Optical Haar wavelet transforms of binary images

Two architectures in space and frequency domains are given to optically implement wavelet transforms (WT) in real time and in 2-D parallel, which in principle can circumvent the 4-D display requirement for 2-D WT. Specifically, we have experimentally performed the 2-D Haar WT of binary images directly in the space domain by means of a shadow-casting system using 2-D lenslet arrays and micropolarizers. Shadowing is natural for scale changes, and polarization encoding is necessary to realize the bipolar nature of Haar wavelets. Haar wavelets have two elementary types in 2-D, a corner mother wavelet and an edge mother wavelet. Both are useful for the real-time feature extraction for multiple-resolution image processing and pattern recognition. Moreover, a holographic processor that implements the 2-D Haar WT through filtering operations in the frequency domain is numerically simulated. The feasibility of both architectures is demonstrated and compared by computer simulations and experiments.