Accelerating the Single Cluster PHD Filter with a GPU Implementation

Transcription

1 Accelerating the Single Cluster PHD Filter with a GPU Implementation Chee Sing Lee, José Franco, Jérémie Houssineau, Daniel Clar Abstract The SC-PHD filter is an algorithm which was esigne to solve a class of multiple object estimation problems where it is necessary to estimate the state of a single-target parent process, in aition to estimating the state of a multiobject population which is conitione on it. The filtering process usually employs a number of particles to represent the parent process, couple each with a conitional PHD filter, which is computationally burensome. In this article, an implementation is escribe which exploits the parallel nature of the filter to obtain consierable spee-up with the help of a GPU. Several consierations nee to be taen into account to mae efficient use of the GPU, an these are also escribe here. I. INTRODUCTION Recent evelopments in multiple target tracing algorithms have spurre the creation of methos which use them as builing blocs to solve more complex problems. For instance, filters base on the finite set statistics (FISST) framewor [1] such as the PHD Filter [2] have been exte to estimate cluster processes [3], perform istribute sensor localization [4], parameter estimation [5], an simultaneous localization an mapping [6], [7], among other applications. Augmenting the sophistication of the tracing systems, however, also ts to lea to a sharp increase in computational buren. However, interesting parallelization opportunities can often be exploite to reuce the overall running times of the algorithm. In particular, the Single Cluster PHD (SC-PHD) filter is suite for tracing problems where the observe targets motion an evolution is epent on some unobserve parent state. The use of this filter has been emonstrate in group target tracing [8], microscope rift estimation [9] an simultaneous localization an mapping [7]. Increasing the spee of inherently parallel programs can be achieve through the use of specialize processors, sometimes calle accelerators. Graphics Processing Units (GPU) are an interesting type of accelerator since they specialize in carrying out massively parallel tass. With increase computational nees spurre by more complex graphics intensive applications, the processing power of GPUs has quicly increase through the years, an there has been a surge of interest in exposing these capabilities for use in a wier range of applications. The CUDA framewor is a recent effort by Nviia corporation to simplify access to these capabilities in Nviia graphics cars for general purpose computation. This wor was supporte by the Engineering an Physical Sciences Research Council (EPSRC) Grant number EP/K014277/1 an the MOD University Defence Research Collaboration in Signal Processing. The authors are with the School of Engineering & Physical Sciences, Heriot-Watt University, Einburgh ( cslee@eia.ug.eu, {jf139,jh207,d.e.clar}@hw.ac.u) The purpose of this article is to provie etails on an implementation of the SC-PHD Filter on the CUDA framewor. In Section II, the SC-PHD Filter is escribe. Section III tals about the particularities of the CUDA Framewor an what consierations must be taen when programming for it, while Section IV etails the implementation of this filter on this architecture. Sample results are shown in Section V, an finally the article conclues in Section VI. II. THE SC-PHD FILTER The PHD Filter is a multi-object estimation filter base on Finite Set Statistics (FISST). In this framewor, the multiobject an measurement spaces are represente as ranom finite sets (RFS), which implies uncertainty both in the states of each of the targets being trace an in the size of the population. The filter propagates the approximation relate to the first moment, or intensity, of the multi-target Bayes filter, which is calle the Probability Hypothesis Density (PHD). This is a function efine over the single-target state space whose integral over a particular region gives the expecte number of targets in that region, an thus gives information not only on the number of objects in the population but also of their location. An interesting class of multiple-target tracing problems eals with tracing a population whose state