A Parametric Level-Set Method for Partially Discrete Tomography

Abstract

This paper introduces a parametric level-set method for tomographic reconstruction of partially discrete images. Such images consist of a continuously varying background and an anomaly with a constant (known) grey-value. We represent the geometry of the anomaly using a level-set function, which we represent using radial basis functions. We pose the reconstruction problem as a bi-level optimization problem in terms of the background and coefficients for the level-set function. To constrain the background reconstruction we impose smoothness through Tikhonov regularization. The bi-level optimization problem is solved in an alternating fashion; in each iteration we first reconstruct the background and consequently update the level-set function. We test our method on numerical phantoms and show that we can successfully reconstruct the geometry of the anomaly, even from limited data. On these phantoms, our method outperforms Total Variation reconstruction, DART and P-DART.

Shape from projections via differentiable forward projector for computed tomography

“…This includes (Elangovan and Whitaker, 2001;Whitaker and Elangovan, 2002;Alvino and Yezzi, 2004) that are based on the Mumford-Shah model (Mumford and Shah, 1989) where boundaries are represented using level-sets (Osher and Fedkiw, 2004). Recently, the parametric level-set method (Aghasi et al, 2011) has been used for tomographic segmentation in (Kadu et al, 2018;Eliasof et al, 2020) where level-sets are represented as an aggregation of radial basis functions. Although the parametric levelset method has fewer unknown variables, its forward projection still depends on a regular grid.…”

DALM, Deformable Attenuation-Labeled Mesh for Tomographic Reconstruction and Segmentation

“…The stated deformable models required dense and regular discretization. To reduce the unknown variables, a parametric level set method was proposed in [19] and used in [20]. Those methods required fewer parameters to represent a level set, which reduced the unknown variables and allows to use efficient second-order optimization methods.…”

Multimodal 3D Shape Reconstruction under Calibration Uncertainty Using Parametric Level Set Methods

“…We leverage a piecewise polynomial Heaviside function as shown in Fig. 3a to improve the condition number of the Hessian and increase the sparsity of the Jacobian [22]. The function ψ : R + → [0, 1] is one of the Wendland RBF functions that appear in Fig.…”

“…As shown in Fig. 3a and mentioned in [22], we wish to make the derivative of this function (the delta function) as flat as possible to render our Gauss-Newton Hessian better conditioned. We define our Heaviside function to be a smoothed linear transition from 0 to 1 using a piecewise polynomial function.…”