Height-Fidelity LiDAR Encoding

• Height-fidelity LiDAR encoding is a strategy that maintains vertical geometry by addressing beam misalignment and quantization errors in LiDAR transformations.
• It employs methods such as explicit geometric decomposition, calibrated projections, and learned regressions to mitigate issues in range images, voxel compression, and occupancy prediction.
• Applications span sensor fusion, terrain mapping, forest phenotyping, and LiDAR super-resolution, achieving improved error metrics and robust height preservation.

Height-fidelity LiDAR encoding is an Editor’s term for representation, projection, quantization, and learning strategies that explicitly preserve vertical geometry when LiDAR data are transformed into compressed geometry streams, range images, voxel hierarchies, canopy-height products, bird’s-eye-view tensors, occupancy priors, or discrete latent codes. Across the literature, the recurring problem is that otherwise efficient encodings can degrade height structure through vertical band misalignment, canopy-top under-estimation, loss of precise continuous height during voxel compression, or ambiguous fixed-range pillar sampling. Representative formulations appear in geometry compression, sensor-intrinsic range-image generation, LiDAR super-resolution, camera–LiDAR fusion, occupancy prediction, phenotyping, forest-height mapping, and dynamic terrain mapping (Zhang et al., 2020, Soutullo et al., 23 Oct 2025, Eskandar et al., 2022, Wang et al., 6 Jul 2025, Wu et al., 6 May 2026).

1. Conceptual basis and recurring failure modes

The literature treats height fidelity as a geometric property of the encoding itself, rather than as a purely downstream evaluation outcome. In spinning LiDAR range images, the problem is beam consistency: if per-beam elevation and azimuth corrections are ignored, points can map to the wrong rows and columns, producing collisions, holes, and vertical band misalignment, so that the vertical row index no longer corresponds to the physical beam (Soutullo et al., 23 Oct 2025). In voxel backbones for multi-modal fusion, the problem is continuous-space preservation: discrete voxelization and merge operations can quantize or average away precise height, which then perturbs image-plane alignment and even the serialized token order consumed by linear-complexity fusion blocks (Wang et al., 6 Jul 2025). In occupancy lifting, the problem is sampling support: uniformly sampling the same vertical pillar range for every BEV cell ignores strong height variation and sparsity, so many projected points fall into sky or other invalid regions (Wu et al., 6 May 2026). In LiDAR super-resolution, the problem is target parameterization: vertical angles are often non-uniform, so reconstructing $z$ from a mis-specified elevation parameter can compound errors (Eskandar et al., 2022).

A second, distinct class of failure arises in environmental height products. In UAV phenotyping, voxel-only encodings tend to under-estimate canopy top heights because averaging within voxels suppresses peaks; raw maxima are more noise-sensitive, so percentile-based top estimates are favored (Dhami et al., 2019). In aerial LiDAR canopy modeling, height saturation is an architectural and target-scaling issue: the training target must preserve the full range of realistic canopy heights without compressing tall crowns into a narrow numerical interval (Wagner et al., 2023, Wagner et al., 17 Jan 2025). In terrain mapping, naive differencing of successive height maps without uncertainty modeling produces false change detection from dust, spurious returns, and pose errors (Bhandari et al., 2024).

These works therefore converge on a narrow technical objective: encode enough vertical structure that the representation remains faithful to beam identity, local height statistics, vertical occupancy, or true physical height above ground, depending on the task.

2. Geometric and signal parameterizations

One major family of methods preserves height through explicit geometric decomposition. In “Linear Model based Geometry Coding for Lidar Acquired Point Clouds” (Zhang et al., 2020), a local line is represented by an anchor point $x_0$ and a unit direction vector $u$, and each point $p_i$ is decomposed into a principal coordinate along the line and an orthogonal offset:

$$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$

This decomposition is central to height-aware quantization because height error can be controlled separately in the anchor, the along-line term, and the orthogonal residuals. The same work defines a vertical metric,

$$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$

and recommends axis-aware and orientation-aware quantization, such as tighter $\Delta_z$ for anchors and offsets, or smaller along-line quantization when $|u_z|$ is large (Zhang et al., 2020).

A second family preserves height through calibrated projection. In ALICE-LRI, a point $p=(x,y,z)$ is converted to spherical variables

$$r=\sqrt{x^2+y^2+z^2},\qquad \alpha=\arctan2\!\big(z,\sqrt{x^2+y^2}\big),\qquad \phi=\operatorname{atan2}(y,x),$$

then assigned to a beam row using inferred per-beam elevation angles and residual vertical offsets, and to a column using inferred per-beam azimuth corrections and a global discrete azimuth grid (Soutullo et al., 23 Oct 2025). The paper’s core claim is that if the residuals are correctly inverted and the width uses the $x_0$0 of per-beam horizontal resolutions, the range image mapping is one-to-one and complete, with zero collisions and zero omissions.

