Fast Inspection of Scattered Regions (FISR) Overview

• FISR is a framework that selectively focuses computational and sensory resources on sparse, disjoint subdomains to reduce latency and redundancy.
• It spans diverse applications—from multi-UAV inspections to LiDAR stream processing and sparse signal search—using tailored decision mechanisms.
• Selective update strategies in FISR enable significant performance gains and efficient resource utilization by avoiding uniform processing of entire domains.

Searching arXiv for recent and directly relevant papers on FISR and adjacent formulations. Fast Inspection of Scattered Regions (FISR) denotes a family of formulations in which computation, sensing, or motion is concentrated on sparse, spatially disjoint, or weakly changing subsets of a larger domain, with the aim of reducing latency, measurement count, travel time, or redundant data collection while preserving task-specific fidelity. The term is introduced explicitly for multi-UAV inspection of disjoint regions of interest (ROIs) in (Gkelios et al., 29 Dec 2025). Closely related formulations appear in LiDAR stream processing, where only changed pillars are recomputed (Dow et al., 9 Dec 2025); in receding-horizon UAV inspection of scattered feature points on a cuboid (Papaioannou et al., 2023); in active search for sparse signals with region sensing (Ma et al., 2016); in Markov-chain inspection of percolation through sparse matrix-vector multiplication (Xu et al., 2014); and in scale-selective blurring of scattered measurements via Gaussian radial basis function interpolation and a fractional Helmholtz operator (Robinson et al., 2019). This suggests that FISR is best understood as a problem class centered on sparsity, locality, and selective update rather than as a single algorithm.

1. Conceptual scope and domain variants

Across the cited literature, the “scattered regions” to be inspected are instantiated in different ways. In multi-UAV inspection they are disjoint polygonal ROIs requiring one representative image each (Gkelios et al., 29 Dec 2025). In cuboid inspection they are finite sets of feature points on planar faces (Papaioannou et al., 2023). In LiDAR perception they are changed pillars or changed feature-map sites within a pseudo-image (Dow et al., 9 Dec 2025). In sparse signal search they are unknown support locations within a discrete domain, observed through region averages (Ma et al., 2016). In percolation they are occupied sites whose connectivity must be inspected between boundaries (Xu et al., 2014). In scattered-data analysis they are measurements at irregular spatial locations whose large-scale or small-scale structure is to be isolated efficiently (Robinson et al., 2019).

Realization	Domain	Core selective mechanism
mUDAI / explicit FISR	Multi-UAV ROI inspection	One optimized viewpoint per ROI, then VRP routing
SSCATeR	LiDAR 3D detection	Recompute only changed pillars and affected feature sites
Receding-horizon inspection	UAV cuboid inspection	MPC/MIQP over scattered feature points
Active region sensing	Sparse signal search	Greedy information gain over rectangular regions
Markov percolation inspection	CNT fiber connectivity	Repeated sparse matrix-vector multiplication
Tunable scattered-data blurring	Scattered measurements	Multiscale filtering via Gaussian mixtures

The unifying feature is not the sensor modality or vehicle type, but the decision to avoid uniform treatment of the entire domain. FISR methods instead maintain an explicit representation of where useful work remains: a set of ROIs, a support posterior, a change map, a detector boundary, or a scale-separated residual field.

2. Canonical structure of a FISR problem

A recurring structural pattern is a decomposition into four elements: spatial representation, admissible observation or update operators, a memory state, and a criterion for speed or efficiency. In (Gkelios et al., 29 Dec 2025), the environment is a planar workspace containing $k$ disjoint polygonal ROIs, and the decision variables are waypoint-based trajectories $\tau_i = [v_1^i,\dots,v_{m_i}^i]$ with viewpoints $v_j^i = [x,y,z,\psi]^\top$. The problem is decomposed lexicographically into viewpoint optimization per ROI and trajectory optimization over the resulting viewpoints. The ground footprint of a camera at altitude $h=z$ is determined by

$$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$

and image quality is evaluated either by the Maximized Coverage Objective,

$$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$

or by the Balanced Coverage Objective,

$\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$

where $A_\cap$, $A_R$, $A_I$, and $\tau_i = [v_1^i,\dots,v_{m_i}^i]$0 denote intersection, ROI, footprint, and union areas respectively (Gkelios et al., 29 Dec 2025).

