Fast, parallel implementation of particle filtering on the GPU architecture
© Gelencsér-Horváth et al.; licensee Springer. 2013
Received: 27 February 2013
Accepted: 8 September 2013
Published: 21 September 2013
In this paper, we introduce a modified cellular particle filter (CPF) which we mapped on a graphics processing unit (GPU) architecture. We developed this filter adaptation using a state-of-the art CPF technique. Mapping this filter realization on a highly parallel architecture entailed a shift in the logical representation of the particles. In this process, the original two-dimensional organization is reordered as a one-dimensional ring topology. We proposed a proof-of-concept measurement on two models with an NVIDIA Fermi architecture GPU. This design achieved a 411- μ s kernel time per state and a 77-ms global running time for all states for 16,384 particles with a 256 neighbourhood size on a sequence of 24 states for a bearing-only tracking model. For a commonly used benchmark model at the same configuration, we achieved a 266- μ s kernel time per state and a 124-ms global running time for all 100 states. Kernel time includes random number generation on the GPU with curand. These results attest to the effective and fast use of the particle filter in high-dimensional, real-time applications.
In applications in the field of image processing [1, 2], navigation , and financial mathematics [4, 5] we deal with non-linear state-space models subject to additive noise which is not restricted to Gaussian noise. Even if each state only depends on the previous state (i.e. the sequence follows the Markov dynamics ), a Kalman filter  is suboptimal for the state estimation due to non-linearity and non-Gaussian noise. Furthermore, an analytic solution is often not available. In contrast, sequential Monte Carlo methods (SMCM) offer a probabilistic framework that is suited to non-linear and non-Gaussian state-space models. In our work, we focus on a particle filter (PF)  which is both part of the SMCM algorithm family and can be considered an extension of a Kalman filter. Our main aim is to introduce a fast and reliable PF on a GPU.
We restrict ourselves to particle filters which use sequential importance resampling (SIR) . The PF algorithm (as described in the next section in detail) has a high running time due to the resampling step - according to the complete cumulative distribution; therefore, an adequate parallel implementation would fetch a remarkable speed-up. The classical resampling algorithm of this process needs N processors to reduce its computational need from O(N) to O(log(N)), and so, this particle filter was considered unsuitable for parallelism. It should be noted that this statement holds as long as the the complete cumulative distribution is required for any particle.
Whilst the price of a device is relatively low, graphics processing units (GPUs) have a high computational efficiency. Therefore, GPUs represent an attractive implementation platform. Since GPUs are spreading fast, thanks to the game industry, and developing rapidly, computational effort is still increasing by leaps and bounds.
There have been some former implementations to parallel architectures [10–19]. In , an implementation strategy is proposed which is parallel; however, it cannot maintain the local connections of the particles. The particles are split into smaller groups (around 100 particles) which perform operations independently. The information exchange among the particle groups is occasional; share ratio is suggested at around 25%. Researchers admit that the reduced flow of information of the distributed particle filter degrades the quality of estimation compared to the original algorithm which resamples according to the complete cumulative distribution.
Besides continuous information sharing, random number generation has a significant effect on the filter result. The simple solution is creating random numbers on the CPU using any of the many implemented reliable random libraries proposed in . However, these researchers are aware of the great disadvantage of this technique, mainly the huge delay raised by data transfer. Hence, they prefer a sufficient GPU random sequence generator instead.
In the resampling step, each particle requires information from all other particles. This adds a high computational delay. Resampling is based on the relative importance of the particles, but the technique is not restricted to uniform sweepstake over the weights (systematic resampling).
Metropolis resampler  in each resampling step iteratively selects B times a candidate according to a given rule for each particle. Since the aforementioned rule is based on pairwise operations, the efficient mapping on many-core architecture is realizable as follows.
For each particle in each iteration, two main parameters, and , are used, where stands for the actual particle and stands for the selected particle; is initialized with particle p. With uniform distribution, we draw a random number on [0,1) and particle from the complete particle set. If the ratio of the weights is over , the selected particle is indicated as actual. This pairwise operation can be performed independently; therefore, efficient implementation is possible on parallel architecture.
