What if every meter of robot motion were optimized for reconstruction?

Excited to share TRACE, our new work on active 3D reconstruction.

Most active methods choose one next-best view at a time, then plan a path to reach it. TRACE instead optimizes the full trajectory for the information gathered along the way.

Paper: https://arxiv.org/abs/2608.02304

Code: https://github.com/spikelab-jhu/trace-active-reconstruction…

Video: https://github.com/spikelab-jhu/trace-active-reconstruction#-video…

TRACE: Ergodic Trajectory Optimization for Active Scene Reconstruction

Source: https://arxiv.org/html/2608.02304 Ziyue Zheng1\equalcontrib, Linli Shi1\equalcontrib, Bingkun He1, Wen Jiang2, Ziyun Wang1\corresponding

Abstract

Existing active reconstruction systems with Gaussian-splatting maps select observations greedily, optimizing a single next-best-view (NBV) at each step and connecting the chosen views by short-horizon path planning. This greedy decoupling disregards the global structure of scene information, producing inefficient trajectories that waste sensing capacity in transit between selected views. In this work, we study active reconstruction as an ergodic coverage problem: the time-averaged spatial statistics of the sensor trajectory should match a target information distribution induced by the current map. Our approach derives this target distribution online from uncertainty and visibility, and calculates ergodic trajectories via a kernel-ergodic horizon planner with gradient flow and footprint depletion, closing the loop between mapping and trajectory optimization. We thoroughly evaluate TRACE on the Replica dataset against the Next-Best-View (NBV) baselines, improving PSNR by 1.5 dB.Code:https://github.com/spikelab-jhu/trace-active-reconstruction.

Figure 1:From discrete viewpoints to ergodic active reconstruction.Next-Best-View (NBV) planners discard information between sparse viewpoints and ignore robot dynamics, producing jerky, infeasible trajectories. Our method instead constructs an information map from streaming RGB-D observations and plans an ergodic search path that continuously gathers information while respecting the robot’s dynamic. This yields smoother trajectories, denser coverage, and 1.5 dB PSNR gain.## 1Introduction

Active 3D reconstruction recovers the geometry of an unknown scene by moving a sensor along a sequence of informative viewpoints, which has long been a core problem in robotic perception(Zenget al.2020; Isleret al.2016; Bircheret al.2016). In such tasks, the robot incrementally builds a high-fidelity model of the scene from its own streaming observations. Recent advances in differentiable rendering, particularly 3D Gaussian Splatting (GS)(Kerblet al.2023)and its surface-aligned variant 2DGS(Huanget al.2024), have raised the achievable reconstruction quality dramatically. The bottleneck has shifted from the map representation to the planner(Jinet al.2025; Chenet al.2025; Liet al.2025): under finite budgets of time, energy, and motion, the agent must decide how to spend each meter of motion across an unknown scene. Therefore, the reconstruction quality depends on both map representations and how the agent chooses to distribute its limited motion budget across the scene.

A common paradigm of active reconstruction with Gaussian-splatting maps is to plan view-by-view: at each step, the planner scores candidate viewpoints by an information-gain proxy, commits to the highest-scoring pose, and connects it to the current location by short-horizon path planning(Jinet al.2025; Chenet al.2025). Recent work introduces a hierarchy over a topological subgraph, but each step still commits to a single node and routes to it via shortest-path planning(Liet al.2025). However, this formulation decouples viewpoint selection from trajectory planning, and the motion between successive views serves only to move the sensor toward the next target rather than to acquire additional information. The decoupling hasthree main issues. First, viewpoint scoring isshort-term: each decision commits to the locally most informative pose without anticipating where the sensor must subsequently travel. Second, the behaviors between selected views do not contribute toefficient information gathering. The resulting trajectories are jagged, redundant, and biased toward isolated high-information viewpoints rather than uniform spatial coverage of the scene. Third, the trajectory formed by the sequence of planned views ignoresreal-world constraintssuch as controllability, actuator limits, and energy efficiency, often producing dynamically infeasible trajectories. In particular, discrete search algorithms used in Next-Best-View (NBV) planning typically optimize over viewpoints rather than executable motions, and therefore cannot directly account for the robot’s continuous-time dynamics.

In this work, we aim to address these issues by proposing the first active 3D reconstruction method that plans continuous and control-feasible trajectories using ergodic search. Instead of modeling the problem as a planning over a set of next poses, we directly optimize a trajectory over the constructed information distribution based on the current state of mapping and exploration. Specifically, our planner optimizes the path so that its time-averaged spatial statistics match a target information distribution: the time the trajectory spends in any region is proportional to the information density there. Instantiating this formulation on a live Gaussian-splatting map raises three technical challenges. First, classical ergodic search assumes a target distribution specified a priori, whereas in active reconstruction the information density is implicit in the map state, encoded in per-Gaussian confidence and evolving occupancy, and must be re-derived online as observations are integrated. Second, the informative regions are scene surfaces that the sensor cannot occupy: the trajectory must remain in free space while its viewing footprint, rather than its position, covers the high-information surfaces. Third, the marginal value of observing a surface diminishes under repeated coverage within a horizon, a time-varying effect that a static target distribution cannot capture.

The target information distribution is derived online from per-Gaussian uncertainty and visibility, while the trajectory is computed by a kernel-ergodic horizon planner with gradient-flow updates and a footprint-depletion mechanism that suppresses re-coverage. Our contributions are summarized as:

• •To the best of our knowledge, we are the first to formulate active reconstruction with Gaussian-splatting maps as a trajectory-level ergodic coverage problem, replacing greedy viewpoint selection with continuous optimization over dynamically feasible trajectories.
• •We adapt kernel-based ergodic search to surface-based reconstruction by diffusing surface information into traversable free space and introducing footprint-aware depletion and gaze objectives for efficient coverage and camera orientation.
• •Under the same mapper, budget, and evaluation protocol, TRACE improves over the strongest NBV baseline by1.51.5dB PSNR across eight Replica scenes and supports direct trajectory execution on quadruped and manipulator (supplementary) without an intermediate path planner.

