Image recognition with the discrete rectangular wave transform

Abstract

The discrete rectangular wave transform (DRWT) is obtained by replacing sines and cosines in the discrete Fourier transform (DFT) by the bipolar rectangular wave function. Thus, DRWT of size N is defined as follows: where x(·) is the input data, and μ(t) is the bipolar rectangular wave function which is 1 for 0 < t mod 1 < 0.5, −1 for 0.5 < t mod 1 < 1, and 0 for t mod 1 = 0 or 0.5. The DRWT matrix contains no numbers other than ±1, ±j, and their combinations. There are also data structures for its fast computation. Since there are add operations only, its electrooptical as well as VLSI implementations are very attractive. DRWT is not an orthogonal transform, but its inverse is well defined, consisting of the complex conjugate transpose of the DRWT matrix followed by a matrix which is the direct sum of blocks of circular correlations. There are no general multiplications in the inverse transform either for N ≤ 32, when N is a power of 2. Since DRWT is both much faster and easier to implement than DFT, it is very attractive to use in applications if it also performs well in terms of various criteria such as accuracy and resolution. Our computer simulations with the method of Fourier descriptors for shape recognition indicate that DRWT performs considerably better than DFT in image recognition where the transform is used for the purpose of feature extraction. The new method using DRWT is obtained by simply replacing DFT with DRWT in the method of Fourier descriptors. In both methods, the same number of largest transform coefficients was kept, and the recognition was based on the Euclidian distances to the set of library features belonging to different reference images. Especially in the presence of white Gaussian noise, the new method is found to be considerably more accurate than the DFT method. We have also developed normalization procedures such that the new transform is invariant under translation, rotation, and scaling of the signal.