is conitional on a single-target ranom variable. This may be ue to, for example, the state of the sensor that observes the population not being accurately nown. The Single Cluster PHD Filter is an extension of the PHD Filter which is esigne to estimate the state not only of the multiple object population through a conitional PHD D (y x), but also of the parent process on which it is conitione. The preiction equation for the SC-PHD filter is given by D 1 (x, y) = s 1 (x )π 1 (x x ) D 1 (y x ) x, (1) where s 1 is the prior istribution of the parent state at time 1, where π 1 is the Marov transition ensity of the parent process, an where D 1 (y x) is the preicte PHD of the aughter process: D 1 (y x) = γ 1 (y x) + D 1 (y x)p S (y x) π 1 (y y, x) y with the following efinitions:

2 γ 1 ( x) PHD for aughter birth process at time, conitione on parent state x D 1 ( x) prior PHD of the aughter state at time 1, conitione on parent state x p S ( x) object survival probability, conitione on x π 1 (, x) single-object Marov transition ensity for the aughter process, conitione on x. The upate equation, in turn, is given by D (x, y) = s 1 (x)l Z (x) s 1 (x )L Z (x ) x D (y x), where s 1 is the preicte PHD of the parent state, where p D ( x) is etection probability of the aughter process given a parent state x, an where L Z (x) the multi-object observation lielihoo of the measurement set Z given a parent state x, efine by L Z (x) = exp ( with η z (x) = κ (z) + ) p D (y x) D 1 (y x) y η z (x), z Z D 1 (y x)p D (y x)g(z y, x) y, where g(, x) is the single-object observation lielihoo given the parent state x an where κ is the PHD of the measurement clutter process. The upate PHD D ( x) of the aughter process is foun to be D (y x) = D 1 (y x) [ (1 p D (y x)) + g(z y, x)p D (y x) η z (y x) z Z Practical implementations of either the PHD or SC-PHD filter require an appropriate representation for the intensity. In this article, the focus will be on accelerating a filter equippe with a particle representation for the istribution s of the parent process an a Gaussian mixture representation for the PHD of the aughter state. To reflect the conitional relationship between the parent an aughter processes, each particle in the parent istribution is associate with its own Gaussian Mixture (GM) representing the aughter PHD conitione on that particle s trajectory. Then at any given timestep, the parent an aughter processes can be state as N s (x) = J (i) D (i) (y x ) = w (i,j) ]. δ(x x(i) ) (2) N (y, µ (i,j), P (i,j) ). (3) Here, N is the number of particles in the representation of the parent state, each with state x (i) an weight, J (i) is the current number of components in the aughter Gaussian mixture, each component having weight w (i,j), mean µ (i,j) an covariance P (i,j). This representation is particularly convenient if the motion an measurement moels are linear an Gaussian, or can be reasonably linearize as such, since it simplifies the obtention of the preicte an upate forms for the Gaussian mixture. To obtain the equation for preiction, equations (2) an (3) are substitute into the SC-PHD preiction equation (1), yieling D 1 (x, y) = N 1 1 π 1(x x (i) 1 ) D 1 (y x (i) 1 ). Next, the Marov transition ensity of the parent is approximate by rawing M samples from it: D 1 (x, y) {x (i,1),, x (i,m) } π 1 (x x (i) N 1 M 1 ) M δ(x x(i,j) ) D 1 (y x (i) 1 ) Measurement-riven births are use, as escribe in [10], so the incorporation of new targets into the aughter PHD is elaye until the upate step. Therefore, the preicte aughter PHD escribes only persisting aughter objects propagate forwar in time: D S, 1 (y x (i) 1 ) = p S (y x (i) 1 ) JS, 1 S, 1 N ( y, S, 1, P(j) S, 1 = w(j) 1 S, 1 = f(µ(j) 1 ) P (j) S, 1 = F P (j) 1 FT + W Q W T S, 1 (i) ) x 1 where f(µ) is the (possibly non-linear) motion moel escribing the motion of aughter objects an F an W are the Jacobians of the motion moel with respect to the aughter feature an process noise respectively. The preicte joint PHD woul now consist of N 1 = M N 1 particles corresponing to the parent state, an the same number of Gaussian mixtures representing the conitional aughter PHDs. This means the number of particles woul grow geometrically with every preiction, but when the particles are resample, only a number of samples equal to the number of original particles is rawn, to mae it feasible to iterate for as many time steps as necessary. For the upate step, each conitional aughter PHD is iniviually upate with the receive measurements: D (y x) = (1 p D(y x)) + z Z J 1 J 1 1 N ( y; 1, P (j) ) 1 N ( y; (z), P(j) (z)) + z Z w γ 0 η z(y x) N ( y; z (x), R ),

