<p>This paper studies convolution Radon sampling and reconstruction in local multiply generated shift-invariant function spaces. We focus on two single-angle convolution Radon sampling schemes: convolution Radon deterministic sampling and convolution Radon random sampling. First, under certain conditions, we prove the stability of two kinds of convolution Radon sampling in local multiply generated shift-invariant function spaces. Second, the main problem of two-dimensional computerized tomography is to reconstruct function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_430_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$f({\textbf{x}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the samples used for this reconstruction process can be determined completely from its single-angle Radon transform. We identify the eligible <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_430_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">${\textbf{p}}=(\cos \theta ,\sin \theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>cos</mo> <mi>θ</mi> <mo>,</mo> <mo>sin</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and sampling set such that all functions <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_430_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$f({\textbf{x}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> belonging to local multiply generated shift-invariant function spaces on the interval <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_430_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$[a,b]^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> can be determined uniquely by its single-angle convolution Radon (w.r.t. <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_430_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${\textbf{p}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">p</mi> </math></EquationSource> </InlineEquation>) samples at sampling set. Meanwhile, we derive an explicit reconstruction formula. This reconstruction formula succeeds under certain conditions of the Radon sampling system.</p>

Single-angle based convolution Radon sampling and reconstruction in local shift-invariant function spaces

期刊论文
发表日期：

2025年6月5日

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Jiao Li,
Wei Li,
Jun Xian

摘要

This paper studies convolution Radon sampling and reconstruction in local multiply generated shift-invariant function spaces. We focus on two single-angle convolution Radon sampling schemes: convolution Radon deterministic sampling and convolution Radon random sampling. First, under certain conditions, we prove the stability of two kinds of convolution Radon sampling in local multiply generated shift-invariant function spaces. Second, the main problem of two-dimensional computerized tomography is to reconstruct function $f({\textbf{x}})$ $f (x)$ , the samples used for this reconstruction process can be determined completely from its single-angle Radon transform. We identify the eligible ${\textbf{p}}=(\cos \theta ,\sin \theta )$ $p = (cos θ, sin θ)$ and sampling set such that all functions $f({\textbf{x}})$ $f (x)$ belonging to local multiply generated shift-invariant function spaces on the interval $[a,b]^{2}$ ${[a, b]}^{2}$ can be determined uniquely by its single-angle convolution Radon (w.r.t. ${\textbf{p}}$ $p$ ) samples at sampling set. Meanwhile, we derive an explicit reconstruction formula. This reconstruction formula succeeds under certain conditions of the Radon sampling system.