How is it possible to process an image with a few 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.