OK, so we now know that convolutions are important in the use of filters. But how does this relate to neural networks?

Recall that a neural network is defined as a linear transform () with a non-linearity applied on it (written as ). Note that x, the input image, is acted upon as a whole. This would be like having a single filter across the entire image. But what if we could process the image one small section at a time?

To add to that, in the preceding section, I showed how a simple filter could be used to blur an image. Filters could also be used to sharpen an image, picking out features that matter and blurring out features that don't. So, what if a machine could learn what filter to create?

That's the reason why we would want to use a convolution in a neural network:

• Convolutions act on small parts of the image at a time, leaving only features that matter
• We can learn the specific filters

This gives a lot of fine-tuned control to the machine. Now, instead of a rough feature detector that works on the whole image at once, we can build many filters, each specializing to a specific feature, thus allowing us to extract the features necessary for the classification of numbers.