Implementation of Particle Filter-based Target …rajbabu/presentations/pf...Implementation of...

Implementation of Particle Filter-based

Target Tracking

V. Rajbabu

[email protected]

VLSI Group Seminar

Dept. of Electrical EngineeringIIT Bombay

November 15, 2007

IntroductionTarget Tracking

Bayesian EstimationParticle Filter

Implementation

Outline

1 Introduction

2 Target Tracking

3 Bayesian Estimation

4 Particle Filter

5 Implementation

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Implementation

Outline

1 Introduction

2 Target Tracking


4 Particle Filter

5 Implementation

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Implementation

Introduction

Problem scenario

Multi-target tracking - batch measurements [Volkan Cevher]

Particle filters - computational complexity

Sensors operating under constrained environment

Problem Illustration

Objective

Efficient digital implementation of particle filters

Real-time

Low-power

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Implementation

Outline

1 Introduction

2 Target Tracking


4 Particle Filter

5 Implementation

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Implementation

Tracking

What is tracking ?

Tracking - process measurements to sequentially estimatehidden states

Target trackingVisual tracking

Applications

Surveillance and monitoring

Medical imaging

Robotics

Motion in sports

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Implementation

Target Tracking

We consider multi-target tracking

State vector - target or vehicle kinematic characteristics, ex.,2−D position and velocity, or motion parameters in polarcoordinatesMeasurements - range or angle with respect to sensor

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Implementation

State-space Formulation

Dynamic state-space problem - sequential

State-space model depends on physics of the problem

System transition equation

xt = ft(xt−1, ut), ut − system noise

ft() − system evolution function (possibly nonlinear)

Observation equation

yt = gt(xt , vt), vt − measurement noise

gt() − measurement function (possibly nonlinear)

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Implementation

Bearings-only Tracking

Automated estimation of a moving target’s state using anglemeasurements (bearings)

State update equation

Xt = FXt−1 + Gut ,

where Xt = [x vx y vy ]Tt , ut = [ux uy ]Tt ,

F =

1 1 0 00 1 0 00 0 1 10 0 0 1

and G =

0.5 01 00 0.50 1

.

Measurement equation

zt = arctan {yt/xt} + rt

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Implementation

Batch-based Tracking – State Vector

Multiple target state vector - independent partitionsConstant velocity during batch period T

Xt = [xT1 (t), xT

2 (t), · · · , xTk (t)]

T

xk(t) =

θk(t)Rk(t)vk(t)φk(t)

, k − target index

θk(t) − direction-of-arrival (DOA)

Rk(t) , log range (range)

vk(t) − velocity

φk(t) − heading direction

2

vT

r(t+T)

r(t)

Y

X

1 φ(t)

θ(t) θ(t+T )

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Implementation

Batch-based Tracking – Observation Model

Image template-matching - batch measurementsDOA measurements from acoustic sensor - beamformingRange measurements from radar sensor

Observability - batch of minimum three measurements

0 T 2T 3TMτ

t

DOA or RangeMissing data

ClutterBatch

yt = {yt+mτ (p)}M−1m=0

t

t+Tt+τ

t+(M−1)τ

DOA or Range

hθ,rmτ (x (j))

hθ,rmτ (x (i))

Figure: Template-matching

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State-Space Model

Batch-based range tracking

Non-linear state transition equation

θk(t+T )

Rk(t+T )

vk(t+T )

φk(t+T )

=

tan−1

eRk sin θk +Tvk sin φk

eRk cos θk +Tvk cos φk

ff

12

log{e2Rk +T 2v2k+2TeRk vk cos(θk−φk)}

vk

φk

+ uk (t)

where uk(t) − Gaussian process noise

Observation likelihood

p(

y(t)|xk(t))

∝M−1Q

m=0

(

1+ 1−κ√2πσ2

r κλ

P

pi

exp

−(hr

mτ (xk (t))−yt+mτ (pi ))2

2σ2r

ff

)



Implementation

Outline

1 Introduction

2 Target Tracking


4 Particle Filter

5 Implementation

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Implementation

Bayesian Estimation

Objective - estimate hidden state xt from observations yt

Probabilistic description - using pdf s

prior distribution - p(x0)state transition - p(xt|xt−1)data likelihood - p(yt |xt)

Bayesian estimation - minimum mean square estimate

Conditional mean E (xt |yt) of the posterior distributionp(xt |yt) ∝ p(yt |xt) · p(xt|xt−1)

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Implementation

Sequential Representation

Cumulative state up to time t: Xt = {xj , j = 0, . . . , t}

Cumulative measurement up to time t:Yt = {yj , j = 0, . . . , t}

Bayesian estimation

Estimate xt based on all available measurements up to t byconstructing the posterior p(Xt |Yt)