2Related Work

2.1Active 3D Reconstruction

Active 3D reconstruction has been studied for decades under thenext-best-view(NBV) formulation(Connolly1985; Isleret al.2016; Bircheret al.2016; Delmericoet al.2018): at each step, the agent selects the viewpoint expected to maximize a chosen information-gain criterion. Sampling-based informative path planning extends NBV to entire trajectories(Hollinger and Sukhatme2014)and long-horizon tree search(Bestet al.2019), while multi-stage aerial pipelines couple it to multi-view-stereo reconstruction(Heppet al.2018). Frontier-based exploration(Yamauchi1997)and occupancy mapping(Hornunget al.2013)provide complementary geometric drivers, and SCONE(Guédonet al.2022)optimizes surface coverage via Monte Carlo volumetric integration.

The emergence of differentiable rendering—NeRF(Mildenhallet al.2020)and subsequent radiance-field variants(Barronet al.2022; Mülleret al.2022; Fridovich-Keilet al.2022; Chenet al.2022)—extended active reconstruction to learned neural map representations. NARUTO(Fenget al.2024), ActiveNeRF(Panet al.2022), ActiveImplicit(Yanet al.2023a), and NeU-NBV(Jinet al.2023)score viewpoints by NeRF uncertainty or implicit-occupancy entropy. FisherRF(Jianget al.2024)uses Fisher information over radiance-field parameters; NVF(Xueet al.2024)composites position-based uncertainty into camera-ray uncertainty; Active Neural Mapping(Yanet al.2023b)measures neural variability under weight perturbation; andSiminget al.(2024)maximize mutual information through a generative NeRF model. GenNBV(Chenet al.2024)trains a generalizable RL policy over a 5-DoF action space, while ACE-NBV(Zhanget al.2023)selects views that improve grasp quality.

Recent Gaussian-splatting maps(Kerblet al.2023; Huanget al.2024)have prompted dedicated active-GS planners. ActiveGS(Jinet al.2025), ActiveGAMER(Chenet al.2025), and ActiveSplat(Liet al.2025)score viewpoints by per-primitive confidence, rendering-based information gain, or frontier cues; ActiveSplat further introduces a local-vs-global hierarchy over a topological subgraph. GauSS-MI(Xieet al.2025)introduces Shannon mutual information over Gaussian-splat appearance, POp-GS(Wilsonet al.2025)reframes information gain through P-Optimality, and Active3D(Liet al.2026)fuses implicit and explicit representations with hierarchical uncertainty quantification. These methods share a common structure: at each step, the planner commits to a best viewpoint and connects it to the current pose by short-horizon path planning, producing trajectories at the viewpoint level rather than the trajectory level.

2.2Ergodic Coverage for Robotic Exploration

Ergodic search formulates information gathering as a trajectory-level coverage problem: the time-averaged spatial statistics of the trajectory should match a target information distribution(Mathew and Mezić2011; Milleret al.2016). This trajectory-centric formulation naturally accommodates finite sensing budgets by optimizing information collection over the entire motion rather than selecting isolated viewpoints. To optimize the ergodic problem, a range of methods optimize the ergodic objective via spectral multi-scale coverage with Fourier analysis(Mathew and Mezić2011), Kullback–Leibler divergence(Abrahamet al.2021), kernel-based ergodic metrics(Sunet al.2025a), flow matching(Sunet al.2025b), receding-horizon control(Mavrommatiet al.2017), potential field approaches(Ivićet al.2016), and LQR-based optimization(Miller and Murphey2013). Subsequent work has extended the framework along several axes, including time-optimal ergodic search(Donget al.2024), dynamic sensor footprints(Zhenget al.2025), probabilistic connectivity(Liu and Ren2025), manipulation(Shettyet al.2022), target localization(Mavrommatiet al.2017)and coverage in constrained domains(Ayvaliet al.2017). Despite these advances, most of these methods typically assume the target distribution is parametrically given a priori from a model of expected information density. In contrast, we derive the target distribution online from a live 2DGS map, where the information density is implicit in per-Gaussian uncertainty and evolving visibility geometry. To our knowledge, this is the first ergodic trajectory optimization formulation for active reconstruction with Gaussian-splatting maps.

3Method

We instantiate the formulation of Sec.1as a horizon-based ergodic trajectory optimization driven by online 2DGS map. Sec.3.1formalizes the problem; Sec.3.2constructs the target information distribution from the map state; Sec.3.3optimizes a kernel-ergodic trajectory by gradient descent on a footprint-depleting information field.

3.1Problem Formulation

We consider the problem of active 2D Gaussian-surfel (2DGS) mapping: an autonomous agent equipped with an RGB-D sensor incrementally builds a 2DGS mapℳ\mathcal{M}of an unknown bounded sceneΩ⊂ℝ3\Omega\subset\mathbb{R}^{3}through a sequence of self-selected viewpoints. We adopt the 2DGS representation for its surface-aligned geometry, which directly supports the mesh-quality metrics commonly used to evaluate active reconstruction(Huanget al.2024). The sensor is modeled as a view cone, rotationally symmetric about its optical axis, so its roll is immaterial and held fixed. At each replanning step (horizon)tt, the planner selects aKK-step trajectoryτt=(𝐩t1,…,𝐩tK)∈S​E​(3)K\tau_{t}=(\mathbf{p}_{t}^{1},\dots,\mathbf{p}_{t}^{K})\in SE(3)^{K}, with positions𝐱tk∈ℝ3\mathbf{x}_{t}^{k}\in\mathbb{R}^{3}and orientations determined by yaw and pitch. The trajectory is parameterized by control𝐮t∈ℝK×5\mathbf{u}_{t}\in\mathbb{R}^{K\times 5}(position velocity, yaw rate, pitch rate) cumulatively integrated under single-integrator dynamics (matching Go2’s interface). The supplementary shows a more complex case on FR3.