3 where η z (y x) = κ(z) + p D ( x) J 1 l=1 g(z y, x) 1 + wγ 0 = w(j) 1 p D( x)g(z y, x) 1 η z(y x) 1 (z) = µ(j) 1 + K(z h(µ(j) 1, x)) P (j) (z) = (I K H )P (j) 1 K = PH T S 1 S = H P 1 H T + R. With measurement riven births, the terms z (x) an R are the mean an covariance of the birth term generate from measurement z. The term z (x) = h 1 (z, x) is chosen such that h(z (x), x) = z, while R can be chosen to be R ( ) (x) = J R ( ) J T if the Jacobian J of the inverse measurement function h 1 is nown. After upating the aughter Gaussian Mixtures, the weights of the parent particles are upate using their multi-object lielihoos L Z (x (i) 1 ): = L Z (x (i) 1 ) = exp L Z (x (i) 1 )ζ(i) 1 N 1, l=0 L Z (x (l) 1 )ζ(l) 1 ( (i) J ) 1 j=0 1 z Z η z (y (i) 1 x(i) 1 ). III. THE CUDA PROGRAMMING MODEL Effective parallel programming with CUDA requires an unerstaning of the capabilities an limitations of the GPU as a computational platform. A CUDA application will generally contain portions which are execute in parallel on the GPU (evice coe), an portions which are execute serially by the CPU (host coe), with program flow alternating between host coe an evice coe. Device coe is execute in parallel by a gri of hunres or thousans of threas on the GPU, which is subivie into threa blocs. Each threa within the gri has access to inex values enoting the particular threa bloc in which it is locate, an its location within that bloc. These inices are use to ifferentiate behavior among threas, for example, accessing ifferent portions of the input an output ata arrays. Every threa receives the same set of instructions in the form of a special C function calle a ernel. In practice, CUDA ernels are not written to execute an entire algorithm on the GPU, but rather to perform smaller, computationally emaning portions of the algorithm that have been previously single out through profiling. Ieal caniates for GPU parallelization are proceures in which very large arrays are processe, with every array element being operate upon with an ientical set of computationally expensive instructions. In other wors, rather than looping serially through an array with a for or while loop, a GPU ernel woul execute all iterations of the loop in parallel, with a single threa hanling each iteration. The most important factor affecting the performance of a CUDA application is memory banwith. Prior to execution of a ernel, the input ata must be transferre from the host to the evice, an the output ata must be transferre from the evice bac to the host following ernel execution. The time to perform these operations is very long relative to the time to execute computational tass, so one must ensure that the computational gains from parallization outweigh the penalties incurre by memory operations. An effective CUDA application will strie a balance between processing small ata arrays in serial on the CPU an processing large arrays in parallel on the GPU. There are many more esign consierations for optimizing the performance of a CUDA ernel, which are outline in the CUDA toolit ocumentation [11]. IV. PARALLEL IMPLEMENTATION The Gaussian mixture SC-PHD filter propagates the parent state as a collection of particles, with each particle being line to a aughter Gaussian mixture PHD conitione on the state an trajectory of that particle. These pseuocoe listings will mae use of the following abbreviations: D (x, y) =. {x (i), ω (i) (i), D (y x)}n D (i) (y x) =. {, P (j), } J (i) Each of the subroutines liste here accepts as input an returns as output a single parent particle an its corresponing PHD. Serial proceures are mare as Functions an parallel proceures meant to be ran on the GPU are enote Kernels. For numerical stability, it is recomme that weights an lielihoos be replace with their log-equivalents, an that the corresponing computations be moifie accoringly. The source coe for this GPU implementation of the SC- PHD filter is freely available uner the Apache 