11,218 research outputs found

    A bounded jump for the bounded Turing degrees

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    We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.Comment: 22 pages. Minor changes for publicatio

    Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions

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    Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general perspective, addressing questions of global solvability and stable inversion. Of special interest are the so-called "near-singular" situations, such as a collision of two closely spaced nodes. We also discuss the problem of reconstructing piecewise-smooth functions from their Fourier coefficients, which is easily reduced by a well-known method of K.Eckhoff to solving a particular Prony system. As we show in the paper, it turns out that a modification of this highly nonlinear method can reconstruct the jump locations and magnitudes of such functions, as well as the pointwise values between the jumps, with the maximal possible accuracy.Comment: arXiv admin note: text overlap with arXiv:1211.068

    Complete Solutions for a Combinatorial Puzzle in Linear Time

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    In this paper we study a single player game consisting of nn black checkers and mm white checkers, called shifting the checkers. We have proved that the minimum number of steps needed to play the game for general nn and mm is nm+n+mnm + n + m. We have also presented an optimal algorithm to generate an optimal move sequence of the game consisting of nn black checkers and mm white checkers, and finally, we present an explicit solution for the general game

    Joint Image Reconstruction and Segmentation Using the Potts Model

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    We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from 77 angular views only. We illustrate the practical applicability on a real PET dataset. As further applications, we consider spherical Radon data as well as blurred data
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