Figure 2:2D illustration of a single planning horizon.Left:voxel classification and our planned path.Right:the corresponding information-map heatmap (Sec.3.2). The planner jointly optimizes the trajectory, allocating sampling effort proportional to the information distribution, dwelling longer and sampling more densely in high-information regions.After executing the full horizon, the agent integrates new RGB-D observations intoℳt\mathcal{M}_{t}and re-plans with𝐮\mathbf{u}warm-started from the previous optimization. Although the voxel map is updated during execution, the planning objective remains fixed until the next horizon. Formally, each replanning step solves

τt⋆=arg⁡minτt∈𝒯t⁡Jt​(τt;ℳt,𝒱t),\tau^{\star}_{t}\;=\;\arg\min_{\tau_{t}\in\mathcal{T}_{t}}\;J_{t}(\tau_{t};\,\mathcal{M}_{t},\mathcal{V}_{t}),(1)where𝒯t\mathcal{T}_{t}is the feasible-trajectory set in horizontt(collision-free, bounded step size) andJtJ_{t}is the ergodic optimal-control objective of Sec.3.3, defined over a target information distributionϕt\phi_{t}derived online from the current map (Sec.3.2). In practice we relax the hard constraints defining𝒯t\mathcal{T}_{t}into the soft penalties of Eq. (7) and optimize the resulting objective over the controls.

3.2Information Map

Ergodic search requires a target information distribution to guide trajectory planning. In our task,ϕt\phi_{t}(information map at horizontt), evolves as new observations are integrated. To introduce the information map, we first present map representation design. We follow the hybrid map representation of ActiveGS(Jinet al.2025): a 2DGS Mapℳt\mathcal{M}_{t}is used for rendering, while a voxel map𝒱t\mathcal{V}_{t}represents the occupancy probability. GS Mapℳt\mathcal{M}_{t}is trained and updated at the end of each horizon, whereas the voxel map𝒱t\mathcal{V}_{t}is updated on the fly following the approach of OctoMap(Hornunget al.2013).

For brevity we suppress the dependence on(ℳt,𝒱t)(\mathcal{M}_{t},\mathcal{V}_{t})in the notation below; all fields are evaluated at the current map state. For each voxelvvin𝒱t\mathcal{V}_{t}, the raw information value is

ϕtraw​(v)\displaystyle\phi_{t}^{\mathrm{raw}}(v)\;=αu​1unexp​(v)+αf​1front​(v)+\displaystyle=\;\alpha_{\mathrm{u}}\,\mathbf{1}_{\mathrm{unexp}}(v)\;+\;\alpha_{\mathrm{f}}\,\mathbf{1}_{\mathrm{front}}(v)\;+\;αb​1unbuilt​(v)+β​(1−c​(v))​1low​(v),\displaystyle\alpha_{\mathrm{b}}\,\mathbf{1}_{\mathrm{unbuilt}}(v)\;+\;\beta\,\bigl(1-c(v)\bigr)\,\mathbf{1}_{\mathrm{low}}(v),(2) Here𝟏unexp​(v)\mathbf{1}_{\mathrm{unexp}}(v),𝟏front​(v)\mathbf{1}_{\mathrm{front}}(v), and𝟏unbuilt​(v)\mathbf{1}_{\mathrm{unbuilt}}(v)indicate, respectively, voxels no depth ray has traversed, free voxels bordering unexplored space, and voxels𝒱t\mathcal{V}_{t}deems occupied but holding no Gaussian ofℳt\mathcal{M}_{t}. Since𝒱t\mathcal{V}_{t}updates on execution butℳt\mathcal{M}_{t}only at horizon boundaries, known-occupied voxels may still hold no Gaussian;𝟏unbuilt\mathbf{1}_{\mathrm{unbuilt}}draws the sensor back to them and reduces holes in the GS map early in the mission. For the confidence term,𝒢v⊆ℳt\mathcal{G}_{v}\subseteq\mathcal{M}_{t}collects the Gaussians whose centers fall in voxelvv,γg∈[0,1]\gamma_{g}\in[0,1]is the per-primitive rendering confidence inherited from ActiveGS(Jinet al.2025), andc​(v)=1|𝒢v|​∑g∈𝒢vγgc(v)=\tfrac{1}{|\mathcal{G}_{v}|}\sum_{g\in\mathcal{G}_{v}}\gamma_{g}is their mean. Intuitively,γg\gamma_{g}grows with the number and angular diversity of viewpoints that have already observed Gaussiangg, so1−c​(v)1-c(v)is large precisely on voxels whose surface fit is still weakly constrained. The gate𝟏low​(v)=1\mathbf{1}_{\mathrm{low}}(v)=1on voxels that contain Gaussians but are not yetwell built(as shown in Fig.3); well-built voxels contribute no mass and stop attracting the sensor. The non-negative weightsαu,αf,αb,β\alpha_{\mathrm{u}},\alpha_{\mathrm{f}},\alpha_{\mathrm{b}},\betacontrol, respectively, exploration of unseen volume, expansion of the frontier, attention to visible-but-unmodeled surfaces, and refinement of low-confidence Gaussians.

