In an approximation problem with a neural network, a low-output root mean square (rms) error is not always a universal criterion. In this paper, we investigate problems where the Jacobians--first derivative of an output value with respect to an input value--of the approximation model are needed and propose to add a quality criterion on these Jacobians during the learning step. More specifically, we focus here on the approximation of functionals A; from a space of continuous functions (discretized in pratice) to a scalar space. In this case, the approximation is confronted with the compensation phenomenon: a lower contribution of one input can be compensated by a larger one of its neighboring inputs. In this case, profiles (with respect to the input index) of neural Jacobians are very irregular instead of smooth. Then, the approximation of A becomes an ill-posed problem because many solutions can be chosen by the learning process. We propose to introduce the smoothness of Jacobian profiles as an a priori information via a regularization technique and develop a new and efficient learning algorithm, called "weight smoothing." We assess the robustness of the weight smoothing algorithm by testing it on a real and complex problem stemming from meteorology: the neural approximation of the forward model of radiative transfer equation in the atmosphere. The stabilized Jacobians of this model are then used in an inversion process to illustrate the improvement of the Jacobians after weight smoothing.

This paper presents in details the algorithm of the watershed, which we have sketched in Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 12, pp 1,163-1,173 and criticized in this issue. First, the formal definition of the flooding list, the key data structure of the algorithm, is given. Then, the construction of this flooding list and of the watershed are described and proved.

The watershed is one of the latest segmentation tools developed in mathematical morphology. In order to prevent its oversegmentation, the notion of dynamics of a minimum, based on geodesic reconstruction, has been proposed. In this paper, we extend the notion of dynamics to the contour arcs. This notion acts as a measure of the saliency of the contour. Contrary to the dynamics of minima, our concept reflects the extension and shape of the corresponding object in the image. This representation is also much more natural, because it is expressed in terms of partitions of the plane, i.e., segmentations. A hierarchical segmentation process is then derived, which gives a compact description of the image, containing all the segmentations one can obtain by the notion of dynamics, by means of a simple thresholding. Finally, efficient algorithms for computing the geodesic reconstruction as well as the dynamics of contours are presented.

Noting that the skeleton is defined by means of the distance function to the nearest point, an analog notion named anti-skeleton is derived when substituting to that distance the distance to the farthest point. A complete parallel is constructed, concerning the mathematical definitions and properties of the skeleton and the anti-skeleton. An example of its use in image analysis is then presented.

This article proposes a coherent method of interpolating 3D data obtained for example by stereo, with a polyhedral surface by means of the Delaunay Triangulation. We first give some theoretical properties concerning the approximations of sampled objects we obtain when the sampling rate increases, based on the study of their skeleton using some tools of Mathematical Morphology. Then, we present the algorithms and their complexity analysis which yield both a surface representation of objects and a volume representation of free space which may be useful in Robotics. This goal is achieved by means of a simple visibility criterion. The method is intrinsically discontinuity preserving and can be used for the integration of multiple viewpoints.