Parametric Level-sets Enhanced To Improve Reconstruction (PaLEnTIR)

TL;DR

PaLEnTIR tackles the challenge of reconstructing piecewise constant images with multiple unknown contrasts using a single level-set. It introduces a smooth transition function and anisotropic basis functions to extend PaLS expressiveness while fixing centers and bounding coefficients, which dramatically improves Jacobian conditioning and accelerates optimization. Across linear and nonlinear inverse problems—including sparse-view and limited-angle X-ray CT and DOT—the method achieves high-fidelity reconstructions with far fewer parameters than pixel-based approaches. The results demonstrate robust multi-contrast recovery, faster convergence, and competitive or superior accuracy, highlighting PaLEnTIR’s practical impact for data-limited imaging tasks.

Abstract

We introduce PaLEnTIR, a significantly enhanced parametric level-set (PaLS) method addressing the restoration and reconstruction of piecewise constant objects. Our key contribution involves a unique PaLS formulation utilizing a single level-set function to restore scenes containing multi-contrast piecewise-constant objects without requiring knowledge of the number of objects or their contrasts. Unlike standard PaLS methods employing radial basis functions (RBFs), our model integrates anisotropic basis functions (ABFs), thereby expanding its capacity to represent a wider class of shapes. Furthermore, PaLEnTIR improves the conditioning of the Jacobian matrix, required as part of the parameter identification process, and consequently accelerates optimization methods. We validate PaLEnTIR's efficacy through diverse experiments encompassing sparse and limited angle of view X-ray computed tomography (2D and 3D), nonlinear diffuse optical tomography (DOT), denoising, and deconvolution tasks using both real and simulated data sets.

Figures (9)

• Figure 1: (a) The plot of the approximate Heaviside function $\hat{H}(x)$ for the zero level-set ($c=0$). (b) The plot of the new transition function $T(x)$ for zero level-set.
• Figure 2: (a) Image of size $256 \times 256$ pixels comprised of five piecewise constant objects with different shapes and contrasts on a zero-contrast background. (b) The image corrupted with additive Gaussian noise (SNR: 22dB). (c) PaLEnTIR recovery of the image with fixed contrast limits. (d) PaLEnTIR recovery using manually adjusted vectors (e) $\mathbf{C}_{H}$ and (f) $\mathbf{C}_{L}$. (g) PaLEnTIR recovery with the parameterized contrast limits, $\mathbf{p}_c \in \mathbb{R}^{2N}$, included in the estimation problem. Estimated parameterized contrast limits (h) $C_{H}(\mathbf{r})$ and (i) $C_{L}(\mathbf{r})$.
• Figure 3: The impact of the parameters on the $c$-level ellipsoidal representation of the new basis function. (a) shows the impact of the stretching parameter $\beta$ and (b) shows the impact of the sliding parameter $\gamma$.
• Figure 4: (a) Reference image of a carved cheese specimen computed using a high-resolution filtered back-projection (FBP) from the 360-projection sinogram. (b) PaLEnTIR recovery from the tomographic X-ray data using only 15 projections. (c) RBF PaLS recovery using the same data as PaLEnTIR.
• Figure 5: (a) Condition number of the residual versus the radius of circular cross sections (corresponding to appropriate level sets) for the two models. The black line (Max) and blue line (Min) represent RBF PaLS; the red line represents PaLEnTIR. (b) Condition number of the residual versus TREGS iterations, and (c) objective function plots over TREGS iterations of the 2D sparse view tomographic X-ray experiment. Blue line is for RBF PaLS and red line is for the new PaLEnTIR. The resultant reconstructions are shown in Figures \ref{['fig:cond_cheese_palentir']} and \ref{['fig:cond_cheese_pals']}.