Table 1:Quantitative comparison of rendering quality on the Replica dataset. We report PSNR, SSIM, and LPIPS across all scenes. Best, second-best, and third-best results are highlighted inred,orange, andyellow, respectively.MethodsMetricsOf0Of2Of3Of4R0R1R2H0NARUTOPSNR↑\uparrow32.9927.6027.4830.2828.4126.7729.7525.49FisherRFPSNR↑\uparrow\cellcolorrankthree36.21\cellcolorrankthree30.33\cellcolorrankthree28.63\cellcolorrankthree32.80\cellcolorrankthree28.88\cellcolorrankthree30.04\cellcolorrankthree32.26\cellcolorrankthree26.79ActiveGSPSNR↑\uparrow\cellcolorranktwo36.78\cellcolorranktwo31.68\cellcolorranktwo32.16\cellcolorranktwo34.08\cellcolorranktwo29.93\cellcolorranktwo31.74\cellcolorranktwo32.35\cellcolorranktwo32.04OursPSNR↑\uparrow\cellcolorrankone38.36\cellcolorrankone32.81\cellcolorrankone33.79\cellcolorrankone34.92\cellcolorrankone31.55\cellcolorrankone32.84\cellcolorrankone34.70\cellcolorrankone33.69ActiveGSSSIM↑\uparrow\cellcolorranktwo0.963\cellcolorranktwo0.926\cellcolorranktwo0.921\cellcolorranktwo0.932\cellcolorranktwo0.891\cellcolorranktwo0.902\cellcolorranktwo0.924\cellcolorranktwo0.937OursSSIM↑\uparrow\cellcolorrankone0.969\cellcolorrankone0.930\cellcolorrankone0.929\cellcolorrankone0.934\cellcolorrankone0.907\cellcolorrankone0.914\cellcolorrankone0.941\cellcolorrankone0.953ActiveGSLPIPS↓\downarrow\cellcolorranktwo0.081\cellcolorranktwo0.136\cellcolorranktwo0.152\cellcolorranktwo0.138\cellcolorranktwo0.184\cellcolorranktwo0.170\cellcolorranktwo0.146\cellcolorranktwo0.139OursLPIPS↓\downarrow\cellcolorrankone0.066\cellcolorrankone0.117\cellcolorrankone0.143\cellcolorrankone0.122\cellcolorrankone0.158\cellcolorrankone0.151\cellcolorrankone0.118\cellcolorrankone0.110We obtainϕt\phi_{t}by box-filteringϕtraw\phi_{t}^{\mathrm{raw}}, masking it to the observed collision-free region of𝒱t\mathcal{V}_{t}, and normalizing to a probability distribution. Masking is what makes the ergodic target realizable: it projects information mass from occupied surfaces onto the reachable free space the sensor can actually occupy. The gaze reward instead uses the unmaskedϕtraw\phi_{t}^{\mathrm{raw}}to preserve absolute scale. Both fields are sampled at continuous locations by trilinear interpolation, so all terms are differentiable in the trajectory.

3.3Kernel-Ergodic Trajectory with Sensor Footprint Depletion

We plan the trajectory with the kernel-ergodic metric ofSunet al.(2025a), which drives a trajectory’s time-averaged positions to match a target distributionϕ\phi. Active reconstruction violates its central premise: the high-ϕ\phimass lies on scene surfaces the robot cannot occupy, so no collision-free trajectory can match it. We resolve this in two ways. First, weprojectthe target into free space—the diffusion and masking of Sec.3.2, so the body covers reachable space rather than the surfaces themselves. Second, we add two footprint-aware mechanisms on top of the position-based metric: afootprint-overlap depletionterm (Eq. (3)) that discourages re-observing covered surfaces, and afootprint gaze reward(Eq. (5)) that aims the camera at high-ϕ\phisurfaces while the body stays in free space. UnlikeZhenget al.(2025), which replaces the point-sensor delta with a footprint distribution, our footprint enters only through the depletion factor while the kernel-ergodic metric stays position-based, and we deriveϕ\phionline from a live 2DGS map rather than a fixed target. Fig.3illustrates the design on an object-centric example.

Figure 3:From reconstruction state to information distribution to ergodic coverage.The current reconstruction has well-built and under-built regions (top left); the voxel map discretizes this state, and the under-built region carries high information (bottom left);Right:the induced information distribution over the object surface and a continuous trajectory whose view cones dwell on high-information regions—the time spent looking at a region is proportional to its information density.#### Time-varyingϕ\phivia footprint-overlap depletion.

In active sensing, the marginal value of observing a region diminishes once prior visits have already covered it;ϕ\phishould therefore decay at already-covered locations as the trajectory progresses. We model this decay by replacing the static evaluationϕt​(𝐱tk)\phi_{t}(\mathbf{x}^{k}_{t})with a coverage-discounted value

ϕtk≡ϕt​(𝐱tk)​∏j<k(1−η⋅vj​k),\phi^{k}_{t}\;\equiv\;\phi_{t}(\mathbf{x}^{k}_{t})\prod_{j<k}\bigl(1-\eta\cdot v_{jk}\bigr),(3)whereη∈(0,1)\eta\in(0,1)is the per-visit discount rate andvj​k∈[0,1]v_{jk}\in[0,1]is a soft footprint-overlap kernel between waypointsj,kj,k. We approximate the sensor footprint at waypointkkbyDDpoints𝐟t,dk=𝐱tk+d​𝐳tk\mathbf{f}^{k}_{t,d}=\mathbf{x}^{k}_{t}+d\,\mathbf{z}^{k}_{t}sampled along the optical axis𝐳tk\mathbf{z}^{k}_{t}at depths{d1,…,dD}\{d_{1},\dots,d_{D}\}, each a candidate surface location at distanceddfrom the camera; the overlap kernel is then

vj​k=maxdk,dj⁡exp⁡(−‖𝐟t,dkk−𝐟t,djj‖22​σfp2).v_{jk}\;=\;\max_{d_{k},\,d_{j}}\,\exp\!\left(-\frac{\|\mathbf{f}^{k}_{t,d_{k}}-\mathbf{f}^{j}_{t,d_{j}}\|^{2}}{2\,\sigma_{\mathrm{fp}}^{2}}\right).(4)This serves as a differentiable proxy for surface co-visibility: two waypoints whose central rays pass through nearby observable points are penalized for re-observing the same region, withσfp\sigma_{\mathrm{fp}}acting as an effective tolerance that softens each ray into a cylinder. The depletion product in Eq. (3) runs over the waypoints of the current horizon; re-coverageacrosshorizons is suppressed separately, by re-derivingϕt\phi_{t}from the updated map at every horizon boundary, where well-built voxels drop out of Eq. (2).

Footprint-based information attraction.

