Self-dual Operator Research Articles

Self-Duality of Markov Processes and Intertwining Functions

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitary equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.

Invariant differential operators in positive characteristic

In 1928, at the IMC, Veblen posed the problem: classify invariant differential operators between spaces of “natural objects” (in modern terms: either tensor fields, or jets) over a real manifold of any dimension. The problem was solved by Rudakov for unary operators (no nonscalar operators except the exterior differential); by Grozman for binary operators. In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case over any field of positive characteristic. Unary invariant operators are known: these are the exterior differential and analogs of the Berezin integral. We construct new binary operators from these analogs and discovered two more (up to dualizations) types of new indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators. Gordan's transvectants, aka Cohen–Rankin brackets, always invariant with respect to the simple 3-dimensional Lie algebra, are also invariant, in characteristic 2, with respect to the whole Lie algebra of vector fields on the line when the height of the indeterminate is equal to 2.

The $SOH$ operator system

In this paper we examine a natural operator system struc- ture on Pisier's self-dual operator space. We prove that this operator system is completely order isomorphic to its dual with the cb-condition number of this isomorphism as small as possible. We examine further properties of this operator system and use it to create a new tensor product on operator systems.

A SCALE-SPACE TOGGLE OPERATOR FOR IMAGE TRANSFORMATIONS

The analysis of different representation levels has been largely used in several image analysis tasks to handle the multiscale nature of image data, allowing the extraction of specific features that become explicit at each scale. In this work, we explore the scale-space properties of a self-dual toggle operator defined on a scaled morphological framework. These properties conduce to a well-controlled image simplification where its maxima and minima interact at the same time during pixels' transformation, in contrast to other approaches that consider these extrema separately. In such a way, it is possible to identify significant image extrema information to be used in several high level tasks. To assess the robustness of our approach, we carry out tests on images of several classes and subjected to different lighting conditions for various applications, including segmentation and binarization.

A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices A and B. Moreover, an interesting generalization of the geometric mean A # B of A and B to convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self‐dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

Self-dual morphological operators and filters

The median operator is a nonlinear image transformation celebrated for its noise cleaning capacities. It treats the foreground and background of an image identically, i.e., it is self-dual. Unfortunately, the median operator has one major drawback: it is not idempotent. Even worse, subsequent iterations of a given image may lead to oscillations. This paper describes a general method for the construction of morphological operators which are self-dual. This construction is based upon the concept of a switch operator. Subsequently, the paper treats a class of operators, the so-called activity-extensive operators, which have the intriguing property that every sequence of iterates of a given image is pointwise monotone and therefore convergent. The underlying concept is that of the activity ordering. Every increasing, self-dual operator can be modified in such a way that it becomes activity-extensive. The sequence of iterates of this modification converges to a self-dual morphological filter.

On the construction of morphological operators which are self-dual and activity-extensive

This paper presents a systematic method to construct self-dual increasing morphological operators on P( Z d) using the concept of a switch operator. Furthermore, it shows how to modify a self-dual operator in such a way that the sequence of iterates of the resulting modified operator converges pointwise monotone to a self-dual morphological filter. The approach is based on the concept of the centre operator and the activity ordering.