device made level-set curveparticles feasible. In that paper, level set functions weremixtures of Gaussian functions. In contrast, we now showthat a representation based on the combination of piecewise-linear functions, in the spirit of the Finite Element Method(FEM), leads at the same time to shorter running times (by aconstant factor) and greater accuracy in the results ( Fig.1).

2. Related work

Level sets [14, 3, 13] describe the boundary B of anevolving (not necessarily connected) region on the plane orin space as the zero crossing of a function φ(�x, t) of spaceand time. While many functions φ share the same zero-crossing B, computational considerations [15] suggest us-ing the signed distance function of B, which is then main-tained in a narrow band [1] around the boundary B. The

978-1-4244-1631-8/07/$25.00 ©2007 IEEE

10 20 30 40 50 60 70 80 90 1000.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of measurement points

Err

or ra

te o

f the

reco

nstru

cted

col

on v

olum

e

Gaussian

Test point for Gaussian

FEM

Test point for FEM

Figure 1. Compare error rates of FEM and Gaussian methods withdifferent number of measurement points for human colon tracking.

motion model is then a partial differential equation (PDE)for φ. The main strength of level sets is that they accounteffortlessly for changes of boundary topology, as exempli-fied in applications to image segmentation [4], object detec-tion [16], tracking [18], shape modeling [12] and medicalimage segmentation [17], among others.

Particle filters [6] have been used in this context mainlyin the area of active contours [9, 2]. These approaches cap-ture the uncertain position of a boundary at time t by aprobability distribution represented by a random sample ofboundaries (particles). Each boundary is represented ex-plicitly, e.g. with splines [5] and propagated forward intime through an assumed, uncertain motion model. Mea-surements update the particles by weighing each particle byits posterior probability given the measurements. Resam-pling draws new particles from the posterior to be readyfor a new step of propagation. This cycle is analogous tothe estimation loop of a Kalman filter [11], but maintains amulti-modal distribution rather than a Gaussian one.

The recent, full-fledged combination of level sets andparticle filtering called “Level-Set Curve Particles” is re-viewed in the next section.

3. Level-set curve particles

In [10] a method was proposed to represent and track adynamic set of closed curves (a “boundary”). This methodcapitalizes on the observation that many of the curves thatpopulate the boundary distribution being tracked are verysimilar to each other. A base curve which is called “mother”particle accounts for the macroscopic shape of the bound-ary. The base curve is then deformed by C base perturba-tions which provide an implicit representation of the 2C de-formations obtained by applying any subset of the C baseperturbations to the base curve. Through this device, anexponential number of curves can be propagated and theirlikelihoods can be computed at linear cost in the frameworkof a particle filter.

More specifically, a particle set χ(t) = {φ(t)c }C

c=0 rep-resents the probability distribution of the boundary B(t) attime t. Each particle is a signed distance function φ

(t)c :

R2 → R whose zero level set denotes an estimate of the

boundary B(t). A weight w(t)c determined by the measure-

ments is associated to each particle. The function φ(t)0 is

the “mother” particle and the other particles {φ(t)c }C

c=1 areexplicit “child” particles. Each particle is associated witha weight w

(t)c . In fact, there can be NM such sets to rep-

resent multi-modal probability distribution of the boundary.For simplicity, only one particle set is discussed in the fol-lowing. The result can be extended to NM particle sets in astraightforward way.

Gaussian mixtures were used in [10] to approximate thedifference Δφ(�x, t) between mother particle and child par-ticles. Assume that Δφ(�x, t) =

∑C

c=1 ac(t)Gc(�x, t) whereGc(�x, t) is initialized as a 2D Gaussian function with ran-domly generated mean and covariance matrix at t = 0 andthen propagated by the motion field over time. ac(t) is ascalar function of t. C is the number of explicit “child”particles. In fact, ac(t) is restricted to be ±1 or 0, therebyleading to 2C combinations of “implicit” child particles. Inother words, the representation of Δφ is simplified as a C-bit binary vector with C functions which are initialized asGaussian functions and changing over time.

The outline of the resulting tracking method is Algo-rithm 1. At each time step, the input is the particle setχ(t−1) from last time step and the output is the new particleset χ(t). Q

(t)m denotes the estimate of a point on boundary

B(t) returned by each of M observers.

