Bidimensional discrete convolutions

The most common type of convolution employed in deep learning is based on bidimensional arrays with any number of channels (such as grayscale or RGB images). For simplicity, let's analyze a single layer (channel) convolution because the extension to n layers is straightforward. If X ∈ ℜw × h and k ∈ ℜn × m, the convolution X ∗ k is defined as (the indexes start from 0):

It's clear that the previous expression is a natural derivation of the continuous definition. In the following graph, there's an example with a 3 × 3 kernel:

Example of bidimensional convolution with a 3x3 kernel

The kernel is shifted horizontally ...

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