Digital Pattern Recognition by Moments

The problem which concerns us may be stated as follows: A digital computer receives information about a plane black-and-white pattern, and is to decide on the basis of this infoim~ation whether the pattern is "similar", in some sense to be specified below, to a given prototype. It is usually assumed that the patterns considered are of bounded size, say each contained in a given rectangle. Thus a pattern may be defined as any subset of the points in the rectangle, namely, the subset of "black" points. The information given to the computer must be of finite length; usually the rectangle is covered by a finite number of cells, and the information to the computer amounts to signals indicating whether any given cell is white or black. Patterns used as prototypes may be, for example, a ring, the letter "A", or the like. The recognition of alphabetic and numerical characters is of particular practical importance. What constitutes "similarity" varies widely from ease to case, and the methods which the machine uses in recognizing similarity must vary accordingly. To give two extreme examples, one might call two patterns similar only if they agree point for point (or rather, cell for cell) ; or else, one might admit as similar two patterns which are topo-logically equivalent (after defining a suitable topology in the space of cells), so that e.g. any simple closed curve would be called similar to a circle. 3/fany intermediate definitions are possible. In particular, for the recognition of printed characters one will wish to admit as similar two patterns if they differ at most in the following respects: (A) Location (B) Size (C) "Stretching" and "Squeezing" in either X-or Y-direction. Mathematically , these are affine transformations preserving the X-and Y-directions, i.e., transformations of the form X* = aX + b, Y* = cY + d. The special ease a = c = 1 corresponds to translations, i.e. to (A). (D) While one might consider rotation through a small angle as admissible, rotations through large angles are obviously not; e.g. the characters "6" and "9" should be recognized as different. It seems preferable to omit rotation entirely and admit in its place "slanting" such as occurs in italic type by comparison with roman. Mathematically, this is characterized as an affine transformation which leaves the points of the X-axis fixed. Combinations of the transformations (A) and (D) ex-240