## Anatomy of the Human Visual system

## Primary visual cortex: Hubel & Wiesel

## Primary visual cortex: Hubel & Wiesel

[Hubel & Wiesel, 1962]

## Convolutional Neural Networks : Mathematics

- One-dimensional discrete convolution (eg in time) with a kernel $g$ of radius $K$:
$$
(f \ast g)[n]=\sum_{m=-K}^{K} f[n-m] \cdot g[m]
$$

## Convolutional Neural Networks : Mathematics

- Convolution of an image (two-dimensional) with a kernel $g$ of radius $K\times K$:

$$
(f \ast g)[x, y] = \sum_{i=-K}^{K} \sum_{j=-K}^{K} f[x-i, y-j] \cdot g[i, j]
$$

## Convolutional Neural Networks : Mathematics

**Cross-correlation** of an image (two-dimensional) with a kernel $g$ of radius $K\times K$:

$$
(f \ast \tilde{g})[x, y] = \sum_{i=-K}^{K} \sum_{j=-K}^{K} f[x+i, y+j] \cdot g[i, j]
$$

## Convolutional Neural Networks : Mathematics

$$
(f \ast \tilde{g})[x, y] = \sum_{c=1}^{C} \sum_{i,j} f[c, x+i, y+j] \cdot g[c, i, j]
$$

## Convolutional Neural Networks : Mathematics

$$
(f \ast \tilde{g})[k, x, y] = \sum_{c=1}^{C} \sum_{i,j} f[c, x+i, y+j] \cdot g[k, c, i, j]
$$

## Convolutional Neural Networks : hierarchy

## Convolutional Neural Networks : Predictive coding

## Convolutional Neural Networks : Topography