The depletion-aware ergodic metric still evaluatesϕ\phiat waypoint𝐱tk\mathbf{x}^{k}_{t}, so its gradient attracts the robot toward high-ϕ\phipositions, which in reconstruction lie on or behind surfaces. We complement this with a footprint reward that attracts the camera’s footprint to high-ϕ\phisurfaces, while the robot position remains governed by the ergodic metric and safety penalties. The reward averages the raw information mapϕtraw\phi_{t}^{\mathrm{raw}}(Eq. (2)) over the same footprint samples used by the depletion kernel, using the raw version to preserve absolute scale across horizons:

Lgaze​(𝐮t)=−1K​D​∑k=1K∑d=1Dwk,d​ϕtraw​(𝐟t,dk),L_{\mathrm{gaze}}(\mathbf{u}_{t})\;=\;-\frac{1}{KD}\sum_{k=1}^{K}\sum_{d=1}^{D}w_{k,d}\,\phi_{t}^{\mathrm{raw}}(\mathbf{f}^{k}_{t,d}),(5)wherewk,d=exp⁡(−κ​dunsafe​(𝐟t,dk))w_{k,d}=\exp\bigl(-\kappa\,d_{\mathrm{unsafe}}(\mathbf{f}^{k}_{t,d})\bigr)attenuates footprint samples in unobservable space. Heredunsafe​(𝐟)d_{\mathrm{unsafe}}(\mathbf{f})is the distance from𝐟\mathbf{f}to the observed collision-free region of𝒱t\mathcal{V}_{t}: zero in free space and growing both inside obstaclesandin the unobserved region behind them, so the reward never credits pointing at high-ϕ\phivoxels visible only through a wall.

Full cost and finite-horizon trajectory optimization.

The depletion-aware ergodic metric is

Ekerneldep​(τt)=\displaystyle E_{\mathrm{kernel}}^{\mathrm{dep}}(\tau_{t})\;=−2K​∑k=1Kϕtk+1K2\displaystyle\;-\frac{2}{K}\sum_{k=1}^{K}\phi^{k}_{t}\;+\;\frac{1}{K^{2}}(6)∑i,j=1Kexp⁡(−‖𝐱ti−𝐱tj‖22​σ2),\displaystyle\sum_{i,j=1}^{K}\exp\!\left(-\frac{\|\mathbf{x}^{i}_{t}-\mathbf{x}^{j}_{t}\|^{2}}{2\sigma^{2}}\right), whereϕtk\phi^{k}_{t}is the depletion-discounted target value of Eq. (3) (information term) and the pairwise term repels nearby waypoints toward uniform coverage (self-correlation term)—the two terms of the kernel-ergodic metric ofSunet al.(2025a), now evaluated on the free-space-projected target of Sec.3.2. The box-filter diffusion plays the mollifying role of the metric’s Gaussian kernel, so evaluating the diffusedϕt\phi_{t}pointwise realizes the information term on a target the sensor can reach: when the diffusion width equals the kernel bandwidth this is the metric ofSunet al.(2025a)exactly, and otherwise a free-space-projected kernel-ergodic objective.

Figure 4:Qualitative results of ActiveGS and our method on the Replica dataset.We show RGB renderings (top two rows) and reconstructed surface meshes (bottom row) across four scenes. Red boxes on the ActiveGS results mark regions of low-quality reconstruction, and blue boxes mark the same regions in our results, which appear sharper and more faithful. By replacing greedy NBV selection with ergodic search over an information map, our method achieves higher-quality reconstruction.The full per-horizon cost combines the depletion-aware ergodic metric with the gaze reward, soft-barrier penaltiesLsafeL_{\mathrm{safe}}for collisions, excessive step size, and out-of-bounds waypoints, and a quadratic control regularizer:

Jt​(𝐮t)=Ekerneldep​(τt​(𝐮t))+λg​Lgaze​(𝐮t)+λs​Lsafe​(𝐮t)+λr​‖𝐮t‖2.\begin{split}J_{t}(\mathbf{u}_{t})={}&E_{\mathrm{kernel}}^{\mathrm{dep}}(\tau_{t}(\mathbf{u}_{t}))+\lambda_{\mathrm{g}}\,L_{\mathrm{gaze}}(\mathbf{u}_{t})\\ &+\lambda_{\mathrm{s}}\,L_{\mathrm{safe}}(\mathbf{u}_{t})+\lambda_{\mathrm{r}}\,\|\mathbf{u}_{t}\|^{2}.\end{split}(7) All terms are differentiable in𝐮t\mathbf{u}_{t}; we optimize with Adam(Kingma and Ba2017), warm-starting from the previous-horizon solution. The agent executes allKKwaypoints before re-planning.

4Experiments

4.1Implementation Details

Dataset and simulator.We evaluate TRACE on eight indoor scenes from the Replica dataset(Straubet al.2019), following the active-reconstruction protocol of ActiveGS(Jinet al.2025). All experiments use the Habitat simulator(Savvaet al.2019)with an RGB-D camera moving freely inℝ3\mathbb{R}^{3}with yaw and pitch control ,[60∘×60∘][60^{\circ}\times 60^{\circ}]FOV,512×512512\!\times\!512resolution,[0.1,5.0][0.1,5.0]m depth range, and Gaussian depth noiseσ=0.01​d\sigma=0.01d. The world is discretized into a2020cm voxel grid.

Real Robot Experiments.We deploy the identical planner on two physical platforms: aUnitree Go2quadruped that reconstructs a room-scale scene, and anFranka FR3arm that reconstructs an object-centric scene. Maps are trained on real RGB-D data. Sec.4.5details the quadruped deployment; the manipulator result is in the supplementary material.

Training and planning schedule.All experiments run on a single RTX 5090. Each mission runs for a time budget of300300s, accumulated astmap+tplan+tflyt_{\text{map}}+t_{\text{plan}}+t_{\text{fly}}, wheretflyt_{\text{fly}}is calculated from the executed path length divided by a constant velocity of11m/s. Per horizon, the planner optimizes a1010-step trajectory with Adam (4040iterations) warm-started from the previous solution; after execution, the Gaussian map is trained for1212gradient steps on a mini-batch of1010frames sampled along the executed path together with1010frames drawn from the observation history.

Evaluation metrics.We report PSNR, SSIM, and LPIPS on a fixed set of10001000test viewpoints sampled uniformly in each scene’s free space,averaged over 5 independent runs. Geometry is examined qualitatively in Sec.4.3. All evaluation settings match primary baseline(Jinet al.2025).

Baselines.We compare against ActiveGS(Jinet al.2025), an NBV planner re-run in our pipeline under matched mapper, budget, and evaluation protocol (same as their setting). For completeness, Table1also reports NARUTO(Fenget al.2024), and FisherRF(Jianget al.2024)with numbers cited fromJinet al.(2025), which already established ActiveGS as the strongest of these earlier methods. Sampling-density ablations are described in Sec.4.4.

