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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301001.png" /> be a compactly supported piecewise-smooth function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301002.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301004.png" /> a bounded domain, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301005.png" /> be its [[Radon transform|Radon transform]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301006.png" /> is the straight line parametrized by the unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301007.png" /> and a scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301008.png" />. The inversion formula which reconstructs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l1301009.png" /> from the knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010011.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010013.png" /> is the unit circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010014.png" />, is known to be:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010016.png" /></td> </tr></table>
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Let $f ( x )$ be a compactly supported piecewise-smooth function, $f ( x ) = 0$ if $x \notin \overline { D } \subset \mathbf{R} ^ { 2 }$, $D$ a bounded domain, and let $\hat { f } ( \alpha , p ) = \int _ { \operatorname { l}_{\alpha p} } f ( x ) d s : = R f$ be its [[Radon transform|Radon transform]], where $\text{l} _ { \alpha p}  : = \{ x : \alpha \cdot x = p \}$ is the straight line parametrized by the unit vector $\alpha$ and a scalar $p$. The inversion formula which reconstructs $f ( x )$ from the knowledge of $\hat { f } ( \alpha , p )$ for all $\alpha \in S ^ { 1 }$ and all $p \in \bf R$, where $S ^ { 1 }$ is the unit circle in $\mathbf{R} ^ { 2 }$, is known to be:
  
It is non-local: one requires the knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010018.png" /> in order to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010019.png" />.
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\begin{equation} \tag{a1} f ( x ) = \frac { 1 } { 4 \pi ^ { 2 } } \int _ { S ^ { 1 } } \int _ { - \infty } ^ { \infty } \frac { \hat { f } _ { p } ( \alpha , p ) } { \alpha . x - p } d \alpha d p, \end{equation}
  
By local tomographic data one means the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010020.png" /> for those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010022.png" /> which satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010024.png" /> is a fixed "point of interest" and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010025.png" /> is a small number. Geometrically, local tomographic data are the values of the integrals over the straight lines which intersect the disc centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010026.png" /> with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010027.png" />. In many applications only local tomographic data are available, while in medical imaging one wants to minimize the radiation dose of a patient and to use only the local tomographic data for diagnostics.
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\begin{equation*} \hat { f } _ { p } : = \frac { \partial \hat { f } } { \partial p }. \end{equation*}
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It is non-local: one requires the knowledge of $\hat { f } ( \alpha , p )$ for all $p$ in order to calculate $f ( x )$.
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By local tomographic data one means the values of $\hat { f } ( \alpha , p )$ for those $\alpha$ and $p$ which satisfy the condition $| \alpha . x _ { 0 } - p | &lt; \delta$, where $x _ { 0 }$ is a fixed "point of interest" and $\delta &gt; 0$ is a small number. Geometrically, local tomographic data are the values of the integrals over the straight lines which intersect the disc centred at $x _ { 0 }$ with radius $\delta$. In many applications only local tomographic data are available, while in medical imaging one wants to minimize the radiation dose of a patient and to use only the local tomographic data for diagnostics.
  
 
Therefore, the basic question is: What practically useful information can one get from local tomographic data?
 
Therefore, the basic question is: What practically useful information can one get from local tomographic data?
  
As mentioned above, one cannot find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010028.png" /> from local tomographic data.
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As mentioned above, one cannot find $f ( x _ { 0 } )$ from local tomographic data.
  
 
What does one mean by "practically useful information" ?
 
What does one mean by "practically useful information" ?
  
By this one means the location of discontinuity curves (surfaces, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010029.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010030.png" /> and the sizes of the jumps of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010031.png" /> across these surfaces.
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By this one means the location of discontinuity curves (surfaces, if $n &gt; 2$) of $f ( x )$ and the sizes of the jumps of $f ( x )$ across these surfaces.
  
