Practical detection of "similarity"? The world needs new mathematical ideas in order to solve the problem of making general purpose image recognition algorithms. What are some new ideas for approaching the problem of determining if two objects are "similar"? There are techniques for detecting precise forms of "similarity" between mathematical objects (for example, "similar triangles", "homomorphic groups".) None of these are very useful for detecting the type of similarity that we see between objects in nature such as two leaves on the same tree or the grain on one area of a board vs the grain in another area. People interested in image recognition have developed techniques for texture detection and recognition. Most are statistical. I have never seen one with broad applicability. In the case of two leaves from the same tree, if we consider a 2D outline of each leaf as a curve, I suppose there are conformal mappings that take one to the other. However, from a practical point of view, this approach begins with a fallacy - namely it assumes that it will be possible to process a typical image to compute bounding curves for such objects. Actual images have occusions where one object obscures part of another. Edge detection methods often fail to detect portions of edges and the subjective ways to fill-in missing edges must be tweaked for particular collections of images. It would please me if effective image recognition algorithms depended on fairly low level ideas, In the total problem of image recognition there must be some dependence on relatively sophisticated knowledge. For example, depth perception (in the sense of the eyes perceiving distance due to seeing different images in each eye) is only effective to about 20 ft or so. So when you "see" a car parked in the distance in front of a telephone pole, it is higher level knowledge about the world that tells you the car is probably in front of the pole instead of the pole being something that sprouts out of the roof of the car. I wonder if the problem of recognizing the similarity between two leaves or two patches of wood grain also requires higher level knowledge.