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2: Raghuveer M. Rao, Manoj K. Arora
where fx (y) is the PDF of the pixel measurement regardless of the class to
which it belongs and P(Cd is the a priori probability that a randomly chosen
pixel belongs to Class 1. A similar relationship obviously holds for P( C2Iy).
Thus, (2.20) yields, in conjunction with Bayes theorem, the MAP classification
rule:
Assign pixel measurement y to Class 1 if
(2.22)
and to Class 2 otherwise.
The MAP rule reduces to the maximum-likelihood rule if the a priori probabilities of belonging to either of the two classes are the same since the relationship in (2.22) then reduces to the one in (2.19).
Since a priori probabilities and conditional probability density functions
are required for application of the MAP classification procedure, one has to
form estimates of these probabilities from actual data. This requires a training
process where the particular class a pixel belongs to is known beforehand (from
the reference data such as field surveys, aerial photographs, existing maps
etc.) during the training period and the various probabilities are estimated.
Thus, this approach falls under the category of a supervised classification
approach. Prior probabilities are more difficult to estimate reliably. If one is
trying to distinguish between vegetation and sand, for example, the percentage
of image area covered by vegetation varies not only by geographic region but
also from image to image in the same geographic region. A conditional PDF
such as fxlCl (y) tends to be more reliable and is estimated through conditional
histograms. Thus, the ML classification rule is easier to apply than the MAP
rule.
Generally, the higher the number of spectral bands, the more difficult the
procedure becomes. For this reason, the process is typically restricted to using
data from a small number of bands (i. e. multispectral data only). A simplification that is often done in lieu of estimating the actual PDF is to regard the PDF
as multivariate Gaussian. This requires estimating the mean vector and covariance matrix of multispectral data from each class. If the classes differ only in
the mean of their measurements and have symmetric covariance matrices l ,
then the ML rule reduces to a pixel being classified as belonging to a particular
class if the pixel measurement is closest in Euclidean distance to the mean
of that class when compared to the distance between the measurement and
the means of other classes. When this rule is adopted for pixel classification
regardless of whether the class distributions are jointly normal with identical,
symmetric distributions but for their means, the approach is called nearest
neighbor (or minimum distance to mean) classification. This rule is very easy
to implement but when the distribution is asymmetric it does not conform to
the maximum likelihood principle.
1 A symmetric distribution about a mean of x implies that the PDF fx (y)is a function only of the
Euclidean distance between y and x.
2: Raghuveer M. Rao, Manoj K. Arora
where fx (y) is the PDF of the pixel measurement regardless of the class to
which it belongs and P(Cd is the a priori probability that a randomly chosen
pixel belongs to Class 1. A similar relationship obviously holds for P( C2Iy).
Thus, (2.20) yields, in conjunction with Bayes theorem, the MAP classification
rule:
Assign pixel measurement y to Class 1 if
(2.22)
and to Class 2 otherwise.
The MAP rule reduces to the maximum-likelihood rule if the a priori probabilities of belonging to either of the two classes are the same since the relationship in (2.22) then reduces to the one in (2.19).
Since a priori probabilities and conditional probability density functions
are required for application of the MAP classification procedure, one has to
form estimates of these probabilities from actual data. This requires a training
process where the particular class a pixel belongs to is known beforehand (from
the reference data such as field surveys, aerial photographs, existing maps
etc.) during the training period and the various probabilities are estimated.
Thus, this approach falls under the category of a supervised classification
approach. Prior probabilities are more difficult to estimate reliably. If one is
trying to distinguish between vegetation and sand, for example, the percentage
of image area covered by vegetation varies not only by geographic region but
also from image to image in the same geographic region. A conditional PDF
such as fxlCl (y) tends to be more reliable and is estimated through conditional
histograms. Thus, the ML classification rule is easier to apply than the MAP
rule.
Generally, the higher the number of spectral bands, the more difficult the
procedure becomes. For this reason, the process is typically restricted to using
data from a small number of bands (i. e. multispectral data only). A simplification that is often done in lieu of estimating the actual PDF is to regard the PDF
as multivariate Gaussian. This requires estimating the mean vector and covariance matrix of multispectral data from each class. If the classes differ only in
the mean of their measurements and have symmetric covariance matrices l ,
then the ML rule reduces to a pixel being classified as belonging to a particular
class if the pixel measurement is closest in Euclidean distance to the mean
of that class when compared to the distance between the measurement and
the means of other classes. When this rule is adopted for pixel classification
regardless of whether the class distributions are jointly normal with identical,
symmetric distributions but for their means, the approach is called nearest
neighbor (or minimum distance to mean) classification. This rule is very easy
to implement but when the distribution is asymmetric it does not conform to
the maximum likelihood principle.
1 A symmetric distribution about a mean of x implies that the PDF fx (y)is a function only of the
Euclidean distance between y and x.