Probably the first empirically found method for finding discontinuities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010032.png" /> from local tomographic data was suggested in [[#References|[a1]]], where the function
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Probably the first empirically found method for finding discontinuities of $f ( x )$ from local tomographic data was suggested in [[#References|[a1]]], where the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} f _ { s \text{l}t } ( x ) : = - \frac { 1 } { 4 \pi } \int _ { S ^ { 1 } } \hat { f } _ { p p } ( \alpha , \alpha \cdot x ) d \alpha, \end{equation}
  
which is the standard local tomography function, was proposed. To calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010034.png" /> one needs to know only the local tomography data corresponding to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010035.png" />. It is proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010037.png" /> have the same discontinuities (but different sizes of the jumps across the discontinuity curves) [[#References|[a11]]]. For various aspects of local tomography, see the references. See also [[Tomography|Tomography]].
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which is the standard local tomography function, was proposed. To calculate $f ( x )$ one needs to know only the local tomography data corresponding to the point $x$. It is proved that $f ( x )$ and $f _ { s \text{l} t } ( x )$ have the same discontinuities (but different sizes of the jumps across the discontinuity curves) [[#References|[a11]]]. For various aspects of local tomography, see the references. See also [[Tomography|Tomography]].
  
In [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], a large family of local tomography functions was proposed. The basic idea here is to establish a relation between hypo-elliptic pseudo-differential operators and a class of linear operators acting on the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010038.png" />.
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In [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], a large family of local tomography functions was proposed. The basic idea here is to establish a relation between hypo-elliptic pseudo-differential operators and a class of linear operators acting on the functions $\hat { f } ( \alpha , p )$.
  
Let a [[Pseudo-differential operator|pseudo-differential operator]] be defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010040.png" /> is the [[Fourier transform|Fourier transform]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010041.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010042.png" /> is a smooth function, which is called the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010045.png" />. If the symbol is hypo-elliptic, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010050.png" /> are positive constants, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010052.png" /> is the [[Wave front|wave front]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010053.png" />. Thus, the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010055.png" /> are the same. One can prove [[#References|[a9]]] the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010058.png" /> is the adjoint to the Radon operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010059.png" /> (cf. also [[Radon transform|Radon transform]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010060.png" /> is the convolution operator, with distributional kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010061.png" /> defined by
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Let a [[Pseudo-differential operator|pseudo-differential operator]] be defined by the formula $B f =\mathcal{ F} ^ { - 1 } [ b ( x , t , \alpha ) \tilde { f } ]$, where $\widetilde { f } : = \mathcal F f$ is the [[Fourier transform|Fourier transform]], $\tilde { f } ( \xi ) = \int _ { \mathbf{R} ^ { n } } f ( x ) e ^ { i \xi x } d x$, and $b ( x , t , \alpha )$ is a smooth function, which is called the symbol of $B$, $\alpha : = \xi / | \xi |$, $t = | \xi |$. If the symbol is hypo-elliptic, that is, $c _ { 1 } | \xi | ^ { m _ { 1 } } \leq | b | \leq c _ { 2 } | \xi | ^ { m _ { 2 } }$, $| \xi | &gt; R$, $x \in D$, $c_1$ and $c_2$ are positive constants, then $\operatorname{WF} ( B f ) = \operatorname{WF} ( f )$, where $ \operatorname {WF} ( f )$ is the [[Wave front|wave front]] of $f$. Thus, the singularities of $B f$ and $f$ are the same. One can prove [[#References|[a9]]] the formula $B f = R ^ { * } ( a _ { \text{e} } \otimes \widehat { f } ) : = A \widehat { f }$, where $R ^ { * } g : = \int _ { S ^ { n - 1 }} g ( \alpha , \alpha . x ) d \alpha $, where $R ^ { * }$ is the adjoint to the Radon operator $R$ (cf. also [[Radon transform|Radon transform]]), and $a \otimes \hat { f } : = \int _ { - \infty } ^ { \infty } a ( x , \alpha , p - q ) \hat { f } ( q ) d q$ is the convolution operator, with distributional kernel $a ( x , \alpha , p - q )$ defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010062.png" /></td> </tr></table>
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\begin{equation*} a ( x , \alpha , p ) : = \frac { 1 } { ( 2 \pi ) ^ { n } } \int _ { 0 } ^ { \infty } t ^ { n - 1 } e ^ { - i t p } b ( x , t , \alpha ) d t, \end{equation*}
  
