A Differential Model of the Complex Cell

TL;DR

This paper introduces a novel complex cell model based on Gaussian derivatives and scale space theory, providing shift-insensitive responses and aligning with biological visual processing.

Contribution

The model uniquely uses Gaussian derivative filters at a single scale and position to approximate complex cell responses, emphasizing scale space and differential structure.

Findings

Model responds to edges and gratings as expected.

Shift insensitivity is demonstrated on natural images.

The approach aligns with cortical image representation theories.

Abstract

The receptive fields of simple cells in the visual cortex can be understood as linear filters. These filters can be modelled by Gabor functions, or by Gaussian derivatives. Gabor functions can also be combined in an `energy model' of the complex cell response. This paper proposes an alternative model of the complex cell, based on Gaussian derivatives. It is most important to account for the insensitivity of the complex response to small shifts of the image. The new model uses a linear combination of the first few derivative filters, at a single position, to approximate the first derivative filter, at a series of adjacent positions. The maximum response, over all positions, gives a signal that is insensitive to small shifts of the image. This model, unlike previous approaches, is based on the scale space theory of visual processing. In particular, the complex cell is built from filters that respond to the \twod\ differential structure of the image. The computational aspects of the new model are studied in one and two dimensions, using the steerability of the Gaussian derivatives. The response of the model to basic images, such as edges and gratings, is derived formally. The response to natural images is also evaluated, using statistical measures of shift insensitivity. The relevance of the new model to the cortical image representation is discussed.