Riemannian mathematical morphology

This paper introduces mathematical morphology operators for real-valued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion. We then extend the canonic case to a most general framework of Riemannian operators based on the notion of admissible Riemannian structuring function. An alternative paradigm of morphological Riemannian operators involves an external structuring function which is parallel transported to each point on the manifold. Besides the definition of the various Riemannian dilation/erosion and Riemannian opening/closing, their main properties are studied. We show also how recent results on Lasry-Lions regularization can be used for non-smooth image filtering based on morphological Riemannian operators. Theoretical connections with previous works on adaptive morphology and manifold shape morphology are also considered. From a practical viewpoint, various useful image embedding into Riemannian manifolds are formalized, with some illustrative examples of morphological processing real-valued 3D surfaces

Adaptive morphological filters based on a multiple orientation vector field dependent on image local features

This paper addresses the formulation of adaptive morphological filters based on spatially-variant structuring elements. The adaptivity of these filters is achieved by modifying the shape and orientation of the structuring elements according to a multiple orientation vector field. This vector field is provided by means of a bank of directional openings which can take into account the possible multiple orientations of the contours in the image. After reviewing and formalizing the definition of the spatially-variant dilation, erosion, opening and closing, the proposed structuring elements are described. These spatially-variant structuring elements are based on ellipses which vary over the image domain adapting locally their orientation according to the multiple orientation vector field and their shape (the eccentricity of the ellipses) according to the distance to relevant contours of the objects. The proposed adaptive morphological filters are used on gray-level images and are compared with spatially-invariant filters, with spatially-variant filters based on a single orientation vector field, and with adaptive morphological bilateral filters. Results show that the morphological filters based on a multiple orientation vector field are more adept at enhancing and preserving structures which contains more than one orientation

Analysis of Amoeba Active Contours

Subject of this paper is the theoretical analysis of structure-adaptive median filter algorithms that approximate curvature-based PDEs for image filtering and segmentation. These so-called morphological amoeba filters are based on a concept introduced by Lerallut et al. They achieve similar results as the well-known geodesic active contour and self-snakes PDEs. In the present work, the PDE approximated by amoeba active contours is derived for a general geometric situation and general amoeba metric. This PDE is structurally similar but not identical to the geodesic active contour equation. It reproduces the previous PDE approximation results for amoeba median filters as special cases. Furthermore, modifications of the basic amoeba active contour algorithm are analysed that are related to the morphological force terms frequently used with geodesic active contours. Experiments demonstrate the basic behaviour of amoeba active contours and its similarity to geodesic active contours.Comment: Revised version with several improvements for clarity, slightly extended experiments and discussion. Accepted for publication in Journal of Mathematical Imaging and Visio

Morphology for matrix data : ordering versus PDE-based approach

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data

Shape Descriptors for classification of functional data

Curve discrimination is an important task in engineering and other sciences. We propose several shape descriptors for classifying functional data, inspired by form anal- ysis from the image analysis eld: statistical moments, coe cients of the components of independent component analysis (ICA) and two mathematical morphology descrip- tors (morphological covariance and spatial size distributions). They are applied to three problems: an arti cial problem, a speech recognition problem and a biomechan- ical application. Shape descriptors are compared with other methods in the literature, obtaining better or similar performance

An investigation of a pattern recognition system to analyse and classify dried fruit