A different but structurally analogous formulation appears in active search with region sensing. There the unknown is a $\tau_i = [v_1^i,\dots,v_{m_i}^i]$1-sparse signal $\tau_i = [v_1^i,\dots,v_{m_i}^i]$2, and each admissible measurement is a noisy average over a contiguous region: $\tau_i = [v_1^i,\dots,v_{m_i}^i]$3 The system maintains a Bayesian posterior over supports $\tau_i = [v_1^i,\dots,v_{m_i}^i]$4 and selects the next region by maximizing mutual information $\tau_i = [v_1^i,\dots,v_{m_i}^i]$5 (Ma et al., 2016). The object being “inspected” is therefore neither an image nor a physical location but posterior uncertainty itself.

The receding-horizon cuboid inspection problem sits between these two formulations. The object is a rectangular cuboid $\tau_i = [v_1^i,\dots,v_{m_i}^i]$6, the scattered targets are feature points $\tau_i = [v_1^i,\dots,v_{m_i}^i]$7 on its faces, and inspection occurs when a point lies within the field-of-view footprint $\tau_i = [v_1^i,\dots,v_{m_i}^i]$8 on the relevant face. The cost

$\tau_i = [v_1^i,\dots,v_{m_i}^i]$9

rewards early inspection of previously uninspected feature points and adds a cost-to-go toward the nearest uninspected one (Papaioannou et al., 2023). In all three cases, the salient mechanism is selective prioritization: image quality before route time, information gain before uniform scanning, and early inspection before deferred inspection.

3. Incremental computational FISR in streaming perception

The most explicit computational realization of FISR as selective update appears in SSCATeR for LiDAR-based 3D object detection (Dow et al., 9 Dec 2025). The scene is discretized into a 2D grid of pillars, exemplified by $v_j^i = [x,y,z,\psi]^\top$0 pillars covering an $v_j^i = [x,y,z,\psi]^\top$1 region with pillar size $v_j^i = [x,y,z,\psi]^\top$2. LiDAR is treated as a continuous stream with a sliding window of $v_j^i = [x,y,z,\psi]^\top$3 advanced every $v_j^i = [x,y,z,\psi]^\top$4. Because each new step shares $v_j^i = [x,y,z,\psi]^\top$5 of data with the previous one, most pillars remain unchanged from frame to frame.

The core state variable is a 2D boolean change map. A pillar is marked changed if points were added or removed during dequeue/enqueue operations on its circular queue. At the Pillar Feature Net stage, only changed pillars require recomputation of arithmetic means, offset features $v_j^i = [x,y,z,\psi]^\top$6, and pooled pillar embeddings. The paper reports that across $v_j^i = [x,y,z,\psi]^\top$7 frames ($v_j^i = [x,y,z,\psi]^\top$8), the mean number of active sites is $v_j^i = [x,y,z,\psi]^\top$9 pillars and the mean number of changed sites is $h=z$0 pillars, so only $h=z$1 of active pillars are changed and computation on the remaining $h=z$2 can be reused (Dow et al., 9 Dec 2025).

The same selective logic is propagated through the backbone. SSCATeR maintains cached previous inputs $h=z$3 and current outputs $h=z$4 for each layer, and updates them via

$h=z$5

In scatter form, only the delta $h=z$6 at changed input sites is scattered to neighboring outputs. Downsampling and upsampling propagate the change map across resolutions, and ReLU marks a site as changed if the post-activation value differs from the previous one. Batch normalization in inference does not alter the change map because parameters are fixed (Dow et al., 9 Dec 2025).

This is an exact incremental evaluation rather than an approximation. The paper states that the feature maps output by SSCATeR are identical to those produced by traditional sparse convolution techniques, and that SSCATeR’s feature maps and outputs are identical to those produced by traditional sparse convolutions while reusing the same weights (Dow et al., 9 Dec 2025). Reported savings include a $h=z$7 reduction in PFN operations versus standard PointPillars PFN, single-layer speedups on Jetson AGX Orin of $h=z$8–$h=z$9 with peak $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$0, and entire-backbone speedups of $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$1 on Orin and $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$2 on Xavier on the Newcastle dataset (Dow et al., 9 Dec 2025). The backbone average time on Orin is $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$3, with a fastest full backbone time of $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$4 (Dow et al., 9 Dec 2025).

The limitations are equally diagnostic of the FISR regime. Runtime increases when dense change events occur, such as wind stirring water reflections and causing many new points. In the extreme, changed sites approximate active sites, and SSCATeR degenerates toward normal sparse convolution cost. Benefits are also reduced in denser scenes such as PandaSet, where many points come from the road surface (Dow et al., 9 Dec 2025).