Highlighted related works
Number ofSMs on GPU
All; spreading-narrowing technique
BOT model,25 time steps
Sampling +weightnormalization +resampling
1,050 particles79.4 ms, positionerror 0.06245; 2,000 particles 124.8 ms,position error0.06226
A factorstochastic volatility model
SMC algorithm:no further details
8,192 particles82 ms; 16,382particles 144 ms;65,536 particles465 ms
All; resamplinguses offline-initiated texture
Skin detection + spreadingregion ofinterest
Object trackingtime; no furtherdetails
1.44 - 13.55 speed-up in fps bcompared to CPU.Best 90 fps formultiple- and 225fps for single-object tracking.
Face trackingalgorithm time;no furtherdetails
No executiontime measurementsfor particle filter
All; randomnumbers fromCPU
BOT model,25 time steps
Sampling +weighting +weightnormalization+ resampling
For 2,048 particles:best time 168.3 ms,position 0.078 -0.083; for 4,096particles: best time 168.0 ms, position0.077 - 0.081
Dynamic equations tomodel arobotic arm
Sum of kernels:random numbergeneration +sampling + local sort + global estimate + exchange + resampling
64,000 particles0.3 ms
GTX 550 Ti
All; all parallel
BOT model,24 time steps
All operations(including memory transfers, PF steps and file I/O)
2,048 particles 77 ms with 0.09 position error; 16,384 particles 77 ms 0.07 position error
We investigated [13–15] from CUDA ZONE, which also addresses particle filtering on GPU or parallel and GPU. In , only the weight calculation is performed on the GPU as the focus of this paper is not particle filtering. However, this work is slightly out of our scope as the aim was the fast estimation of face tracking with PF; the contribution of parallelism on the GPU is relevant in the total speed-up. Three different case studies are presented for Monte Carlo methods in . They found out that global resampling has a significant influence on the runtime, and they achieved 10 to 37 times speed-up compared to a single threaded CPU implementation. The measurements were made with the use of a factor stochastic volatility model; therefore, direct comparison to our benchmark model is troublesome. However the computations run on the GPU, the resampling - unlike in our proposed work - is not parallel but sequential. In , resampling is performed with a technique based on using an offline-created and offline-uploaded texture of uniformly distributed random numbers. The focus of  was single- and multiple-object tracking, using skin detection and spreading the region of interest; therefore, the lack of common model encumbers the direct comparison of the result to our proposed method.
However, in , the GPU device is different (making the exact time comparison difficult), and measurements were made using the BOT model. The proposed particle filter method is parallel, still the random numbers are generated on the CPU, and there is no information share among local processes (e.g. in resampling). The position errors are in the same range and almost identical. Execution time in  is expressed as the sum of sampling, weighting, weight normalization and resampling times, whilst our presented execution time includes all operations (file I/O, memory transfers, etc). A real-world problem is presented with a particle filtering method in . PF is implemented on the GPU with a distributed resampling. The work is based on sub-filters which have a limited information share among themselves. Communication can be represented as a graph where sub-filters correspond to nodes and edges are defined arbitrarily (as an attribute). Before resampling, a particle exchange step is performed among neighbouring graph nodes. However, this approach is not completely local; the amount of the exchanged particles is relatively small. Therefore, information flow is not as complete as in the case of the sequential (cumulative distribution-based original algorithm) or even as in our proposed method. Finally, we would like to mention  and  which both present parallel but non-GPU particle filters. In , the particles are split to subsets. Similar to , each subset performs the sub-steps of PF independently; however, there is no information share among the subsets. Central estimation is calculated using the results of subsets. In , three different techniques are presented, and locally distributed particle filter is considered as giving the best speed-up and estimation. Also, this is the closest from the three presented methods to our approach; however, operations are performed without any information share and only simulation results are given for a discrete time non-linear dynamic model of nearly constant turn. For the summary of related work, please see Table 1 where we highlighted the most relevant GPU-related works. The table also reveals the difficulty of direct comparison of the results due to the different models, data and GPU devices.