Includes bibliographical references.Both the declining cost and increasing capabilities of specialised computer hardware for image processing have enabled computer vision systems to become a viable alternative to human visual inspection in industrial applications. In this thesis a vision system that will analyse and classify dried fruit is investigated. In human visual inspection of dried fruit, the colour of the fruit is often the main determinant of its grade; in specific cases the presence of blemishes and geometrical fault are also incorporated in order to determine the fruit grade. A colour model that would successfully represent the colour variations within dried fruit grades, was investigated. The selected colour feature space formed the basis of a classification system which automatically allocated a sample unit of dried fruit to one specific grade. Various classification methods were investigated, and that which suited the system data and parameters was selected and evaluated using test sets of three types of dried fruit. In order to successfully grade dried fruit, a number of additional problems had to be catered for: the red/brown coloured central core area of dried peaches had to be removed from the colour analysis, and Black blemishes upon dried pears had to be isolated and sized in order to supplement the colour classifier in the final classification of the pear. The core area of a dried peach was isolated using the Morphological Top-Hat transform, and Black blemishes upon pears were isolated using colour histogram thresholding techniques. The test results indicated that although colour classification was the major determinant in the grading of dried fruit, other characteristics of the fruit had to be incorporated to achieve successful final classification results; these characteristics may be different for different types of dried fruit, but in the case of dried apricots, dried peaches and dried pears, they include the: peach core area removal, fruit geometry validation, and dried pear blemish isolation and sizing

( max , min )-convolution and Mathematical Morphology

International audienceA formal denition of morphological operators in (max, min)-algebra is introduced and their relevant properties from an algebraic viewpoint are stated. Some previous works in mathematical morphology have already encountered this type of operators but a systematic study of them has not yet been undertaken in the morphological literature. It is shown in particular that one of their fundamental property is the equivalence with level set processing using Minkowski addition and subtraction. Theory of viscosity solutions of the Hamilton-Jacobi equation with Hamiltonians containing u and Du is summarized, in particular, the corresponding Hopf-Lax-Oleinik formulas as (max, min)-operators. Links between (max, min)-convolutions and some previous approaches of unconventional morphology, in particular fuzzy morphology and viscous morphology, are reviewed

Morphological PDE and dilation/erosion semigroups on length spaces

International audienceThis paper gives a survey of recent research on Hamilton-Jacobi partial dierential equations (PDE) on length spaces. This theory provides the background to formulate morphological PDEs for processing data and images supported on a length space, without the need of a Riemmanian structure. We first introduce the most general pair of dilation/erosion semigroups on a length space, whose basic ingredients are the metric distance and a convex shape function. The second objective is to show under which conditions the solution of a morphological PDE in the length space framework is equal to the dilation/erosion semigroups

Compensated convexity on bounded domains, mixed Moreau envelopes and computational methods

Compensated convex transforms have been introduced for extended real valued functions defined over Rn. In their application to image processing, interpolation and shape interrogation, where one deals with functions defined over a bounded domain, one was making the implicit assumption that the function coincides with its transform at the boundary of the data domain. In this paper, we introduce local compensated convex transforms for functions defined in bounded open convex subsets Ω of Rn by making specific extensions of the function to the whole space, and establish their relations to globally defined compensated convex transforms via the mixed critical Moreau envelopes. We find that the compensated convex transforms of such extensions coincide with the local compensated convex transforms in the closure of Ω. We also propose a numerical scheme for computing Moreau envelopes, establishing convergence of the scheme with the rate of convergence depending on the regularity of the original function. We give an estimate of the number of iterations needed for computing the discrete Moreau envelope. We then apply the local compensated convex transforms to image processing and shape interrogation. Our results are compared with those obtained by using schemes based on computing the convex envelope from the original definition of compensated convex transforms

Morphological Scale-Space Operators for Images Supported on Point Clouds

International audienceThe aim of this paper is to develop the theory, and to propose an algorithm, for morphological processing of images painted on point clouds, viewed as a length metric measure space $(X,d,\mu)$. In order to extend morphological operators to process point cloud supported images, one needs to define dilation and erosion as semigroup operators on $(X,d)$. That corresponds to a supremal convolution (and infimal convolution) using admissible structuring function on $(X,d)$. From a more theoretical perspective, we introduce the notion of abstract structuring functions formulated on length metric Maslov idempotent measurable spaces, which is the appropriate setting for $(X,d)$. In practice, computation of Maslov structuring function is approached by a random walks framework to estimate heat kernel on $(X,d,\mu)$, followed by the logarithmic trick