On the Converse to Pompeiu's Problem

This is a reprint of a 1976 paper that appears in an inaccessible Brazilian journal and has become very looked after. It deals with the problem of determining a convex plane domain from the existence of infinitely many over determined Neumann eigenvalues. Recent related work in magneto hydrodynamics of Vogelius and other applications are closely related to this result. The more general result appears in J. Analyse Math 1980 and Crelle l987. See Zalcmain's bibliographic survey of pompeiu problem for other references

Harmonic Functions and Inverse Conductivity Problems on Networks

In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do this we introduce an elliptic operator Dw and an w-harmonic function on thegraph, with its physical interpretation been the diffusion equation on the graph, which models an electric network. After deriving the basic properties of w-harmonic functions, we prove the solvability of (direct) problems such as the Dirichlet and Neumann boundary value problems.Our main result is the global uniqueness of the inverse conductivity problem for a network under a suitable monotonicity condition

Radon transform on spaces of constant curvature

A correspondence among the totally geodesic Radon transforms---as well as among their duals---on the constant curvature spaces is established, and is used here to obtain various range characterizations

Quotient Signal Decomposition and Order Estimation

In this paper we propose a method for blind signal decomposition that does not require the independence or stationarity of the sources. This method, that we consider a simple instance of non-linear projection pursuit, is based on the possibility of recovering the areas in the time-frequency where the original signals are isolated or almost isolated with the use of suitable quotients of linear combinations of the spectrograms of the mixtures.We then threshold such quotients according to the value of their imaginary part to prove that the method is theoretically sound under mild assumptions on the mixing matrix and the sources. We study one basic algorithm based on this method. Moreover we propose a practical measure of separation for the sources in a given time frequency representation.The algorithm has the important feature of estimating the number of sources with two measurements, it then requires n-2 additional measurements to provide a reconstruction of n sources. Experimental results show that the method works even when severalshifted version of the same source are mixed

Network Tomography

While conventional tomography is associated to the Radon transform in Euclidean spaces, electrical impedance tomography or EIT is associated to the Radon transform in the hyperbolic plane. We discuss some recent work on network tomography that can be associated to a problem similar to EIT on graphs and indicate how in some sense it may be also associated to the Radon transform on trees

Structurally Robust Weak Continuity

Building on earlier work, we pose the following optimization: Given a sequence of finite extent, find a finite-alphabet sequence of finite extent, which satisfies a hard structural (syntactic) constraint (e.g., it is piecewise constant of plateau run-length > M, or locally monotonic of a given lomo-degree), and which minimizes the sum of a per-letter fidelity measure, and a first-order smoothness-complexity measure. This optimization represents the unification and outgrowth of several digital nonlinear filtering schemes, including the digital counterpart of the so-called Weak Continuity (WC) formulation of Mumford-Shah and Blake-Zisserman, the Minimum Description Length (MDL) approach of Leclerc, and previous work by the first author in so- called VORCA filtering and Digital Locally Monotonic Regression. It is shown that the proposed optimization admits efficient Viterbi-type solution, and overcomes a shortcoming of WC, while preserving its unique strengths. Similarly, it overcomes a drawback of VORCA and Digital Locally Monotonic Regression, while maintaining robustness to outliers.<P

Further Results on MAP Optimality and Strong Consistency of Certain Classes of Morphological Filters

In two recent papers [1], [2], Sidiropoulos et al. have obtained statistical proofs of Maximum A Posteriori} (MAP) optimality and strong consistency of certain popular classes of Morphological filters, namely, Morphological Openings, Closings, unions of Openings, and intersections of Closings, under i.i.d. (both pixel-wise, and sequence-wide) assumptions on the noise model. In this paper we revisit this classic filtering problem, and prove MAP optimality and strong consistency under a different, and, in a sense, more appealing set of assumptions, which allows the explicit incorporation of geometric and Morphological constraints into the noise model, i.e., the noise may now exhibit structure; Surprisingly, it turns out that this affects neither the optimality nor the consistency of these field-proven filters.<P