We can say that resampling is a key point in SIR particle filtering. Approaches to deal with SIR resampling try to optimize the speed quality trade-off for the given set-up. Cellular particle filter(CPF)  introduces a third approach for the resampling problem by changing the logic representation of PF to a two-dimensional (2D), locally connected grid inspired by cellular neural network (CNN) architecture . Each element in the grid is connected to each of its eight neighbours enabling rapid local information flow. The critical resampling step can then be performed on a subset in an r radius neighbourhood.
Our proposed algorithm is based on cellular particle filter , using the idea of local neighbourhoods. Due to the CNN-type representation and the decreased dimension of resampling sets, CPF offers a solution for the problem of reduced local information change, which is stated in . However, this representation is not optimal for a GPU architecture. Hence, we made some further modifications to achieve an efficient implementation which exploits the characteristics of GPUs. Besides, one of our principles is to generate random sequences on the GPU since NVIDIA SDK Mersenne Twister proved to be insufficient at low numbers. Therefore, we explored possible solutions and finally propose two different approaches for random number generation.
This paper is organized as follows: the ‘Background and theory’ section describes the necessary background and theory for hidden Markov models (HMMs), particle filters, especially cellular particle filter, and architectural perspectives. In the ‘Our proposed method’ section, we introduce our proposed method in detail. The ‘Evaluation and results’ section provides information about applied measurement techniques and our results. Finally, The ‘Conclusions’ section delivers our conclusion.
Background and theory
Hidden Markov models and particle filtering
These notable extensions transfer the resolution beyond the Kalman filter  to the scope of particle filtering. In case of state estimation, it is considered that φ,ψ, functions and distributions of e 1(t) and e 2(t) are given. For more information about hidden Markov models, see .
A particle filter is a tool for estimating the hidden states based on the observation. It is not an analytical calculation but the use of a set of particles at each time step that follows the model dynamics. The algorithm is built up from three main steps in each time t (i.e. state).
where are the resampled particles, . Particle filtering technique has been used since 1962 , and SIR particle filter has been used since 1993 . Still, the proof of convergence was published only 18 years later . More information about particle filters can be found in [23, 26]. Henceforward, ‘original algorithm’ stands for the algorithm described in this section.
Cellular particle filter
In the resampling step, for each retake, we have to use the whole particle set. This is highly time-consuming and for a long time was considered not parallelizable. Cellular particle filter  offers a solution for this problem, and in contrast to other distributed particle filters , it maintains local connectivity, which allows for each particle to access the information of its neighbours in each time t. This ensures the same or, at some parameters, even better quality of approximation. To provide theoretical proofs for our concept is beyond this articles’ scope, but see  for some experimental validation.
where j∈N i .
Now, the resampling step can be performed for each i particle simultaneously within the local N i neighbourhood according to the local distribution of the weights. Fetching the weights is realized by local communication on the physical device, therefore, it is fast. The random take on all subsets around each i particle produces N resampled particles, respectively. Hence, the time-consuming part is parallelized, and the required computational effort is essentially independent of the number of particles. This method might seem to be similar to distributed particle filters , but there are no communication limits among the subsets in any time states.
CPF is suited to GPU architecture due to its parallel, locally connected nature. Our aim is to ensure efficient computation and therefore to exploit the properties of the GPU in our adaptation.
In the logical sense, the kernel function is the function executed on the device. It is executed simultaneously by threads. Threads are organized into blocks (typically 32 to 512 in each, based on the current task), in a one-, two-, or three-dimensional array. Blocks are organized in a grid in a one- or two-dimensional array. The number of threads per block and block per grid is called execution configuration.
