## M1 N1

^ 0 otherwise.

A useful image enhancement operation is convolution using local operators, also known as kernels. Considering a kernel w(fc, Z) to be an array of (2K + 1 x 2L + 1) coefficients where the point (fc, Z) = (0,0) is the center of the kernel, convolution of the image with the kernel is defined by:

k l g (m, n) = w (fc, Z)*/ (m, n)= ^^ ^^ w (fc, Z) • / (m - fc, n - Z), fc=-k z=-l where g (m, n) is the outcome of the convolution or output image. To convolve an image with a kernel, the kernel is centered on an image pixel (m, n), the point-by-point products of the kernel coefficients and corresponding image pixels are obtained, and the subsequent summation of these products is used as the pixel value of the output image at (m, n). The complete output image g (m, n) is obtained by repeating the same operation on all pixels of the original image [4, 5, 13]. A convolution kernel can be applied to an image in order to effect a specific enhancement operation or change in the image characteristics. This typically results in desirable attributes being amplified and undesirable attributes being suppressed. The specific values of the kernel coefficients depend on the different types of enhancement that may be desired.

Attention is needed at the boundaries of the image where parts of the kernel extend beyond the input image. One approach is to simply use the portion of the kernel that overlaps the input image. This approach can, however, lead to artifacts at the boundaries of the output image. In this chapter we have chosen to simply not apply the filter in parts of the input image where the kernel extends beyond the image. As a result, the output images are typically smaller than the input image by the size of the kernel.

The Fourier transform F (m, v) of an image / (m, n) is defined as

m = 0,1,2,..., M - 1 v = 0,1,2,..., N - 1, where m and v are the spatial frequency parameters. The Fourier transform provides the spectral representation of an image, which can be modified to enhance desired properties. A spatial-domain image can be obtained from a spectral-domain image with the inverse Fourier transform given by

0 0