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Recently, in the compressed sensing framework we found that a two-dimensional interior region-of-interest (ROI) can be exactly reconstructed via the total variation minimization if the ROI is piecewise constant (Yu and Wang, 2009). Here we present a general theorem charactering a minimization property for a piecewise constant function defined on a domain in any dimension. Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009).

While in general an interior region-of-interest (ROI) cannot be uniquely reconstructed from projection data only associated with lines through the ROI [

For piecewise constant or piecewise smooth functions, it is natural to use the space of functions of bounded variation [

We that assume

Let

After an integration by parts and some rearrangement, we have, for any

For the opposite inequality, we first consider the case where

For

As an example of (

Some comments on the name of our approach “CS-based interior tomography” are in order. In the strict sense, compressed sensing refers to situations where the sampling scheme is built (often with random techniques) to achieve specific properties for satisfactory recovery of an underlying signal, rather than imposed by a specific detector arrangement as in limited data tomography. However, in a broad sense, compressed sensing can be interpreted as achieving better reconstruction from less data relative to the common practice. Hence, while the current name is not far off, an alternative phrase for our approach can be “total variation minimization-based interior tomography.”

In conclusion, we have extended the total variation minimization property of a piecewise function from two-dimensions to any dimensionality in the Sobolev space, which can be used for exact reconstruction of any piecewise function on an ROI by minimizing its total variation under the constraint of the truncated projection data through the ROI. Previously, we implemented an alternating iterative reconstruction algorithm to minimize the total variation, which is time-consuming and needs improvement. Under the guidance of the theoretical finding presented here, we are working to develop a multidimensional ROI reconstruction algorithm for better performance. Clearly, major efforts are still needed in this direction.

This work is partially supported by NIH/NIBIB Grants (EB002667, EB004287, EB007288) and a Grant from Toshiba Medical Research Institute USA, Inc.