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Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the <formula formulatype="inline"><tex Notation="TeX">$\ell_{1}$</tex></formula> norm of the image gradient magnitude, and reducing the <formula formulatype="inline"><tex Notation="TeX">$\ell_{1}$</tex></formula> norm is known to encourage sparsity in its argument. Recently, there has been interest in employing nonconvex <formula formulatype="inline"><tex Notation="TeX">$\ell_{p}$</tex></formula> quasinorms with <formula formulatype="inline"><tex Notation="TeX">$0<p<1$</tex></formula> for sparsity exploiting image reconstruction, which is potentially more effective than <formula formulatype="inline"><tex Notation="TeX">$\ell_{1}$</tex></formula> because nonconvex <formula formulatype="inline"><tex Notation="TeX">$\ell_{p}$</tex></formula> is closer to <formula formulatype="inline"><tex Notation="TeX">$\ell_{0}$</tex></formula>—a direct measure of sparsity. This paper develops algorithms for constrained minimization of the total <formula formulatype="inline"><tex Notation="TeX">$p$</tex></formula>-variation <formula formulatype="inline"><tex Notation="TeX">$({\rm T}p{\rm V})$</tex></formula>, <formula formulatype="inline"> <tex Notation="TeX">$\ell_{p}$</tex></formula> of the image gradient. Use of the algorithms is illustrated in the context of breast CT—an imaging modality that is still in the research phase and for which constraints on X-ray dose are extremely tight. The <formula formulatype="inline"><tex Notation="TeX">${\rm T}p{\rm V}$</tex></formula>-based image reconstruction algorithms are demonstrated on computer simulated data for exploiting gradient magnitude sparsity to reduce the projection view angle sampling. The proposed algorithms are applied to projection data from a realistic breast CT simulation, where the total X-ray dose is equivalent to two-view digital mammography. Following the simulation survey, the algorithms are then demonstrated on a clinical breast CT data set.

ieee journal of translational engineering in health and medicine

Constrained <formula formulatype="inline"><tex Notation="TeX">${\rm T}p{\rm V}$</tex> </formula> Minimization for Enhanced Exploitation of Gradient Sparsity: Application to CT Image Reconstruction

Constrained ${\rm T}p{\rm V}$ Minimization for Enhanced Exploitation of Gradient Sparsity: Application to CT Image Reconstruction