The receptive fields of simple cells in the visual cortex can be understood as linear filters. These filters can be modeled by Gabor functions or gaussian derivatives. Gabor functions can also be combined in an energy model of the complex cell response. This letter 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 2D 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 neural implementation and predictions of the model are discussed.

The problem of multimodal clustering arises whenever the data are gathered with several physically different sensors. Observations from different modalities are not necessarily aligned in the sense there there is no obvious way to associate or compare them in some common space. A solution may consist in considering multiple clustering tasks independently for each modality. The main difficulty with such an approach is to guarantee that the unimodal clusterings are mutually consistent. In this letter, we show that multimodal clustering can be addressed within a novel framework: conjugate mixture models. These models exploit the explicit transformations that are often available between an unobserved parameter space (objects) and each of the observation spaces (sensors). We formulate the problem as a likelihood maximization task and derive the associated conjugate expectation-maximization algorithm. The convergence properties of the proposed algorithm are thoroughly investigated. Several local and global optimization techniques are proposed in order to increase its convergence speed. Two initialization strategies are proposed and compared. A consistent model selection criterion is proposed. The algorithm and its variants are tested and evaluated within the task of 3D localization of several speakers using both auditory and visual data.