Thresholding implemented in the frequency domain

Image processing procedures are usually carried out in the spatial domain where the images are acquired and presented/utilized. The linear nature of the Fourier transform allows only those operations that are linear to be mapped into the frequency domain. In contrast, nonlinear operations and manipulations cannot be realized directly in the frequency domain. One of these nonlinear operations is thresholding. When operating in the spatial domain to segment image contents into object and background, thresholding is simple and efficient. However, it has no obvious representation in the frequency domain and cannot be carried out there in a straightforward fashion. In this paper, a means to relax the rigid linear limitation of the Fourier transform was investigated. A novel approach was established to achieve spatial thresholding using only frequency domain operations. The spatial grayscale or scalar data set (two-dimensional (2-D) image or three-dimensional (3-D) volume) was expanded into a binary volume in hyperspace having one more dimension than the original data set. The extended dimension is the gray level of the original data. Manipulating only on that dimension produces the effect of thresholding.