281.

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statistics through the use of a moving window. The Frost filter is formed according to the following adaptive impulse response (Frost et al., 1982):

(1.59)

where K₃ is a normalising constant.

To process the image at location (i₀, j₀), the parameters σ_z and μ_z are approximated using the pixels within a window centred at (i₀, j₀). The Frost filter then outputs the estimate by taking the weighted average of data in the neighbourhood of (i₀, j₀). The weight for the pixel at address t is m_t, as shown in Equation (1.59). The term in this equation represents the distance between the window’s centre pixel and its neighbouring pixel at address t. For example, if the window is centred on the pixel , the pixel at address has

1.8.3.5 Refined gamma MAP filter

Under the multiplicative noise assumption of Equation (1.36), the maximum a posteriori (MAP) estimate (Chapter 2) is obtained by maximising the Bayesian criterion with respect to a noise-free signal x given observation z:

 

or, equivalently,

(1.60)

where is modelled by a gamma probability density function (p.d.f.), which explains the origin of the term gamma MAP. It is known that, within a radar image, forests, agricultural areas, and oceans can be well modelled by a gamma distribution. Hence, the gamma MAP filter generally can produce satisfactory speckle suppression results.

Maximising Equation (1.60) requires that the following condition must be met (Lopes et al., 1990):

(1.61)

where ∂ denotes the partial derivative. For an N-look SAR image, P(z|x) is expressed as a gamma p.d.f. (Touzi et al., 1988):

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