View source for Pseudo-local tomography

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301401.png" /> be a piecewise smooth, compactly supported function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301402.png" />. The [[Radon transform|Radon transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301403.png" /> is defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301404.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301405.png" /> is a straight line parametrized by a unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301407.png" /> is the unit circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p1301409.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014010.png" />. By local tomographic data one means the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014013.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014015.png" /> is a given point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014016.png" /> is a small number. Thus, local tomographic data are the line integrals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014017.png" /> for the lines intersecting the  "region of interest" , the disc centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014018.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014019.png" /> (cf. also [[Local tomography|Local tomography]]; [[Tomography|Tomography]]).

It is not possible, in general, to find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014020.png" /> from the local tomographic data [[#References|[a2]]]. What practically useful information about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014021.png" /> can one get from these data? Information, very useful practically, is the location of discontinuity curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014022.png" /> and the sizes of the jumps of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014023.png" /> across these curves.

Pseudo-local tomography solves the problem of finding the above information from the local tomographic data.

This is done by computing the pseudo-local tomography function, introduced in [[#References|[a2]]]:

<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/p/p130/p130140/p13014024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014025.png" />. The inversion formula reads:

<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/p/p130/p130140/p13014026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

so that (a1) is based on the following idea: Keep a small neighbourhood of the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014027.png" /> in (a1) and neglect the rest of the [[Cauchy integral|Cauchy integral]] in (a1).

By definition, one needs only the local tomographic data to calculate the pseudo-local tomography function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014028.png" />.

The basic result is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014029.png" /> is a [[Continuous function|continuous function]] [[#References|[a2]]], [[#References|[a1]]].

Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014031.png" /> have the same discontinuity curves and the same sizes of the jumps across discontinuities.

It is also proved in [[#References|[a2]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014033.png" /> is an open set, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014034.png" /> has the following properties:

<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/p/p130/p130140/p13014035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

and the convergence in (a3) is uniform on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014038.png" /> is a smooth discontinuity curve of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014039.png" />, then

<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/p/p130/p130140/p13014040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014041.png" /> are the limiting values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014042.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014043.png" /> from different sides of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014044.png" /> along a path non-tangential to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014045.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014046.png" /> is a unit vector normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014047.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014048.png" />, then for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014050.png" />, one has

<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/p/p130/p130140/p13014051.png" /></td> </tr></table>

where

<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/p/p130/p130140/p13014052.png" /></td> </tr></table>

and

<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/p/p130/p130140/p13014053.png" /></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/p/p130/p130140/p13014054.png" /></td> </tr></table>

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014055.png" />, is monotonically decreasing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014057.png" />,

<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/p/p130/p130140/p13014058.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014059.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014060.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014061.png" />th order derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014062.png" /> exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014063.png" />, some of them being discontinuous across <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014064.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014065.png" /> is piecewise-smooth in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014067.png" />.

Other properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014068.png" /> can be found in [[#References|[a2]]], which also contains a general method for constructing a family of pseudo-local tomography functions, that is, functions which are computable from local tomographic data and having the same discontinuities and the same sizes of the jumps as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130140/p13014069.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Ramm,   A. Katsevich,   "Pseudolocal tomography"  ''SIAM J. Appl. Math.'' , '''56''' :  1  (1996)  pp. 167–191</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Ramm,   A. Katsevich,   "The Radon transform and local tomography" , CRC  (1996)</TD></TR></table>