In the physical sense, the device is built up from streaming multiprocessors (SMP). Each SMP consists of an SD RAM, a number of cuda cores and a scheduling unit. The SD RAM is an on-chip memory, with a few tens of clock cycle delays, its size is 64 kB, and it is divided to L1 cache and shared memory. The GPU has an off-chip global memory to be accessed by each SMP. Its size is usually around 1 to 4 GB, depending on the type of the particular device. Its delay is 400 to 600 clock cycles. Additionally, there are two other types of memory spaces that both reside off-chip and are cached on-chip. The first one is texture memory, and the second one is constant memory. The latter’s size is 64 kB.
Blocks are mapped to SMPs. Shared memory of block B i can only be accessed by the threads which reside in B i . The communication and data share among the blocks are performed through the global memory. A fixed collection of threads is called warp. Currently, the number of threads in a warp (warp size) is 32, which is physically executed simultaneously. Besides proper memory usage, warp conflict avoidance is essential . The vendor suggests block sizes multiple of the warp size to achieve the most efficient computation. However, in extreme situations, optimal block size can differ from the advised values . If some threads in the warp choose different branches of operation based on the processed data, the threads within the warp may diverge. This is called warp desynchronization and results in the serialization of some threads which adds runtime delay. As a rule of thumb, the block size should be a multiple of 32 and each multiprocessor should execute at least six warps at the same time because the pipeline is six levels deep; therefore, 8×32=256 may be an ideal thread number.
To achieve a high throughput, on-chip memory (shared memory) should be used if threads require frequent data access. Therefore, the main computational tasks are performed block-wise, in the shared memory of the blocks, and only necessary global synchronization is performed through global memory. Although the two-dimensional texture and surface memory would also be feasible, shared memory throughput can be optimized for a one-dimensional array type representation; thus, structural aspects in CPF algorithm were reconsidered.
In the shared memory, we use arrays with the size 512 for the particles, the error terms, the normalizing sums and the uniform random numbers. In our case study, when taking the highest neighbourhood size and at a single precision 10,250 bytes are occupied. Each shared memory can access a 48-kB memory/multiprocessor at compute capability 2.x . Global memory in recent GPUs is at least 1 GB which restricts the number of states in the observation and estimation, but both memory access is high enough for our computations.
Our proposed method
Random number generation
The SIR in each t time step requires the same amount of random numbers as the number of particles. We intend to generate these random sequences on the GPU device instead of the CPU to spare repetitive data transfer between the main memory of the system and the global memory of the device as recognized in . The distribution of the random numbers in the resampling is critical on the quality of the estimation. If it is not uniform, though the drawing of particles should depend on the weights exclusively, then it would be biased.
Recently GPU random number generation for various purposes has become widely investigated and well tested (see [29–31]). NVIDIA provides two solutions for random number generation. The first option was the Mersenne Twister in the SDK. Unfortunately, we observed that the generated distribution is inappropriate for a small set (hundreds or thousands) of numbers and is primarily admissible for around two million numbers and above. We made delicate modifications to get admissible distribution (see Appendix for details). The second option was curand, which proved to be fast and appropriate.
GPU CPF algorithm
The local connectivity is preserved by choosing each shared memory array size higher than the thread number exactly with the size of the required neighbourhood. Each thread with index i can obtain its k neighbours at indexes .
In the following, we specify some implementation details. Although the operations are mainly performed in the shared memory because of the synchronization and CPU-GPU data transfer, the following variables are global memory arrays: the observation sequence (Y), the state estimation (X) and the set of particles x particles. The number of particles is denoted by N. Y is naturally given, X is empty, and x particles is initialized with N samples of the same distribution as n t described in Equation 10. The number of threads in a block was set to 256. The size of the neighbourhood is r, meaning each particle is connected to exactly r+1 particle (every particle is connected to itself). In each block, two shared memory arrays of size 256+r are created for particle states x shared and fitness values L shared; additionally, two arrays whose size equal the number of threads in a block are allocated for uniform pseudo-random numbers U shared and normalizing weights w shared.
