Sparse Coding in First- and Second-Order Filtered Images.

Abstract

Most natural images contain regions of interest, interspersed with redundant uniform information. This redundancy may be reduced by using a sparsely-coded image representation, in which only a small number of neurons are activated. Here we simulate the responses of filters for first-order (luminance-modulated) and second-order (texture and contrast-modulated) information to examine the sparseness of their activation. First-order linear filtering was implemented as convolutions of cosine-windowed images with oriented Gabor functions, with gains scaled to give equal amplitude response across spatial frequency to random fractal images. Second-order operators were a pair of such Gabors in a filter-rectify-filter cascade. We obtained unsigned responses across a range of filter parameters (spatial frequency: 2–64 cycles/image, orientation: 0–180 deg., phase: sine and cosine). The sparseness of each filter's response was measured as its kurtosis (fourth-order cumulant). Both first- and second-order outputs exhibit a high kurtosis to natural but not phase-scrambled images. For first-order information, high spatial frequency filters (>32 cycles/image) orientated around the horizontal and vertical show the greatest sparseness. This effect can be removed by inverse-scaling the cosine window size with spatial frequency, as previously hypothesized (Thompson, 2001). For second-order information, there is no orientation or spatial frequency dependence, but kurtosis is always significantly larger for natural images than for their phase-scrambled counterparts. A number of different nonlinearities in the FRF cascade were studied, with sparseness being maximized using a power function with an n= 3 or 4. Finally, for a given image, those filters which respond strongest to first- and second-order information, also show the strongest correlation, and are the best at maximizing sparseness.