4.2Quantitative Analysis

TRACE achieves higher PSNR than ActiveGS on all scenes (Table1), with a mean gain of1.5\mathbf{1.5}dB; LPIPS and SSIM show the same pattern. The lead arises from two mechanisms: continuous-pose optimization places the camera at any high-ϕ\phipose along the trajectory rather than the nearest discrete candidate, and depletion suppresses re-coverage, spreading subsequent waypoints over low-confidence surfaces. The PSNR gain is positive on every scene, but its composition differs by scene type. On the office scenes, SSIM is comparable, while LPIPS improves by𝟔\mathbf{6}–𝟏𝟗%\mathbf{19\%}, indicating that the gain is concentrated in fine-scale appearance. We attribute this to continuous-pose optimization, which reaches poses unavailable to discrete candidates. On the room and hotel scenes, the SSIM gap widens to0.012\mathbf{0.012}–0.017\mathbf{0.017}, indicating that structural differences also emerge. These scenes contain heavier occlusion from furniture, where surfaces missed by discrete viewpoint selection require deliberate coverage, which the ergodic trajectory provides by continuously varying its heading. The gain is not explained by frame count alone: Sec.4.4shows thatdensifying the baseline’s sampling along its path does not close the gap.

4.3Qualitative Results

Figure4compares reconstructions from TRACE and ActiveGS on Replica scenes. The most pronounced differences lie in surface fidelity: in the highlighted regions, TRACE (blue boxes) recovers sharper texture, consistent lighting, and finer geometric detail on furniture and wall surfaces, while ActiveGS (red boxes) produces flatter, lower-fidelity reconstructions on the same regions. These fidelity gains arise from the same trajectory-level coupling that drives the PSNR lead: depletion keeps the camera looking at under-confident surfaces rather than re-sampling already-covered ones. The regions that differ visually are also where LPIPS separates most, confirming that the gain is concentrated in high-frequency texture rather than spread uniformly over the image. The mesh row shows the same pattern, where ActiveGS leaves holes on surfaces its path merely passes, while TRACE recovers a continuous surface in the same regions.

Table 2:Ablation study on Replica (PSNR).ActiveGS-Random and ActiveGS-Uniform observe at1010random / uniformly-interpolated poses along ActiveGS’s path;Ours-Kernel ESremoves the footprint mechanism of Sec.3.3.MethodOf0Of2Of3Of4R0R1R2H0ActiveGS\cellcolorranktwo36.78\cellcolorranktwo31.68\cellcolorrankthree32.1634.08\cellcolorrankthree29.93\cellcolorranktwo31.7432.35\cellcolorrankthree32.04ActiveGS-Random\cellcolorrankthree36.70\cellcolorrankthree31.37\cellcolorranktwo32.70\cellcolorranktwo34.35\cellcolorranktwo30.09\cellcolorrankthree31.29\cellcolorrankthree32.79\cellcolorranktwo32.11ActiveGS-Uniform35.8629.9830.4932.3827.6330.9531.6528.24Ours-Kernel ES35.6330.7831.83\cellcolorrankthree34.0929.9131.15\cellcolorranktwo33.0429.39Ours\cellcolorrankone38.36\cellcolorrankone32.81\cellcolorrankone33.79\cellcolorrankone34.92\cellcolorrankone31.55\cellcolorrankone32.84\cellcolorrankone34.70\cellcolorrankone33.69

4.4Ablation Study

To examine whether TRACE’s gain simply comes from denser sampling along the trajectory, and to validate the effectiveness of footprint depletion, we design the following ablations. For the first, we implement two ActiveGS variants: ActiveGS-Random scatters1010random observations per horizon (a baseline without planner), and ActiveGS-Uniform captures1010uniformly-interpolated poses along its original path (the information reachable from the trajectory geometry alone). Table2reports that ActiveGS-Random matches ActiveGS within noise(+0.08\mathbf{+0.08}dB), and ActiveGS-Uniform drops(−1.7\mathbf{-1.7}dB). Neither closes the gap to TRACE.Simply densifying observations performs even worse, as redundant views dilute the training weight of informative ones.For the second, we drop the footprint mechanism in Sec.3.3, leaving the original kernel-ergodic metric fromSunet al.(2025a): This variant loses2.1\mathbf{2.1}dBand falls below ActiveGS on six of eight scenes. NBV replans after every view, and its confidence updates implicitly deplete visited regions. Ergodic search alone optimizes the whole horizon against a frozen information map, yielding redundant views. Depletion is therefore essential to our framework.

4.5Executing Trajectories in Real World

We deploy TRACE on two physical platforms:a quadruped (Unitree Go2)exploring a room-scale scene with the same planner as in Sec.3.3and aFranka arm for object reconstruction (supplementary materials). Since the planned trajectory is itself the optimization variable, its waypoints are handed to each platform’s controller directly, with no intermediate path planner. This is a practical payoff of trajectory-level planning. An NBV planner commits to discrete viewpoints and delegates the connecting motion to a separate path planner, which can issuedynamically infeasiblecommands on a physical platform. TRACE instead optimizes the trajectory under the platform’s own dynamics, so every planned waypoint isdynamically consistent by construction. As shown in the supplementary materials, TRACE achieved a𝟏𝟎𝟎%\mathbf{100\%}success rate, whereas our NBV baseline made minor contact with the environment in every trial.

Figure 5:TRACE on Unitree Go2.Top:the Unitree Go2 executes a planned ergodic trajectory in a laboratory scene (composite with time-colored path); the dashed boxes mark the start and final poses and a region that the NBV baseline could not traverse safely (inset).Middle:Visualization of the reconstruction quality.Bottom:the ergodic trajectory turns smoothly and continuously, whereas NBV produces rapid heading reversals between committed viewpoints.#### Room-scale exploration (Unitree Go2).