- 2.x shared[i loc+r]←x particles[i gl].
If i loc<r, then x shared[i loc]←x particles[i gl−r+i loc].
If i gl<r, then x shared[i loc]←x particles[N−r+i gl].
- IIError calculation:
L shared[i loc+r]←l(Y[t]−x shared[i loc+r]).
If i loc<r, then L shared[i loc]←l(Y[t]−x s [i loc]).
- 3.get normalization sums for each particle.
If i loc==0, refresh seed value of the block.
Fill U shared with uniform random numbers.
Iteratively sum L shared[j loc]/w shared[j loc] where . Stop if adding an L shared[k loc]/w shared[k loc] term affects the sum to exceed U shared[i loc] for the first time.
From the neighbourhood of i loc, the corresponding k particle is selected to . Estimation is calculated from these particle values.
- IVIteration on particles (generating samples for next state):
Fill U shared with normally distributed random numbers.
x shared[r+i loc]←φ(x shared[r+i loc],n t ).
x particles[i gl]←x shared[i loc+r].
The estimation is performed by two kernel calls. The first kernel calculates the sum for each shared memory arrays to a global memory array gSum. The second kernel takes the average value of gSum with respect to the number of particles.
Besides, the following optimization techniques were used to achieve optimal efficiency on the GPU: (1) random number generation, resampling and average calculation are performed only on the relevant part of the shared memory to spare time; (2) the number of if statements is minimized as possible and are transformed to ternary expressions; and (3) the shared memory arrays are reused; therefore, even some parameter passes can be spared.
Evaluation and results
This model is non-autonomous, non-linear and has a continuous state space; thus, linear tools for state estimation are not applicable. The state of the system is x t , and the observation is y t ; n t and u t are IID Gaussian sequences, n t ∼ N(0,10) and u t ∼ N(0,1).
There are two aspects of the measurements, namely, the average quality and the required time for one estimation. These two quantities were monitored with different configurations (number of particles and radius of neighbourhood) of the filter. The number of particles were swept through the following values: 2,048, 4,096, 8,192, and 16,384, while the radius of neighbourhood took the following values: 32, 64, 128, and 256. Consequently, 24 pairs of these are composed; in our terminology, these are called configurations (i.e. N,r pairs). For the first model, the input observation trajectories (y t ) during the tests were exactly the same as the ones used in  to ensure a fair comparison. For the BOT model, we generated 100 trajectories over 24 time steps based on the given state transition equations in  and, respectively, the observations.
Measurements were done on a PC with Intel i5-660 (3.33 GHz, 4-MB cache) 4 CPU with 4-GB system memory running Ubuntu Linux 11.04 with kernel version 2.6.38-15 (amd64). We used an NVIDIA GeForce GTX 550 Ti GPU with 1-GB GDDR memory with CUDA toolkit 4.1 with 295.49 driver version. The following nvcc compiler options were used to drive the GPU binary code generation: -arch=sm_20;-use_fast_math. We also made some measurements with -arch=sm_13. The host c code was compiled with gcc 4.5; the compiler flag was -O2. GPU kernel running times were measured with the official profiler provided by the toolkit, and the global times were measured by the OS’s own timer. The kernel time measurements include the particle filtering kernel of a single time step; the global times include all operations during the execution for all states (file I/O, memory allocations, computational operations, etc.).
The quality of estimation was measured by the mean square error (MSE). In the case as the first model, 1,000 estimations for each configuration of 100 time step long HMMs were made. In the case of the BOT model, the generated 100 trajectories were estimated with each configuration.
The quality of estimation for the first model was measured by the MSE between each hidden and estimated trajectories, for the BOT model by the position error (ie. Euclidean distance). We used 1,000 and 100 different state sequences (i.e. observation sequences) for each configuration in the first model and the BOT model, respectively.
