1962 IRE TRANSACTIONS ON INFORMATION THEORY 179 Visual Pattern Recognition by Moment Invariants” MING-KUEI HUt SENIOR MEMBER, IRE Summary-In this paper a theory of two-dimensional moment invariants for planar geometric figures is presented. A fundamental theorem is established to relate such moment invariants to the well- known algebraic invariants. Complete systems of moment invariants under translation, similitude and orthogonal transformations are derived. Some moment invariants under general two-dimensional linear transformations are also included. Both theoretical formulation and practical models of visual pattern recognition based upon these moment invariants are discussed. A simple simulation program together with its perform- ance are also presented. It is shown that recognition of geometrical patterns and alphabetical characters independently of position, size and orientation can be accomplished. It is also indicated that generalization is possible to include invariance with parallel pro- jection. I. INTRODUCTION I% ECOGNITION of visual patterns and characters independent of position, size, and orientation in the visual field has been a goal of much recent research. To achieve maximum utility and flexibility, the methods used should be insensitive to variations in shape and should provide for improved performance with re- peated trials. The method presented in this paper meets a.11 these conditions to some degree. Of the many ingeneious and interesting methods so far devised, only two main categories will be mentioned here: 1) The property-list approach, and 2) The statistical approach, including both the decision theory and random net approaches.’ The property-list method works very well when the list is designed for a particular set of pat- terns. In theory, it is truly position, size, and orientation independent, and may also allow for other variations. Its severe limitation is that it becomes quite useless, if a different set of patterns is presented to it. There is no known method which can generate automatically a new property-list. On the other hand, the statistical approach is capable of handling new sets of patterns with little difficulty, but it is limited in its ability to recognize pat- terns independently of position, size and orientation. This paper reports the mathematical foundation of two- dimensional moment invariants and their applications to visual information processing.’ The results show that recognition schemes based on these invariants could be truly position, size and orientation independent, and also flexible enough to learn almost any set of patterns. In classical mechanics and statistical theory, the con- * Received by the PGIT, August 1, 1961. t Electrical Engineering Department, Syracuse University, Syracuse, N. Y. 1 M. Minsky, “Steps toward artificial intelligence,” PROC. IRE, vol. 49, pp. 830; January, 1961. Many references to these methods can be found in the Bibliography of M. Minsky’s article. 2 M-K. Hu, Pattern recognition by moment invariants,” PROC. IRE (Correspondence), vol. 49, p. 1428; September, 1961. cept of moments is used extensively; central moments, size normaliza,tion, and principal axes are also used. To the author’s knowledge, the two-dimensional moment invariants, absolute as well as relative, that are to be presented have not been studied. In the pattern recogni- tion field, centroid and size normalizatfion have been exploited3-5 for “preprocessing.” Orientation normaliza- tion has also been attempted.5 The method presented here achieves orientation independence without ambiguity by using either absolute or relative orthogonal moment invariants. The method further uses “moment invariants” (to be described in III) or invariant moments (moments referred to a pair of uniquely determined principal axes) to characterize each pattern for recognition. Section II gives definitions and properties of two- dimensional moments and algebraic invariants. The mo- ment invariants under translation, similitude, orthogonal transformations and also under the general linear trans- formations are developed in Section III. Two specific methods of using moment invariants for pattern recogni- tion are described in IV. A simulation program of a simple model (programmed for an LGP-SO), the performance of the program, and some possible generalizations are described in Section V. II. MOMENTSANDALGEBRAIC INVARIANTS A. A Uniqueness Theorem Concerning Moments In this paper, the two-dimensional (p + n)th order moments of a density distribution function p(z, y) are defined in terms of Riemann integrals as m m m,, = ss xpYaPb, Y) &J dY, -m -m p, q = 0,1,2, *-* . (1) If it is assumed that p(z, y) is a piecewise continuous therefore bounded function, and that it can have nonzero values only in the finite part of the xy plane; then moments of all orders exist and the following uniqueness theorem can be proved. Uniqueness Theorem: The double moment sequence {m,,] is uniquely determined by p(s, y); and conversely, p(z, y) is uniquely determined by {m,,) . It should be noted that the finiteness assumption is important; otherwise, the above uniqueness theorem might not hold. 3 W. Pitts and W. S. McCulloch, “How to know universals,” Bull. Math. Biophys., vol. 9, pp. 127-147; September, 1947. * L. G. Roberts, “Pattern recognition with an adaptive network,” 1960 IRE INTERNATIONAL CONVENTION RECORD, pt. 2, pp. 66-70. 6 Minsky, op. cit., pp. 11-12. Authorized licensed use limited to: The University of Utah. Downloaded on February 4, 2010 at 13:21 from IEEE Xplore. Restrictions apply.180 IRE TRANSACTIONS ON INFORMATION THEORY February B. Characteristic Function and Moment Generating Function in terms of the ordinary moments. For the first four The