 
and with
 
and with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010063.png" /></td> </tr></table>
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\begin{equation*} a _ { e } ( x , \alpha , p ) : = \frac { a ( x , \alpha , p ) + a ( x , - \alpha , - p ) } { 2 } \end{equation*}
  
the even part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010064.png" />.
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the even part of $a ( x , \alpha , p )$.
  
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010065.png" /> is called a local tomography operator if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010066.png" /> uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010068.png" />.
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An operator $A$ is called a local tomography operator if and only if $\operatorname { supp } a _ { \operatorname {e} } ( x , \alpha , p ) \subset [ - \delta , \delta ]$ uniformly with respect to $x \in D$ and $\alpha \in S ^ { n - 1 }$.
  
A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010069.png" /> to be a local tomography operator is given in [[#References|[a9]]]: The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010070.png" /> is an [[Entire function|entire function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010071.png" /> of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010072.png" /> uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010074.png" />.
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A necessary and sufficient condition for $A$ to be a local tomography operator is given in [[#References|[a9]]]: The kernel $b ( x , t , \alpha ) t _ { + } ^ { n - 1 } + b ( x , - t , - \alpha ) t ^ { n - 1 }_-$ is an [[Entire function|entire function]] of $t$ of exponential type $\leq \delta$ uniformly with respect to $x \in D$ and $\alpha \in S ^ { n - 1 }$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Vainberg, I. Kazak, V. Kurczaev, "Reconstruction of the internal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010075.png" />D structure of objects based on real-time integral projections" ''Soviet J. Nondestr. Test.'' , '''17''' (1981) pp. 415–423 (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Faridani, E. Ritman, K. Smith, "Local tomography" ''SIAM J. Appl. Math.'' , '''52''' (1992) pp. 459–484 {{MR|1174054}} {{MR|1154783}} {{ZBL|0777.65076}} {{ZBL|0758.65081}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Katsevich, "Local tomography for the generalized Radon transform" ''SIAM J. Appl. Math.'' , '''57''' : 4 (1997) pp. 1128–1162 {{MR|1462054}} {{ZBL|0897.65084}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Katsevich, "Local tomography for the limited-angle problem" ''J. Math. Anal. Appl.'' , '''213''' (1997) pp. 160–182 {{MR|1469368}} {{ZBL|0894.65065}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A. Katsevich, "Local tomography with nonsmooth attenuation II" A.G. Ramm (ed.) , ''Inverse Problems, Tomography, and Image Processing'' , Plenum (1998) pp. 73–86 {{MR|1625256}} {{ZBL|0981.44002}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Katsevich, "Local tomography with nonsmooth attenuation" ''Trans. Amer. Math. Soc.'' , '''351''' (1999) pp. 1947–1974 {{MR|1466950}} {{ZBL|0924.35207}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A.G. Ramm, "Optimal local tomography formulas" ''PanAmer. Math. J.'' , '''4''' : 4 (1994) pp. 125–127 {{MR|1310327}} {{ZBL|0847.44001}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> A.G. Ramm, "Finding discontinuities from tomographic data" ''Proc. Amer. Math. Soc.'' , '''123''' : 8 (1995) pp. 2499–2505 {{MR|1273517}} {{ZBL|0830.44001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> A.G. Ramm, "Necessary and sufficient conditions for a PDO to be a local tomography operator" ''C.R. Acad. Sci. Paris'' , '''332''' : 7 (1996) pp. 613–618 {{MR|1386462}} {{ZBL|0848.35147}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A.G. Ramm, "New