However, for the BOT model estimations, where the particle number is more or equal to 2,048 (alike [16, 17]), provide a fair result. The position error is in the same range as in  and our proposed method. The results suggest that the proportion of the neighbourhood size to the particle number realizes an information sharing ratio among the particles. This can be seen in Figure 10: the optimal ratio for the configuration is when the position error is minimal, typically marked with squares except 2,048 particles when marked with triangle.
Due to the logic of physical mapping of blocks to multiprocessors, the GPU is under-utilized for particle numbers under 2,048. Above this particle number, the scaling of the execution time is nicely illustrated in Figure 11.
For various neighbourhood sizes, we can say that the required time is proportionally increasing to the number of R. This phenomenon is due to the resampling step as it examines the candidates sequentially for resampling. Even if the proper particle is found, the loop does not terminate until the current particle is compared to all of its neighbours to avoid warp desynchronization.
A key point of our proposed algorithm is the local resampling technique which has a high influence on the estimation quality. This key point can be viewed as diffusion of information inasmuch as every particle with relatively high likelihood attracts all particles which has this likely particle in its neighbourhood. In this way, the other particles not having this very particle in their neighbourhood are not affected in this state estimation time step. Although it is not the traditional full resampling, it enables the algorithm to be sufficient even at high-uncertainty dynamic models. The BOT model is not a highly uncertain model as sharp changes are unlikely, but in the one dimension benchmark model (as you can see in Figure 7), rapid and significant changes are typical. Cellular resampling preserves the diversity of the particles to avoid quality loss. The estimation error for a given particle number changes within a narrow range around the optimal, depending on the neighbourhood size. This modulation is not in direct or inverse ratio to the number of used neighbours but follows a descending and then increasing characteristic (like the shape of letter ‘U’, see Figure 10). This indicates that for a given model, at any particle number, there exists an optimal share ratio range among particles to achieve the lowest error. In our proposed method, the information sharing ratio is tunable and may be modulated adaptively; therefore, it broadens the range of options than using a predefined information share value. For further details and reasoning, see .
To ensure the local diffuse information share, we used shared memory arrays. Due to the high number of writing and reading data arrays (particle samples, weights, likelihoods, resampled values etc), if it were performed in global memory alone, the performance would be worse. However, the use of global memory cannot be totally evaded as synchronization and regular information sharing among blocks are essential. Without this synchronization, a similar information loss and quality degradation would appear as in the case of the distributed particle filter presented in  though this synchronization is a time quality trade-off. However, using constant memory (which is cached) would expose an attainable solution; only the observation values could be stored in the constant memory since all the other values are generated during kernel execution. Additionally, it would not improve the performance as reading from the constant memory space requires a fetch from the off-chip memory to cache the value of the current observation (like the current fetch from the global memory to the shared memory), and the access time of the constant cache is similar to the access time of the shared memory.
Two different models were used in this work. The first one is a synthetic benchmark model. It does not model any physical system of practical interest. It is just a widely used highly non-linear model since both the observation and the state transition is non-linear, unlike the other model (BOT model) which has a linear state transition. This second model describes a bearing-only tracking of an object in the two-dimensional (x−y) plane, with a fixed observer position, where observation (z) is the bearing of the object trajectory. Through the BOT model, we can compare the estimation quality of CPF to GPU particle filters in [11, 16, 17]. We can see that the error is in the same range with  and is better than error in . According to the error and time measurements, we can state that this is a feasible mapping from a virtual machine (CNN UM-inspired architecture) to a state-of-the-art architecture with a mature ecosystem available at a low cost.
In this algorithm, each thread executes roughly 300 to 2,100 floating point operations. This depends on the neighbourhood size. Those operations which are performed through the neighbourhood are additions and unfortunately divisions (calculating the actual weights with the norming sums in the resampling). This amount of divisions are clearly one bottleneck. The speed-up achieved with fast_math also supports this explanation. The other bottleneck of the algorithm is the shared memory size and access pattern. If more threads could reside in a block, then the ratio of the overlay among blocks due to the neighbourhood size would be less. In the resampling, branches are unavoidable, and fork and join of threads within a warp are necessary. There are a number of for-loops which iterate through the given number of iterations where the end index is unknown at the time of host code compilation. In our framework, it is only known in runtime. Still, for a given application, optimal parameters can be set directly in the code based on measurements, and therefore, loop unroll can be applied. Another approach can be just-in-time (JIT) kernel compilation. This method is effective only if the JIT compilation time is less than the cumulative gain from the loop unrolling through the state predictions.