Algorithm 1 : Tracking algorithm

INPUT: χ(t−1) (t > 0)OUTPUT: χ(t).(1) Propagate χ(t−1) to χ(t);(2) Take measurements {Q(t)

m }Mm=1;

(3) Update χ(t) to χ(t);(4) Resample χ(t) and generate χ(t).

Initialization: The first iteration of Algorithm 1 requiresχ(0) which is initialized as follows. The mother particle ofχ(0) is defined as φ

(0)0 . Its explicit child particles are initial-

ized by choosing C different Gaussian functions and com-bining them with φ

(0)0 . Specifically, the initial particle set

is denoted as χ(0) = {φ(0)0 , {W

(0)c , Δφ

(0)c }C

c=1, {w(0)c }C

c=0}

where Δφ(0)c represents the difference between mother and

child c which is only nonzero inside window W(0)c and w

(0)c

is the weight of particle c. All particles have the same ini-tial weight. If the region of the plan of interest is discretizedinto a grid of size N , initialization takes O(CN) time.

Propagation: Suppose the motion field at each time stepis externally given by a smooth velocity field �V (�x) on theplane. Both mother and explicit child particles are propa-gated by the level set equation:

φt = −�V · ∇φ (1)

which takes O(CN) time. The particle set becomes χ(t) =

{φ(t)0 , {W

(t)c , Δφ

(t)c }C

c=1, {w(t−1)c }C

c=0} where φ(t)0 is the

propagated mother and the difference between mother andchild c is still represented by Δφ

(t)c within window W

(t)c .

The weights do not change during propagation. Because ofthe linearity of the gradient operator ∇, implicit child parti-cles are implicitly propagated for free.

Update: Before update, each observer returns a noisymeasurement Q

(t)m which is the position of the closest point

on the boundary. The likelihood function used to evaluatethe particles is defined as follows:

Λ(t)(φ) =

M∏

m=1

G(φ(Q(t)m )) (2)

where G is a Gaussian function whose standard deviationζ depends on the noise statistics of the measurements. Themother’s weight is updated as w

(t)0 = Λ(t)(φ

(t)0 ) · w

(t−1)0 .

Since φ(t)0 is always maintained as a signed distance func-

tion, φ(t)0 (Q

(t)m ) can be obtained as a simple lookup. Update

of the mother particle takes O(M) time. The likelihoodsand weights for explicit child particles can be obtained bymodifying those of mother particle based on the differencesΔφ

(t)c at a cost of O(CM).A small boundary component is added to the mother par-

ticle whenever the measurement information implies that anew boundary component has appeared. This has a worst-time cost O(MN).

Resampling: To resample all explicit and implicit childparticles, all their weights must be known and thereforeevaluation of the weights of implicit children is necessary.Two cases are discussed in [10]: disjoint windows and inter-secting windows. For the latter case, the following lemmais proven in [10]:

Lemma 1 Let K(t)c =

w(t)c

w(t)0

and K(t)c =

w(t)c

w(t)0

.

Given Z ⊆ {1, . . . , C}, suppose that the com-

bined particle φ(t)Z has weight w(φ

(t−1)Z ) = w

(t−1)0 ·∏

c∈Z K(t−1)c

∏j,k∈Z,j �=k S

(t−1)jk before propagation. Af-

ter update, the weight of φ(t)Z is w(φ

(t)Z ) = w

(t)0 ·∏

c∈Z K(t)c

∏j,k∈Z,j �=k S

(t)jk where S

(t)jk = S

(t−1)jk · I

(t)jk and

I(t)jk = exp(

∑Q

(t)m ∈W

(t)j

T

W(t)k

−2Δφ(t)j

(Q(t)m )Δφ

(t)k

(Q(t)m )

ζ2 ).

This lemma tells us that besides χ(t), it is also necessaryto maintain an intersection factor set {S(t)

jk } for every pairof explicit child particles over time. This factor is de-fined recursively by the lemma itself, as S

(t)jk is updated as

S(t)jk = S

(t−1)jk · I

(t)jk . With S

(t)jk , the weights of implicit par-

ticles can be obtained. Then the goal of resampling implicitparticles can be achieved by changing the weights of ex-plicit particles. Resampling costs O(2C) but the constantfactor that multiplies 2C is small. For the case of intersect-ing windows, the cost of maintaining intersection factors{S

(t)jk } is O(MC2).

Complexity: The total complexity for Algorithm 1 isO(2C + CN + MN + MC2). A detailed representationof shape variations requires C = O(N), so the asymptoticcost of Algorithm 1 is O(2N + MN2).

