Basis Functions in Image Reconstruction From Projections: A Tutorial Introduction

Herman, Gabor T.

doi:10.1007/s11220-015-0107-2

Cited by 6 publications

(7 citation statements)

References 33 publications

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“…"Blob" basis functions, called "radial basis functions" in approximation theory, see, e.g., [56], are spherically symmetric basis functions for CT reconstruction that were suggested in the field of image reconstruction from projections by Lewitt in the early 1990s [57], [58]. These blobs are generalizations of a well-known class of functions used in digital signal processing called Kaiser-Bessel window functions and they yield excellent results in multiple imaging modalities, see, e.g., the enlightening tutorial by Herman [59]. Matej and Lewitt [60], [61] provided a careful investigation of how the blob basis functions should be chosen when they are used in the context of image reconstruction from projections.…”

Section: G Blob Basis Functions For Pct Image Reconstructionmentioning

confidence: 99%

Developments in Mathematical Algorithms and Computational Tools for Proton CT and Particle Therapy Treatment Planning

Censor¹,

Schubert²,

Schulte³

2021

Preprint

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We summarize recent results and ongoing activities in mathematical algorithms and computer science methods related to proton computed tomography (pCT) and intensitymodulated particle therapy (IMPT) treatment planning. Proton therapy necessitates a high level of delivery accuracy to exploit the selective targeting imparted by the Bragg peak. For this purpose, pCT utilizes the proton beam itself to create images. The technique works by sending a low-intensity beam of protons through the patient and measuring the position, direction, and energy loss of each exiting proton. The pCT technique allows reconstruction of the volumetric distribution of the relative stopping power (RSP) of the patient tissues for use in treatment planning and pre-treatment range verification. We have investigated new ways to make the reconstruction both efficient and accurate. Better accuracy of RSP also enables more robust inverse approaches to IMPT. For IMPT, we developed a framework for performing intensity-modulation of the proton pencil beams. We expect that these developments will lead to additional project work in the years to come, which requires a regular exchange between experts in the fields of mathematics, computer science, and medical physics. We have initiated such an exchange by organizing annual workshops on pCT and IMPT algorithm and technology developments. This report is, admittedly, tilted toward our interdisciplinary work and methods. We offer a comprehensive overview of results, problems, and challenges in pCT and IMPT with the aim of making other scientists wanting to tackle such issues and to strengthen their interdisciplinary collaboration by bringing together cutting-edge know-how from medicine, computer science, physics, and mathematics to bear on medical physics problems at hand.

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Section: G Blob Basis Functions For Pct Image Reconstructionmentioning

confidence: 99%

Developments in Mathematical Algorithms and Computational Tools for Proton CT and Particle Therapy Treatment Planning

Censor¹,

Schubert²,

Schulte³

2021

Preprint

View full text Add to dashboard Cite

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“…This implies that for taking s independent of T , the spaces V T,s,ϕ have a limited approximation capability in the sense that for a typical function f , the approximation error min u∈V T,s,ϕ f − u 2 L 2 does not converge to zero as T → 0 and s is kept fixed. Despite these negative results, radially symmetric basis functions are of great popularity in computed tomography (see for example, [23,33,34,39,44,57,56]). In this paper we therefore propose to also allow the shift parameter s to be variable.…”

Section: Shift Invariant Spaces In Computed Tomographymentioning

confidence: 99%

“…The pixel basis (also called voxel basis in the case d > 2) has been frequently used for image representation in early tomographic image reconstruction (see, for example [13,23,25]). It consists of scaled and translated version of the indicator function of the hyper-cube…”

Section: Example: the Pixel Basismentioning

confidence: 99%

“…This in turn is the case if and only if s = 2 −m for some m ∈ N. The case s = 1 seems the most natural one, since it uses non-overlapping basis functions filling the whole space R d . The non-overlapping case is in fact used in existing tomographic image reconstruction algorithms; see [13,23,25]. Further, note that the number of basis elements χ k T,s for which its center m k := T sk is contained in the unit cube [−1, 1] d is given by (2/(T s) + 1) d and that T is inversely proportional to the essential bandwidth of the basis function.…”

Section: Example: the Pixel Basismentioning

confidence: 99%

“…Note that shift invariant spaces are frequently employed in computed tomography and include splines spaces, spaces of bandlimited functions, or spaces generated by Kaiser-Bessel functions. In this paper we will especially use Kaiser-Bessel functions which are often considered as the most suitable basis for computed tomography [23,34,39,44]. For the use in PAT they have first been proposed in [56].…”

Section: Proposed Galerkin Least Squares Approachmentioning

confidence: 99%

See 2 more Smart Citations

A Galerkin Least Squares Approach for Photoacoustic Tomography

Schwab¹,

Pereverzyev²,

Haltmeier³

2018

SIAM J. Numer. Anal.

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The development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography. In this paper we address this issue for photoacoustic computed tomography in circular geometry. We investigate the Galerkin least squares method for that purpose. For approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single function. This includes many systems that have previously been employed in PAT such as generalized Kaiser-Bessel basis functions or the natural pixel basis. By exploiting an isometry property of the forward problem we are able to efficiently set up the Galerkin equation for a wide class of generating functions and devise efficient algorithms for its solution. We establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm.

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A Semi-Discrete Landweber–Kaczmarz Method for Cone Beam Tomography and Laminography Exploiting Geometric Prior Information

Vogelgesang

Schorr

2016

Sens Imaging

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Basis Functions in Image Reconstruction From Projections: A Tutorial Introduction

Cited by 6 publications

References 33 publications

Developments in Mathematical Algorithms and Computational Tools for Proton CT and Particle Therapy Treatment Planning

Developments in Mathematical Algorithms and Computational Tools for Proton CT and Particle Therapy Treatment Planning

A Galerkin Least Squares Approach for Photoacoustic Tomography

A Semi-Discrete Landweber–Kaczmarz Method for Cone Beam Tomography and Laminography Exploiting Geometric Prior Information

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