Nonlinear Combined
Spatial-Frequency Domain
Two-Stage Image Filtering (nonlinear image processing)

The purpose of the filtering operation is assumed to be an effective elimination or attenuation of the noise that is corrupting the signal.

A fundamental difference between the behavior of linear and nonlinear image filters is the following. The impulse response (that is, the output signal when the input signal is equal to 1 at time instant 0 and zero otherwise) of a nontrivial time-invariant linear filter cannot be a zero-valued sequence. This means that a linear filter always smoothes the signal. A nonlinear image processing filter preserves the signal carefully, and usually it is more effective for noise removal despite the fact that it can be idle only for the partial signals.

At the same time, despite the fact that nonlinear image processing spatial domain filters remove noise effectively and preserve image boundaries with higher accuracy, the resulting image always becomes smoothed after filtering. Of course, it may not be so smoothed as after linear filtering, but the latter case some important details (especially the smallest ones) may be lost.

From the first point of view it is possible to solve the problem by moving to frequency domain filtering. But on the other hand, a classical frequency domain filtering (e.g. Wiener filtering) is not a simple problem from the computational point of view.

So an appropriate solution may be found using spatial and frequency domain filtering in combination. This means that a noisy image may be processed by an appropriate spatial domain filter in order to reduce noise as strongly as possible. Then a spectrum of the resulting image is corrected in order to eliminate smoothing of image boundaries. This procedure should be repeated several times, if necessary. In this way filtering is organized as iterative process of spatial domain nonlinear filtering and further correction of the resulting image spectrum.

The simplest means of spectrum correction from our point of view is correction of the spectrum amplitude and preservation of the spectrum phase. We propose such a solution precisely because distortion of the amplitude involves image smoothing, and also because a major part of noise is concentrated in the amplitude.

It is also necessary to note that a similar problem exists in image sharpening. Actually, sharpening is a kind of frequency correction or so-called anti-filtering. Sharpening effect is achieved by amplification of high frequencies. This operation may be called high frequency correction. Nonlinear image processing spatial domain filters (multi-valued frequency correction and median frequency correction) for solving the frequency correction problem are effective for solving this problem, but at the same time they have a common disadvantage: these spatial domain filters shift image boundaries. A natural way to overcome this disadvantage is to work in the frequency domain. But as for noise reduction, solving the problem in the frequency domain is complicated. Thus we combine spatial and frequency domain solutions.

Examples

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Speckle Noise Filtering

Original radar satellite noisy image	Combined Filter With Rank-Order KNV, image boundaries are preserved with very high accuracy	Enhanced difference of the filtered image from the original image
Original radar satellite noisy image	Combined Filter With Rank-Order EV	Enhanced difference of the filtered image from the original image