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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Convolution for optical broad-beam responses in scattering media ： ウィキペディア英語版 Convolution for optical broad-beam responses in scattering media Photon transport theories, such as the Monte Carlo method, are commonly used to model light propagation in tissue. The responses to a pencil beam incident on a scattering medium are referred to as Green's functions or impulse responses. Photon transport methods can be directly used to compute broad-beam responses by distributing photons over the cross section of the beam. However, convolution can be used in certain cases to improve computational efficiency. ==General convolution formulas== In order for convolution to be used to calculate a broad-beam response, a system must be time invariant, linear, and translation invariant. Time invariance implies that a photon beam delayed by a given time produces a response shifted by the same delay. Linearity indicates that a given response will increase by the same amount if the input is scaled and obeys the property of superposition. Translational invariance means that if a beam is shifted to a new location on the tissue surface, its response is also shifted in the same direction by the same distance. Here, only spatial convolution is considered. Responses from photon transport methods can be physical quantities such as absorption, fluence, reflectance, or transmittance. Given a specific physical quantity, ''G(x,y,z)'', from a pencil beam in Cartesian space and a collimated light source with beam profile ''S(x,y)'', a broad-beam response can be calculated using the following 2-D convolution formula: :$C(x,y,z)=\backslash int\_^\backslash infty\; \backslash int\_^\backslash infty\; \backslash \; G(\; x-x\text{'},y-y\text{'},z)S(x\text{'},y\text{'})\backslash ,\; dx\text{'}\backslash ,dy\text{'}.\; \backslash qquad(1)$ Similar to 1-D convolution, 2-D convolution is commutative between ''G'' and ''S'' with a change of variables $x\text{'}\text{'}=x-x\text{'}\backslash ,$ and $y\text{'}\text{'}=y-y\text{'}\backslash ,$: :$C(x,y,z)=\backslash int\_^\backslash !\backslash int\_^\backslash \; G(\; x\text{'}\text{'},y\text{'}\text{'},z)S(x-x\text{'}\text{'},y-y\text{'}\text{'})\backslash ,\; dx\text{'}\text{'}\backslash ,dy\text{'}\text{'}.\; \backslash qquad(2)$ Because the broad-beam response $C(x,y,z)\backslash ,$ has cylindrical symmetry, its convolution integrals can be rewritten as: :$C(r,z)\; =\; \backslash int\_0^\backslash infty\; \backslash \; S(r\text{'})r\text{'}\; \backslash left\; (\backslash int\_^\; \backslash \; G\backslash left\; (\backslash sqrt,z\; \backslash right\; )\backslash ,\; d\backslash phi\text{'}\; \backslash right)dr\text{'}\; \backslash qquad(3)$ :$C(r,z)=\backslash int\_0^\backslash infty\; G(r\text{'}\text{'},z)r\text{'}\text{'}\; \backslash left\; (\backslash int\_^\; S\; \backslash left\; (\; \backslash sqrt\; \backslash right\; )\; \backslash ,\; d\backslash phi\text{'}\text{'}\; \backslash right)dr\text{'}\text{'}\backslash qquad(4)$ where $r\text{'}=\backslash sqrt$. Because the inner integration of Equation 4 is independent of ''z'', it only needs to be calculated once for all depths. Thus this form of the broad-beam response is more computationally advantageous. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Convolution for optical broad-beam responses in scattering media」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.