278.

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Page 49

likely to be outliers (noise) and should be excluded from the averaging process.

The mean of the window centred on a pixel at location (i, j) with value z_ij is regarded as the mean of noise-free signal x. According to Lee (1986), the two-sigma range can be expressed as , or equivalently, using Equation (1.39). In a window of size (2n+1, 2n+l), the estimated value of x_ij is (Lee, 1986):

(1.49)

where

(1.50)

A problem arises when the number of pixels outside the two-sigma range is large. A threshold r can be used to tackle this problem. If the number of pixels within two-sigma range is less than r then x_ij is replaced by the average of its four nearest neighbours (i.e. its first-order neighbourhood, described in Chapter 6). Lee (1986) also suggest that r should be smaller than 3 in the case of 5×5 window, and less than 4 if a 7×7 window is used because the use of a high threshold (r) can cause blurring effects.

1.8.3.3 The Kuan filter

Kuan et al. (1987) describe a filter that adopts a linear constraint on the model estimation mechanism, and it outputs a noise-free estimate from an observation z, such that the mean square error is a minimum. According to Kuan et al. (1987), Lopes et al. (1990) and Wu and Maitre (1992), the estimated noise-free signal using the Kuan filter is expressed as:

(1.51)

where the weighting coefficient k is defined as:

(1.52)

Combining Equations (1.51) and (1.52) gives (Baraldi and Parmiggiani, 1995b):

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