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Meta-optimization of the Extended Kalman filter’s parameters
for improved feature extraction on hyper-temporal images.
B.P. Salmon1,2* , W. Kleynhans1,2, F. van den Bergh2, J.C. Olivier1,
W.J. Marais3 and K.J. Wessels2
1. Department of Electrical Engineering, University of Pretoria, South Africa
2. Remote Sensing Research Unit, Meraka, CSIR, South Africa
3. Space Science and Engineering Center, University of Wisconsin-Madison, Wisconsin, USA
* Presenting author
Overview
• Problem statement – Reliable surveying of land cover and transformation
• Discuss the importance of time series analysis
• Study area: Gauteng province, South Africa
• Using the EKF as feature extractor from time series data
• Meta-optimization of EKF’s parameters
• Results: Land cover classification
• Conclusions
Problem Statement
Reliable surveying of land cover and transformation
Year
Estimated Population
Change
2000
8,038,200
-
2001
8,243,719
2.56%
2002
8,499,900
3.11%
2003
8,775,200
3.23%
2004
8,851,455
0.87%
2005
9,002,534
1.71%
2006
9,193,800
2.12%
2007
9,665,841
5.13%
2008
10,450,000
8.11%
2009
10,531,300
0.77%
Time Series Analysis
Band 2
Separation
Band 2
Separation
Band 2
Vegetation
Band 2
Settlement
MODIS Band 2
Band 1
Separation
Band11
MODIS Band
Separation
Band 1
Vegetation
Band 1
Settlement
Objective
Time series can be modulated with a triply modulated cosine function [1].
[1] W.
Kleynhans et. al, 'Improving land cover class separation using an extended Kalman filter on MODIS NDVI
time-series data', IEEE Geoscience and Remote Sensing Letters, vol. 6, no. 4. April 2010
Objective
Parameters of a triply modulated cosine can be used to distinguish
between several different land cover classes.
Parameters derived using a EKF framework has been proven as a
feasible solution.
Introduce a meta-optimization approach for setting the parameters of a
Extended Kalman filter to rapidly estimate better features for a triply
modulated cosine function.
Triply modulated time series
• Time series modelled as a triply modulated cosine function
• Where
= Mean
= Seasonal cycle (8/365)
= Amplitude
= Phase
= Angular frequency
= Noise
= Spectral band
= Pixel index
= Time index
Extended Kalman Filter Framework
Mean
• State vector
• Process model
• Observation model
Amplitude Phase
Modelling the time series
Unstable parameter
Mean
Unstable parameter
Unstable parameter
Amplitude
Phase
Tuneable parameters
• Process model
• Observation model
Initial estimates of state vector
Process covariance matrix
Observation noise covariance matrix
Tuneable parameters
Initial estimates of state vector
Process covariance matrix
Observation noise covariance matrix
Tunable parameters
Where j denotes the epoch number
What do we want?
Tunable parameters
Absolute Error
Mean
Amplitude
Phase
Creating extreme conditions
Tunable parameters
Absolute Error
Set
Capture a probability density function (PDF) for each time
increment k using all the pixels and if ideal will be denoted by
Creating extreme conditions
Tunable parameters
Mean
Set
Capture a probability density function (PDF) for each time
Increment k using all the pixels and if ideal will be denoted by
Creating extreme conditions
Tunable parameters
Amplitude
Set
Capture a probability density function (PDF) for each time
Increment k using all the pixels and if ideal will be denoted by
Creating extreme conditions
Tunable parameters
Phase
Set
Capture a probability density function (PDF) for each time
Increment k using all the pixels and if ideal will be denoted by
Creating a metric
• Set an initial (candidate) state as
• Calculated the f-divergent distance as
Absolute error
Mean
Amplitude
Phase
Define a comparison metric
• Create a vector containing all the f-divergent distances as
• Define a metric for an unbiased Extended Kalman filter
• Optimize the vector using comparison metric
Iterative updates
Results: Standard deviation for MODIS
spectral band 1
Mean
Amplitude
Absolute Error
1142 MODIS pixels = 285.5km2
Results: Standard deviation for MODIS
spectral band 2
Mean
Amplitude
Absolute Error
1142 MODIS pixels = 285.5km2
Results: Standard deviation for MODIS bands
1142 MODIS pixels = 285.5km2
Results: Classification on labelled data
K-means (Band 1, Band 2)
Vegetation Accuracy
Settlement accuracy
1142 MODIS pixels = 285.5km2
Results: Accuracy for MODIS bands
1142 MODIS pixels = 285.5km2
Results: Gauteng province settlements
23.16% Settlement
78704 MODIS pixels = 19676km2
Conclusions
• Temporal property is of high importance in remote sensing
• A meta-optimization for the EKF using a spatio-temporal window was proposed.
• Proper feature analysis can greatly enhance analysis of data.
• Presentation of features to any machine learning algorithm
Questions?
Mining
Informal settlements
Expansion of irrigation
Commercial forestry
Alien tree removal
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