Download Indicator kriging in feature space, PPT

A Feature-Space-Based Indicator
Kriging Approach for Remote Sensing
Image Classification
鄭克聲
台灣大學生物環境系統工程學系
Image Classification
Statistical pattern recognition techniques
are widely used for landuse/land cover
classification.
Some supervised classification algorithms
Parametric Approach
Maximum likelihood classifier
Bayes classifier
Non-parametric Approach
Nearest Neighbour classifier
Artificial Neural network classifier
ANN Classifiers
ANN classifiers do not consider
classification features as having probability
distributions, and therefore, classification is
not explicitly probability-based.
In a loosely defined sense, ANN
classification is a process of searching
optimal solution of weight vector that
minimizes the sum of squared errors
between network and desired output
responses.
ANN Classifiers
It has been shown that the output of a
backpropagation network can approximate the
posterior density function, if its activation
function is capable of representing the a
posteriori probability function and the number of
training samples is sufficiently large (Lee et al.,
1991).
Manry et al. (1996) also showed that a neural
network can approximate the minimum mean
square estimator arbitrarily well, provided that it
is of adequate size and is well-trained.
ANN Classifiers
Egmont-Petersen et al. (2002) point out that
ANNs suffer from what is known as the
black-box problem: given any input a
corresponding output is produced, but it
cannot be elucidated why this decision was
reached, how reliable it is, etc.
Image Classification
The work of image classification can be
considered as partitioning a hyperspace
using discriminant rules established by
samples.
Each sample point in feature space is
labeled a class index.
Difficulties Encountered In Application of
Parametric Approaches
Application of parametric approaches
require knowledge of probability
distribution of classification features.
Classification features often have finite
mixture distributions (multi-modal class
densities).
The class distribution may be non-Gaussian.
Geostatistical Approach of
Spatial Estimation
Geostatistics is a set of techniques, often
referred to as kriging methods, which utilize
the spatial covariance function or the
semivariogram for spatial data analysis.
Ordinary kriging yields best linear unbiased
estimator (BLUE).
Indicator kriging yields estimate of the
probability distribution at specified
locations.
Since probability density and correlation
structure between classification features are
insightful, probability-based classification
methods are appealing to many researchers
and practitioners.
The work of probability-based classification
can be conceived as a spatial estimation
problem for which the estimates are
probabilities that a pixel with certain
feature-vector belongs to different classes.
Ordinary Kriging
Ordinary kriging assumes second-order
stationary properties for the random field
{Z(x), x}
Properties of OK Estimates
Unbiased
i.e.
Minimum variance of estimation error
Conditional minimization
Minimizing
Ordinary Kriging System
Semi-variogram
Typical Form of A Variogram
Variogram characterizes the spatial
variation of a random field.
Matrix Form of OK System
Indicator Kriging
Indicator kriging is a method of spatial estimation
that yields an estimate of probability distribution
function of the random variable of interest.
Consider a random field
of k
classes where Ω represents the spatial domain of
the random field. A total number of n features are
used for classification of the k classes. For
convenience of illustration, let’s assume k = 3 and
n = 2. From a set of training pixels, we first
establish the k-class scatter plot in feature space.
Scatter Plot in Feature Space
Indicator Variable
For a continuous random field, the indicator
variable can be used to estimate the
distribution of the random variable by using
a set of cutoff values.
The indicator variable at location x is
defined as
where
is a selected cutoff value.
The weighted average of indicator variables
is an estimate of the cumulative probability,
i.e.,
If Z(xj), j = 1, 2, …, N, are mutually
independent, then j = 1/N .
For a random field with spatial autocorrelation characteristics, indicator
variogram must be established and used to
estimate the cumulative probability of the
random variable at unobserved locations.
Indicator Variable for
Categorical Random Field
Similar to the case of continuous random field, the
indicator variable can be used to estimate the
probability that a pixel belongs to a certain class for
categorical random field. Let the indicator variable
be defined as
wherere
presents the j-th class and
represents
the pixel at location x.
is the value of the
indicator variable related to the j-th class.
The weighted average of values of indicator
variables is an estimate of the probability
that a pixel
belongs to the j-th class, i.e.
Class-specific Indicator Variable
Scatter Plot in Feature Space
Three-class scatter plot of indicator
variables in two dimensional
feature space.
Class-specific Scatter plot of indicator variables
(Binary Scatter Plots)
Class-1
Class-2
Class-3
For each binary scatter plot, we consider the variation
of indicator variables as a random field associated
with that particular class.
By conducting ordinary kriging, for each class, of
indicator variables in feature space, we obtain the
probability that the pixel of interest belongs to each
individual class.
Class assignment of the pixel of interest is done based
on the following criterion
If
max
, then assign
to class .
Study Area and Data
An area of approximately 70 km2 in Central
Taiwan is selected as our study site.
The study area includes a small township and
nearby mountainous area. A major river flows
westward along the northern edge of the area.
Five landcover classes (water, built-up, forest,
crop, and bare land) are identified in this study.
SPOT satellite image acquired on September
21, 2001 was used for landuse classification.
SPOT Image of the Study Area
Three classification features (green, red and
near infrared bands) were used.
A total of 1886 training pixels and 732
verification pixels were selected.
Confusion Matrix – ML (Training)
Confusion Matrix – IK (Training)
Confusion Matrix – IK
(Verification)
A Further Test Case
Further Considerations
Replicates in feature space.
Anisotropic variation in feature space.
Conclusions
Indicator kriging approach is distributionfree; therefore, it does not require the
knowledge of distribution types.
IK algorithm achieves high classification
accuracies.
References
Bierkens, M. F. and P. A. Burrough , 1993.The
indicator approach to categorical soil
data.Ι.Theory .J. of Soil Science, 44, pp. 361-368.
Bierkens, M. F. and P.A. Burrough , 1993.The
indicator approach to categorical soil data. II.
Application to mapping and land use suitability
analysis. J. of Soil Science, 44, pp. 369-381.
Meer, F. V. D., 1996. Classification of remotelysensed imagery using an indicator kriging
approach: application to the problem of calcite
dolomite mineral. Int. J. of remote sensing. Vol. 17,
no. 6, pp. 1233-1249.
References
Journel, A. G., 1983. non-parametric estimation of
spatial distributions, math. Geol. 15445-468.
Lillesand, T. M. , Johnson, W. L. , Deueil, R.
L.,O.M. Lixdstrom and D.E. Meisner,1983.Use of
Landsat data to predict the trophic state of
Minnesota lakes. Photogrammetric Engineering &
Remote Sensing, Vol. 49,No.2, pp. 219~229.
Lillesand, Thomas M. and Kiefer, Ralph W. ,1994.
Remote sensing and image interpretation.