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Implementation

Recursive Filter

Consists of two stepsPrediction step: p(xt−1|Yt−1) → p(xt |Yt−1)

Update step: p(xt |Yt−1), yt → p(xt |Yt)

Given the pdf sPrediction step:

p(xt |Yt−1) =

∫

p(xt |xt−1)p(xt−1|Yt−1)dxt−1

Update step:

p(xt |Yt) =p(yt |xt)p(xt |Yt−1)

p(yt |Yt−1)

where, p(yt |Yt−1) =∫

p(yt |xt)p(xt |Yt−1)

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Implementation

Recursive Filter

Recursive application of prediction and update steps providesthe optimal Bayesian solution

Minimum mean square estimate (MMSE) is the conditionalmean - E (xt |Yt)

Analytically intractable - integrals and pdf s

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Implementation

Kalman Filter

Optimal solution for the recursive problem exists

Kalman filter - optimal solution if

state and measurement models - linear, andstate and measurement noises - Gaussian

Extended Kalman filter (EKF) - extension of Kalman filter

state and/or measurement models - nonlinear, andstate and measurement noises - Gaussian

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Implementation

Outline

1 Introduction

2 Target Tracking


4 Particle Filter

5 Implementation

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Implementation

Particle Filter

Suboptimal filter - for nonlinear systems and non-Gaussiannoise

handles nonlinearity as such - without linearisationhandles multimodal distributions

Based on Monte Carlo methods

Monte Carlo - “randomly chosen”

Other names

Sequential Monte Carlo (SMC) methodsBootstrap filterSequential Importance Sampling (SIS) filterCONDENSATION - CONditional DENSity propogATION

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Implementation

Monte Carlo Integration

To numerically evaluate: I =∫

q(x)dx where x ∈ Rnx

Monte Carlo (MC) method

Factorize: q(x) = f (x) · π(x), s.t., π(x) is a pdf. We can drawN samples {x i ; i = 1, . . . , N}

MC estimate of I =∫

f (x)π(x)dx is the sample mean

IN =1

N

N∑

i=1

f (x i)

What if it is difficult to draw samples from π(x) ?

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Implementation

Importance Sampling

Choose an Importance distribution g(x), such that:

π(x) > 0 ⇒ g(x) > 0 for all x ∈ Rnx .

and is easy to draw samples from g().

MC estimate - generate samples {x (i)} ∼ g(x)

IN =1

N

N∑

i=1

f (x (i))w̃ (x (i))

where,w̃(x (i)) =π(x (i))

g(x (i)).

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Implementation

Importance Sampling

Random Support N = 20

x

x (j)∼N (µ,σ2)

Target distribution π(x)

Proposal distribution g(x)

Unnormalized weights : wju = π(x (j))

1√(2πσ2)

exp

− (x(j)−µ)2

2σ2

ff

Target distribution’s expectation:

Eπ{F (x)} ≈∑20

j=1 F (x (j))w ju

∑20j=1 w

(j)u

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Implementation

Particle Filter - (1/2)

Use randomly chosen “particles” to represent posteriordistribution

{

x(i)t , w

(i)t

}N

i=1,

x(i)t : support points

w(i)t : associated weights

N − number of particles

Discrete weighted approximation to the posterior

p(xt |Yt) ≈N

∑

i=1

w(i)t δ(X − X

(i)t )

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Implementation

Particle Filter- (2/2)

Discrete approximation of distribution using

Importance Sampling

Particles - determine the ’support’ regionWeights - proportional to probabilitiesProposal or importance function plays a critical part

Recursive update by propagating ’particles’ and ’weights’

Use updated distributions to obtain estimates

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Implementation

Resampling

Degeneracy of particles - after a few iterations most particleshave negligible weights

ResamplingEliminate or replicate particles depending on their importanceweights

Image adapted from Dr.Volkan Cevher

Weights

Do estimation

Resampling

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Implementation

Sequential Importance Resampling (SIR) Particle Filter

Given the observed data yk at k, do

1. For i = 1, 2, . . . , N,

Sample particles: x(i)k ∼ g(xk |x (i)

k−1, yk).

2. For i = 1, 2, . . . , N,

Calculate the importance weights: w(i)k =

p(yk |x(i)k

)p(x(i)k

|x(i)k−1)

g(x(i)k

|x(i)k−1,yk)

.

For i = 1, 2, . . . , N,

Normalize the weights: w̃(i)k =

w(i)k

PNj=1 w

(i)k

.

3. Calculate the state estimates: E{f (xt)} =∑N

i=1 w̃(i)t f (x

(i)t ).

4. Resample{

x(i)k , w̃

(i)k

}

to obtain new set of particles{

x(j)k , w

(j)k = 1

N

}

.

