2D convolution

Convolution involving one-dimensional signals is referred to as 1D convolution or just convolution. Otherwise, if the convolution is performed between two signals spanning along two mutually perpendicular dimensions (i.e., if signals are two-dimensional in nature), then it will be referred to as 2D convolution. This concept can be extended to involve multi-dimensional signals due to which we can have multi-dimensional convolution.

In the digital domain, convolution is performed by multiplying and accumulating the instantaneous values of the overlapping samples corresponding to two input signals, one of which is flipped. This definition of 1D convolution is applicable even for 2D convolution except that, in the latter case, one of the inputs is  flipped twice.

This kind of operation is extensively used in the field of digital image processing wherein the 2D matrix representing the image will be convolved with a comparatively smaller matrix called 2D kernel. the kernel is called the image filter and the process of applying this kernel to the given image is called image filtering. The output obtained after applying the kernel to the image is called the filtered image. Depending on the values in the kernel, it performs different functions such as blurring, detecting edges, and so on.

As for one-dimensional signals, images also can be filtered with various low- pass filters (LPF), high-pass filters (HPF), etc. A LPF helps in removing noise, or blurring the image. A HPF helps in finding edges in an image. OpenCV provides a function, cv2.filter2D(), to convolve a kernel with an image. As an example, we will try an averaging filter on an image. A 3x3 averaging filter kernel can be defined as follows :

                                          | [ 1 1 1] |

                               k =1/9 | [1 1 1] |

                                           | [1 1 1] |