A quadruped carries a front-mounted RGB-D camera (Intel D435). The trajectory is constrained to the traversable plane with yaw aligned to the direction of motion, and waypoints are tracked by a velocity-level controller built on the manufacturer’s high-level interface, without an intermediate path planner. Over55-minute missions in a4242m2scene, every planned horizon executes without infeasible commands, and observations are integrated into the 2DGS map. In Fig.5, the time-colored ergodic path (top) sweeps the room as a single continuous trajectory; the dashed inset marks a cluttered region theNBV baseline cannot traverse without collision, while TRACE threads the free space safely.The reconstruction (middle) is clean and complete. The heading traces (bottom) show our heading varying continuously, whereas NBV snaps between discrete targets with rapid reversals, exactly the motion a viewpoint-level planner cannot penalize.

5Conclusion

We presented TRACE, an ergodic-trajectory formulation for active 2D Gaussian-Surfel reconstruction. Instead of committing to discrete next-best-views, TRACE derives a target information distribution online from voxel-level information and per-Gaussian uncertainty, and optimizes a continuous, control-feasible trajectory with a kernel-ergodic horizon planner and footprint-overlap depletion. Across all eight Replica scenes, TRACE improves over our baseline (ActiveGS) by+1.5+1.5dB PSNR on average, and its trajectories execute directly on real physical robot without an intermediate path planner. By recasting active reconstruction as ergodic coverage, TRACE plans information gathering and feasible motion jointly within a single objective.

6Acknowledgments

This work was also supported in part by funding from the Johns Hopkins Data Science and AI Institute. Thanks Rex for insightful discussion on our physical experiments.

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TRACE: Ergodic Trajectory Optimization for Active Scene Reconstruction Supplementary Material

Appendix APer-Horizon Algorithm

Algorithm1summarizes one plan-and-execute horizon of TRACE; the notation follows Sec.3.

Input:Live 2DGS map

ℳt\mathcal{M}_{t}, voxel map

𝒱t\mathcal{V}_{t}, current pose

𝐩t\mathbf{p}_{t}, warm-start

𝐮prev\mathbf{u}_{\mathrm{prev}} ϕt←BuildInfoMap​(ℳt,𝒱t)\phi_{t}\leftarrow\textsc{BuildInfoMap}(\mathcal{M}_{t},\mathcal{V}_{t});

//(2)

1

𝐮←WarmStart​(𝐮prev)\mathbf{u}\leftarrow\textsc{WarmStart}(\mathbf{u}_{\mathrm{prev}});

2for*i=1,…,Niteri=1,\dots,N_{\mathrm{iter}}*do

3

τ←Rollout​(𝐩t,𝐮)\tau\leftarrow\textsc{Rollout}(\mathbf{p}_{t},\mathbf{u});

4Compute

J​(𝐮)J(\mathbf{u})via Eqs. (3)–(7);

5

𝐮←AdamStep​(𝐮,∇𝐮J)\mathbf{u}\leftarrow\textsc{AdamStep}(\mathbf{u},\nabla_{\mathbf{u}}J);

6

7end for

8

τt←Rollout​(𝐩t,𝐮)\tau_{t}\leftarrow\textsc{Rollout}(\mathbf{p}_{t},\mathbf{u});

9Execute

τt\tau_{t}; integrate RGB-D into

ℳt+1,𝒱t+1\mathcal{M}_{t+1},\mathcal{V}_{t+1};

10

𝐮prev←𝐮\mathbf{u}_{\mathrm{prev}}\leftarrow\mathbf{u};

Algorithm 1TRACE per-horizon plan-and-execute

Appendix BJoint-Space Ergodic Search on the FR3

On the Franka FR3 arm, we plan directly in joint space. A single integrator sufficed for the Go2 because it has a velocity interface; the arm does not. The trajectory variable is instead a sequence of joint configurations, and a differentiable forward-kinematics chain maps each to the camera pose, which determines where the world-frame information mapϕ\phiis sampled. This makes reachability intrinsic: with joint limits as box constraints, every planned view is reachable, and the plan runs withno inverse kinematics, no candidate viewpoint set, and no feasibility repair.Collisions with the table, wall, and base column are penalized in the same objective, on forward-kinematic body points. Here the ergodic objective of Sec.3.3is optimized through the nonlinear forward kinematics rather than over camera poses directly, yet every executed waypoint is dynamically consistent.

Setup.

The arm’s state is a joint configurationq∈𝒬⊂ℝ7q\in\mathcal{Q}\subset\mathbb{R}^{7}. Working within a single horizon, we drop the horizon indextt; the planner optimizes aKK-step joint trajectory𝐪=(q1,…,qK)\mathbf{q}=(q^{1},\dots,q^{K})from joint-velocity controls𝐮\mathbf{u}, withqk=qk−1+Δ​t​ukq^{k}=q^{k-1}+\Delta t\,u^{k}. The camera is mounted on the end-effector, so its pose follows from the configuration through forward kinematics.

Definition 1(Kinematic Sensor Map).

LetFK:𝒬→S​E​(3)\mathrm{FK}:\mathcal{Q}\to SE(3)be the forward-kinematics map to the camera pose,FK​(q)=Tee​(q)​Tcam\mathrm{FK}(q)=T_{\mathrm{ee}}(q)\,T_{\mathrm{cam}}, withTcamT_{\mathrm{cam}}the fixed hand–eye transform. We writeFK​(q)=(R​(q),𝐱​(q))\mathrm{FK}(q)=(R(q),\mathbf{x}(q)), withR​(q)∈S​O​(3)R(q)\in SO(3)the camera orientation and𝐱​(q)∈ℝ3\mathbf{x}(q)\in\mathbb{R}^{3}its position.

A configuration fixes both where the camera sits, through𝐱​(q)\mathbf{x}(q), and where it looks, through the optical axis𝐳​(q)=R​(q)​𝐞3\mathbf{z}(q)=R(q)\mathbf{e}_{3}. We model its view as samples along this axis.

Figure 6:TRACE on a Franka FR3.The arm executes the planned ergodic trajectory directly, with no path planner. We overlay the executed end-effector (EE) path, its camera frustums, and the start and final poses (green circle, red square). Planning in joint space makes every waypoint dynamically consistent.The supplementary video shows this in full.###### Definition 2(Sensor Footprint).

