Where Does the Uncertainty Come From? Four-Channel Information Decomposition for Active 3D Capture

Active 3D reconstruction usually picks the next viewpoint by a scalar expected information gain, which says how much a view teaches but not which part of the scene it constrains or whether that information belongs to the persistent scene. We introduce PRISM, a task-aware next-best-view method for Gaussian Splatting that decomposes a candidate view's persistent information gain into geometry and opacity, base appearance, and view-dependent appearance. Under a local linear-Gaussian approximation, a Shapley allocation yields non-negative attributions that sum exactly to the nuisance-marginalized gain, while transient content enters as a separate conditional-information term that cannot inflate the persistent shares. Task weights recombine the attributions for different objectives over one candidate pool, and the same predictive covariances turn the residuals of arriving images into nuisance evidence for later rounds. On surface reconstruction, low-light capture, and reflective scenes, each task's utility beats total-information, Fisher-information, and mutual-information planners on its own metric, most inside the regions the task cares about; on a fixed drone route it slows where its evidence lies; and in a cluttered scene, nuisance-aware scoring cuts the share of selected rays on moving objects from 0.31 to 0.12.