methods for finding discontinuities of functions from local tomographic data" ''J. Inverse Ill-Posed Probl.'' , '''5''' : 2 (1997) pp. 165–174 {{MR|1452016}} {{ZBL|0881.44003}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A.G. Ramm, A.I. Katsevich, "The Radon transform and local tomography" , CRC (1996) {{MR|1384070}} {{ZBL|0863.44001}} </TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> E. Vainberg, I. Kazak, V. Kurczaev, "Reconstruction of the internal $3$D structure of objects based on real-time integral projections" ''Soviet J. Nondestr. Test.'' , '''17''' (1981) pp. 415–423 (In Russian)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> A. Faridani, E. Ritman, K. Smith, "Local tomography" ''SIAM J. Appl. Math.'' , '''52''' (1992) pp. 459–484 {{MR|1174054}} {{MR|1154783}} {{ZBL|0777.65076}} {{ZBL|0758.65081}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> A. Katsevich, "Local tomography for the generalized Radon transform" ''SIAM J. Appl. Math.'' , '''57''' : 4 (1997) pp. 1128–1162 {{MR|1462054}} {{ZBL|0897.65084}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A. Katsevich, "Local tomography for the limited-angle problem" ''J. Math. Anal. Appl.'' , '''213''' (1997) pp. 160–182 {{MR|1469368}} {{ZBL|0894.65065}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> A. Katsevich, "Local tomography with nonsmooth attenuation II" A.G. Ramm (ed.) , ''Inverse Problems, Tomography, and Image Processing'' , Plenum (1998) pp. 73–86 {{MR|1625256}} {{ZBL|0981.44002}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> A. Katsevich, "Local tomography with nonsmooth attenuation" ''Trans. Amer. Math. Soc.'' , '''351''' (1999) pp. 1947–1974 {{MR|1466950}} {{ZBL|0924.35207}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> A.G. Ramm, "Optimal local tomography formulas" ''PanAmer. Math. J.'' , '''4''' : 4 (1994) pp. 125–127 {{MR|1310327}} {{ZBL|0847.44001}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> A.G. Ramm, "Finding discontinuities from tomographic data" ''Proc. Amer. Math. Soc.'' , '''123''' : 8 (1995) pp. 2499–2505 {{MR|1273517}} {{ZBL|0830.44001}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> A.G. Ramm, "Necessary and sufficient conditions for a PDO to be a local tomography operator" ''C.R. Acad. Sci. Paris'' , '''332''' : 7 (1996) pp. 613–618 {{MR|1386462}} {{ZBL|0848.35147}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> A.G. Ramm, "New methods for finding discontinuities of functions from local tomographic data" ''J. Inverse Ill-Posed Probl.'' , '''5''' : 2 (1997) pp. 165–174 {{MR|1452016}} {{ZBL|0881.44003}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> A.G. Ramm, A.I. Katsevich, "The Radon transform and local tomography" , CRC (1996) {{MR|1384070}} {{ZBL|0863.44001}} </td></tr></table>

Revision as of 16:59, 1 July 2020

Let $f ( x )$ be a compactly supported piecewise-smooth function, $f ( x ) = 0$ if $x \notin \overline { D } \subset \mathbf{R} ^ { 2 }$, $D$ a bounded domain, and let $\hat { f } ( \alpha , p ) = \int _ { \operatorname { l}_{\alpha p} } f ( x ) d s : = R f$ be its Radon transform, where $\text{l} _ { \alpha p} : = \{ x : \alpha \cdot x = p \}$ is the straight line parametrized by the unit vector $\alpha$ and a scalar $p$. The inversion formula which reconstructs $f ( x )$ from the knowledge of $\hat { f } ( \alpha , p )$ for all $\alpha \in S ^ { 1 }$ and all $p \in \bf R$, where $S ^ { 1 }$ is the unit circle in $\mathbf{R} ^ { 2 }$, is known to be:

\begin{equation} \tag{a1} f ( x ) = \frac { 1 } { 4 \pi ^ { 2 } } \int _ { S ^ { 1 } } \int _ { - \infty } ^ { \infty } \frac { \hat { f } _ { p } ( \alpha , p ) } { \alpha . x - p } d \alpha d p, \end{equation}

\begin{equation*} \hat { f } _ { p } : = \frac { \partial \hat { f } } { \partial p }. \end{equation*}

It is non-local: one requires the knowledge of $\hat { f } ( \alpha , p )$ for all $p$ in order to calculate $f ( x )$.