4. Robotic inspection planning and the explicit FISR problem

The multi-UAV Disjoint Areas Inspection method mUDAI is the source in which FISR is formalized as a distinct inspection problem (Gkelios et al., 29 Dec 2025). Its target is not complete coverage but one high-quality, representative image of each among many scattered ROIs, as fast as possible, under realistic flight and sensing constraints. The formulation includes per-UAV battery or flight-time limits, maximum speed, altitude bounds, and sensor specification or footprint constraints. The trajectory stage minimizes the makespan, $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$5, subject to visiting the optimized viewpoint of every ROI (Gkelios et al., 29 Dec 2025).

The viewpoint stage is solved independently for each ROI using dual-annealing over $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$6. The route stage is cast as a Vehicle Routing Problem and solved with Google OR-Tools using Path Cheapest Arc and Guided Local Search. When a returned route exceeds battery capacity, it is split and routing is recomputed with additional segments. Multi-UAV collision avoidance is handled by assigning each UAV a unique cruising altitude and inserting intermediate waypoints so that transitions occur with altitude separation (Gkelios et al., 29 Dec 2025).

The reported trade-off between the two quality objectives is explicit. Over 50 random polygons, MCO yields recall $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$7, precision $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$8, GSD $$d_h = 2h \tan(\mathrm{HFOV}/2), \qquad d_v = 2h \tan(\mathrm{VFOV}/2),$$9, and $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$0 iterations, whereas BCO yields recall $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$1, precision $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$2, GSD $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$3, and $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$4 iterations (Gkelios et al., 29 Dec 2025). Simulation with $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$5 to $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$6 ROIs and swarm sizes from $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$7 to $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$8 UAVs shows large reductions in mission time with increasing swarm size, but also diminishing returns once each UAV serves only a small subset of viewpoints (Gkelios et al., 29 Dec 2025). In the Chortiatis real-world deployment, mCPP with 2 UAVs requires $$\mathcal{J}_{MCO}(y) = \begin{cases} \frac{A_\cap}{A_R}, & \text{if } \frac{A_\cap}{A_R} < 1 \ \frac{A_\cap}{A_R} + \frac{1}{A_I}, & \text{if } \frac{A_\cap}{A_R} = 1, \end{cases}$$9 min with $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$0 recall, $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$1 precision, and theoretical average GSD $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$2, whereas mUDAI with 1 UAV requires $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$3 min with $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$4 recall, $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$5 precision, and average GSD $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$6, and mUDAI with 2 UAVs requires $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$7 min with the same recall, precision, and GSD (Gkelios et al., 29 Dec 2025). Orthomosaic generation from mCPP data took more than $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$8 hours, illustrating the post-processing burden of redundant coverage (Gkelios et al., 29 Dec 2025).

The 2023 receding-horizon cuboid inspection paper provides a closely related precursor from a control-theoretic perspective (Papaioannou et al., 2023). There, the scattered inspection targets are finite feature points on the surface of a cuboid-like structure, the camera footprint side length satisfies $\mathcal{J}_{BCO}(y)=\frac{A_\cap}{A_\cup},$9, and the UAV is modeled by linear discrete-time dynamics

$A_\cap$0

Visibility, face selection, point inclusion in the footprint, memory of previously inspected points, and collision avoidance are encoded by binary variables inside a mixed-integer quadratic program. Monte Carlo runtime studies on a 2 GHz laptop with 8 GB RAM and Gurobi 9 report, for example, average solve times of $A_\cap$1 for 20 points with $A_\cap$2, $A_\cap$3 for 20 points with $A_\cap$4, $A_\cap$5 for 20 points with $A_\cap$6, and $A_\cap$7 for 32 points with $A_\cap$8 (Papaioannou et al., 2023). In the reported simulation, 20 feature points are inspected within 66 time steps with $A_\cap$9 (Papaioannou et al., 2023).

Together, these two papers distinguish an explicit robotic meaning of FISR from classical coverage path planning. The defining shift is from full-coverage trajectories toward selective, quality-constrained visits to sparse targets.

5. Information-theoretic, graph-theoretic, and multiscale formulations

Active search for sparse signals with region sensing provides an information-theoretic formulation of FISR (Ma et al., 2016). The domain is a finite set of $A_R$0 locations, the unknown support has size $A_R$1, and each measurement is a noisy average over an interval or hyper-rectangle. The policy greedily maximizes expected information gain,

$A_R$2

under a posterior over supports. In 1D, the paper shows that the algorithm requires

$A_R$3

measurements to recover all $A_R$4 signal locations with small Bayes error (Ma et al., 2016). It also shows that active designs can be fundamentally more efficient than passive designs with region sensing (Ma et al., 2016). In the FISR vocabulary, this is a search-based rather than route-based formulation: speed is measured in measurements rather than flight time or FLOPs.

