How many neurons does the CNN input layer have?

Question

In all the literature they say the input layer of a convnet is a tensor of shape (width, height, channels). I understand that a fully connected network has an input layer with the number of neurons same as the number of pixels in an image(considering grayscale image). So, my question is how many neurons are in the input layer of a Convolutional Neural Network? The below imageseems misleading(or I have understood it wrong) It says 3 neurons in the input layer. If so what do these 3 neurons represent? Are they tensors? From my understanding of CNN shouldn't there be just one neuron of size (height, width, channel)? Please correct me if I am wrong

Answer 1

It seems that you have misunderstood some of the terminology and are also confused that convolutional layers have 3 dimensions.

EDIT: I should make it clear that the input layer to a CNN is a convolutional layer.

The number of neurons in any layer is decided by the developer. For a fully connected layer, usually it is the case that there is a neuron for each input. So as you mention in your question, for an image, the number of neurons in a fully connected input layer would likely be equal to the number of pixels (unless the developer wanted to downsample at this point of something). This also means that you could create a fully connected input layer that takes all pixels in each channel (width, height, channel). Although each input is received by an input neuron only once, unlike convolutional layers.

Convolutional layers work a little differently. Each neuron in a convolutional layer has what we call a local receptive field. This just means that the neuron is not connected to the entire input (this would be called fully connected) but just some section of the input (that must be spatially local). These input neurons provide abstractions of small sections of the input data that when taken together over the whole input we call a feature map.

An important feature of convolutional layers is that they are spatially invariant. This means that they look for the same features across the entire image. After all, you wouldn't want a neural network trained on object recognition to only recognise a bicycle if it is in the bottom left corner of the image! This is achieved by constraining all of the weights across the local receptive fields to be the same. Neurons in a convolutional layer that cover the entire input and look for one feature are called filters. These filters are 2 dimensional (they cover the entire image).

However, having the whole convolutional layer looking for just one feature (such as a corner) would massively limit the capacity of your network. So developers add a number of filters so that the layer can look for a number of features across the whole input. This collection of filters creates a 3 dimensional convolutional layer.

I hope that helped!

EDIT- Using the example the op gave to clear up loose ends:

OP's Question: So imagine we have (27 X 27) image. And let's say there are 3 filters each of size (3 X 3). So there are totally 3 X 3 X 3 = 27 parameters (W's). So my question is how are these neurons connected? Each of the filters has to iterate over 27 pixels(neurons). So at a time, 9 input neurons are connected to one filter neuron. And these connections change as the filter iterates over all pixels.

Answer: First, it is important to note that it is typical (and often important) that the receptive fields overlap. So for an overlap/stride of 2 the 3x3 receptive field of the top left neuron (neuron A), the receptive field of the neuron to its right (neuron B) would also have a 3x3 receptive field, whose leftmost 3 connections could take the same inputs as the rightmost connections of neuron A.

That being said, I think it seems that you would like to visualise this so I will stick to your example were there is no overlap and will assume that we do not want any padding around the image. If there is an image of resolution 27x27, and we want 3 filters (this is our choice). Then each filter will have 81 neurons (9x9 2D grid of neurons). Each of these neurons would have 9 connections (corresponding to the 3x3 receptive field). Because there are 3 filters, and each has 81 neurons, we would have 243 neurons.

I hope that clears things up. It is clear to me that you are confused with your terminology (layer, filter, neuron, parameter etc.). I would recommend that you read some blogs to better understand these things and then focus on CNNs. Good luck :)

Answer 2

First, lets clear up the image. The image doesn't say there are exactly 3 neurons in the input layer, it is only for visualisation purposes. The image is showing the general architecture of the network, representing each layer with an arbitrary number of neurons.

Now, to understand CNNs, it is best to see how they will work on images.

Images are 2D objects, and in a computer are represented as 2D matrices, each cell having an intensity value for the pixel. An image can have multiple channels, for example, the traditional RGB channels for a colored image. So these different channels can be thought of as values for different dimensions of the image (in case of RGB these are color dimensions) for the same locations in the image.

On the other hand, neural layers are single dimensional. They take input from one end, and give output from the other. So how do we process 2D images in 1D neural layers? Here the Convolutional Neural Networks (CNNs) come into play.

One can flatten a 2D image into a single 1D vector by concatenating successive rows in one channel, then successive channels. An image of size (width, height, channel) will become a 1D vector of size (width x height x channel) which will then be fed into the input layer of the CNN. So to answer your question, the input layer of a CNN has as many neurons as there are pixels in the image across all its channels.

I think you have confusion on the basic concept of a neuron:

From my understanding of CNN shouldn't there be just one neuron of size (height, width, channel)?

Think of a neuron as a single computational unit, which cant handle more than one number at a time. So a single neuron cant handle all the pixels of an image at once. A neural layer made up of many neurons is equipped for dealing with a whole image.

Hope this clears up some of your doubts. Please feel free to ask any queries in the comments. :)

Edit:

So imagine we have (27 X 27) image. And let's say there are 3 filters each of size (3 X 3). So there are totally 3 X 3 X 3 = 27 parameters (W's). So my question is how are these neurons connected? Each of the filters has to iterate over 27 pixels(neurons). So at a time, 9 input neurons are connected to one filter neuron. And these connections change as the filter iterates over all pixels.

Is my understanding right? I am just trying to visualize CNNs as neurons with the connections.

A simple way to visualise CNN filters is to imagine them as small windows that you are moving across the image. In your case you have 3 filters of size 3x3.

1. We generally use multiple filters so as to learn different kinds of features from the same local receptive field (as michael_question_answerer aptly puts it) or simpler terms, our window. Each filters' weights are randomly initialised, so each filter learns a slightly different feature.

2. Now imagine each filter moving across the image, covering only a 3x3 grid at a time. We define a stride value which specifies how much the window shifts to the right, and how much down. At each position, the filter weights and image pixels at the window will give a single new value in the new volume created. So to answer your question, at an instance a total of 3x3=9 pixels are connected with the 9 neurons corresponding to one filter. The same for the other 2 filters.

Your approach to understanding CNNs by visualisation is correct. But you still need the brush up your basic understanding of terminology. Here are a couple of nice resources that should help: http://cs231n.github.io/convolutional-networks/

https://adeshpande3.github.io/A-Beginner%27s-Guide-To-Understanding-Convolutional-Neural-Networks/

Hope this helps. Keep up the curiosity :)