ES_SAA_OG_PF_ECCTD_Pos
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GeneralizedDivision-FreeArchitectureand CompactMemoryStructureforResampling inParticleFilters Syed Asad Alam and Oscar Gustafsson {syed.asad.alam, oscar.gustafsson}@liu.se Department of Electrical Engineering, Linköping University, Sweden Aims and Objectives • The most basic form of resampling algorithm in a particle filter has a high hardware cost • Requires normalized and ordered data set for implementation → Multinomial resampling • Alternative algorithms used to avoid multinomial → Stratified, systematic resampling • Aim – Architecture for multinomial resampling free from the need of ordering and normalization – Memory optimization for the weights and random function Particle Filters • Model based filtering – State transition and observation models are non-linear and noise non-gaussian • Purpose → Estimation of a state from a set of observations corrupted by noise • Applications → Target tracking, computer vision, channel estimation . . . • Steps → Time-update, weight computation and resampling estimate ResamplingTime update computation Weight Input observations xn yn wn ˜xn, ˜wn Output Figure 1: Overall structure of particle filter. Comparator Memory Weight Normalized Replicated− Factors Replicated/ Memory Discarded/ Normalization and Cumulative Sum Unit Generation Number Random Particle Memoryand Time Update Sample (i) Time Update Unit Unit Control Unit (iii) Resampling (ii) Weight Memory and Random Number Generation Memory address wK W ′ K U′ K Figure 2: Basic architecture of particle filter. Resampling Resampling prevents degeneration of particles and improves estimation by eliminating and replicating particles having low and high particle weights respectively. The total number of particles, M, remains the same. 0 1 Systematic Stratified Multinomial Figure 3: Uniformly distributed samples for M = 10. Standard algorithms for resampling available in the literature • Multinomial → Uniform random numbers – U[0, 1) • Stratified → Partition U[0, 1) into M regions, one sample from each interval with a random offset • Systematic → Similar to stratified resampling, offset is fixed Proposed Idea Background • Complexity of multinomial resampling can be reduced from M2 to approximately 2M by generating ordered random numbers → High hardware cost • Accumulation and normalization will provide an intrinsic ordering, reducing the hardware cost • The comparison, needed for replication and discarding particles, can be formulated as: WK WM ⋚ UK UM (1) where WK = K j=0 wj and UK = K j=0 uj, are the running sums, WM and UM are the cumulative sum. Division Free Architecture Reformulation of (1) gives: WK × UM W ′ K ⋚ UK × WM U′ K (2) • No normalization required • Equally efficient for non-powers-of-two M • Independent of generating ordered random numbers • Can be used for stratified and systematic resampling with appropriate random number generators REG REG Weight Memory Accumulator Memory Random value Accumulator From Control Unit From Control Unit uK wK WM Bw + log2 M Br + log2 M UM U′ K W ′ K Figure 4: Memory and data generation for resampling with stored cumulative sum. Memory Optimization • Storing cumulative sum of data increases the word length requirement for the two memories • Can be reduced from Bw + log2 M and Br + log2 M to Bw and Br respectively by on-the-fly accumulation • Accumulators are placed after each memory • Extra hardware cost of multiplexer and associated control logic Weight Memory REG Accumulator Memory Random value REG Accumulator Unit Unit From Control From Control uK wK WMUM Bw Br U′ K W ′ K Figure 5: Memory and data generation for resampling with on-the-fly cumulative sum. Results Complexity – Standard Cells Table 1: Complexity, in terms of Area (mm2 ), of architectures based on stored and online sum. Particle count Bit growth Stored sum Online sum Savings (%) 10 4 0.022 0.014 36.36 20 5 0.035 0.019 45.71 100 7 0.112 0.088 21.43 128 7 0.114 0.088 22.81 200 8 0.220 0.153 30.45 256 8 0.220 0.154 30.00 512 9 0.441 0.291 34.01 1000 10 0.833 0.550 33.97 1024 10 0.857 0.555 35.24 2000 11 1.703 1.103 35.23 2048 11 1.731 1.105 36.16 Complexity – FPGA Number of particles, M 512 1k 1024 2k 2048 3k 4k 8k 10k 15k 16k 20k NumberofLUTs 0 50 100 150 200 250 300 350 400 450 500 Stored Online Figure 6: Look-up table used by architectures based on stored and online sum. Number of particles, M 512 1k 1024 2k 2048 3k 4k 8k 10k 15k 16k 20k Numberof36kBRAM 0 10 20 30 40 50 60 70 Stored Online Figure 7: Memory used by architectures based on stored and online sum. Summary • Proposed a generalized division free architecture for the resampling stage • Independent of the non-powers-of-two number of particles, normalization and generation of ordered random numbers • Achieved by use of double multipliers and accumulators • Memory optimization results in reduction of area and memory usage up to 45% and 50% respectively • Achieved by on-the-fly accumulation of particle weights and random numbers • Each memory holds the original particle weight and random number • Reduces the word length required for each memory