2.0 license at the following aress: The top-level filter iteration shoul loo familiar to those experience with other Bayesian filter methos such as the EKF. The upate step is split into two subroutines: PreUpate computes auxilliary terms which are neee for the computation of the SC-PHD upate an Upate implements the actual upate equations. In this wor the PruneAnMerge subroutine was implemente as escribe in [12], but an alternative Gaussian mixture reuction algorithm can be substitute at the user s iscretion. The same may be sai about the Resample subroutine for resampling the parent particles. In the GPU implementation, the joint PHD consists of one array of particle states representing the parent, an another array of Gaussians representing all conitional aughter PHDs concatenate together. Typically, the parent state consists of several hunre particles, each represente by a vector of less than 10 imensions. The aughter array is several orers of magnitue larger. Each aughter PHD consists of several hunre Gaussian terms, each comprise of a scalar weight, mean vector, an covariance matrix. In general, operations

4 Function SC-PHD(D 1, Z ) Input: Prior joint state estimate, current measurements Output: Posterior joint state estimate, measurements for i = 1... N 1 o D (i) 1 =Preict(D(i) 1 (x, y)) ˆD (i) 1 =PreUpate(D(i) 1 (x, y), Z ) (i) [ D, L (i) (i) Z (X)] = Upate( D 1, ˆD 1, Z ) ω (i) = ω (i) L Z (X) D (i) (i) = PruneAnMerge( D ) // [12, Table II] // Particle resampling accoring to [13, Algorithm 2] x (1...N ) = Resample(x (1...N ), ω (1...N ) ) ω (1...N ) = 1/K return D = {x (i), ω (i), (i) D (y x)}n Function Preict(D 1 ) Input: Prior parent state particle an conitional aughter PHD Output: Preicte sensor state an object PHD // Sample sensor transition ensity for n = 1... M o x (n) 1 π( x x) ω (n) 1 ω // Preict aughter PHD forall the { 1, P(j) 1, w(j) 1 } in D 1 o 1 = w(j) 1 1 = f(µ(j) 1 ) P (j) 1 = F(j) P (j) 1 F(j),T + W (j) Q (j) W (j),t D (n) 1 = {µ(j) 1, P(j) 1 }J return D 1 = {x (n) 1, ω(n) (n) 1, D 1 }M n=1 involving the aughter PHD array will be carrie out on the GPU, while operations involving only the parent array will be carrie serially with the CPU, as the parent array is too small to overcome the memory performance bottlenec. By parallelizing the filter, the for loop in the above pseuocoe is eliminate. A. Threa gri imensioning an inexing The threas relate to the aughter Gaussian mixture of a single particle nee to share information in orer to perform the upate, so we assign one threa bloc per parent particle. This raises two esign challenges which must be ealt with: 1) threa blocs must be uniform in size, but the ifferent conitional aughter PHDs will iffer in size, an, 2) the number of Gaussian terms in a single conitional aughter may excee the maximum allowable threa bloc size. To aress both issues, an array is passe to the GPU where each element inicates the number of terms in the corresponing GM. A prefix sum of this array yiels an inexing offset pointing to the beginning of the appropriate Gaussian mixture in the concatenate array. Moreover, nowing the number of Gaussians permits the construction of a for loop which processes the input ata blocdim threas at a time. B. Preiction The preiction stage involves propagating both the parent istribution an aughter PHDs forwar in time. The parent preiction is performe serially while the aughter PHD is preicte in parallel on the GPU. Implementing a per-threa ranom number generator woul be prohibitively complicate, so the ranom noise samples are generate by the host, an passe to the ernel as input arguments. C. Pre-Upate In the PreUpate subroutine, single-object measurement lielihoos for each measurement an each term within the aughter Gaussian mixture are compute. The Gaussian mixture terms are also upate with the measurements via a Kalman upate. Function CUDAPreict(priorParent, priordaughter) preicteparent = hostarray(sizeof(priorparent)) for i = 0,..., np articles o processnoise = maenoise() preicteparent[i] = computeparentmotion(priorparent[i],processnoise) preictedaughter = evicearray(sizeof(priordaughter)) noisearray = maenoise() aughterpreictkernel (priordaughter,preictedaughter,noisearray) Let np reict be the total number of terms in the concatenate preicte aughter