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<p>The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies—for data lying in the range of the ADRT—that the transform of an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N \times N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> image can be formally inverted with complexity <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}(N^2 \log ^2 N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> <msup> <mo>log</mo> <mn>2</mn> </msup> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime.</p>

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An explicit spectral decomposition of the ADRT

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月26日

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  • Weilin Li,
  • Karl Otness,
  • Kui Ren,
  • Donsub Rim

摘要

The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies—for data lying in the range of the ADRT—that the transform of an \(N \times N\) N × N image can be formally inverted with complexity \(\mathcal {O}(N^2 \log ^2 N)\) O ( N 2 log 2 N ) . We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime.