The exponential chirp transform for log-polar sampled images

Vision architectures which are space-variant, i.e., the local image resolution is a function of space, are of great importance in both biological and machine vision. In the present study we focus on a particular map, the log-polar map. According to this map function, objects located in the fovea are sampled with higher frequency, compared to those present in the periphery. When such a mapping is applied directly to a traditional image, the resulting image contains a very small number of pixels, compared to the original one, while maintaining locally the initial resolution and visual field. In this study we show that using such a map-function, we are able to design a pattern recognition system capable of recognizing an object despite its position in space, rotation and scaling. This is achieved by introducing a new linear transform, called the Exponential Chirp Transform (ECT), which provides frequency domain estimation of log-polar warped images. The need for such a tool derives from the extremely distorted representation of images and performing even the simplest object recognition task, such as detection of the presence and position of an object in an image already segmented, becomes a very complicated task. The ECT can be computed very efficiently by introducing a log-polar coordinate transformation in frequency, as done for the Mellin-Fourier Transform, through a cross-correlation with complexity ;We demonstrate the use of the fast ECT with many examples, and show its ability to perform traditional Fourier-based processing: cross-correlation (template matching), autocorrelation and filtering which are now space-variant. In this study also a biological model based on space-variant cross-correlation is proposed as a basic pattern recognition algorithm biologically plausible for vertebrate's visual system. Finally, we outline a one-dimensional transform, along with examples of logarithmic sampling in time, which could be used in applications of acoustic environment simulation.;This work provides, for the first time, a conceptual basis for combining global spatial frequency methods with space-variant mappings (i.e., non-uniform sampling) in a way which is generalizable for any type of mapping.