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the surrounding neighborhood. As a filter, it boasts a smooth transition from passing low-frequency structure to effectively blocking high-frequency detail. Because of this, the Gaussian kernel produces the smoothest smoothing.

It is very important to understand that the effect of a filter depends critically on the sharpness of detail in the image. If the image contains significant detail on one-pixel-to-the-next scale, then applying a smoothing filter will reduce the detail present. However, if the image has already been convolved with a smoothing function such as atmospheric turbulence, then the appropriate smoothing filter— one in which the radius of the smoothing kernel is about 0.6 times the full-width half-maximum of the point-spread function of the image—can reduce image noise without producing a significant loss in the (already reduced) image sharpness.

14.2.3.2 Sharpening Kernels (High-Pass Filters)

Sharpening kernels increase the difference between a central pixel and the pixels in its immediate neighborhood. Sharpening enhances the contrast of detail in an image, but also increases the visibility of noise in it. Most sharpening filters are designed so that the sum of the elements of the kernel is 1. A useful measure of the filter's strength is its contrast enhancement, which equals the value of the central pixel divided by the sum of the elements in the filter.

Technically, sharpening filters and high-pass filters are not identical. A high-pass filter passes high spatial frequencies and blocks low ones, but a sharpening filter detects high spatial frequencies and adds them to the existing image. The output of a high-pass filter is an image containing only high frequencies, whereas the output of a sharpening filter is a copy of the source with the high spatial frequencies enhanced so that the image looks sharper and more detailed.

The effect of a high-pass filter depends critically on the sharpness of detail in the image. If the image contains significant detail on a one-pixel-to-the-next scale, then applying a sharpening filter enhances the contrast of the detail markedly. However, if the width of the filter kernel is larger than the point-spread function inherent in the image, the convolution increases the visibility of the noise in the image without producing an improvement in the contrast of image detail.

The classic sharpening kernel:

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