How is it possible to process an image with a few neurons? An 1024*1024 pixel image has around one million pixels. If I would like to connect each pixel to an R,G,B input neuron, then more than 3 million neurons are needed.

It would be really hard, to train a neural network, which has millions of inputs. How is it possible, to reduce the number of neurons?
There are several ways to make this big number trainable:


Use CNNs
Auto-Encoders (see Reducing the Dimensionality of Data with Neural Networks)


Dimensionality reduction of the input


Scale the image down
PCA / LDA


Troll-Answer

If you really meant "only a few neurons" then you might want to have a look at Spiking neural networks. Those are incredibly computationally intensive, need a lot of hand-crafting and still get worse performance than normal neural networks for most tasks ... but you only need very little of them.