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International audience; This paper deals with the theory and applications of spatially-variant discrete mathematical morphology. We review and formalize the definition of spatially variant dilation/erosion and opening/closing for binary and gray-level images using exclusively the structuring function, without resorting to complement. This theoretical framework allows to build morphological operators whose structuring elements can locally adapt their shape and orientation across the dominant direction of the structures in the image. The shape and orientation of the structuring element at each pixel are extracted from the image under study: the orientation is given by means of a diffusion process of the average square gradient field, which regularizes and extends the orientation information from the edges of the objects to the homogeneous areas of the image; and the shape of the orientated structuring elements can be linear or it can be given by the distance to relevant edges of the objects. The proposed filters are used on binary and gray-level images for enhancement of anisotropic features such as coherent, flow-like structures. Results of spatially-variant erosions/dilations and openings/closings-based filters prove the validity of this theoretical sound and novel approach.

The first subject of this paper is the estimation of a high resolution directional field of fingerprints. Traditional methods are discussed and a new method, based on principal component analysis, is proposed. The method not only computes the direction in any pixel location, but its coherence as well. It is proven that this method provides exactly the same results as the averaged square-gradient method that is known from literature. Undoubtedly, the existence of a completely different equivalent solution increases the insight into the problem's nature. The second subject of this paper is singular point detection. A very efficient algorithm is proposed that extracts singular points from the high-resolution directional field. The algorithm is based on the Poincare index and provides a consistent binary decision that is not based on post-processing steps like applying a threshold on a continuous resemblance measure for singular points. Furthermore, a method is presented to estimate the orientation of the extracted singular points. The accuracy of the methods is illustrated by experiments on a live-scanned fingerprint database.