Robust and stable region-of-interest tomographic reconstruction using a robust width prior

Figure 1.  Illustrations of recoverable regions for truncated projection data: (a) initial DBP methods [25,7,35] require at least one projection view in which the complete object is covered, (b) interior reconstruction is possible given a known subregion [20,34,30,17] and (c) no assumptions are made other than that the shape of the ROI is convex and approximate sparsity within a ridgelet domain (this paper). The gray dashed line indicates the measured area on the detector array for one particular source angle

Figure 2.  Illustration of ROI tomographic reconstruction problem. (a) In ROI tomography, only projections intersecting the Region-of-Interest $ S $ are known. (b) In the Radon-transform domain, the region $ P(S) $ corresponding to $ S $ is a strip-like domain

Figure 3.  Schematic illustration of the solution space for $ y $, given a current estimate $ f $, as intersection of the balls $ \left\Vert My-y_{0}\right\Vert ^{2}\leq \alpha $ and $ \left\Vert y-Wf\right\Vert ^{2}\leq \beta $, as well as solutions favored by the sparsity prior $ \left\Vert {y}\right\Vert _{\sharp} $ (see Sec. 3). Data consistent solutions may have a non-zero data fidelity, while data fidelity solutions are in general not consistent. We control the reconstruction error by combining fidelity and consistency constraints with the additional sparsity assumption

Figure 4.  Commutative diagram of the measurement operator $ \Phi $ and the restricted measurement operator $ \Phi ' $ (see Theorem 3.5). The following relationship holds: $ P_{\tilde{\mathcal{H}}'}\Phi = \Phi 'P_{\mathcal{H}'} $.

Figure 5.  Full LSCG reconstruction of the fan-beam data sets: (a) Preclinical - lungs, (b) Preclinical - abdomen. Images are cropped for visualization purposes

Figure 6.  PSNR results for 2D fan-beam ROI reconstruction with increasing radius using (a) lungs and (b) adbomen. The PSNR is calculated inside the ROI

Figure 7.  ROI reconstruction results for a fixed radius of $ 64 $ pixels (or 3.2 mm). (a), (b): SIRA with different values of $ \alpha $, $ \beta $ and $ u = \Vert (I-\tilde{W}\tilde{W}^{+})y_{0}\Vert _{2}^{2} $; (c): LSCG; (d): CS-TV, (e): CS-ridgelet, (f) Full view LSCG reconstruction (used for computing PSNR)

Figure 8.  (a)-(c): Reconstruction results for different ROI radii using SIRA with fixed parameters $ \alpha = 0 $, $ \beta = 0.25\left\Vert \left(I-\tilde{W}\tilde{W}^{+}\right)y_{0}\right\Vert _{2}^{2}. $ (d)-(f): Reconstruction results using LSCG for same ROI radii