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Implementation

PF Flow - Suboptimal Proposal Function

InitializationProposeParticles

Measurements

EvaluateWeights

w∗(i)t =w

(i)t−T

p(yt |X(i)t )

NormalizeWeights

ResampleParticles

EstimateStates

NP

i=1w

(i)t X

(i)t

OutputProposal function - stateupdate

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Implementation

PF Flow - Optimal Proposal Function

InitializationProposeParticles

Measurements

EvaluateWeights

w∗(i)t =w

(i)t−T

p(yt |x(i)t )p(x

(i)t |xt−T )

Q

k gk (x(i)k

(t)|yt ,x(i)k

(t−T ))

NormalizeWeights

ResampleParticles

EstimateStates

NP

i=1w

(i)t X

(i)t

OutputProposal function -full-posterior

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Implementation

Example: 1D Estimation

Estimate states of a nonlinear, non-stationary state space model

State dynamic equation

xk = 0.5xk−1 +25xk−1

(1 + x2k−1)

+ cos(1.2(k − 1)) + wk

Measurement equation

yk =x2

k

20+ vk

wk and vk are zero-mean, Gaussian white noise.

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Implementation

Example - 1D Estimation

0 5 10 15 20 25 30 35 40 45 50−20

−10

0

10

20

30

Time

Sta

te e

stim

ate

State (xk) of a 1D model

True value

Posterior mean estimate

−20−10

010

20

010

2030

4050

0

100

200

300

Sample space

Posterior using Particle Filters

Time

Po

ste

rio

r d

en

sity

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Implementation

Target Tracking Results

Tracking result

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Implementation

Outline

1 Introduction

2 Target Tracking


4 Particle Filter

5 Implementation

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Implementation

Design Hierarchy

Various stages in developing and implementing a particle filteralgorithm

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Implementation

Design Hierarchy

Various stages in developing and implementing a particle filteralgorithm

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Implementation

DSP Implementation

Floating-point DSP – TI C6713

Internal memory (IRAM) or External memory (SDRAM)

SpeedSize

Sampling rate of ∼ 0.3 seconds, for K = 1, N = 1000,225 MHz

Table: Memory sizes in the C6713-DSK - Single-target, N = 1000

Type Section Available Occupied

Internal IRAM 192 KB 190.16 KB

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Implementation

FPGA Realization

Matlab simulation

Finite word length identified from Matlab fixed-pointsimulation

Xilinx System generator - Simulink based tool

Xilinx blocks - bit and cycle true FPGA codeNonlinear functions using CORDIC algorithmDevice: Xilinx Virtex II Pro

Xilinx Embedded Development Kit (EDK)

To use MicroBlaze soft-core and Power PC hard-core

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Particle State Update

Particle Proposal Stage

Data Likelihood Evaluation

Particle Weight Evaluation Stage

Reampling Stage

Resampling

Compute number of replications based on particle weight

Control circuitry and logic functions

Figure: Residual Systematic Resampling

1: U ∼ U [0, 1]2: K = M/Wn

3: indr = 0, indd = M − 14: for m=1 to M do

5: temp = w(m)n .K − U

6: r indr = ⌈temp⌉7: U = temp − r indr

8: if r indr > 0 then

9: i(indr )r = m, indr = indr + 1

10: else

11: i(indd )d = m, indd = indd − 1

12: end if

13: end for



Implementation

Approximation for Data Likelihood

Data likelihood

p(y(t)|xk (t))∝M−1Q

m=0

(

1+ 1−κ√2πσ2

θκλ

P

pi

exp

(

−(hθ

mτ (xk (t))−yt+mτ (pi ))2

2σ2θ

))

DOA component of nonlinear state transition

hθmτ (xk(t))=arctan

sin(θk (t))+T cos(φk (t))e(Qk(t))

cos(θk (t))+T sin(φk (t))e(Qk(t))

Laplace method to approximate distribution

Newton search – Jacobians and Hessians

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Implementation

Newton search in FPGA (1/2)

Hardware implementation of square-root and division useNewton-Raphson search

Difficulty in using Newton search to identify mode

Cost function , Hessian, and Gradient evaluation - nonlinearfunctions

Software implementation in floating-point - MicroBlaze

soft-core or Power PC hard-core processor

Avoids sensitivity to word length

Accelerated by configurable hardware

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Implementation

Newton search in FPGA (2/2)

MicroBlaze - soft-core IP (processor)

Configurable - has 64 kB RAM for code and dataFloating-point unit (FPU) - accelerates floating-pointoperations

Power PC 405 - hard-core IP (processor)

Non-Configurable - has 128 kB for data and 64 kB for codeFloating-point operations are emulated in software

