274.

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Page 45

(1.38)

where E denotes the expected value. Suppose that a window is placed over a homogeneous area, as described above, then one can set because over the area the pixel values should be close to each other. Equation (1.38) can be further derived as:

(1.39)

Based on Equation (1.37), μ_x can be substituted by μ_z/μ_v, which gives the following relationship (assuming that the local statistics are extracted from a homogeneous area):

(1.40)

where C_z is the coefficient of variation of the noise fading signal z. The reader should not confuse C_z with CoV in Equation (1.34), since both denote different quantities. Since μ_v=1 as specified previously, it follows that σ_v=C_z. In other words, σ_v is equal to the coefficient of variation of pixels over a homogeneous area.

The noise standard derivation σ_v can also be defined on the basis of theoretical studies. In general, σ_v is set as 1/N (N is the number of looks). However, σ_v could also take other values dependent on different radar image formation characteristics. Ulaby et al. (1986b) provide further details.

Recall from Equations (1.36) and (1.37) that the variance of x can be derived as (Lee, 1986):

(1.41)

Note again that in Equation (1.41) the terms μ_z and are approximated by the local mean and local variance (according to the third assumption) within a predefined window. Equation (1.36) implies that an observed pixel z can be linearised by using a first-order Taylor series expansion about the point (μ_x, μ_v) (Lee, 1986):

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