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Mathematical Morphology (MM) is a general framework for studying mappings between complete lattices. In particular, mappings between binary images that are translation invariant and locally defined within a window are of special interest in MM. They are called W-operators. A key aspect of MM is the representation of W-operators in terms of dilations, erosions, intersection, union, complementation and composition. When W-operators are expressed in this form, they are called morphological operators. An implementation of this decomposition structure is called morphological machine (MMach). A remarkable property of this decomposition structure is that it can be represented efficiently by graphs called Binary Decision Diagrams (BDDs). In this paper, we propose a new architecture for MMachs that is based on BDDs. We also show that reduced and ordered BDDs (ROBDDs) are non-ambiguous schemes for representing W- operators and we present a method to compute them. This procedure can be applied for the automatic proof of equivalence between morphological operators, since the W-operator they represent are equal if and only if they have the same ROBDD.

This paper presents morphological operators with non-fixed shape kernels, or amoebas, which take into account the image contour variations to adapt their shape. Experiments on grayscale and color images demonstrate that these novel filters outperform classical morphological operations with a fixed, space-invariant structuring element for noise reduction applications.

Binary morphological transformations based on the residues (ultimate erosion, skeleton by openings, etc.) are extended to functions by means of the transformation definition and of its associated function based on the analysis of the residue evolution in every point of the image. This definition allows to build not only the transformed image itself but also its associated function, indicating the value of the residue index for which this evolution is the most important. These definitions are totally compatible with the existing definitions for sets. Moreover, they have the advantage of supplying effective tools for shape analysis on one hand and, on the other hand, of allowing the definition of new residual transforms together with their associated functions. Two of these numerical residues will be introduced, called respectively ultimate opening and quasi-distance and, through some applications, the interest and efficiency of these operators will be illustrated.

Path openings and closings are algebraic morphological operators using families of thin and oriented structuring elements that are not necessarily perfectly straight. These operators are naturally translation invariant and can be used in filtering applications instead of operators based on the more standard families of straight line structuring elements. They give similar results to area or attribute-based operators but with more flexibility in the constraints.

Trivial implementations of this idea using actual supreme or infima of morphological operators with paths as structuring elements would imply exponential complexity. Fortunately a linear complexity algorithm exists in the literature, which has similar running times as an efficient implementation of algebraic operators using straight lines as structuring elements.

However even this implementation is sometimes not fast enough, leading practitioners to favour some attribute-based operators instead, which in some applications is not optimal.

In this paper we propose an implementation of path-based morphological operators which is shown experimentally to exhibit logarithmic complexity and comparable computing times with those of attribute-based operators.

This paper deals with the combination of classical morphological tools and motion compensation techniques. Morphological operators have proven to be efficient for filtering and segmenting still images. For video sequences however, using motion information to modify the morphological processing is necessary. In previous work, iterative frame by frame segmentation using motion information has been developed in various forms. In this paper, motion is used at a very low level, by locally modifying the shape of the structuring element in a video sequence considered as a 3D data block. Motion adapted morphological tools are described and their use is demonstrated on video sequences. Moreover, the features of the motion model best suited to our purpose are also discussed.

Interpolation is an important step in many applications of image processing. This paper presents a morphological interpolation technique for binary images based on the median set concept. A characteristic of our method is that it treats recursively the connected components of input slices. This technique uses the minimal skeleton by pruning (MSP) as reference points for translating connected components; this fact guarantees the non-empty intersection between them.

The work presented in this paper introduces a novel method for second-order connected attribute filtering using Max-Trees. The proposed scheme is generated in a recursive manner from two images, the original and a modified copy by an either extensive or an anti-extensive operator. The tree structure is shaped by the component hierarchy of the modified image while the node attributes are based on the connected components of the original image. Attribute filtering of second-order connected sets proceeds as in conventional Max-Trees with no further computational overhead.

In this paper, a class of transformations with reconstruction criteria, derived from the reconstruction transformations, is investigated. The idea to build these transformations consists in stopping the reconstruction process according to a size criterion. This class of transformations was initially proposed for obtaining intermediate results between the morphological opening and the opening by reconstruction. Here, the transformations are presented in the general case, as in the reconstruction transformations case, by imposing some conditions on the marker. We show that the set of markers for the transformations with reconstruction criteria is given by the set of dilated images. The interest of these transformations in image segmentation is shown. Also the notion of granulometry and the alternating sequential filters are investigated.

In this paper connected operators from mathematical morphology are extended to a wider class of operators, which are based on connectivities in higher dimension spaces, similar to scale spaces which will be called attribute spaces. Though some properties of connected filters are lost, granulometries can be defined under certain conditions, and pattern spectra in most cases. The advantage of this approach is that regions can be split into constituent parts before filtering more naturally than by using partitioning connectivities.

A variant of morphological attribute filters is developed, in which the attribute on which filtering is based, is no longer a scalar, as is usual, but a vector. This leads to new granulometries and associated pattern spectra. When the vector-attribute used is a shape descriptor, the resulting granulometries filter an image based on a shape or shape family instead of one or more scalar values.