Newton search for identifying mode - code size > 100 kB

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FPGA - Resource Utilization

Batch-based tracker utilizes at least four times more resources

Excluding Newton-search

Requires large, recent FPGA device

Table: Resource utilization - Batch-based tracker

ResourcePF Stage

Proposal Weight evaluation Resampling Overall

# Slices†a 7803 8869 374 17046†a For comparison, a 16 bit multiplier uses 153 slices

Table: Resource utilization - Bearings-only tracker

ResourcePF Stage

Proposal Weight evaluation Resampling Overall

# Slices 2700 1215 374 4635



Implementation

FPGA - Latency

Table: Update rate using FPGA Clock frequency of 100 MHz

Batch-based tracker Bearings-only tracker

Latency 3N + 307 3N + 50

Update rateN = 200 N = 1000 N = 200 N = 1000

9 µs 33 µs 3.5 µs 30 µs

Update rate of 9 µs is sufficient to generate estimatesevery T = 1 s

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DSP Implementation Results

Complete batch-based particle filter algorithm on TI C6713

Figure: Target DOAs

0 5 10 15 20−50

0

50

100

θin

[◦

]

time [sec]

Figure: Target x -y tracks

−50 0 50 100 150 200 250 300 350−150

−100

−50

0

50

100

150

200

250

300

Floating point DSP

Matlab

Generic C

Ground−truth

x

y

FPGA Implementation Results

Matlab ModelSim

Proposeparticles

Evaluateweights

Measurements

Resampleparticles

Estimatestates

Figure: FPGA simulation setup.

FPGA Implementation Results

FPGA implementation of data likelihood evaluated in ModelSim

Figure: Target DOAs

0 5 10 15 20

−150

−100

−50

0

50

100

150

θin

[◦

]

time [sec]

Figure: Target x -y tracks

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

FPGA estimate

Matlab estimate

Ground−truth

Sensor

x

y



Implementation

Comparison

Comparison of implementation strategies for particle filters

Characteristic DSP FPGA

Accuracy High High

Speed Medium High

Power dissipation High High

ImplementationHigh Medium

flexibility

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Implementation

Discussion (1/2)

Particle filter characteristics

Number of particles N can be large - complexity

Parallel computations for individual particles

Resampling can prevent parallelization - parallel resampling

Nonlinear functions

Constraints

Real-time operation

Low power

Accuracy - word length

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Implementation

Discussion (2/2)

Digital implementation of particle filter

Multiple FPGAs

SIMD architectures

General purpose Graphical processing units (GP-GPUs)

Random number generators

Analog or mixed-mode implementation of particle filter

Nonlinear functions in analog - arctan, Gaussian

Random number or noise generation

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Acknowledgements and References

Thanks to

Advisor Prof.James H. McClellan, Georgia Tech.

Dr.Volkan Cevher, Rice University.

SMC Webpage: http://www-sigproc.eng.cam.ac.uk/smc/

A. Doucet and N. Freitas and N. GordonSequential Monte Carlo Methods in Practice.Springer-Verlag, 2001.

V. Cevher, R. Velmurugan, and J. H. McClellanAcoustic Multi-Target Tracking using Direction-of-Arrival BatchesIEEE Tran. Signal Processing, June 2007.

http://www-sigproc.eng.cam.ac.uk/smc/



Implementation

Thanks !

Questions

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Sequential Importance Sampling (SIS) Particle Filter

Illustration

Source: Michael Isard’s CONDENSATION demoshttp://www.robots.ox.ac.uk/~misard/condensation.html

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http://www.robots.ox.ac.uk/~misard/condensation.html

CORDIC Algorithm

COordinate Rotation DIgital Computer (CORDIC) algorithm

xn+1 = xn − mdnyn2−σ(n)

yn+1 = yn + dnxn2−σ(n)

zn+1 = zn − wσ(n)

Type m σn dn = signzn dn = −signyn

circular1 n

xn → K(x0 cos z0 − y0 sin z0) xn → Kp

x20 + y2

0

yn → K(y0 cos z0 + x0 sin z0) yn → 0

tan−1 2−k zn → 0 zn → z0 − tan−1 y0x0

hyperbolic−1 n − k

xn → K ′(x1 cosh z1 + y1 sinh z1) xn → Kp

x20 + y2

0

yn → K ′(y1 cosh z1 + x1 sinh z1) yn → 0

tanh−1 2−k zn → 0 zn → z0 − tanh−1 y0x0

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FPGA Device Detail

Table: Xilinx Virtex II Pro FPGA - XC2VP30

Resource # Logic cells†a # Slices†b Block RAM Clock frequency

30816 13696 2448 kb 100 MHz

†a Logic cell ≈ one 4-input LUT + one Flip-Flop + Carry logic†b Each slice includes two 4-input function generators, carry logic, arithmeticlogic gates, wide function multiplexers, and two storage elements.

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