Local tomography

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