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InverseRadonTransform[expr,{p,ϕ},{x,y}]

给出 expr 的逆拉东变换.

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范例  
基本范例  
范围  
基本用法  
高斯函数  
分段和广义函数  
应用  
属性和关系  
巧妙范例  
参见
相关指南
历史
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InverseRadonTransform

InverseRadonTransform[expr,{p,ϕ},{x,y}]

给出 expr 的逆拉东变换.

更多信息和选项

范例

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基本范例  (1)

计算一个函数的逆拉东变换：

Wolfram Language code: InverseRadonTransform[E^-p^2 (Cos[ϕ]^2 + 2 p^2 Sin[ϕ]^2) , {p, ϕ}, {x, y}]

绘制函数及其逆变换：

Wolfram Language code: {DensityPlot[E^-p^2 (Cos[ϕ]^2 + 2 p^2 Sin[ϕ]^2), {ϕ, -π / 2, π / 2}, {p, -3, 3}, Frame -> None, Ticks -> None], DensityPlot[%, {x, -2, 2}, {y, -2, 2}, Frame -> None, Ticks -> None]}

范围  (5)

基本用法  (1)

计算参数值为符号的函数的逆拉东变换：

Wolfram Language code: InverseRadonTransform[E^-p^2 p Cos[ϕ], {p, ϕ}, {x, y}]

对参数使用精确值：

Wolfram Language code: InverseRadonTransform[E^-p^2 p Cos[ϕ], {p, ϕ}, {1 / 3, 1 / 5}]

对参数使用不精确值：

Wolfram Language code: InverseRadonTransform[E^-p^2 p Cos[ϕ], {p, ϕ}, {0.3, 0.7}]

高斯函数  (2)

高斯函数的逆拉东变换：

Wolfram Language code: InverseRadonTransform[E^-p^2, {p, ϕ}, {x, y}]

绘制函数及其逆变换：

Wolfram Language code: {DensityPlot[E^-p^2, {ϕ, -π / 2, π / 2}, {p, -10, 10}, Frame -> None, Ticks -> None], DensityPlot[%, {x, -2, 2}, {y, -2, 2}, Frame -> None, Ticks -> None]}

高斯多项式函数：

Wolfram Language code: InverseRadonTransform[E^-p^2 (1 + 2 p^2) , {p, ϕ}, {x, y}]
Wolfram Language code: {DensityPlot[E^-p^2 (1 + 2 p^2) , {ϕ, -π / 2, π / 2}, {p, -10, 10}, Frame -> None, Ticks -> None], DensityPlot[%, {x, -2, 2}, {y, -2, 2}, Frame -> None, Ticks -> None]}

高斯多项式函数与三角函数的乘积：

Wolfram Language code: InverseRadonTransform[ E^-p^2 p (-3 + 2 p^2) Sin[ϕ] Sin[2 ϕ], {p, ϕ}, {x, y}]
Wolfram Language code: {DensityPlot[ E^-p^2 p (-3 + 2 p^2) Sin[ϕ] Sin[2 ϕ], {ϕ, -π / 2, π / 2}, {p, -2, 2}, Frame -> None, Ticks -> None, PlotPoints -> 200], DensityPlot[%, {x, -2, 2}, {y, -2, 2}, Frame -> None, Ticks -> None]}

分段和广义函数  (2)

分段函数的逆拉东变换：

Wolfram Language code: InverseRadonTransform[Piecewise[{{2*Sqrt[1 - p^2], -1 < p < 1}}, 0], {p, ϕ}, {x, y}]
Wolfram Language code: {DensityPlot[ Piecewise[{{2*Sqrt[1 - p^2], -1 < p < 1}}, 0], {ϕ, -π / 2, π / 2} , {p, -2, 2}, Frame -> None, Ticks -> None, PlotPoints -> 200], DensityPlot[%, {x, -2, 2}, {y, -2, 2}, Frame -> None, Ticks -> None]}

含有 DiracDelta 的表达式的逆拉东变换：

Wolfram Language code: InverseRadonTransform[DiracDelta[p - a Cos[ϕ] - b Sin[ϕ]], {p, ϕ}, {x, y}]

应用  (2)

计算函数的符号逆拉东变换：

Wolfram Language code: InverseRadonTransform[E^-p^2 , {p, ϕ}, {x, y}]
Wolfram Language code: {DensityPlot[E^-p^2 , {ϕ, -π / 2, π / 2}, {p, -5, 5}, Frame -> None, Ticks -> None], DensityPlot[%, {x, -2, 2}, {y, -2, 2}, Frame -> None, Ticks -> None]}

用 InverseRadon 获取同样的结果：

Wolfram Language code: disk = Image[DiskMatrix[50, 201], "Bit"];
Wolfram Language code: {rad = Radon[disk, {201, 201}], InverseRadon[rad, {201, 201}]}

