Pattern recognition by affine moment invariants

-The paper deals with moment invariants, which are invariant under general affine transformation and may be used for recognition of affine-deformed objects. Our approach is based on the theory of algebraic invariants. The invariants from secondand third-order moments are derived and shown to be complete. The paper is a significant extension and generalization of recent works. Several numerical experiments dealing with pattern recognition by means of the affine moment invariants as the features are described. Feature extraction Affine transform Algebraic invariants Moment invariants Pattern recognition Image matching I. I N T R O D U C T I O N A feature-based recognition of objects or patterns independent of their position, size, orientation and other variations has been the goal of much recent research. Finding efficient invariant features is the key to solving this problem. There have been several kinds of features used for recognition. These may be divided into four groups as follows: (1) visual features (edges, textures and contours); (2) transform coefficient features (Fourier descriptors, ~''2~ Hadamard coefficients; ~3~ (3) algebraic features (based on matrix decomposition of image, see reference (4) for details); and (4) statistical features (moment invariants). In this paper, attention is paid to statistical features. Moment invariants are very useful tools for pattern recognition. They were derived by Hu tsl and they were successfully used in aircraft identification, ~61 remotely sensed data matching ~7~ and character recognition. ~s~ Further studies were made by Maitra ~m and Hsia "°~ in order to reach higher reliability. Several effective algorithms for fast computat ion of moment invariants were recently described in references (11-13). All the above-mentioned features are invariant only under translation, rotation and scaling of the object. In this paper, our aim is to find features which are invariant under general affine transformations and which may be used for recognition of affine-deformed objects. Our approach is based on the theory of algebraic invariants. "4~ The first attempt to find affine invariants in this way was made by Hu, ~s~ but his affine moment invariants were derived incorrectly. Several correct affine moment invariants are derived in Section 2, and their use for object recognition and scene matching is experimentally proved in Section 3. 2. A F F I N E M O M E N T INVARIANTS The affine moment invariants are derived by means of the theory of algebraic invariants. They are invariant under general affine transformation u = a 0 + a ] x + a2), v = bo + b l x + b2y. (1) The general two-dimensional (p + q)th order moments of a density distribution function p ( x , y ) are defined as: mpq = fS xPyqP(X'y) dx dy p,q = 0, !, 2 . . . . (2) For simplicity we deal only with binary objects in this paper, then p is a characteristic function of object G, and mpq=S~xPk ,qdxd) , , p,q =0 , 1,2 . . . . 13) G It is possible to generalize all the following relations and results for grey-level objects. The affine transformation (1) can be decomposed into six one-parameter transformations: 1. u = x + 7 2. u = x 3. u = ~ ' x v = y v = y + / ~ v = co.y 4. u = f i ' x 5. u = x + t ' y 6. u = x v = y t , = y v = t " x + y. Any function F of moments which is invariant under these six transformations will be invariant under the general affine transformation (1). From the requirement of invariantness under these transformations we can derive the type and parameters of the function F. If we use central moments instead of general moments (2) or (3), any function of them will be invariant