By local tomographic data one means the values of $\hat { f } ( \alpha , p )$ for those $\alpha$ and $p$ which satisfy the condition $| \alpha . x _ { 0 } - p | < \delta$, where $x _ { 0 }$ is a fixed "point of interest" and $\delta > 0$ is a small number. Geometrically, local tomographic data are the values of the integrals over the straight lines which intersect the disc centred at $x _ { 0 }$ with radius $\delta$. In many applications only local tomographic data are available, while in medical imaging one wants to minimize the radiation dose of a patient and to use only the local tomographic data for diagnostics.

Therefore, the basic question is: What practically useful information can one get from local tomographic data?

As mentioned above, one cannot find $f ( x _ { 0 } )$ from local tomographic data.

What does one mean by "practically useful information" ?

By this one means the location of discontinuity curves (surfaces, if $n > 2$) of $f ( x )$ and the sizes of the jumps of $f ( x )$ across these surfaces.

Probably the first empirically found method for finding discontinuities of $f ( x )$ from local tomographic data was suggested in [a1], where the function

\begin{equation} \tag{a2} f _ { s \text{l}t } ( x ) : = - \frac { 1 } { 4 \pi } \int _ { S ^ { 1 } } \hat { f } _ { p p } ( \alpha , \alpha \cdot x ) d \alpha, \end{equation}

which is the standard local tomography function, was proposed. To calculate $f ( x )$ one needs to know only the local tomography data corresponding to the point $x$. It is proved that $f ( x )$ and $f _ { s \text{l} t } ( x )$ have the same discontinuities (but different sizes of the jumps across the discontinuity curves) [a11]. For various aspects of local tomography, see the references. See also Tomography.

In [a7], [a8], [a9], a large family of local tomography functions was proposed. The basic idea here is to establish a relation between hypo-elliptic pseudo-differential operators and a class of linear operators acting on the functions $\hat { f } ( \alpha , p )$.

Let a pseudo-differential operator be defined by the formula $B f =\mathcal{ F} ^ { - 1 } [ b ( x , t , \alpha ) \tilde { f } ]$, where $\widetilde { f } : = \mathcal F f$ is the Fourier transform, $\tilde { f } ( \xi ) = \int _ { \mathbf{R} ^ { n } } f ( x ) e ^ { i \xi x } d x$, and $b ( x , t , \alpha )$ is a smooth function, which is called the symbol of $B$, $\alpha : = \xi / | \xi |$, $t = | \xi |$. If the symbol is hypo-elliptic, that is, $c _ { 1 } | \xi | ^ { m _ { 1 } } \leq | b | \leq c _ { 2 } | \xi | ^ { m _ { 2 } }$, $| \xi | > R$, $x \in D$, $c_1$ and $c_2$ are positive constants, then $\operatorname{WF} ( B f ) = \operatorname{WF} ( f )$, where $ \operatorname {WF} ( f )$ is the wave front of $f$. Thus, the singularities of $B f$ and $f$ are the same. One can prove [a9] the formula $B f = R ^ { * } ( a _ { \text{e} } \otimes \widehat { f } ) : = A \widehat { f }$, where $R ^ { * } g : = \int _ { S ^ { n - 1 }} g ( \alpha , \alpha . x ) d \alpha $, where $R ^ { * }$ is the adjoint to the Radon operator $R$ (cf. also Radon transform), and $a \otimes \hat { f } : = \int _ { - \infty } ^ { \infty } a ( x , \alpha , p - q ) \hat { f } ( q ) d q$ is the convolution operator, with distributional kernel $a ( x , \alpha , p - q )$ defined by

