Data at multiple scales

SIFT is a decent way to extract features. The features are scale-invariant, and they're at least partially rotation invariant. They're translation invariant, and changes in lighting, etc, don't strongly affect the features.

FAST is also a pretty decent algorithm, though its main purpose is just for identifying points of interest (corners), whereas SIFT identifies points of interest then extracts features at those locations. The corner detection is also able to transcend scale reasonably well, and at least in terms of finding the same corner (though not necessarily knowing it's the same corner) it's rotation and translation invariant.

Yann LeCunn's LeNet exhibits a lot of translation and rotation invariance, and scale invariance.

If you want a bit of insight, you might want to look for the "Regularization in NN" in Bishop:

You have some options when using NN and scaled data (or even rotated)

1. Create a larger database
2. Create a regularizer that penalizes changes in the output when the input is transformed.
3. You can try and extract features that are invariant, like colors
4. You can build this properties in the structure of your net

If you use Tangent propagation, you are using approach 2, where you penalize different outputs for scaled inputs.

Le Cun used the 4th approach, he used convolutional networks, where you share some weights in an extra layer called the convolutional layer.

In Bishop's chapter 5.5 it is pretty well explained