Convolution is one of the concepts in signal processing. Signal processing refers to operations performed on signals to extract meaningful data. Mathematicians use convolution techniques to construct the output signals for any random input signal. For this, they make use of the impulse response provided by the system. Functional analysis, electrical engineering, probability and image and signal processing applications are areas that use two-dimensional convolution techniques. Convolution represents a mathematical operation on two functions, and it results in a third function which evolves as the result of the modification of the original functions.
Deriving 2-D Convolution from 1-D Convolution
Two-dimensional convolutions are the most time-consuming parts in an application. Mathematicians arrive at two-dimensional convolutions by broadening the mathematical treatment from one dimension to two dimensions. Two-dimensional convolutions are extensions of one-dimensional convolutions, where convolution happens in both horizontal and vertical directions over a two-dimensional space. The mathematicians refer to the impulse response provided by a system in 2-D convolution as a kernel or a filter.
Application of a 2-D Convolution
The mathematicians generalize the convolution theory that they developed for one-dimensional signal to two dimensions for applying to images. They treat each of these two dimensions separately in most cases. To apply two-dimensional convolution to the signal f(x, y), the mathematicians first apply Fourier Transform for the variable x while keeping y fixed, and then for the variable y while keeping x fixed. Thus, they get a function of two frequency variables. However, it is not possible to decompose the functions in two dimensions into two separate sequences of one-dimensional operations. Every two-dimensional convolution results in a single series, unless it is possible to separate one of the input signals.
2-D Convolution in Image and Signal Processing
Image and signal processing applications make use of two-dimensional convolution techniques. During image processing using 2-D convolution techniques, the convolution operator helps to vary the image characteristics, thereby acting as a filter. For instance, it helps to smooth and sharpen the image edges, smudges or distorts the image or helps to eliminate the noises of different frequencies that may be present with the image. When applied to signal processing, 2-D convolutions help to restrain and remove the unnecessary parts of the signal. It also helps to split the signal into multiple parts. Seismic processing also makes use of 2-D convolution techniques.
Convolution of an Image
Every image has two dimensions, thus image processing can be done using 2-D convolutions. During an image convolution, the mathematicians associate each pixel of the image with a filter array subset that contains filter elements of the same size. Every step of the convolution involves processing of the color components of each pixel associated with an array element. Further, the filter elements scale each of the corresponding image components. After each step in the convolution process, the filter position shifts by one, and so do the pixels corresponding to the input image. In this manner, convolution of the entire image happens.
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