Computational imaging can reduce the acquisition burden when the system is designed around task-relevant information rather than complete image recovery. We investigate this framework for sparse-view planar X-ray tomography (XCT) of layered objects, using measurement of the axial termination gap (ATG) of back-drilled printed-circuit-board (PCB) vias as a representative task. Instead of using 3D reconstruction, the proposed method estimates ATG directly from polychromatic projection data through a physics-based sensitivity model. A gated hierarchical estimator first fits ATG together with radiometric perturbations and selectively introduces global geometry and specimen-warp models only when a reduced chi-squared adequacy test and Bayesian information criterion justify the added degrees of freedom. The same low-dimensional perturbation model is incorporated into marginalized Fisher-information analyses, yielding task-specific conditional and calibration-aware Cram'er-Rao bounds. Numerical experiments show that, under a fixed total photon budget, task-driven metrology changes little over 12-64 views, while reconstruction-driven metrology is strongly view-dependent with much larger errors. The direct estimator also follows the expected photon-limited scaling. These results demonstrate a computational sensing strategy in which XCT acquisition, physical modeling, and inference are evaluated according to the quantitative information required by the task rather than for general-purpose volumetric image fidelity.

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