meta-optimization of the extended kalman filter’s parameters for improved feature extraction on...
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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 Africa2. Remote Sensing Research Unit, Meraka, CSIR, South Africa3. 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
MODIS Band 1
MODIS Band 2
Band 2 Separation
Band 1 Separation
Band 2 Separation
Band 1 Separation
Band 2 Vegetation
Band 1 Vegetation
Band 2 Settlement
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.
• Time series modelled as a triply modulated cosine function
• Where
= Mean
= Amplitude
= Angular frequency
= Spectral band
= Time index
Triply modulated time series
= Seasonal cycle (8/365)
= Phase
= Noise
= Pixel index
• State vector
• Process model
• Observation model
Extended Kalman Filter Framework
Mean Amplitude Phase
Modelling the time series
Unstable parameter
Unstable parameter
Unstable parameter
Mean
Amplitude
Phase
• Process model
• Observation model
Tuneable parameters
Observation noise covariance matrix
Process covariance matrix
Initial estimates of state vector
Tuneable parameters
Observation noise covariance matrix
Process covariance matrix
Initial estimates of state vector
Tunable parameters
Where j denotes the epoch number
Creating extreme conditions
Absolute Error
Tunable parameters
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 timeIncrement k using all the pixels and if ideal will be denoted by
Creating extreme conditions
Tunable parameters
Set
Capture a probability density function (PDF) for each timeIncrement k using all the pixels and if ideal will be denoted by
Amplitude
Creating extreme conditions
Tunable parameters
Set
Capture a probability density function (PDF) for each timeIncrement k using all the pixels and if ideal will be denoted by
Phase
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
Results: Standard deviation for MODIS spectral band 1
1142 MODIS pixels = 285.5km2
Mean
Amplitude
Absolute Error
Results: Standard deviation for MODIS spectral band 2
1142 MODIS pixels = 285.5km2
Mean
Amplitude
Absolute Error
Results: Classification on labelled data K-means (Band 1, Band 2)
1142 MODIS pixels = 285.5km2
Vegetation Accuracy
Settlement accuracy
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
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