A third family reparameterizes the target coordinates themselves. HALS regresses polar coordinates $x_0$1, with $x_0$2, instead of spherical $x_0$3, and reconstructs

$x_0$4

so that height is predicted directly rather than being recovered through an elevation angle (Eskandar et al., 2022). The same work adds a Virtual Normal Loss over sampled point triplets to preserve vertical and planar structure.

A fourth family operates at the sensing stage. BE-ToF decomposes delay into a coarse burst-scale phase and a fine intra-cycle phase, estimating

$x_0$5

with unambiguous range set by the burst period and precision set by the short modulation period (Bao et al., 28 May 2025). This is not a point-cloud codec, but it is an encoding of depth information whose stated purpose is “high-fidelity, long-distance depth imaging.”

3. Compression, projection fidelity, and lossless geometry preservation

In geometry compression, height fidelity is usually expressed as a rate–distortion problem with explicit control over vertical error. The linear-model codec of (Zhang et al., 2020) is implemented on top of MPEG G-PCC reference software and selects between linear-model coding and standard octree coding through a Lagrangian objective. On the Ford LiDAR sequences “Ford_01_q_1mm”, “Ford_02_q_1mm”, and “Ford_03_q_1mm”, it reports average BD-rate reductions of $x_0$6 for D1 and $x_0$7 for D2, with gains most pronounced at middle and lower bitrates. The same paper argues that axis-aware quantization can bound vertical error more tightly than an unconstrained octree-only coder at similar bitrates (Zhang et al., 2020).

ALICE-LRI addresses a different compression bottleneck: the projection itself. Its premise is that approximate spherical or cylindrical projections introduce irreversible losses before any actual coding stage. On KITTI and DurLAR, ALICE-LRI reports zero points lost across all clouds, with $x_0$8 for every frame at native widths, average $x_0$9 and $u$0 for KITTI, and $u$1 and $u$2 for DurLAR (Soutullo et al., 23 Oct 2025). In its RTST case study, replacing only the projection and unprojection stages reduced SE from approximately $u$3 to approximately $u$4 at low error thresholds.

ELiC preserves height by keeping geometry lossless at the chosen grid while improving entropy modeling for high-depth occupancy prediction. It factorizes each octant occupancy label into two 4-bit quadrants, propagates features from denser lower depths to sparser higher depths, and preserves Morton order across depth transitions so that no per-level sorting is required (Kim et al., 18 Nov 2025). Because high bit-depth levels are precisely where thin poles, façade edges, and other vertical elements are represented, this cross-bit-depth propagation is height-relevant. ELiC reports BD-rate reductions versus G-PCC v30 of $u$5 on Ford and $u$6 on SemanticKITTI for its standard model, and $u$7 and $u$8 for ELiC-Large, with average runtime of $u$9 encode and $p_i$0 decode per frame (Kim et al., 18 Nov 2025).

A distinct but related compression-oriented representation appears in UltraLiDAR. It voxelizes points at $p_i$1, treats height as BEV feature channels, downsamples by $p_i$2 before quantization, and uses a learnable codebook of size $p_i$3 with $p_i$4-dimensional embeddings (Xiong et al., 2023). In this design, each code summarizes an $p_i$5 BEV patch with its full vertical stack, so vertical geometry is preserved within the latent token rather than discarded before quantization.

4. Height maps, canopy products, and terrain grids

In environmental mapping, height-fidelity encoding is usually formulated as the faithful derivation of a surface or per-cell height variable from LiDAR points. In UAV phenotyping, the workflow is plot-local: LiDAR scans are merged into a 3D farm map, cropped, clustered into plots, and each plot is assigned a local ground plane via RANSAC and linear least-squares. Height above ground for a point $p_i$6 is taken as

$p_i$7

and the canopy-top statistic is derived either by a raw maximum or a robust percentile (Dhami et al., 2019). The paper explicitly favors the $p_i$8th percentile, $p_i$9, and reports, on a wheat field with 112 plots, that raw $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$0th percentile height achieves $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$1, whereas voxel-only max height has error $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$2 with underestimation bias.

The same top-surface problem appears at larger scale in forest mapping. The California U-Net work defines

$$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$3

stores the CHM as 8-bit after multiplication by $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$4, and trains a regression U-Net on the normalized target $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$5 with heights clipped to $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$6 (Wagner et al., 2023). The model reports mean error approximately $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$7 statewide, preserves canopy heights up to about $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$8 without saturation, and delays saturation until about $$s_i = u^T (p_i - x_0), \qquad o_i = (I - u u^T)(p_i - x_0), \qquad \hat p_i = x_0 + s_i u + o_i.$$9. The Amazon NICFI study follows the same LiDAR-informed U-Net pattern but at $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$0 optical resolution, using median resampling from 1 m CHMs and a LiDAR-coverage-weighted MSE. It reports validation MAE $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$1 and successful estimation up to $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$2–$$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$3 without much saturation (Wagner et al., 17 Jan 2025).

