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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" />.
 
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.
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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?
 
Therefore, the basic question is: What practically useful information can one get from local tomographic data?
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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.
 
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" ?
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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.
 
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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====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</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>
+
<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>

Revision as of 16:59, 15 April 2012

Let be a compactly supported piecewise-smooth function, if , a bounded domain, and let be its Radon transform, where is the straight line parametrized by the unit vector and a scalar . The inversion formula which reconstructs from the knowledge of for all and all , where is the unit circle in , is known to be:

(a1)

It is non-local: one requires the knowledge of for all in order to calculate .

By local tomographic data one means the values of for those and which satisfy the condition , where is a fixed "point of interest" and is a small number. Geometrically, local tomographic data are the values of the integrals over the straight lines which intersect the disc centred at with radius . 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 from local tomographic data.

What does one mean by "practically useful information" ?

By this one means the location of discontinuity curves (surfaces, if ) of and the sizes of the jumps of across these surfaces.

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

(a2)

which is the standard local tomography function, was proposed. To calculate one needs to know only the local tomography data corresponding to the point . It is proved that and 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 .

Let a pseudo-differential operator be defined by the formula , where is the Fourier transform, , and is a smooth function, which is called the symbol of , , . If the symbol is hypo-elliptic, that is, , , , and are positive constants, then , where is the wave front of . Thus, the singularities of and are the same. One can prove [a9] the formula , where , where is the adjoint to the Radon operator (cf. also Radon transform), and is the convolution operator, with distributional kernel defined by

and with

the even part of .

An operator is called a local tomography operator if and only if uniformly with respect to and .

A necessary and sufficient condition for to be a local tomography operator is given in [a9]: The kernel is an entire function of of exponential type uniformly with respect to and .

References

[a1] E. Vainberg, I. Kazak, V. Kurczaev, "Reconstruction of the internal 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