4. FEM curve particles

Experiments on tracking the boundary of a colon in to-mographic imagery from sparse edge measurements showthe promise of the above approach. However, the track-ing results are “bumpy” compared to the ground truth. Tomake the tracking results smoother and the tracking ap-proach more efficient, we replace Gaussian perturbationswith finite element perturbations.

The finite-element method (FEM) [8] originated fromthe needs for solving complex elasticity, structural analysisproblems in civil engineering and aeronautical engineering.It can be used for finding approximate solution of PDE aswell as of integral equations such as the heat transport equa-tion. We are first inspired to use finite element perturbationsbecause of their linearity between the nodal points. If thepropagations of the grid points between the nodal points canbe implicitly interpolated by the propagations of the nodalpoints, a significant amount of time can be saved. Second,the approximation error for FEM is inherently better under-stood than that of mixtures of Gaussian functions.

In this section, we show that FEM is a better way thanGaussian functions to represent the differences Δφ(t) be-tween mother and child particles.

Assume that Δφ(�x, t) =∑E

e=1 de(t)Te(�x) where Te(�x)denotes the time-independent finite element function andthe coefficient de(t) is a time-dependent real number. Eis the total number of elements. This representation es-sentially projects Δφ onto E elements linearly and reducesthe boundary tracking problem to that of estimating a time-dependent vector {de(t)}

Ee=1 such that the boundary un-

der tracking can be represented by the zero crossing ofφ(�x, t) = φ0(�x, t) +

∑E

e=1 de(t)Te(�x).There are several choices for 2D finite elements includ-

ing the bilinear quadrilateral element, isoparametric ele-ment, linear triangular element, etc. [8]. For convenience,

we choose the linear triangular element. The N grid points

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(a)

R

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1

(b) (c)

p

p

p23

1

Figure 2. Illustration of the linear triangular element. (a) Triangu-lar domains. (b) The element shape function. (c) The hexagonal“tent” Te formed by the six element shape functions that share anodal point.

is divided into R × R rectangles each of which has N/R2

grid points and is cut by its diagonal into two triangles(Fig. 2 (a)). Each triangle is a domain which has threenodes. For example, the triangle Δp1p2p3 is a domain. Theelement shape function is illustrated in Fig. 2 (b). There are(R − 1)2 nodal points on the plane excluding those on thebounding box. Each nodal point has six connected neigh-bors and is adjacent to six element domains. The combina-tion of six element shape functions which share same nodalpoint looks like a tent in Fig. 2 (c). These tents are the el-ements Te which will be used to represent Δφ. There areE = (R − 1)2 elements on the plane each of which canperturb the mother particle in a different area (Fig. 3).

Now the particle set χ(t) can be viewed as one motherparticle φ

(t)0 plus a set of C E-dimensional vectors each of

which corresponds to an explicit child particle. Let a C byE matrix D(t) represent the vectors s.t. for every explicitchild particle φ

(t)c = φ

(t)0 +

∑E

e=1 D(t)(c, e) · Te. Sincethe E elements are fixed, the particle set can be written asχ(t) = {φ

(t)0 , D(t), {w

(t)c }C

c=0}. Compared to [10] , onlythe representation of the explicit child particles is changed.The way to combine them is same and there are still 2C

implicit child particles.Based on the new representation, Algorithm 1 can be

implemented with the following differences from previous

work. First of all, the particle set χ(0) is initialized bychoosing C elements from the above E elements and addingthem to φ(0). In particular, C ≤ (R − 3)2 because only thenodal points in the middle of the plane have all the neigh-bors around them. See the red points in Fig. 2 (a). Each el-ement perturbs χ(0) in a limited area. Fig. 3 shows how thearray of tent functions influence the boundary when R = 8and C = 12. The blue curve is the mother particle and thered curves are child particles. The red window denotes theposition of each tent. After initialization, the particle setis χ(0) = {φ

(0)0 , D(0), {w

(0)c }C

c=0} where D(0) is the rep-

resentation of the C explicit child particles and w(0)c is the

initial weight which is same for all particles.Second, the propagation of each tent function Te can

be approximated by only propagating function values at itsnodal points because the tent function is piecewise linearand the motion field �V (�x) is smooth. Specifically, use thelevel set equation Eqn. (1) to calculate the function valuechange for each nodal point p in the nodal point set NP (e)of Te. After propagating Te, construct a matrix U of size Eby E such that each row of U represents how the propagatedTe is approximated by its neighboring tent functions and it-self, i.e., Te− �V ·∇Te ≈