On the one hand, if we would perform the algorithm as purely sequential, the order would be O(N×R) without the random number generation which (as mentioned) is not part of the basic task. On the other hand, with a virtual GPU on which all blocks are active simultaneously (totally utilized), the order of the algorithm would be O(R) as the threads are independent from each other, not counting the synchronization points. However, on GTX 550, the time increase starts after 1,536 threads (on 6 SMPs × 256 threads). On GTX 580, there are 16 multiprocessors in comparison with the 6 multiprocessors of GTX 550; therefore, under 4,096 threads, the GPU would be under-utilized (i.e. GTX 580 is the top GPU of the Fermi GPUs).
A direct comparison to a CPU, sequential particle filter, is not entirely adequate; still, we would like to mention differences in the running time. In , particle filter was realized and used as a reference for time measurements of the proposed algorithm; therefore, we compare our results to this also. However, time measurements are not presented for many particle numbers; measurements of 4,096 particles already indicates the benefits of our implementation compared to the CPU version. Execution time of the algorithm was 16.417 s for the first model on a dual-core processor PC (Intel T6570) with 2.1 GHz. Compared to our 100-ms execution time, we can say that a 164x speed-up is achieved. We provide all our measurement data as Additional file 1.
In this paper, we introduced the first adaptation of CPF to GPU architecture. Compared to , we measured the performance on a real hardware. The strength of this approach is to maintain the local connectivity to prevent information loss, while the position error and the execution time are comparable to those of  if we assume that their measurements also include all operations (file I/O, random number generation, etc.). We utilize architectural features: whilst CPF algorithm would demand two-dimensional representation, (e.g. texture or surface), we modified the algorithm to enable one-dimensional processing and still kept the local connectivity and the local neighbourhood-based reduced and parallel processing.
The compute capability of the GPU determines the maxima of the number of threads that can be handled at each state. The sequential operations are performed in the shared memory thus shall make no effect on running time when we increase the number of particles. Still, as the shared memory blocks are arranged to multiprocessors as defined on the device, there is a scheduling which introduces a delay. Therefore, however parallel, the algorithm still will require more time at more particles.
The proposed method is independent from the random generator if the quality of the uniform and normal distribution is acceptable. As the first approach, we used NVIDIA Mersenne Twister for which the corrections to achieve adequate distribution, see details in the Appendix. In the second approach, we used curand which generates appropriate distribution quickly. This adaptation of CPF by delicate modifications presents GPU as an excellent platform to solve problems that could not be solved real-time previously.
Modification of NVIDIA SDK Mersenne twister
To achieve an adequate distribution for the resampling, we made the modifications as follows based on  and our empirical experiences. The degree of recursion was changed from 19 to 397; the middle term, from 9 to 624; and the shift value u, from 12 to 11 based on the original values from . Besides, in the tempering transformation, we used the originally defined masks instead of the loaded ones from a predefined file, with hexadecimal values in the bitwise operations. For each thread, the first element of the state array is calculated with a thread and current time-based seed value based on the thread ID and the current system time. The final value is calculated with initializing on the first element of the bit vector for each thread.
This research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4.A/-11-1-2012-0001 (National Excellence Program). The support of Furukawa Electric Institute of Technology Ltd. and the grants TÁMOP-4.2.1.B-11/2/KMR-2011-0002 and TÁMOP-4.2.2/B-10/1-2010-0014 are also gratefully acknowledged. The authors would like to thank Miklós Rásonyi for his help and suggestions.
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