kalman filter partilce tracking
TRANSCRIPT
Kalman Filtering
Presented by
Muhammad Irfan Anjum
Introduction
Dynamical Signal Models
Scalar Kalman Filter
Vector Kalman Filter
Extended Kalman Filter
Simulation results
Outline
Introduction
• Uses a series of measurements over time, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone.
• Operates recursively on streams of noisy input data to produce a statistically optimal estimate of the underlying system state.
• Two step process– Estimates of current state variables with their uncertainties.– Estimates are updated using weighted average after observing
output.• Operates on real time data, no additional past information is required.
Dynamical Signal Models
][][][ nwnAnx
][][ˆ nxnA
][][ nwAnx
Gauss Markov Process
1st order Gauss Markov Process:
Vector Gauss-Markov Model:
n
k
kn nBuAsAns0
1 ]1[]1[][
0],[]1[][ nnBunAsns
][]1[][ nunasns
n
k
kn nuasans0
1 ]1[]1[][
snansE 1])[(
1Az
][ns][nu
B
Scalar Kalman Filter
1az
(a) Dynamical Model
][ns][nu
(b) Kalman Filter
]1|[̂ nns
][nx
1az
][ˆ nu ]|[̂ nns][nK
][~ nx
][nw
][ns
Scalar Kalman Filter
][]1[][ nunasns
]1|[
]1|[][ 2
nnM
nnMnK
w
22 ]1|1[]1|[ unnMannM
]1|1[̂]1|[̂ nnsanns
][][][ nwnsnx
]1|[])[1(]|[ nnMnKnnM
])1|[̂][]([]1|[̂]|[̂ nnsnxnKnnsnns
Transmitted Signal:
Received Signal:
Minimum Prediction MMSE:
Minimum MSE:
Correction:
Kalman Gain:
Prediction:
Vector Kalman Filter
1Az
][ˆ nu ]|[̂ nns
]1|[̂ nns
][nK][~ nx
][nx
][nw
][ns
][nh
][nh
1Az
][ns][nu
B
Scalar state Vector Kalman Filter
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])1|[̂][][]([]1|[̂]|[̂
][]1|[][
][]1|[][
]1|1[]1|[
]1|1[̂]1|[̂
][][][][
][]1[][
2
nnMnhnKInnM
nnsnhnxnKnnsnns
nhnnMnh
nhnnMnK
BQBnnAMnnM
nnsAnns
nwnsnhnx
nBunAsns
T
T
Tn
T
T
Transmitted Signal:
Received Signal:
Prediction:
Minimum Prediction MMSE:
Kalman Gain:
Correction:
Minimum MSE:
Vector state Vector Kalman Filter
]1|[])[][(]|[
])1|[̂][][]([]1|[̂]|[̂
][]1|[][][
][]1|[][
]1|1[]1|[
]1|1[̂]1|[̂
][][][][
][]1[][
nnMnHnKInnM
nnsnHnxnKnnsnns
nHnnMnHnC
nHnnMnK
BQBnnAMnnM
nnsAnns
nwnsnHnx
nBunAsns
T
T
T
Transmitted Signal:
Received Signal:
Prediction:
Minimum Prediction MMSE:
Kalman Gain:
Correction:
Minimum MSE:
Extended Kalman Filter
][])1[(][ nBunsans
Vector Kalman Filter
Extended Kalman Filter
][])[(][ nwnshnx
][][][][
][]1[][
nwnsnHnx
nBunAsns
]1|1[ˆ]1[|]1[
])1|1[̂(])1[(
nnsnsns
annsansa
]1|[ˆ][|][
])1|[̂(])[(
nnsnsns
hnnshnsh
]1|1[ˆ]1[|]1[
]1[
nnsnsns
anA ]1|1[ˆ][|
][][
nnsnsns
hnH
Extended Kalman Filter
]))1|[̂(][]([]1|[̂]|[̂ nnshnxnKnnsnns
TT BQBnAnnMnAnnM ]1[]1|1[]1[]1|[
])1|1[̂(]1|[̂ nnsanns
][]1|[][][
][]1|[][
nHnnMnHnC
nHnnMnK
T
T
]1|[])[][(]|[ nnMnHnKInnM
Particle Tracking using Scalar Kalman filter
MMSE in Scalar Kalman filter particle tracking
Particle Tracking using Vector Kalman filter
Particle Tracking using Extended Kalman filter