PHD, an nm easure be the number of receive measurements. The preupate will compute np reupate = np reict nm easure single-object lielihoos an upate Gaussians. D. Upate The remainer of the SC-PHD upate is performe by the Upate subroutine. Here, the preicte an preupate augther PHD terms, along with measurement-riven birth terms are concatenate an reweighte to form the final upate PHD. The upate aughter PHD will be the preupate array concatenate with np reict terms representing misse etection terms, an nm easure np articles terms representing measurement-riven birth. Within each threa bloc, share memory is use to compute weight normalization terms an the multi-object lielihoo value to re-weight the parent particle. V. RESULTS To analyze the performance gains that can be obtaine using the GPU implementation of the filter, a simple tracing scenario was generate consisting on targets moving in a plane accoring to Brownian motion. The sensor that observes this population also moves accoring to Brownian motion, an thus

5 Kernel aughterpreictkernel(prior,noise,preictearray) for j = 0,nThreas,2nThreas,...,nFeatures o ix = ti + j preictearray[ix] = computemotion (prior[ix],noise[ix]) Function PreUpate(D 1, Z ) Input: Preicte sensor particle an multi-object PHD, measurements Output: Pre-upate terms for SC-PHD aughter upate forall the {, P (j), } in D 1 o ẑ = h 1 (X, ) // Preicte measurement S = HP (j) H T + R // Innovation covariance K = P (j) H T S 1 // Kalman Gain P (j ) = (I KH)P // Upate covariance for i = 1... Z o // single-object lielihoo p(z (i), y) = N ( z (i) ; ẑ, S ) µ (j i) = µ + K(z ẑ) // Upate mean ˆD o s = {µ (j i), P (j i) }...J... Z return p(, x), ˆD 1 the acquire measurements exhibit ranom rift, which also nees to be estimate. Let the parent state be efine by a bias vector X = [x, y]. Given noise parameters σ x, σ y for the parent motion moel, the parent particles are preicte as follows; X 1 = X 1 + v v N ( [ X; [ 0 0 ], σ 2 ] x ) 0 0 σ 2 y Concurrently with the parent particles, the terms in the aughter Gaussian mixtures are preicte using a Brownian motion moel with noise parameters σ r, σ s. The preicte mean an covariance are compute as follows: [ ] σ 2 r µ 1 = µ 1 P 1 = P σ 2 s Kernel preupatekernel(preictedaughterarray, preicteparentarray,measurements,offsets,lielihoos, preupatedaughter) parent = preicteparentarray[blocix.x] ; preicte = preictedaughterarray[threaix.x] z = measurements[threaix.y] preupate.weight = computesingleobjectlielihoo(z,preicte,parent) preupate.mean,preupate.cov = KalmanUpate(z,preicte,parent) ; preupatedaughter[threaix.x,threaix.y] = preupate Function Upate(D 1, ˆD 1, p z µ,x ) Input: Preicte multi-object PHD an pre-upate terms Output: Upate multi-object PHD forall the, P (j), in D 1 o // Non-etection terms n = µ(j) ; P (j) n = P(j) ; w n = // Detection terms for i = 1... Z o µ (j i) = µ (j i) ; P (j i) = P (j i) ; // from ˆD 1 w (j i) = p Dp z s,o (z i, y) // Measurement-erive birth terms for i = 1... Z o µ (i) 0 = h 1 (y, z i); P (i) 0 = R ; w (i) 0 = w 0 // Normalize weights // compute multi-object lielihoo Ñ = J w(j) L Z (X) = exp(ñ) for i = 1... Z o η zi = κ(z i) + J w(j i) + 2w 0 for j = 1... J o w (j i) /= η zi w (i) 0 /= ηz i L Z (X) = L Z (X) η zi // Concatenate terms µ = [µ (1...J) n, µ (1...J Z ), µ (1... Z ) 0 ] P = [P (1...J) n, P (1...J Z ), P (1... Z ) 0 ] w = [w (1...J) n, w (1...J Z ), w (1... Z ) 0 ] return {µ, P, w, L Z (X)} The measurement moel is simply an ientity function on the aughter state, biase by the parent particle state. ẑ(µ X) = µ 1 + X 1 Fig. 1. Simulate measurements, with x versus time above an y versus time below. The conitional single object measurement lielihoo is compute by evaluating the following multi-variate Gaussian: p(z µ, X) = N ( z; ẑ, S ) S = P 1 + R where R is the measurement noise covariance matrix. This moel is similar to the simultaneous protein tracing an

