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Using filtered backprojection (FBP) and an analytic continuation approach, we prove that exact interior reconstruction is possible and unique from truncated limited-angle projection data, if we assume a prior knowledge on a subregion or subvolume within an object to be reconstructed. Our results show that (i) the interior region-of-interest (ROI) problem and interior volume-of-interest (VOI) problem can be exactly reconstructed from a limited-angle scan of the ROI/VOI and a 180 degree PI-scan of the subregion or subvolume and (ii) the whole object function can be exactly reconstructed from nontruncated projections from a limited-angle scan. These results improve the classical theory of Hamaker et al. (1980).

The
importance of performing exact image reconstruction from the minimum amount of
data has been recognized for a long time. The first landmark achievement is the
well-known fan-beam half-scan formula [

Using the analytic continuation technique, here we further extend our exact interior reconstruction results to the case of a truncated limited-angle scan. The paper is organized as follows. In the next section, we summarize the relevant notations and key theorem. In the third section, we prove our theorem in the filtering backprojection (FBP) framework. In the fourth section, we will discuss relevant ideas and conclude the paper.

The basic setting of our previous work is
cone-beam scanning along a general smooth trajectory

Basic setting for exact 3D interior reconstruction from truncated limited-angle datasets.

Assume that there are
three points

Let us
remark on the conditions for Theorem

Based on Katsevich's work [

For a fixed point

Variable change from

From (

Complex coordinate system for the analytic continuity.

Now we return to (

Back to (

Because the exact interior
reconstruction is unique from truncated limited-angle data according to Theorem

As an inspiring case, let us
consider the 2D ROI-focused scan illustrated in Figure

(a) Illustration of the subregion/volume half-scan ROI problem; (b) the 1D coordinate system along the X-ray path indicated in (a).

Furthermore, let us revisit the so-called nontruncated
limited-angle scanning problem. For clarity, we only consider the 2D case as
illustrated in Figure

(a) Illustration of nontruncated limited-angle scanning problem; (b) the 1D coordinate system along the X-ray path indicated in (a).

Although our work has been done within the X-ray
CT framework, our results can be directly applied to other tomographic
modalities that share similar imaging models such as MRI, ultrasound imaging,
PET, and SPECT. By similarity between imaging models, we underline that the
exponential Radon transform is a particular attractive area since ageneralized Hilbert transform theory has been reported for exact reconstruction from
transversely truncated data [

In
conclusion, we have proved that the exact interior reconstruction is theoretically
solvable. Theorem

This work is partially supported by NIH/NIBIB Grants EB002667, EB004287, and EB007288.