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Retraining maximum likelihood classifiers using a low-rank model

Arnt-Børre Salberg

Norwegian Computing Center

Oslo, Norway

IGARSS July 25, 2011

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Introduction► Challenge: Dataset shift problem:

▪ Training data match the test data poorly due to atmospherical, geographical, botanical and phenological variations in the image data→ reduced classification performance

▪ Class-dependent data distribution varies ◦ between training images◦ between test and training images

► Goal: Develop a method that re-estimates the parameters such that classifier possess a good fit to the test data

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Introduction

► Many surface reflectance algorithms often requires data from external sources▪ LEDAPS (Landsat):

◦ ozone and water vapor measurements

► Phenological, botanical and geographical variation in addition to atmospherical makes the calibration problem even harder

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An existing method…

► Models the test image as a mixture distribution and estimates all parameters using the EM-algorithm, with estimated parameters from training data as initial values

► To many degrees of freedom. Statistic fit is excellent, but class labels get mixed.

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Low-rank parameter modeling► Training image k:

▪ Class mean vector and covariance matrix (class i)

► Class mean vector and covariance matrix model for the test image

α and β are unknown parameter vectors to be estimated from the data

)( and 1 ,, DD) (D kiki ×× Σμ

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Low-rank data modeling

► The proposed method for modeling the test data is a low-rank approach since the number of parameters in α is L<D. ▪ This is much less than estimating all C·D

parameters i µ i, i=1,…,C

► By using a low-rank estimation of the class mean vectors of the test data, the spectral differences between the classes is in larger degree maintained

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Parameter estimation► Procedure for estimating α

and β:▪ Select N random

samples {y1, y2,… yN} from the test image

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Parameter estimation► Procedure for estimating α and β:

▪ Select N random samples {y1, y2,… yN} from the test image

▪ Model them using a Gaussian mixture distribution

► Estimate the parameters by solving the likelihood

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Experiment 1:Cloud detection in optical images

► 15 different QuickBird and WorldView-2 images covering 7 different scenes in Norway

► Features▪ Band 2 (green)▪ Band 3 (red)

► Classes▪ clouds, cloud shadows, vegetation,

concrete/asphalt/etc., haze and water

► Resolution down-sampled to 19.2 m (16.0 m)

► 4 different training (sub)images

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Experiment 1:Cloud detection in optical images

► Model

δ i is the eigenvector corresponding to the largest eigenvalue νi of the matrix

average

Test

eigenvector

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Experiment 1:Cloud detection in optical images

► Parameter estimation. At iteration l+1:

where

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Results:Cloud detection in optical images

Without retraining With retraining

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Results:Cloud detection in optical images

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Results:Cloud detection in optical images

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Experiment 2:Tree cover mapping of tropical forest

► 13 different Landsat TM images covering an area nearby Amani, Tanzania (path/row 166/063)

► Features▪ Band 1-5 and 7

► Classes▪ Forest, spares forest, grass and soil

► Two training images (1986-10-06 and 2010-02-10)

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Experiment 2:Tree cover mapping of tropical forest

► Model

α constrained to contain only positive elements

► Solution found using non-negative least-squares in combination with iterative maximum-likelihood estimation

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Experiment 2:Tree cover mapping of tropical forest

► Parameter estimation: At iteration l+1

where

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Results:Tree cover mapping of tropical forest

► *

Without retraining With retraining

February 2010 July 2009

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Summary and conclusion

► Proposed a simple method for handling the dataset shift between training and test data

► Cloud detection: Evaluated successfully on a many different Quickbird and WorldView-2 images. ▪ Haze versus clouds▪ Confuses snow and clouds

► Guidelines on how to select the low-rank modeling functions is needed

► EM-algorithm and local minima problem

► More testing and evalidation of the method is necessary