The footprint atqqis the set of world points

𝐟d​(q)=𝐱​(q)+d​𝐳​(q),d∈{d1,…,dD},\mathbf{f}_{d}(q)=\mathbf{x}(q)+d\,\mathbf{z}(q),\qquad d\in\{d_{1},\dots,d_{D}\},(8)sampled along the optical axis at depthsd1,…,dDd_{1},\dots,d_{D}.

Figure 7:Real scene versus TRACE reconstruction on the Unitree Go2.The room-scale reconstruction, produced from the directly executed ergodic trajectory, recovers the scene geometry and appearance.The supplementary video shows this in full.

Objective.

We keep the position-space kernel-ergodic metric ofSunet al.(2025a), which scores a trajectory by how closely its time-averaged occupancy matches the target distribution and splits into an information and a uniform-coverage term. Evaluated on the camera positions𝐱​(qk)\mathbf{x}(q^{k}), it reads

Ekerneldep​(𝐪)=\displaystyle E_{\mathrm{kernel}}^{\mathrm{dep}}(\mathbf{q})=−2K​∑k=1Kϕk+1K2\displaystyle-\frac{2}{K}\sum_{k=1}^{K}\phi^{k}+\frac{1}{K^{2}}(9)∑i,j=1Kexp⁡(−‖𝐱​(qi)−𝐱​(qj)‖22​σ2).\displaystyle\sum_{i,j=1}^{K}\exp\!\left(-\frac{\|\mathbf{x}(q^{i})-\mathbf{x}(q^{j})\|^{2}}{2\sigma^{2}}\right).The first term is the information term: it evaluates the depletion-discounted mapϕk=ϕ​(𝐱​(qk))​∏j<k(1−η⋅vj​k)\phi^{k}=\phi(\mathbf{x}(q^{k}))\prod_{j<k}(1-\eta\cdot v_{jk})at each camera position and pulls the trajectory toward informative regions, wherevj​kv_{jk}is the footprint-overlap kernel (Eqs. (3)–(4)). The second is the coverage term: a Gaussian kernel between camera positions that penalizes revisiting and spreads the trajectory out. They reproduce the information-and-coverage decomposition ofSunet al.(2025a), now on the camera positions the configuration produces.

Position alone does not fix what the camera sees, so orientation enters through the footprint. The gaze term evaluates the raw mapϕraw\phi^{\mathrm{raw}}over the footprint samples,

Lgaze​(𝐪)=−1K​D​∑k=1K∑d=1Dwk,d​ϕraw​(𝐟d​(qk)),L_{\mathrm{gaze}}(\mathbf{q})=-\frac{1}{KD}\sum_{k=1}^{K}\sum_{d=1}^{D}w_{k,d}\,\phi^{\mathrm{raw}}\!\big(\mathbf{f}_{d}(q^{k})\big),(10)withwk,dw_{k,d}down-weighting occluded samples; its gradient turnsR​(q)R(q)toward high-information surfaces. Position and orientation are therefore optimized together, both as functions of the single mapFK\mathrm{FK}.

Problem 1(Joint-Space Ergodic Search).

Given an information mapϕ\phi, minimize the objective over joint-velocity controls subject to the arm’s kinematics:

min𝐮\displaystyle\min_{\mathbf{u}}\ \;Ekerneldep​(𝐪)+λg​Lgaze​(𝐪)+λs​Lsafe​(𝐪)+λr​‖𝐮‖2\displaystyle E_{\mathrm{kernel}}^{\mathrm{dep}}(\mathbf{q})+\lambda_{\mathrm{g}}L_{\mathrm{gaze}}(\mathbf{q})+\lambda_{\mathrm{s}}L_{\mathrm{safe}}(\mathbf{q})+\lambda_{\mathrm{r}}\|\mathbf{u}\|^{2}s.t.qk=qk−1+Δ​t​uk,qk∈[qmin,qmax],\displaystyle q^{k}=q^{k-1}+\Delta t\,u^{k},\qquad q^{k}\in[q_{\min},q_{\max}],whereLsafeL_{\mathrm{safe}}is the soft-barrier penalty of the full cost (7), evaluated on forward-kinematic body points.

Proposition 1(Feasibility and Generality).

The joint-space problem has three properties. First, the objective is differentiable in𝐮\mathbf{u}: the forward kinematics is differentiable and the kernel-ergodic and gaze terms are differentiable in the camera pose, so the arm reuses the quadruped’s gradient-based optimizer unchanged. Second, every trajectory that respects the joint limitsqk∈[qmin,qmax]q^{k}\in[q_{\min},q_{\max}]maps to an achievable camera poseFK​(qk)\mathrm{FK}(q^{k}), so every planned waypoint is executable without inverse kinematics or feasibility repair. Third, when the decision variable is the camera pose itself, the forward-kinematics map drops out and the problem reduces to the camera-pose objective of the main paper, recovering the quadruped case.

Table 3:Executing ActiveGS viewpoints directly on the FR3 (simulation).Across1010paired trials with diverse start poses under a shared mapper and budget, we command each planner’s output on the arm without a path planner. Every ActiveGS trial collides with the scene and knocks the object off the table; TRACE stays collision-free.MethodCollision-free trialsActiveGS0/100/10\cellcolorrankoneTRACE (Ours)\cellcolorrankone𝟏𝟎/𝟏𝟎\mathbf{10/10}Figure6shows a representative mission on the real arm. TRACE follows the joint-space trajectory in one continuous sweep around the object, and every view executes without collision or intervention. Table3quantifies the contrast in simulation, where an unrealizable plan can be run safely. Across1010paired trials under a shared mapper and budget, the camera poses ActiveGS selectscollide with the scene on every trialand knock the object off the table, while TRACE, planning in joint space,stays collision-free throughout.

Appendix CRoom-Scale Reconstruction on the Go2

Figure7compares the reconstruction TRACE produces on the Unitree Go2 with photo of the same scene. Over a55-minute mission the quadruped executes the planned ergodic trajectory directly, and the online 2DGS map fills in furniture and wall surfaces across the room. The reconstruction is clean and complete, recovering geometry and texture on the surfaces the trajectory sweeps, which confirms the executed path is bothfeasible and information-rich at room scale.