Mathematical Morphology

Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: “architectural requirements” like locality, recursivity and causality in the scale space, “stability requirements” like the comparison principle and “morphological requirements”, which correspond to shape-preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.

This paper deals with a class of morphological operators called connected operators. These operators filter the signal by merging its flat zones. As a result, they do not create any new contours and are very attractive for filtering tasks where the contour information has to be preserved. This paper shows that connected operators work implicitly on a structured representation of the image made of flat zones. The max-tree is proposed as a suitable and efficient structure to deal with the processing steps involved in antiextensive connected operators. A formal definition of the various processing steps involved in the operator is proposed and, as a result, several lines of generalization are developed. First, the notion of connectivity and its definition are analyzed. Several modifications of the traditional approach are presented. They lead to connected operators that are able to deal with texture. They also allow the definition of connected operators with less leakage than the classical ones. Second, a set of simplification criteria are proposed and discussed. They lead to simplicity-, entropy-, and motion-oriented operators. The problem of using a nonincreasing criterion is analyzed. Its solution is formulated as an optimization problem that can be very efficiently solved by a Viterbi (1979) algorithm. Finally, several implementation issues are discussed showing that these operators can be very efficiently implemented

A hierarchical image fusion scheme is presented that preserves those details from the input images that are most relevant to visual perception. Results show that fused images present a more detailed representation of the scene and provide information that cannot be obtained by viewing the input images separately. Detection, recognition, and search tasks may therefore benefit from this fused image representation.

A multiresolution image representation is presented in which iterative morphological filters of many scales but identical shape serve as basis functions. Structural pattern decomposition is achieved by subtracting successive layers in the multiresolution representation. The representation differs from established techniques in that the code elements have a well defined location and size. The resulting image description provides a useful basis for multiresolution shape analysis and is well suited for VLSI implementation.