# Convolutional neural network

Convolutional Neural Network

Primarily used for image tasks such as computer vision or image generation, though they can be used anywhere you have a rectangular grid with spatial relationship among your data.

Typically convolutional layers are using in blocks consisting of the following:

- Conv2D layer.
- Usually stride 2 for encoders, stride 1 for decoders.
- Often includes some type of padding such as zero padding.

- Upscale layer (for decoders only).
- Normalization or pooling layer (e.g. Batch normalization or Max Pool).
- Activation (typically ReLU or some variant).

More traditionally, convolutional blocks came in blocks of two conv layers

- Conv2D layer.
- Activation.
- Conv2d Layer.
- Activation.
- Max pool or Avg pool

Upsampling blocks also have a transposed convolution or a bilinear upsample in the beginning.

The last layer is typically just a Conv2D with a possible Sigmoid.

## Contents

## Motivation

Convolutional neural networks leverage the following properties of images:

*Stationarity*or shift-invariance - objects in an image should be recognized regardless of their position*Locality*or local-connectivity - nearby pixels are more relevant than distant pixels*Compositionality*- objects in images have a multi-resolution structure.

## Convolutions

Pytorch Convolution Layers

Types of convolutions animations

Here, we will explain 2d convolutions.

Suppose we have the following input image:

1 2 3 4 5 6 7 8 2 3 2 5 9 9 5 4 8 8 2 1

and the following 3x3 kernel:

1 1 1 2 1 2 3 2 1

For each possible position of the 3x3 kernel over the input image,
we perform an element-wise multiplication (\(\displaystyle \odot\)) and sum over all entries to get a single value.

Placing the kernel in the first position would yield:

\(\displaystyle
\begin{bmatrix}
1 & 2 & 3\\
6 & 7 & 8\\
2 & 5 & 9
\end{bmatrix}
\odot
\begin{bmatrix}
1 & 1 & 1\\
2 & 1 & 2\\
3 & 2 & 1
\end{bmatrix}
=
\begin{bmatrix}
1 & 2 & 3\\
12 & 7 & 16\\
6 & 10 & 9
\end{bmatrix}
\)

Summing up all the elements gives us \(\displaystyle 66\) which would go in the first index of the output.

The formula for the output resolution of a convolution is: \(\displaystyle \frac{x-k+2p}{s}+1 \) where:

- \(x\) is the input resolution
- \(k\) is the kernel size (e.g. 3)
- \(p\) is the padding on each side
- \(s\) is the stride

Typically a \(3 \times 3\) conv layer will have a padding of 1 and stride of 1 to maintain the same size. A stride of \(2\) would halve the resolution.

### Stride

How much the kernel moves along. Typically 1 or 2.

### Padding

Convolutional layers typically yield an output smaller than the input size.
We can use padding to increase the input size.

See Machinecurve: Using Constant Padding, Reflection Padding and Replication Padding with Keras

- Types of padding

- Zero/Constant
- Mirror/Reflection
- Replication

With convolution layers in tensorflow and other libraries you often see these two types of padding:

`VALID`

- Do not do any padding`SAME`

- Apply zero padding such that the output will have resolution \(\lfloor x/stride \rfloor\).- \(p=\frac{k-1}{2}\)
- Exact behavior varies between ML frameworks. Only apply this when you know you will have symmetric padding on both sides.

### Dilation

Space between pixels in the kernel

A dilation of 1 will apply a 3x3 kernel over a 5x5 region. This would be equivalent to a 5x5 kernel with odd index weights (\(\displaystyle i \% 2 == 1\)) set to 0.

### Groups

## Other Types of Convolutions

### Transpose Convolution

See Medium: Transposed convolutions explained.

Instead of your 3x3 kernel taking 9 values as input and returning 1 value: \(\sum_i \sum_j w_{ij} * i_{i+x,j+y}\), the kernel now takes 1 value and returns 9 values: \(w_{ij} * i_{x,y}\).

This is sometimes misleadingly referred to as deconvolution.

Oftentimes, papers apply a 2x bilinear upsampling rather than using transpose convolution layers.

### Gated Convolution

See Gated Convolution (ICCV 2019)

Given an image, we have two convolution layers \(\displaystyle k_{feature}\) and \(\displaystyle k_{gate}\).

The output is \(\displaystyle O = \phi(k_{feature}(I)) \odot \sigma(k_{gate}(I))\)

## Pooling

Pooling is one method of reducing and increasing the resolution of your feature maps.

You can also use bilinear upsampling or downsampling.

### Unpooling

During max pooling, remember the indices where you pulled from in "switch variables".

Then when unpooling, save the max value into those indices. Other indices get values of 0.

## Spherical Images

There are many ways to adapt convolutional layers to spherical images. Below are just a few that I've seen.

- Learning Spherical Convolution for Fast Features from 360 Imagery (NIPS 2017) proposes using different kernels with different weights and sizes for different altitudes \(\phi\).
- Circular Convolutional Neural Networks (IV 2019) proposes padding the left and right sides of each input and feature map using pixels such that the input wraps around. This works since equirectangular images wrap around on the x-axis.
- SpherePHD (CVPR 2019) proposes using faces of an icosahedron as pixels. They propose a kernel which considers the neighboring 9 triangles of each triangle.