Reconstruction of piecewise constant functions from x-ray data

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

Download Full-text

Cut Pursuit: Fast Algorithms to Learn Piecewise Constant Functions on General Weighted Graphs

SIAM Journal on Imaging Sciences ◽

10.1137/17m1113436 ◽

2017 ◽

Vol 10 (4) ◽

pp. 1724-1766 ◽

Cited By ~ 9

Author(s):

Loic Landrieu ◽

Guillaume Obozinski

Keyword(s):

Fast Algorithms ◽

Weighted Graphs ◽

Piecewise Constant ◽

Piecewise Constant Functions

Download Full-text

Approximation of Periodic Functions of Many Variables in Metric Spaces by Piecewise-Constant Functions

Ukrainian Mathematical Journal ◽

10.1007/s11253-014-0871-5 ◽

2014 ◽

Vol 65 (10) ◽

pp. 1447-1459

Author(s):

T. A. Agoshkova

Keyword(s):

Metric Spaces ◽

Periodic Functions ◽

Piecewise Constant ◽

Piecewise Constant Functions ◽

Approximation Of Periodic Functions

Download Full-text

Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

Fractal and Fractional ◽

10.3390/fractalfract5040216 ◽

2021 ◽

Vol 5 (4) ◽

pp. 216

Author(s):

Shahram Rezapour ◽

Mohammed Said Souid ◽

Sina Etemad ◽

Zoubida Bouazza ◽

Sotiris K. Ntouyas ◽

...

Keyword(s):

Existence Of Solutions ◽

Coincidence Degree ◽

Degree Theory ◽

Variable Order ◽

Mawhin’S Continuation Theorem ◽

Piecewise Constant ◽

At Resonance ◽

Piecewise Constant Functions ◽

The Given ◽

Continuation Technique

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

Download Full-text

Parseval frames of piecewise constant functions

Operators and Matrices ◽

10.7153/oam-2020-14-25 ◽

2020 ◽

pp. 317-331

Author(s):

Dorin Ervin Dutkay ◽

Rajitha Ranasinghe

Keyword(s):

Piecewise Constant ◽

Piecewise Constant Functions ◽

Parseval Frames

Download Full-text

Joint image reconstruction and segmentation: Comparison of two algorithms for few-view tomography

Computer Optics ◽

10.18287/2412-6179-2019-43-6-1008-1020 ◽

2019 ◽

Vol 43 (6) ◽

pp. 1008-1020 ◽

Cited By ~ 1

Author(s):

V.V. Vlasov ◽

A.B. Konovalov ◽

S.V. Kolchugin

Keyword(s):

Image Reconstruction ◽

High Frequency ◽

Reconstruction Technique ◽

Adaptive Segmentation ◽

Reconstruction Accuracy ◽

Piecewise Constant ◽

X Ray ◽

Exact Reconstruction ◽

Tv Regularization ◽

Good Potential

Two algorithms of few-view tomography are compared, specifically, the iterative Potts minimization algorithm (IPMA) and the algebraic reconstruction technique with TV-regularization and adaptive segmentation (ART-TVS). Both aim to reconstruct piecewise-constant structures, use the compressed sensing theory, and combine image reconstruction and segmentation procedures. Using a numerical experiment, it is shown that either algorithm can exactly reconstruct the Shepp-Logan phantom from as small as 7 views with noise characteristic of the medical applications of X-ray tomography. However, if an object has a complicated high-frequency structure (QR-code), the minimal number of views required for its exact reconstruction increases to 17–21 for ART-TVS and to 32–34 for IPMA. The ART-TVS algorithm developed by the authors is shown to outperform IPMA in reconstruction accuracy and speed and in resistance to abnormally high noise as well. ART-TVS holds good potential for further improvement.

Download Full-text

Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000543 ◽

2018 ◽

Vol 168 (1) ◽

pp. 29-41 ◽

Cited By ~ 1

Author(s):

JOONAS ILMAVIRTA ◽

JERE LEHTONEN ◽

MIKKO SALO

Keyword(s):

Higher Dimensions ◽

Uniqueness Result ◽

Constant Function ◽

Piecewise Constant Function ◽

Two Dimensional ◽

Local Uniqueness ◽

Piecewise Constant ◽

Convex Boundary ◽

X Ray ◽

Strictly Convex

AbstractWe show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

Download Full-text