## H

This filter formulation is computationally quite expensive since the filter changes from pixel to pixel in the image.

Lee derived an efficient implementation of a noise-adaptive Wiener filter by modeling the signal locally as a stationary process [14,3] that results in the filter f(x) = mf (x) + Jf{Xl (g (x) — mf (x)) (32)

af(x) + an where mf is the local mean of the signal g, and aj is the local signal variance. A local mean operator can be implemented by a normalized lowpass filter where the support region of the filter defines the locality of the mean operator. It is interesting to note that this formulation gives an expression very similar to the expression of unsharp masking in Eq. (11), although the latter is not locally adaptive. Figure 5 shows the result of Lee's filter on MR data through the pelvis.

### 4.2 Nonlinear Extension by a Visibility Function

The Wiener filter that results from a minimization based on the MSE criterion can only relate to second-order statistics of the input data and no higher [1]. The use of a Wiener filter or a linear adaptive filter to extract signals of interest will therefore yield suboptimal solutions. By introducing nonlinearities in the structure, some limitations can be taken care of.

Abramatic and Silverman [12] modified the stationary white noise Wiener solution (Eq. (30)) by introducing a visibility function a, 0 < a < 1, which depends on the magnitude of the image gradient vector, where a is 0 for "large" gradients and 1 in areas of no gradient. They showed that the generalized Backus-Gilbert criterion yields the solution methods are small. A stable solution is almost inevitably smooth.

Equation (33) shows a very explicit trade-off between resolution and stability. a = 0 gives a filter that is the identity mapping (maximum resolution) and a = 1 gives the smoother Wiener solution. By choosing a to be spatially variant, i.e., a function of the position in the image, a = a(x), a simple adaptive filter is obtained. For large gradients, the alpha function cancels the noise term and the function H becomes the identity map. This approach has the undesired feature that the filter changes from point to point in a way that is generally computationally burdensome. Abramatic and Silverman [12] proposed an "signal equivalent'' approach giving a filter that is a linear combination of the stationary Wiener filter and the identity map:

Note that Ha equals the Wiener solution (Eq. (30)) for a = 1, and for a = 0 the filter becomes the identity map. It is interesting to note that Eq. (34) can be rewritten as

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