Jean Serra

Flat zones filtering, connected operators, and filters by reconstruction

This correspondence deals with the notion of connected operators. Starting from the definition for operator acting on sets, it is shown how to extend it to operators acting on function. Typically, a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions. It is shown that from any connected operator acting on sets, one can construct a connected operator for functions (however, it is not the unique way of generating connected operators for functions). Moreover, the concept of pyramid is introduced in a formal way. It is shown that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Filters by reconstruction are defined and their main properties are presented. Finally, some examples of application of connected operators and use of flat zones are described.

Introduction to mathematical morphology

Transformations morphologiques, morphologie euclidienne, erosion et notions derivees, ouvertures et fermetures, amincissements et epaississements

An overview of morphological filtering

This paper consists of a tutorial overview of morphological filtering, a theory introduced in 1988 in the context of mathematical morphology. Its first section is devoted to the presentation of the lattice framework. Emphasis is put on the lattices of numerical functions in digital and continuous spaces. The basic filters, namely the openings and the closings, are then described and their various versions are listed. In the third section morphological filters are defined as increasing idempotent operators, and their laws of composition are proved. The last sections are concerned with two special classes of filters and their derivations: first, the alternating sequential filters allow us to bring into play families of operators depending on a positive scale parameter. Finally, the center and the toggle mappings modify the function under study by comparing it, at each point, with a few reference transforms.

Connectivity on Complete Lattices

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. A nontopological axiomatics has been proposed by Matheron and Serra. The present paper extends it to complete sup-generated lattices. A connection turns out to be characterized by a family of openings labelled by the sup-generators, which partition each element of the lattice into maximal terms, of zero infima. When combined with partition closings, these openings generate strong sequential alternating filters. Starting from a first connection several others may be designed by acting on some dilations or symmetrical operators. When applying this theory to function lattices, one interprets the so-called connected operators in terms of actual connections, as well as the watershed mappings. But the theory encompasses the numerical functions and extends, among others, to multivariate lattices.

A Lattice Approach to Image Segmentation

After a formal definition of segmentation as the largest partition of the space according to a criterion σ and a function f, the notion of a morphological connection is reminded. It is used as an input to a central theorem of the paper (Theorem 8), that identifies segmentation with the connections that are based on connective criteria. Just as connections, the segmentations can then be regrouped by suprema and infima. The generality of the theorem makes it valid for functions from any space to any other one. Two propositions make precise the AND and OR combinations of connective criteria.The soundness of the approach is demonstrated by listing a series of segmentation techniques. One considers first the cases when the segmentation under study does not involve initial seeds. Various modes of regularity are discussed, which all derive from Lipschitz functions. A second category of examples involves the presence of seeds around which the partition of the space is organized. An overall proposition shows that these examples are a matter for the central theorem. Watershed and jump connection based segmentations illustrate this type of situation. The third and last category of examples deals with cases when the segmentation occurs in an indirect space, such as an histogram, and is then projected back on the actual space under study.The relationships between filtering and segmentation are then investigated. A theoretical chapter introduces and studies the two notions of a pulse opening and of a connected operator. The conditions under which a family of pulse openings can yield a connected filter are clarified. The ability of segmentations to generate pyramids, or hierarchies, is analyzed. A distinction is made between weak hierarchies where the partitions increase when going up in the pyramid, and the strong hierarchies where the various levels are structured as semi-groups, and particularly as granulometric semi-groups.The last section is based on one example, and goes back over the controversy about “lattice” versus “functional” optimization. The problem is now tackled via a case of colour segmentation, where the saturation serves as a cursor between luminance and hue. The emphasis is put on the difficulty of grouping the various necessary optimizations into a single one.

Morphological filtering: an overview

Abstract This paper is an overview on the concept of morphological filtering. Starting from openings and the associated granulometries, we discuss the notion and construction of morphological filters. Then the major differences between the ‘morphological’ and the ‘linear’ approaches are highlighted. Finally, the problem of optimal morphological filtering is presented.

The flat zone approach: a general low-level region merging segmentation method

Abstract This paper presents a segmentation method, the flat zone approach, that avoids some limitations of the watershed-plus-markers method. The watershed-plus-markers approach, which is the traditional segmentation technique in mathematical morphology, has two inherent problems: (1) the possible separation of a piecewise-constant region of the input image into several regions in the output partition, and (2) the problem of obtaining markers (connected components of pixels signaling significant regions) for features that are one or two pixels wide. These problems are related to the limited resolution power (for feature extraction) of gradient operators. The flat zone approach extends the region marker concept (to contain the entire (and not part of) regions) and requires neither the computation of a gradient function nor the modification of the support of the input image in order to increase the size of the features. Our approach works on the graph formed by the image flat zones (or piecewise-constant regions). This fact ensures that the input image regions are not broken and can consider all input flat zones regardless their size. An inclusion relationship between the flat zones of the input image and the regions of the output partition is imposed. That is, a flat zone segmentation method is a region (flat zone) merging method and behaves like a connected operator. Our method is robust (in the sense that it is invariant under certain intensity value transformations) and uses a hierarchical waiting queue algorithm that makes it extremely efficient.

Morphological multiscale image segmentation

This paper deals with a morphological approach to the problem of unsupervised image segmentation. The proposed technique relies on a multiscale approach which allows a hierarchical processing of the data ranging from the most global scale to the most detailed one. At each scale, the algorithm relies on four steps: preprocessing, feature extraction, decision and quality estimation. The goal of the preprocessing step is to simplify the original signal which is too complex to be processed at once. Morphological filters by reconstruction are very attractive for this purpose because they simplify without corrupting the contour information. The feature extraction intends to extract the pertinent parameters for assessing the degree of homogeneity of the regions. To this goal, morphological techniques extracting flat or contrasted regions are very efficient. The decision step defines precisely the contours of the regions. This decision is achieved by a watershed algorithm. Finally, the quality estimation is used to compute the information that has to be further processed by the next scale to improve the segmentation result. The estimation is based on a region modeling procedure. The resulting segmentation is very robust and can deal with very different types of images. Moreover, the first levels give segmentation results with a few regions; but precisely located contours.

Color segmentation by ordered mergings

The paper deals with the use of the various color pieces of information for segmenting color images and sequences with mathematical morphology operators. It is divided in four parts. The first one is concerning the choice of the color space suitable for morphological processing. The choice of a connection which induces a specific segmentation is discussed. The authors then present the color segmentation approach which is based on a nonparametric pyramid of watersheds, with a comparative study of different color gradients. Another multiscale color segmentation algorithm is then introduced, relying on the merging of chromatic-achromatic partitions ordered by the saturation component.

Connections for sets and functions

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. An algebraic definition, called connection, has been proposed by Serra to extend the notion of connectivity to complete sup-generated lattices. A connection turns out to be characterized by a family of openings parameterized by the sup-generators, which partition each element of the lattice into maximal components. Starting from a first connection, several others may be constructed; e.g., by applying dilations. The present paper applies this theory to numerical functions. Every connection leads to segmenting the support of the function under study into regions. Inside each region, the function is r-continuous, for a modulus of continuity r given a priori, and characteristic of the connection. However, the segmentation is not unique, and may be particularized by other considerations (self-duality, large or low number of point components, etc.). These variants are introduced by means of examples for three different connections: flat zone connections, jump connections, and smooth path connections. They turn out to provide remarkable segmentations, depending only on a few parameters. In the last section, some morphological filters are described, based on flat zone connections, namely openings by reconstruction, flattenings and levelings.