The Markov stochastic theory for site percolation in carbon nanotube fibers gives a reachability-based formulation (Xu et al., 2014). A modified FCC lattice with coordination number $A_R$5 represents the fiber, metallic sites form the active subset, and percolation inspection is recast as first passage to a detector state in a Markov chain. The core recurrence is

$A_R$6

so percolation inspection reduces to repeated sparse matrix-vector multiplication. The method scales to lattices with up to $A_R$7 million sites and yields an infinite-system site-percolation threshold

$A_R$8

for the modified FCC CNT-fiber lattice (Xu et al., 2014). Here FISR appears as a decision procedure over sparse connectivity rather than over visual or geometric targets.

The tunable blurring algorithm for scattered data supplies a multiscale signal-processing formulation (Robinson et al., 2019). Scattered observations $A_R$9 at locations $A_I$0 are interpolated by a Gaussian radial basis function field and blurred by approximating the inverse of the fractional bound-state Helmholtz operator

$A_I$1

whose spectrum is

$A_I$2

The blur operator on discrete observations is

$A_I$3

and, with appropriate fast solvers, time and memory can be $A_I$4 for fixed dimension, scales, and target accuracy (Robinson et al., 2019). The method is used to separate large-scale and small-scale components in scattered oceanographic float measurements and radiosonde data (Robinson et al., 2019). In a FISR reading, the selective act is scale selection: inspection is focused on structures at spatial scales encoded by $A_I$5.

These three formulations differ in objective and machinery, but all replace exhaustive examination with selective inference over a sparse or aggregated representation. One operates by posterior entropy reduction, one by sparse linear algebra over a detector graph, and one by low-pass separation in irregularly sampled fields.

6. Operational regimes, misconceptions, and limitations

Several recurring misconceptions are corrected by the cited work. First, FISR is not synonymous with coverage path planning. The explicit FISR formulation in (Gkelios et al., 29 Dec 2025) is defined precisely against dense overlapping coverage, and it requires one representative image per ROI rather than orthomosaic-style completeness. Second, FISR is not inherently approximate. SSCATeR states that its feature maps and outputs are identical to those produced by traditional sparse convolutions, so the gain is from bypassing redundant recomputation rather than changing the model output (Dow et al., 9 Dec 2025). Third, under region-limited sensing, adaptivity is not negligible: active region sensing is shown to be fundamentally more efficient than passive designs (Ma et al., 2016).

The favorable regime is consistent across domains. Speedups are strongest when ROIs are spatially disjoint, when only a small subset of active sites changes between time steps, when feature points are few relative to the environment, or when the sought signal is sparse. This suggests that FISR is especially appropriate when the domain contains large irrelevant or unchanged regions. The unfavorable regime is equally clear. Dense change events in LiDAR erode SSCATeR gains (Dow et al., 9 Dec 2025); denser point-cloud scenes reduce sparsity benefits (Dow et al., 9 Dec 2025); longer MPC horizons and larger numbers of feature points increase MIQP solve times (Papaioannou et al., 2023); single-image-per-ROI assumptions create a coverage-versus-resolution trade-off for elongated or complex polygons (Gkelios et al., 29 Dec 2025); passive region-sensing designs waste measurements on empty areas (Ma et al., 2016); and extreme clustering or ill-conditioning in scattered data complicates Gaussian RBF interpolation (Robinson et al., 2019).

Memory, battery, and computational overhead also remain central. SSCATeR requires temporal buffers for previous inputs and outputs per layer, though it avoids rulebooks and is reported to use less memory than rulebook-based sparse convolution (Dow et al., 9 Dec 2025). mUDAI must split routes when battery constraints are violated and assumes a centralized planner with reliable communication and a static environment (Gkelios et al., 29 Dec 2025). The receding-horizon cuboid framework assumes a known static map, idealized face-normal camera alignment, and simple point-mass UAV dynamics (Papaioannou et al., 2023). The scattered-data blurring method depends on solver conditioning and on approximation choices for Gaussian mixture truncation (Robinson et al., 2019).

Viewed across these literatures, FISR is a unifying strategy for selective processing of sparse, disjoint, or weakly varying subsets of a larger space. Its concrete realizations differ—incremental convolution, region-sensing search, MIQP or VRP-based routing, sparse Markov propagation, and multiscale filtering—but each uses an explicit mechanism for deciding where computation or sensing should occur, and each derives its efficiency from refusing to inspect the whole domain uniformly.