6 Kernel PHDUpate(preicteGaussians,offsets,measurements) [ixin,ixout,npreict,blocoffset] = computeinices(offsets,threaix,blocix) preicte = preictegaussians[ixin] i = ixout - blocoffset if i >npreict then non-etection weight = preicte.weight upategaussians[ixout] = preicte else if npreict <= i <(npreict*(1 + nmeasurements)) then etection m = floor((i - npreict)/npreict) upate = preupate(preicte,measurements[m]) weight = upate.weight upategaussians[ixout] = upate else if i >= (npreict*(1 + nmeasurements)) then birth m = i - (npreict*(1 + nmeasurements)) birth = computebirth(measurements[m]) sharemem[threaix] = weight normalizer = sum(sharemem) upategaussians[ixout] /=normalizer Time (secons) # particles Fig. 2. Execution time of the GM-SCPHD filter. The ashe line inicates the running time for the serial implementation, an the soli the parallel one. The orinate is the logarithm of the running time. microscope rift estimation problem presente in [9]. For simulation inputs, a 20 time step scenario was generate for which 10 Monte Carlo runs were execute with varying parent particle counts: N = {50, 100, 200, 500, 1000, 2000, 5000, 10000}. The test ata can be visualize in Figure 1. The timing results are shown in Figure 2. These results show that while the GPU implementation is significantly faster than the serial implementation, the execution time still increases proportionally to the number of parent particles. Further profiling reveale that the majority of the computation time occurs within the Gaussian mixture reuction subroutine. Because it is an inherently sequential operation, the GM reuction was left as a serial subroutine to be run on the CPU. As the CPU loops over each parent particle to reuce its corresponing Gaussian mixture, it is to be expecte that time woul increase with a higher number of particles. VI. CONCLUSIONS The SC-PHD filter is a useful tool when solving a wie class of estimation problems that consist on jointly tracing a parent single-target state an a aughter multi-object state that is conitione on the parent. Although this filter has been wiely shown to perform well when solving this problem, serial implementations are slow since they o not exploit the inherent parallelism offere by this filter. In this article, a parallel version of the filter was etaile, an an implementation exploiting the high-performance capabilities of GPUs was escribe. This implementation was shown to perform well on simulate ata, proviing consierable spee-up compare to the serial version of the same algorithm. REFERENCES [1] R. P. Mahler, Statistical multisource-multitarget information fusion. Artech House ˆ eboston Boston, 2007, vol [2], Multitarget bayes filtering via first-orer multitarget moments, Aerospace an Electronic Systems, IEEE Transactions on, vol. 39, no. 4, pp , [3] A. Swain an D. Clar, Bayesian estimation of the intensity for inepent cluster point processes: An analytic solution, Proceia Environmental Sciences, vol. 7, pp , [4] M. Uney, B. Mulgrew, an D. Clar, Cooperative sensor localisation in istribute fusion networs by exploiting non-cooperative targets. ICASSP, [5] B. Ristic, D. E. Clar, an N. Goron, Calibration of multi-target tracing algorithms using non-cooperative targets, Selecte Topics in Signal Processing, IEEE Journal of, vol. 7, no. 3, pp , [6] C. S. Lee, D. Clar, an J. Salvi, Slam with single cluster ph filters, in Robotics an Automation (ICRA), 2012 IEEE International Conference on, May 2012, pp [7] C. S. Lee, D. E. Clar, an J. Salvi, Slam with ynamic targets via single-cluster ph filtering, Selecte Topics in Signal Processing, IEEE Journal of, vol. 7, no. 3, pp , [8] A. Swain an D. E. Clar, First-moment filters for spatial inepent cluster processes, in SPIE Defense, Security, an Sensing. International Society for Optics an Photonics, 2010, pp I I. [9] J. Franco, J. Houssineau, D. Clar, an C. Ricman, Simultaneous tracing of multiple particles an sensor position estimation in fluorescence microscopy images, in Control, Automation an Information Sciences (ICCAIS), 2013 International Conference on. IEEE, 2013, pp [10] J. Houssineau an D. Laneuville, Ph filter with iffuse spatial prior on the birth process with applications to gm-ph filter, in Information Fusion (FUSION), th Conference on. IEEE, 2010, pp [11] Nviia Corporation, CUDA toolit ocumentation, [Online]. Available: [12] B.-N. Vo an W.-K. Ma, The gaussian mixture probability hypothesis ensity filter, Signal Processing, IEEE Transactions on, vol. 54, no. 11, pp , [13] M. S. Arulampalam, S. Masell, N. Goron, an T. Clapp, A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracing, Signal Processing, IEEE Transactions on, vol. 50, no. 2, pp , 2002.