∑p∈NP (e) U(e, p) ·Tp where both

e and p are the indices of the nodal points correspondingto the tent functions. The changes for nodal points on thebounding box is set to zero. If the gradient at a nodal pointis not well-defined, take the average of gradients from dif-ferent directions. Because of the linearity of the ∇ operator,we can propagate each explicit child particle by taking theproduct of D(t−1) and U as D(t) = D(t−1) ·U . After propa-gation, the particle set is χ(t) = {φ

(t)0 , D(t), {w

(t−1)c }C

c=0}.Third, to update each explicit child particle, compute its

signed distance function φ(t)c and then φ

(t)c (Q

(t)m ) becomes

available by lookup. Evaluate the likelihood of the particleby Eqn. (2) and update its weight.

Last, to maintain the intersection factor set {S(t)jk }, define

I(t)jk =

∏M

m=1 exp(−2Δφ(t)j (Q

(t)m )Δφ

(t)k (Q

(t)m ))/ζ2). Note

that child particles are allowed to intersect in any combina-tion. Even so, thanks to the quadratic form of the exponentsin the Gaussians of the likelihood function, only pairwiseintersection factors need to be maintained.

The computational complexity of the algorithm is ana-lyzed as follows. For initialization, the mother particle andtent functions cost O(N), and the matrix D costs O(C).Propagation advances the mother particle by the motionfield, at a cost O(N) and E tent functions by O(E) becauseeach tent only takes constant time. Then propagation of ex-plicit child particles requires O(CE) time. For update, themother particle takes O(M) time and the explicit childrentake O(NC + MC). The cost of maintaining intersectionfactors S(t) is O(MC2). Resampling still costs O(2C). Thegeneration of new components takes at most O(MN) time.

Figure 3. Explicit particle set generated by adding tent functions to a mother particle (blue). Resulting perturbations are in red.

Therefore, the total complexity for the proposed trackingalgorithm is O(2C + CE + NC + MC2 + MN). Asymp-totically, E = O(C) = O(N), so the cost is O(2N +MN2)which is same as for the Gaussian approach.

If we had used a standard particle filter with 2C =O(2N ) explicit particles, the cost for propagation and up-date would be O(2N (N + M)) which is much higher thanthat for the proposed approach. If we use NM particlesets each of which has a different mother particle to rep-resent the probability distribution of the boundary, bothcomplexities are scaled by NM because the algorithm isrun NM times for each of the particle set. If we use theso-called narrow-band version of LSM [1] , assume thatthe number of grid points in the narrow band is NB andthe number of related elements is EB . The total cost isO(2C + CEB + CNB + MC2 + MNB). Asymptotically,EB = O(C) = O(NB), so the cost is O(2NB + MN2

B).If we had used a standard particle filter with 2C explicitparticles the cost for propagation and update would beO(2NB (NB + M)), which is much higher.

5. Experiments

We implemented the proposed FEM-based level-setcurve particles on two different data sets: human colon dataand sea surface temperature data, and compare the resultswith those of Gaussian-based level-set curve particles.

5.1. Colon Tracking

The algorithm in Section 4 has been run on the dataset used in [10] which is a sequence of 355 slices from

a Computerized Axial Tomography (CAT) scan of a humancolon. Results were compared to those reported in [10].The boundaries being tracked are the cross-sections of thecolon with the horizontal slices of the CAT volume. Bound-aries appear and disappear as from one slice to the next (SeeFig. 4 (a)). In the experiments, each boundary was first re-constructed by a standard edge detector and the resultingboundaries were taken as ground truth. The tracker on theother hand can only access the ground truth through 100measurement points in each slice. Fig. 4 (b) is the 3D re-construction of the tracking results from the proposed FEMapproach and Fig. 4 (c) is the 3D reconstruction with Gaus-sian perturbations from [10]. Both tests use 10 explicitchild particles and 100 measurement points per slice. Thegrid size is 512 × 512 and the element array size is 8 × 8.Although the asymptotic complexities for both approachesare same, practical running times are different. The averagerunning time for the FEM approach is about 11s per slicecompared to 20s per slice for the Gaussian approach.

