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Mathematics-Online course: Linear Algebra - Linear Systems of Equations - Approximation Problems

Tomography

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As illustrated in the figure, in computer tomography the density $ x(u,v)$ of cell tissue is reconstructed from measured data of the loss of intensity of X-rays along $ k$ bundles of $ \ell$ parallel lines

$\displaystyle \mathcal{R}_i:\ (u_i,v_i) + \mathbb{R}(\cos\vartheta_i,\sin\vartheta_i), \quad i=1,\ldots,m=k\ell\, . $

The density distribution is approximated by piecewise constant functions on a grid of squares $ \mathcal{Q}_j$ and the line integrals resulting from the intensity measurements are replaced by the approximations

$\displaystyle b_i = \int_{\mathbb{R}} x(u_i+t\cos\vartheta_i,v_i+t\sin\vartheta_i)\, dt \approx \sum_{j=1}^n a_{i,j}x_j\, , $

where $ x_j$ is an approximation of $ x(u,v)$ on $ \mathcal{Q}_j$, and

$\displaystyle a_{i,j}=\vert\mathcal{R}_i\cap\mathcal{Q}_j\vert $

denotes the length of the intersection of line $ \mathcal{R}_i$ and square $ \mathcal{Q}_j$. Since, in general, the number of measurements $ m$ is much larger than the number of squares $ n$ and the integrals are not calculated exactly, we obtain a least squares problem to determine $ x$ from the data $ b$.

\includegraphics[width=\moimagesize]{b_tomographie}

The right side of the figure shows an approximation for the density distribution, depicted on the left, calculated on an $ 11\times 11$-grid. In this case, for the angles

$\displaystyle \vartheta = 0,\pi/16,\pi/8,\ldots , $

$ 11$ parallel scan directions with distance equal to the grid width were used.
  automatically generated 4/21/2005