用拉东变换求解泊松方程：

Wolfram Language code: peqn = Laplacian[u[x, y], {x, y}] == E^-x^2 - y^2 y (-2 + x^2 + y^2);

对方程应用 RadonTransform：

Wolfram Language code: RadonTransform[peqn, {x, y}, {p, ϕ}]

用 DSolveValue 求解常微分方程：

Wolfram Language code: DSolveValue[% /. {RadonTransform[u[x, y], {x, y}, {p, ϕ}] -> f[p]}, f[p], p]

把解中的任意常数设为 0：

Wolfram Language code: % /. {C[i_] -> 0}

用 InverseRadonTransform 获取原来方程的解：

Wolfram Language code: sol = InverseRadonTransform[%, {p, ϕ}, {x, y}]

验证解：

Wolfram Language code: peqn /. {u -> Function[{x, y}, Evaluate[sol]]}//Simplify

绘制解：

Wolfram Language code: Plot3D[sol, {x, -3, 3}, {y, -3, 3}, PlotRange -> All]

属性和关系  (3)

InverseRadonTransform 和 RadonTransform 互为逆：

Wolfram Language code: RadonTransform[InverseRadonTransform[F[p, ϕ], {p, ϕ}, {x, y}], {x, y}, {p, ϕ}]
Wolfram Language code: InverseRadonTransform[RadonTransform[f[x, y], {x, y}, {p, ϕ}], {p, ϕ}, {x, y}]

InverseRadonTransform 是线性运算符：

Wolfram Language code: InverseRadonTransform[a f[p, ϕ] + b g[p, ϕ], {p, ϕ}, {x, y}]

用傅立叶变换计算逆拉东变换：

Wolfram Language code: f[p_, ϕ_] := E^-p^2 p Sqrt[π] (Cos[ϕ] + Sin[ϕ])

求关于 p 的傅立叶变换：

Wolfram Language code: FourierTransform[f[p, ϕ], p, k, FourierParameters -> {0, -2π}]

用单位向量 ξ = { u1,u2} 表示结果，假定 k=TemplateBox[{xi}, Abs]：

Wolfram Language code: Simplify[% /. {Sin[a_] :> TrigExpand[Sin[a]]} /. {Cos[ϕ] -> u1 / k, Sin[ϕ] -> u2 / k} /. {k -> Sqrt[u1 ^ 2 + u2 ^ 2]}, Im[u1] == 0 && Im[u2] == 0]

计算关于 { u1,u2} 的傅立叶逆变换：

Wolfram Language code: InverseFourierTransform[%, {u1, u2}, {x, y}, FourierParameters -> {0, -2 * Pi}]

用 InverseRadonTransform 直接获取同样的结果：

Wolfram Language code: InverseRadonTransform[f[p, ϕ], {p, ϕ}, {x, y}]

巧妙范例  (1)

创建逆拉东变换表：

Wolfram Language code: flist = {Sqrt[Pi] / E ^ p ^ 2, (p * Sqrt[Pi] * Cos[ϕ]) / E ^ p ^ 2, ((1 / 2) * Sqrt[Pi] * (Cos[ϕ] ^ 2 + 2 * p ^ 2 * Sin[ϕ] ^ 2)) / E ^ p ^ 2, ((1 / 2) * (1 + 2 * p ^ 2) * Sqrt[Pi]) / E ^ p ^ 2, ((1 / 4) * (3 + 4 * (p ^ 2 + p ^ 4)) * Sqrt[Pi]) / E ^ p ^ 2, Piecewise[{{2 * Sqrt[1 - p ^ 2], -1 < p < 1}}, 0], ((1 / 2) * (-1 + 2 * p ^ 2) * Sqrt[Pi] * Cos[2 * ϕ]) / E ^ p ^ 2};
Wolfram Language code: Grid[Prepend[{#, InverseRadonTransform[#1, {p, ϕ}, {x, y}]}& /@ flist, {"Function", "Inverse Radon Transform"}], IconizedObject[«Grid options»]]//TraditionalForm
Wolfram Research (2017)，InverseRadonTransform，Wolfram 语言函数，https://reference.wolfram.com/language/ref/InverseRadonTransform.html.

文本

Wolfram Research (2017)，InverseRadonTransform，Wolfram 语言函数，https://reference.wolfram.com/language/ref/InverseRadonTransform.html.

CMS

Wolfram 语言. 2017. "InverseRadonTransform." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/InverseRadonTransform.html.

APA

Wolfram 语言. (2017). InverseRadonTransform. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/InverseRadonTransform.html 年

BibTeX

@misc{reference.wolfram_2026_inverseradontransform, author="Wolfram Research", title="{InverseRadonTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/InverseRadonTransform.html}", note=[Accessed: 25-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_inverseradontransform, organization={Wolfram Research}, title={InverseRadonTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/InverseRadonTransform.html}, note=[Accessed: 25-October-2026]}