6.5.2 Kernel PCA Texture Feature
The likelihood function is realized as the distribution of KPCA features. In this
section, we introduce in detail the proposed KPCA features. This section is organized as follows. First we briefly introduce KPCA approach. Then we explain the
extraction of texture features based on KPCA approach as well as the interpretation
of the extracted features. Lastly, we explore the compatibility of KPCA texture
features with K-Means algorithm.
6.5.2.1 Kernel Principal Component Analysis
Similar to classic PCA, the KPCA approach intends to obtain a series of orthogonal
directions that explain most of data variance. However, KPCA works in nonlinear
feature space rather than the original space (Sch€ olkopf et al. 1998). As such, the
KPCA is supposed to be more powerful in terms of discovering meaningful patterns
hidden in the dataset. The nonlinear transformation is achieved by a mapping
function Φ (Á) that maps the original space to feature space. Then KPCA can be
achieved by performing classic PCA in nonlinear feature space. Alternatively,
KPCA can be implemented by using kernel function without explicitly exploring
the form of mapping function (Sch€ olkopf et al. 1998). This purpose of approach is
mainly to avoid the complexity of nonlinear mapping operation. In this chapter, we
adopt the logarithmic function as the mapping function in order to take into
consideration the multiplicative nature of SAR speckle noise. And we employ the
mapping function instead of the kernel function, considering that the mapping
function here is not complex and does not increase the dimensionality of the data.
6.5.2.2 KPCA Local Texture Features
Texture features are usually predefined linear or nonlinear transformations of
original image pixels. However, instead of using these predefined texture features,
we design a set of totally data-driven local texture features based on KPCA
technique. These adaptive features are capable of revealing meaningful information
hidden in image patches, while achieving Gaussian-like noise characteristics.
Assuming that the speckle noise is fully developed, a SAR image patch variable
can be modeled as (Hoekman 2001):
y ¼ x 1 n 1 , x 2 n 2 , . . . , x p n p
À
Á T
ð6:9Þ
where x i and n i are respectively the terrain backscatter intensity and the speckle
intensity of the ith pixel in image patch. For fully developed speckle noise, x i and n i
are independent, and n i is spatially uncorrelated. Accordingly, we denote the SAR
image as a collection of all the image patches by a data matrix:
122
L. Xu and J. Li
The likelihood function is realized as the distribution of KPCA features. In this
section, we introduce in detail the proposed KPCA features. This section is organized as follows. First we briefly introduce KPCA approach. Then we explain the
extraction of texture features based on KPCA approach as well as the interpretation
of the extracted features. Lastly, we explore the compatibility of KPCA texture
features with K-Means algorithm.
6.5.2.1 Kernel Principal Component Analysis
Similar to classic PCA, the KPCA approach intends to obtain a series of orthogonal
directions that explain most of data variance. However, KPCA works in nonlinear
feature space rather than the original space (Sch€ olkopf et al. 1998). As such, the
KPCA is supposed to be more powerful in terms of discovering meaningful patterns
hidden in the dataset. The nonlinear transformation is achieved by a mapping
function Φ (Á) that maps the original space to feature space. Then KPCA can be
achieved by performing classic PCA in nonlinear feature space. Alternatively,
KPCA can be implemented by using kernel function without explicitly exploring
the form of mapping function (Sch€ olkopf et al. 1998). This purpose of approach is
mainly to avoid the complexity of nonlinear mapping operation. In this chapter, we
adopt the logarithmic function as the mapping function in order to take into
consideration the multiplicative nature of SAR speckle noise. And we employ the
mapping function instead of the kernel function, considering that the mapping
function here is not complex and does not increase the dimensionality of the data.
6.5.2.2 KPCA Local Texture Features
Texture features are usually predefined linear or nonlinear transformations of
original image pixels. However, instead of using these predefined texture features,
we design a set of totally data-driven local texture features based on KPCA
technique. These adaptive features are capable of revealing meaningful information
hidden in image patches, while achieving Gaussian-like noise characteristics.
Assuming that the speckle noise is fully developed, a SAR image patch variable
can be modeled as (Hoekman 2001):
y ¼ x 1 n 1 , x 2 n 2 , . . . , x p n p
À
Á T
ð6:9Þ
where x i and n i are respectively the terrain backscatter intensity and the speckle
intensity of the ith pixel in image patch. For fully developed speckle noise, x i and n i
are independent, and n i is spatially uncorrelated. Accordingly, we denote the SAR
image as a collection of all the image patches by a data matrix:
122
L. Xu and J. Li