In this paper, both non-mixing and mixing local minima of the entropy are analyzed from the viewpoint of blind source separation (BSS); they correspond respectively to acceptable and spurious solutions of the BSS problem. The contribution of this work is twofold. First, a Taylor development is used to show that the exact output entropy cost function has a non-mixing minimum when this output is proportional to any of the non-Gaussian sources, and not only when the output is proportional to the lowest entropic source. Second, in order to prove that mixing entropy minima exist when the source densities are strongly multimodal, an entropy approximator is proposed. The latter has the major advantage that an error bound can be provided. Even if this approximator (and the associated bound) is used here in the BSS context, it can be applied for estimating the entropy of any random variable with multimodal density

This work provides the expansion up to second order of the (differential) entropy of a random variable contaminated with another small variable. It is motivated by the blind source separation problem, in which it is of interest to consider the entropy of a mixture of random variables and study how it changes with respect to the mixing proportions.

Recent simulation results have indicated that spurious minima in information-theoretic criteria with an orthogonality constraint for blind source separation may exist. Nevertheless, those results involve approximations (e.g., density estimation), so that they do not constitute an absolute proof. In this letter, the problem is tackled from a theoretical point of view. An example is provided for which it is rigorously proved that spurious minima can exist in both mutual information and negentropy optima. The proof is based on a Taylor expansion of the entropy.

This letter introduces a flexible parametrization for the nonlinear compensator in the postnonlinear mixture model. The idea is to parametrize not the compensator but a set of quantiles of the distribution of its output. They are specified by the density-quantiles which are parametrized through a spline basis. Good results can be achieved with only a few splines.

This paper presents a unified approach to the problem of blind separation of sources, based on the concept of mutual information. This concept is applied to the whole source sequences as stationary processes and thus provides a universal contrast applicable to both the instantaneous and convolutive mixture cases. For practical implementation, we introduce several degraded forms of this contrast, computable from a finite-dimensional distribution of the reconstructed source processes only. From them, we derive several sets of estimating equations, generalizing those considered earlier.

A new approach for gray-level image segmentation is presented using a probabilistic fuzzy c-means clustering algorithm. This approach combines the spatial probabilistic information and the fuzzy membership function in the clustering process. The proposed probabilistic fuzzy c-means method can deal effectively with image segmentation in a noisy environment.

The authors present a method for robustly and accurately estimating the rotation and translation between a camera and a 3-D object from point and line correspondences. First they devise an error function and then show how to minimize this error function. The quadratic nature of this function is made possible by representing rotation and translation with a dual number quaternion. A detailed account is provided of the computational aspects of a trust-region optimization method. This method compares favourably with Newton's method, which has extensively been used to solve the problem, and with Faugeras-Toscani's linear method (1986) for calibrating a camera. Some experimental results are presented which demonstrate the robustness of the method with respect to image noise and matching errors.>

The problem of designing a finite impulse response filter devoted to the joint process problem, and in particular to noise cancellation, is investigated. The goal is restricted to how to choose an optimal order of the FIR filter. The analysis is based on the maximization of a noise reduction criterion which is in fact perfectly matched to the desired performance, namely the minimization of residuals. The criterion includes estimation errors introduced by the substitution of the true covariance matrices for estimates. Since true covariance matrices again enter the noise reduction criterion, they must be replaced by estimates. The criterion is then modified in order to take this fact into account, but the scheme is not iterated any further. Simulations are presented for various simple cases.>