Good representations enable the recognition of real world objects. They make it possible to compute differences between objects. If the differences between similar objects are always small, they constitute the basis for a generalization. This is the case for a continuous mapping of the original objects, what has been called a compact representation. It may, however, yield similar representations for dissimilar objects, like by the use of features. This is avoided by what has been defined as a true representation. Does it exist? Is it possible to give an example?

In a true representation small distances between represented objects correspond to similar objects. This is true for a proper pixel representation of a 2-dimensional object like a handwritten digit. Another almost identical set of pixels represents an almost identical object. In is not possible that such an object is entirely different if there are no hidden features like color or weight. The pixel values are all there is to say about the object. If another set is not very different the objects must be similar.

Is the pixel representation also compact? Does it hold that minor changes in the object always result in a similar representation? If the digit is shifted a little bit or if it is given a small rotation then suddenly some pixel turn from black to white and the other way around. If the pixels constitute a vector space the object jumps there: small changes may cause big jumps. This can be avoided by oversampling the digit heavily by which pixels on the edge of the digit become gray. Now a sufficiently small shift or rotation will just change the intensities of the border pixels gradually.

It can be concluded that a pixel representation is both, compact as well as true, if the original sampling is sufficiently high. Low spatial resamplings are also possible afterwards if the (gray) pixel intensities are represented with a high accuracy.

What is described here for digits may hold for any sampling of objects, 2-dimensional, 3-dimensional, gray-value and color images, but also for spectra and histograms. The condition is that it is possible to  reconstruct the object from the pixels (samples) in such a way that the human recognition is not affected.

The Pixel Space
True Representations

Filed under: PR SystemRepresentation