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\iteman{io-port 01085464}
\itemau{Caponnetto, A.; Bertero, M.}
\itemti{Tomography with a finite set of projections: Singular value decomposition and resolution.}
\itemso{Inverse Probl. 13, No.5, 1191-1205 (1997).}
\itemab
In some applied problems of mammography, one models the density function $f(x)$ by a linear combination of delta functions: $$f(x)= \sum^m_{j= 1}c_j\delta(x- x_j).\tag1$$ The Radon transform is defined by the formula $$Rf(\alpha, p):= \int_{l_{\alpha p}} f(x)ds,\tag2$$ where $l_{\alpha p}$ is the straight line $\alpha\cdot x= p$, $\alpha\in S'$ is a unit vector and $p$ is a real number. If $f(x)$ is defined by (1), then (2) can be written as $$Rf(\alpha, p)= \sum^m_{j= 1}c_j\delta(p- \alpha\cdot x_j).$$ The authors develop methods for finding $c_j$ and $x_j$. First they consider the case where the locations $x_j$, $j=1,2,\dots, m$ are known. The value of $c_j$ is estimated from $q(\alpha):= \int^\infty_{-\infty}(Rf)(\alpha, p)h(p)dp$, where $h$ is a function and $\alpha$ is a non-generated projection direction such that $\alpha\cdot x_j\ne \alpha\cdot x_i$ for $j\ne i$. Then they derive a method for the general case: the number $m$ of $\delta$ functions, their localization and their intensity are estimated from the data $Rf(\alpha, p)$. They show that this new method is more efficient than the filtered backprojection when the resolution in the variable $p$ of the sinogram is high.
\itemrv{R.S.Dahiya (Ames)}
\itemcc{}
\itemut{tomography; singular value decomposition; mammography; Radon transform; filtered backprojection}
\itemli{doi:10.1088/0266-5611/13/5/006}
\end