View source for Local tomography

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:

<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>

<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>

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" />.

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.

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.

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.

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

<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>

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]].

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" />.

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

<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>

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>

the even part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130100/l13010064.png" />.

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" />.

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" />.

====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</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</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</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</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</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</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</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</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</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)</TD></TR></table>