Multidimensional Systems and Signal Processing, 2, 421-436 (1991) © 1991 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Gaussian Wavelet Transform: Two Alternative Fast Implementations for Images RAFAEL NAVARRO AND ANTONIOTABERNERO Instituto de Optica "Daza do Valdbs" (CSIC), Serrano 121. 28006 Madrid, Spain Received February 26, 1991, Accepted April 5, 1991 Abstract. A series of schemes for pyramid multiresolution image coding has been proposed, all of them based on sets of orthogonal functions. Several of them are implementable in the spatial domain (such as wavelets), whereas others are more suitable for Fourier domainimplementation(as for instancethe cortextransform). Gabor functionshavemany importantadvantages,allowingeasyand fast implementationsin either domain, but are usually discarded by their lack of orthogonalitywhichcauses_incomplete transforms. In this paper we quantifysuch effect, showing a Gaussian WaveletTransform, GWT, with quasiorthogonal Gabor functions, which allows robust and efficient coding. Our particular GWT is based on a human visual model. Its incompletenesscauses small amounts of reconstruction errors (due to small indentations in the MTF), which, however, are irrelevant under criteria based on visual perception. Keywords. Gaussian wavelets, Gabor functions, image coding, completeness, space and frequency domains implementations 1. Introduction Multiresolution signal decomposition is being commonly accepted as an optimal solution to processing, coding, and analyzing signals (see for instance [Rosenfeld 1984]). In this sense, the functional organization of the human visual system is based in this kind of image decom- position since individual cells in the cortex are tuned for specific bands of spatial frequen- cies and orientations [De Valois, Albrecht and Thorell 1982; Campbell and Kulikowski 1966]. In fact, the visual system performs a multiresolution, and also a multiorientation, parallel processing of images. Both neurophysiological [Marcelja 1980] and psychophysical [Daugman 1984] studies agree that the receptive fields, or frequency channels in the visual system are very accurately modelable by Gabor functions. From a point of view of compact coding design, however, Gabor functions are not com- monly accepted. Probably this is due to two reasons: first Gabor, in his early work [Gabor 1946] did not really address multiresolution pyramid coding. Second, and most important, Gaussian functions (wave packets) do not directly yield a complete (exact) mapping of the signal. Although this can be overcome by more or less sophisticated methods [Daugman 1988], many other complete transforms have been proposed, such as the Cortex Transform [Watson 1987]; the generalized Gabor scheme [Bastiaans 1981; Porat and Zeevi 1988]; wavelets [Morlet, Arens, Forgeau and Giard 1982; Mallat t989]; QMF filters [Woods and Nell 1986; Simoncelli and Adelson 1990] etc. Most of those transforms (wavelets, general- ized Gabor, QMF, etc.) have been designed to constitute a compact nonredundant coding, 67