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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 tomographic reconstruction ： ウィキペディア英語版 tomographic reconstruction Tomographic imaging is applied in Computed Tomography to obtain cross-sectional images of patients. This article applies in general to tomographic reconstruction for all kinds of tomography, but some of the terms and physical descriptions refer directly to X-ray computed tomography. The mathematical basis for tomographic imaging was laid down by Johann Radon. == Description == The projection of an object, resulting from the tomographic measurement process at a given angle $\backslash theta$, is made up of a set of ''line integrals'' (see Fig. 1). As set of many such projections under different angles organized in 2D is called sinogram (see Fig. 3). In X-ray CT, the line integral represents the total attenuation of the beam of x-rays as it travels in a straight line through the object. As mentioned above, the resulting image is a 2D (or 3D) model of the attenuation coefficient. That is, we wish to find the image $\backslash mu(x,y)$. The simplest and easiest way to visualise the method of scanning is the system of parallel projection, as used in the first scanners. For this discussion we consider the data to be collected as a series of parallel rays, at position $r$, across a projection at angle $\backslash theta$. This is repeated for various angles. Attenuation occurs exponentially in tissue: :$I\; =\; I\_0\backslash exp\backslash left(\backslash right)$ where $\backslash mu(x)$ is the attenuation coefficient at position $x$ along the ray path. Therefore generally the total attenuation $p$ of a ray at position $r$, on the projection at angle $\backslash theta$, is given by the line integral: :$p(r,\backslash theta)\; =\; \backslash ln\; (I/I\_0)\; =\; -\backslash int\backslash mu(x,y)\backslash ,ds$ Using the coordinate system of Figure 1, the value of $r$ onto which the point $(x,y)$ will be projected at angle $\backslash theta$ is given by: :$x\backslash cos\backslash theta\; +\; y\backslash sin\backslash theta\; =\; r\backslash$ So the equation above can be rewritten as :$p(r,\backslash theta)=\backslash int^\backslash infty\_\backslash int^\backslash infty\_f(x,y)\backslash delta(x\backslash cos\backslash theta+y\backslash sin\backslash theta-r)\backslash ,dx\backslash ,dy$ where $f(x,y)$ represents $\backslash mu(x,y)$. This function is known as the Radon transform (or ''sinogram'') of the 2D object. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object, $f(x,y)$. So to get $f(x,y)$ back, from the above equation means finding the inverse Radon transform. It is possible to find an explicit formula for the inverse Radon transform. However, the inverse Radon transform proves to be extremely unstable with respect to noisy data. 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「tomographic reconstruction」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.