Convolution is a process used for applying general-purpose filter effects like blurring, sharpening, embossing, edge detection, and more.

To understand convolutions we must first understand what a *convolution matrix* is, also referred to as *kernel*. Take for example the blurring filter. In blur filter, we set each pixel to the mean of its neighbouring pixels. Take a look at the below averaging operation. We replaced 227 with 225.

This is a simple blur filter. We repeat this for all pixels in the image. For corner pixels, we either ignore it or use the immediate neighbour pixels. The matrix representation of each 3×3 window on which we apply our filter looks like:

The blur operation can be represented as

h0 = (h0 + h1 + h2 + h3 + h4 + h5 + h6 + h7 + h8 ) / 9

Now if we were to represent the above operation using a matrix, we would get the below representation:

What happened above is, we multiplied each pixel to its corresponding pixel (which in this case is 1/9) in the convolution matrix, then summed it up to get the value of the centre pixel. This process is called Convolution represented by the symbol * with a circle around it. The second matrix is called a kernel, mask or a convolution matrix.

The above convolution operation can be represented mathematically as below:

where, *F(x,y)* is the output image, *f(x,y)* is the original image, *h(i,j)* is the kernal.

The below code perform convolutions in python.

import numpy as np from scipy.ndimage.filters import convolve a = np.array([[228, 227, 222],[228, 227, 222],[228, 227, 222]]) b = np.array([[1/9, 1/9, 1/9],[1/9, 1/9, 1/9],[1/9, 1/9, 1/9]]) print(convolve(a, b)) -------- #the output would be like array([[227, 225, 223], [227, 225, 223], [227, 225, 223]]) #the corner pixels used only 4 or 6 pixels.

For coloured images, we perform convolution on each colour space separately and then merge them together. Applying the blurring convolution on a full image we get the below result.

The code used for the above processing is:

import matplotlib.pyplot as plt import numpy as np from skimage.io import imread from scipy.ndimage.filters import convolve img = imread("beautiful-woman.jpg") box_blur_kernel = np.array([[1/9, 1/9, 1/9], [1/9, 1/9, 1/9], [1/9, 1/9, 1/9]]) image_copy = img.copy() image_copy[:,:,0] = convolve(image_copy[:,:,0], box_blur_kernel) image_copy[:,:,1] = convolve(image_copy[:,:,1], box_blur_kernel) image_copy[:,:,2] = convolve(image_copy[:,:,2], box_blur_kernel) fig, ax = plt.subplots(ncols=2) ax[0].set_title("original") ax[0].imshow(img) ax[1].set_title("box blurred") ax[1].imshow(image_copy)

There are many other kernels for various image processing operations.

#### Gaussian blur

#### Edge detection

This one will require the image to converted to grayscale first before convolution.

#### Sharpen image

#### Emboss Image

###### References:

- https://en.wikipedia.org/wiki/Kernel_(image_processing)
- https://en.wikipedia.org/wiki/Gaussian_blur

###### Sample Image:

- https://www.pexels.com/photo/close-up-photo-of-woman-with-her-eyes-closed-2351707/