More importantly, the FEM approach yields more accu-rate results. Accuracy is measured by the error rate definedas the ratio of the symmetric difference between the 3D re-constructions of the tracking results and the ground truth

over the volume of the ground truth. Specifically, let B(t)

denote the area enclosed by the tracking result boundary andB(t) denote the area enclosed by the ground truth boundaryat time step t respectively. Then the error rate is defined as

ErrorRate =Σt|(B

(t)\B(t)) ∪ (B(t)\B

(t))|

Σt|B(t)|

(3)

Figure 4. Compare 3D reconstructions of ground truth (a), FEM tracking results (b) and Gaussian tracking results (c).

where | · | means the size of the area or the number of gridpoints inside the area in discrete case.

Our results are compared to the previous Gaussian ap-proach with the same number of child particles in Fig. 1.With decreasing number of measurement points, the errorrate of FEM approach is increasing more slowly than thatof the Gaussian approach.

5.2. Sea Surface Temperature Tracking

Sea surface temperature (SST) is the water temperatureat the sea surface. SSTs above 26.5 degrees C are favor-able for the formation and sustaining of tropical cyclones.In general, the higher the SST, the stronger the storm. Sotracking the isotherms of high SSTs is very helpful in fore-casting tropical cyclones and hurricanes.

We tracked the SST isotherms in a data set where denseinformation of the boundary is available, but we simulatethe sparseness of observers by withholding the dense infor-mation from our tracking algorithm. Specifically, we ob-tain NOAA ERSST V2 data 1which is an extended recon-struction of historical SST monthly mean values using im-proved statistical methods from 1854 to present and take itas ground truth. In particular, we define the boundary undertracking is the isotherm at 25 degrees C. One example of theboundary (Jan. 1990) is shown in Fig. 5. The observers aresimulated by the points on the map and each observer canreport its closest point on the boundary. In real applications,the observers could be a set of ships or buoys with tempera-ture sensors. Since SST changes periodically, for simplicity,a motion model is learned for each calendar month (Jan. toDec.) from the historical data between 1990 and 1999. Forexample, to get the motion model for Jan., we compare thedata in Jan. and Feb. in every year and find a motion modelby the optical flow method [7], and then take the average of

1provided by NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, fromhttp://www.cdc.noaa.gov/

the motion models over the ten years (Fig. 5).

5 10 15 20 25 30 35 40 45 500.1

0.15

0.2

0.25

0.3

0.35

Number of measurement points

Err

or ra

te

FEMGaussian

Figure 6. Compare error rates of FEM and Gaussian methods withdifferent number of measurement points for SST tracking.

We started tracking from Jan. 2000 and ended in Dec.2004 (60 months). The particle set was initialized by per-turbations based on the boundary in Dec. 1999. The pertur-bations are generated by linear combinations of the elementfunctions. The number of explicit child particles is 50. Thegrid size is 89 × 180 (2 degree latitude ×2 degree longi-tude, global grid:88N − 88S, 0E − 358E). The elementarray size is 9× 18. We have applied both the FEM methodand Gaussian method. The error rate is defined as Eqn. (3).The comparison of the accuracy of these two methods isshown in Fig. 6. The improvement of the accuracy by theFEM approach is obvious when M > 5.

6. Summary and future work

We have introduced “Finite-Element Level-Set CurveParticles” that combine piecewise-linear functions to makethe merge of level sets and particle filters more efficient.The source of greater efficiency is that FEM nodal pointscarry all the information about the piecewise-linear func-tions between them. In addition, the approximation error

Figure 5. The red curve is the isotherm of SST at 25 degrees C in Jan. 1990 and the blue arrows denote the learned motion model for theisotherm at 25 degrees C in Jan. from the SST data between 1990 and 1999. The black dots denote the land.

for FEM is inherently better understood than for mixturesof Gaussian functions. The new method simplifies the rep-resentation of the child particles as well as their propaga-tion. The results show improvements in both running time(by a constant factor) and accuracy compared to Gaussianlevel-set curve particles.

For future work, we are interested in improving the com-putation efficiency for maintaining the weights of implicitchild particles and resampling which takes exponential cost.We were puzzled by the fact that in the colon experiment theimprovement introduced by the FEM method increases withfewer observers, while it increases with more observers inthe temperature experiment. Maybe the more accurate mo-tion model available in the latter experiment is a factor inthis phenomenon, but we intend to investigate this furtherfor a more definite explanation.

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