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

AmplitudePhase

• 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

What do we want?

Mean

Amplitude

Phase

Absolute Error

Tunable parameters

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

Iterative updates

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: Standard deviation for MODIS bands

1142 MODIS pixels = 285.5km2

Results: Classification on labelled data K-means (Band 1, Band 2)

1142 MODIS pixels = 285.5km2

Vegetation Accuracy

Settlement accuracy

Results: Accuracy for MODIS bands

1142 MODIS pixels = 285.5km2

Results: Gauteng province settlements

78704 MODIS pixels = 19676km2

23.16% Settlement

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?

Expansion of irrigationCommercial forestry

Mining Informal settlements

Alien tree removal