A hexagonal orthogonal-oriented pyramid as a model of image representation in visual cortex.

Retinal ganglion cells represent the visual image with a spatial code, in which each cell conveys information about a small region in the image. In contrast, cells of primary visual cortex employ a hybrid space-frequency code in which each cell conveys information about a region that is local in space, spatial frequency, and orientation. Despite the presumable importance of this transformation, we lack any comprehensive notion of how it occurs. Here we describe a mathematical model for this transformation. The hexagonal orthogonal-oriented quadrature pyramid (HOP) transform, which operates on a hexagonal input lattice, employs basis functions that are orthogonal, self-similar, and localized in space, spatial frequency, orientation, and phase. The basis functions, which are generated from seven basic types through a recursive process, form an image code of the pyramid type. The seven basis functions, six bandpass and one low-pass, occupy a point and a hexagon of six nearest neighbors on a hexagonal sample lattice. The six bandpass basis functions consist of three with even symmetry, and three with odd symmetry. The three even kernels are rotations of 0, 60, and 120 degrees of a common kernel; likewise for the three odd kernels. At the lowest level, the inputs are image samples. At each higher level, the input lattice is provided by the low-pass coefficients computed at the previous level. At each level, the output is subsampled in such a way as to yield a new hexagonal lattice with a spacing square root 7 larger than the previous level, so that the number of coefficients is reduced by a factor of seven at each level. In the biological model, the input lattice is the retinal ganglion cell array. The resulting scheme provides a compact, efficient code of the image and generates receptive fields that resemble those of the primary visual cortex.