Wavelet KSigma Filtering

In the section above, we saw that wavelet filtering can remove noise from images—providing it is well-behaved Gaussian noise. However, CCD and digital camera images usually contain a mixture of Poisson noise, Gaussian noise, and complex noise types generated in dark subtraction, flat-fielding, and image stack-

Light Noise Texture

Figure 18.11 Wavelet kCS filtering filters images with any noise type, but requires that the user specify the acceptable noise level at each wavelet scale. In this example, a standardized "strong" filter smoothed away background noise without significant blurring or softening of image detail.

Figure 18.11 Wavelet kCS filtering filters images with any noise type, but requires that the user specify the acceptable noise level at each wavelet scale. In this example, a standardized "strong" filter smoothed away background noise without significant blurring or softening of image detail.

ing. The complexity of the noise makes it difficult to estimate the effective gain, readout noise, and bias of the processed image required for constant sigma scaling.

Furthermore, operations used in stacking images such as translation, rotation, scaling, and resampling act as blur filters, reducing the amplitude of the wavelet scale j= 1 coefficients with little effect on higher wavelet scales.

Taken together, calibration and stacking deprive us not only of the ability to gauge the total noise in an image, but also of the ability to predict the noise at the high scales based on the noise measured in scale j= 1 . And without a priori knowledge of how much noise to expect at each scale, iterative solutions are prone to failure. Unfortunately, almost all CCD images and many digital camera images have been calibrated and many have been stacked. What to do?

Wavelet ka (k-sigma) filtering is designed to deal with the messy noise types found in such images. Rather than assuming well-behaved noise, the wavelet kc filter requires the observer to specify an acceptable noise threshold, kj, for every wavelet scale. In operation, the filter then determines the expected noise at each scale, o;, and truncates wavelet coefficients smaller than kp]. After truncation, an inverse wavelet transform reconstructs a filtered image.

The ko filter is extremely versatile. For example, by setting kj equal to 3 for wavelet Scales 1, 2, and 3, the user can reject noise that is probably similar to Gaussian noise in the lower levels. Since a ko filter determines Cj independently for each level, it doesn't matter whether the noise behaves like Gaussian noise at all scales. In the higher levels, which normally contain little noise, then taper the kj values so that k4= 2 , k5= 1, and k6, , and ks all equal 0. The effect will be that the filter removes 99.7% of the "noise" it finds in Scales 1, 2, and 3,95.4% of the "noise" in Scale 4, 68% in Scale 5, and none in Scales 6, 7, and 8. However, significant wavelet coefficients—defined as those coefficients greater than kjOj —will pass through the filter undiminished.

With ordinary CCD images, it is useful to define standard filter types that produce good results with a wide variety of images, and also to allow the user to develop custom filters. Here are some proven kj values for wavelet ko filters with effects from gentle to strong:

Wavelet Scale

Gentle

Normal

Strong

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