\begin{equation*} a ( x , \alpha , p ) : = \frac { 1 } { ( 2 \pi ) ^ { n } } \int _ { 0 } ^ { \infty } t ^ { n - 1 } e ^ { - i t p } b ( x , t , \alpha ) d t, \end{equation*}

and with

\begin{equation*} a _ { e } ( x , \alpha , p ) : = \frac { a ( x , \alpha , p ) + a ( x , - \alpha , - p ) } { 2 } \end{equation*}

the even part of $a ( x , \alpha , p )$.

An operator $A$ is called a local tomography operator if and only if $\operatorname { supp } a _ { \operatorname {e} } ( x , \alpha , p ) \subset [ - \delta , \delta ]$ uniformly with respect to $x \in D$ and $\alpha \in S ^ { n - 1 }$.

A necessary and sufficient condition for $A$ to be a local tomography operator is given in [a9]: The kernel $b ( x , t , \alpha ) t _ { + } ^ { n - 1 } + b ( x , - t , - \alpha ) t ^ { n - 1 }_-$ is an entire function of $t$ of exponential type $\leq \delta$ uniformly with respect to $x \in D$ and $\alpha \in S ^ { n - 1 }$.

References

[a1] E. Vainberg, I. Kazak, V. Kurczaev, "Reconstruction of the internal $3$D structure of objects based on real-time integral projections" Soviet J. Nondestr. Test. , 17 (1981) pp. 415–423 (In Russian)
[a2] A. Faridani, E. Ritman, K. Smith, "Local tomography" SIAM J. Appl. Math. , 52 (1992) pp. 459–484 MR1174054 MR1154783 Zbl 0777.65076 Zbl 0758.65081
[a3] A. Katsevich, "Local tomography for the generalized Radon transform" SIAM J. Appl. Math. , 57 : 4 (1997) pp. 1128–1162 MR1462054 Zbl 0897.65084
[a4] A. Katsevich, "Local tomography for the limited-angle problem" J. Math. Anal. Appl. , 213 (1997) pp. 160–182 MR1469368 Zbl 0894.65065
[a5] A. Katsevich, "Local tomography with nonsmooth attenuation II" A.G. Ramm (ed.) , Inverse Problems, Tomography, and Image Processing , Plenum (1998) pp. 73–86 MR1625256 Zbl 0981.44002
[a6] A. Katsevich, "Local tomography with nonsmooth attenuation" Trans. Amer. Math. Soc. , 351 (1999) pp. 1947–1974 MR1466950 Zbl 0924.35207
[a7] A.G. Ramm, "Optimal local tomography formulas" PanAmer. Math. J. , 4 : 4 (1994) pp. 125–127 MR1310327 Zbl 0847.44001
[a8] A.G. Ramm, "Finding discontinuities from tomographic data" Proc. Amer. Math. Soc. , 123 : 8 (1995) pp. 2499–2505 MR1273517 Zbl 0830.44001
[a9] A.G. Ramm, "Necessary and sufficient conditions for a PDO to be a local tomography operator" C.R. Acad. Sci. Paris , 332 : 7 (1996) pp. 613–618 MR1386462 Zbl 0848.35147
[a10] A.G. Ramm, "New methods for finding discontinuities of functions from local tomographic data" J. Inverse Ill-Posed Probl. , 5 : 2 (1997) pp. 165–174 MR1452016 Zbl 0881.44003
[a11] A.G. Ramm, A.I. Katsevich, "The Radon transform and local tomography" , CRC (1996) MR1384070 Zbl 0863.44001
How to Cite This Entry:
Local tomography. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_tomography&oldid=24499
This article was adapted from an original article by A.G. Ramm (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article