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Frequency Domain
Nonlinear Median-Like Filter
for Quasi-Periodic and
Periodic Noise Image Filtering

The theory of linear filtering gives optimal noise reduction methods, when the corruption can be modeled as a Gaussian process and the mean square error is the criterion for accuracy. These assumptions are reasonable in most applications. But in digital systems, errors are often caused by bit changes, and the resulting distribution is far from Gaussian. Also in many cases, where visual quality is the ultimate criterion, the mean square error is not a realistic criterion, and images with large mean square error can be visually much better than ones with a small mean square error. This is where nonlinear methods prove most useful.

Removal of half-tone patterns from photographs is often an important problem to address, which occupies many news group discussions (usually with misinformation). Processing in the frequency domain is a much better solution than operations such as blurring that can hide the dots but also reduce edge sharpness. The resolution of the scanner used to input the image also affects the high frequency noise pattern in the acquired image and can produce additional moire patterns. This artifact is also characteristic of gray scale images obtained from single-chip video cameras.

Usually such quasi-periodic and periodic noise in image results peaks in its spectrum. In case of periodic nose, these peaks are even visible to the eye, while peaks corresponding to quasi-periodic noise are hidden and itТs quite hard to detect them. A good idea is to find some mean to detect such peaks. To filter images corrupted by the quasi-periodic noise, for example, Wiener filter can be used, but itТs slow from the computational point of view. It works with a whole spectrum at once and it needs a mask, which we cannot supply without detecting the needed coefficients. Thus we again come to the problem of peak detection.

We utilize a new filter for the detection, reduction and even removal (filter) of the quasi-periodic and periodic noise. It is quite fast and it allows achieving a very good result.

The main idea of the filter is to remove peaks from the image spectrum, which are replaced by the median calculated in the local window. Fourier, Cosine and Walsh (with Walsh ordering) spectra can be used to achieve the result. The Fourier basis gives the best results, while correction of the Cosine and Walsh spectra for removing quasi-periodic noise adds the white one (Cosine and Walsh spectra can be used for truly periodic noise filtering). To be more specific, the correction of the spectrum using the proposed filter is concentrated in the correction of the spectrum amplitude. It is analyzed using a local window of NxN size. To decide whether the current spectral coefficient needs to be modified, the coefficients that belong to the local window around it are taken. A median value is calculated within this window and then the central coefficient is compared to the median value. Depending on the comparison result, the central value is replaced by the median one. Mutual dependence of the central and median values is stable for the non-corrupted spectral coefficients independently of the frequencies to which they correspond. So it is independent of the position of the peaks in the spectrum amplitude.

Using this filter, it is possible to filter periodic noise, and to reach a significant reduction of quasi-periodic noise, completely preserving the image boundaries.

Examples

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Periodic Noise Filtering

Original Filtered

Quasi-periodic and Impulsive noise filtering

Input infra-red image
Quasi-periodic noise filtering and impulsive noise filtering after the noise detection Difference between the input and the resulting image: image boundaries are preserved completely

 

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Last Updated
Mon, November 01, 2009 13:18