Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography
Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead o...
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/23/4757 |
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| author | Vangelis Marinakis Athanassios S. Fokas George A. Kastis Nicholas E. Protonotarios |
| author_facet | Vangelis Marinakis Athanassios S. Fokas George A. Kastis Nicholas E. Protonotarios |
| author_sort | Vangelis Marinakis |
| collection | DOAJ |
| description | Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points along a line, Chebyshev interpolation involves the roots of Chebyshev polynomials, known as Chebyshev nodes, corresponding to equally spaced points along the unit semicircle. By reviewing prior research on the applications of Chebyshev interpolation, it becomes apparent that this interpolation is rather impractical for medical imaging. Especially in clinical positron emission tomography (PET) and in single-photon emission computerized tomography (SPECT), the so-called <i>sinogram</i> is <i>always</i> calculated at equally spaced points, since the detectors are almost always uniformly distributed. We have been able to overcome this difficulty as follows. Suppose that the function to be interpolated has compact support and is known at <i>q</i> equally spaced points in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mfenced></semantics></math></inline-formula>. We extend the domain to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mo>−</mo><mi>a</mi><mo>,</mo><mi>a</mi></mfenced></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, and select a sufficiently large value of <i>a</i>, such that <i>exactly</i><i>q</i> Chebyshev nodes are included in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mfenced></semantics></math></inline-formula>, which are <i>almost</i> equally spaced. This construction provides a generalization of the concept of standard Chebyshev interpolation to almost equally spaced points. Our preliminary results indicate that our modification of the Chebyshev method provides comparable, or, in several cases including Runge’s phenomenon, superior interpolation over the standard Chebyshev interpolation. In terms of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula> norm of the interpolation error, a decrease of up to 75% was observed. Furthermore, our approach opens the way for using Chebyshev polynomials in the solution of the inverse problems arising in PET and SPECT image reconstruction. |
| first_indexed | 2024-03-09T01:46:36Z |
| format | Article |
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| last_indexed | 2024-03-09T01:46:36Z |
| publishDate | 2023-11-01 |
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| series | Mathematics |
| spelling | doaj.art-0e32dd8f7c06497e880ac7a8475a675b2023-12-08T15:21:41ZengMDPI AGMathematics2227-73902023-11-011123475710.3390/math11234757Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission TomographyVangelis Marinakis0Athanassios S. Fokas1George A. Kastis2Nicholas E. Protonotarios3Department of Civil Engineering, University of the Peloponnese, 26334 Patras, GreeceDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UKMathematics Research Center, Academy of Athens, 11527 Athens, GreeceMathematics Research Center, Academy of Athens, 11527 Athens, GreeceSince their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points along a line, Chebyshev interpolation involves the roots of Chebyshev polynomials, known as Chebyshev nodes, corresponding to equally spaced points along the unit semicircle. By reviewing prior research on the applications of Chebyshev interpolation, it becomes apparent that this interpolation is rather impractical for medical imaging. Especially in clinical positron emission tomography (PET) and in single-photon emission computerized tomography (SPECT), the so-called <i>sinogram</i> is <i>always</i> calculated at equally spaced points, since the detectors are almost always uniformly distributed. We have been able to overcome this difficulty as follows. Suppose that the function to be interpolated has compact support and is known at <i>q</i> equally spaced points in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mfenced></semantics></math></inline-formula>. We extend the domain to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mo>−</mo><mi>a</mi><mo>,</mo><mi>a</mi></mfenced></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, and select a sufficiently large value of <i>a</i>, such that <i>exactly</i><i>q</i> Chebyshev nodes are included in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mfenced></semantics></math></inline-formula>, which are <i>almost</i> equally spaced. This construction provides a generalization of the concept of standard Chebyshev interpolation to almost equally spaced points. Our preliminary results indicate that our modification of the Chebyshev method provides comparable, or, in several cases including Runge’s phenomenon, superior interpolation over the standard Chebyshev interpolation. In terms of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula> norm of the interpolation error, a decrease of up to 75% was observed. Furthermore, our approach opens the way for using Chebyshev polynomials in the solution of the inverse problems arising in PET and SPECT image reconstruction.https://www.mdpi.com/2227-7390/11/23/4757Chebyshev polynomialsChebyshev nodespolynomial interpolationmedical image reconstructionemission tomographyPET |
| spellingShingle | Vangelis Marinakis Athanassios S. Fokas George A. Kastis Nicholas E. Protonotarios Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography Mathematics Chebyshev polynomials Chebyshev nodes polynomial interpolation medical image reconstruction emission tomography PET |
| title | Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography |
| title_full | Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography |
| title_fullStr | Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography |
| title_full_unstemmed | Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography |
| title_short | Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography |
| title_sort | chebyshev interpolation using almost equally spaced points and applications in emission tomography |
| topic | Chebyshev polynomials Chebyshev nodes polynomial interpolation medical image reconstruction emission tomography PET |
| url | https://www.mdpi.com/2227-7390/11/23/4757 |
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