THREASURE-Net formalizes the reference raster differently: it computes per-pixel $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$4th percentile height from LiDAR HD over Metropolitan France, using vegetation-class points and excluding crop vegetation below $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$5 in crop parcels (Kalinicheva et al., 12 Dec 2025). Its three variants achieve MAE $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$6 at $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$7, $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$8 at $$RMS_z = \sqrt{\frac{1}{N}\sum_{i=1}^N (z_i - \hat z_i)^2},$$9, and $\Delta_z$0 at $\Delta_z$1, with a loss that combines patch-wise MAE and a Weighted Gradient Difference Loss to preserve canopy edges.

Dynamic terrain mapping uses a different encoding again. The mining-shovel height grid discretizes each cell’s admissible height interval into $\Delta_z$2 bins and assigns a Hidden Markov Model to each cell, with transition matrix diagonal $\Delta_z$3, off-diagonal $\Delta_z$4, and Gaussian observation likelihood over the current LiDAR-derived height (Bhandari et al., 2024). Height fidelity here is confidence-aware: a state change is accepted only if the maximum posterior exceeds a threshold $\Delta_z$5, so dust and transient artifacts do not trigger immediate terrain updates.

5. Learned encodings for fusion, occupancy, super-resolution, and real-time detection

Height fidelity has become especially prominent in learned multi-modal and BEV pipelines, where small vertical errors alter alignment before any final task head is applied. MambaFusion defines initial and downsampled voxel coordinates directly in continuous space using scatter-mean:

$\Delta_z$6

and supplements this with a conflict test for generated voxels (Wang et al., 6 Jul 2025). Its stated motivation is that preserving precise continuous height improves camera–LiDAR alignment and serialized sequence order in global fusion. On nuScenes validation, MambaFusion-Base achieves NDS $\Delta_z$7 and mAP $\Delta_z$8.

HiPR turns LiDAR height into a conditioning prior for camera-only or camera–LiDAR occupancy lifting. It constructs a BEV height map by taking the highest occupied vertical index in each pillar, maps this to metric height, masks empty pillars, and replaces the conventional fixed pillar range $\Delta_z$9 with a per-cell range $|u_z|$0 (Wu et al., 6 May 2026). The sampling positions become

$|u_z|$1

and invalid pillars skip image-plane aggregation altogether. On Occ3D, HiPR reports mIoU $|u_z|$2, while its ablations show that LiDAR max-height conditioning outperforms mean-height conditioning.

HALS addresses the inverse problem of generating denser vertical samples from sparse scans. Its height-aware generator uses multiple upsampling branches with different receptive fields and learns confidence masks for fusion, reflecting the observation that upper rows and lower rows in range images have different range distributions (Eskandar et al., 2022). On KITTI Raw $|u_z|$3 upsampling, replacing spherical regression with polar regression improves EMD/CD from $|u_z|$4 to $|u_z|$5, and adding Virtual Normal Loss further improves to $|u_z|$6.

TriBand-BEV compresses the full point cloud into a three-channel BEV tensor by partitioning height into three bands, storing per-band maximum corrected reflectance, and training with a small global vertical re-bin and global reflectance jitter (Khoshkdahan et al., 12 May 2026). It explicitly treats 3D detection as 2D BEV detection followed by 3D reconstruction using distance-adaptive dilation and an interquartile range filter over the top and bottom $|u_z|$7 samples within each predicted footprint. On KITTI it reports pedestrian BEV AP of $|u_z|$8 for easy, moderate, and hard at $|u_z|$9.

UltraLiDAR occupies an intermediate position between compression and generative perception. By treating $p=(x,y,z)$0 as channels in BEV and quantizing $p=(x,y,z)$1 patches into a VQ-VAE codebook, it preserves layered elevations, thin tall objects, and multi-level occlusion while enabling completion and generation (Xiong et al., 2023). On SemanticKITTI it reports occupancy IoU $p=(x,y,z)$2, and on KITTI-360 generation it reports MMD $p=(x,y,z)$3 and JSD $p=(x,y,z)$4.

6. Evaluation regimes, limitations, and technical implications

The literature does not use a single evaluation protocol for height fidelity. Geometry compression papers emphasize D1 and D2 PSNR or BD-rate (Zhang et al., 2020, Kim et al., 18 Nov 2025). Projection-preserving work uses Chamfer Distance, PSNR, and Sampling Error, often with exact point-preservation checks (Soutullo et al., 23 Oct 2025). Environmental mapping papers rely on RMSE, MAE, percentile errors, and sometimes explicitly vertical metrics such as $p=(x,y,z)$5 or plot-level percent error (Dhami et al., 2019, Zhang et al., 2020, Wagner et al., 2023, Wagner et al., 17 Jan 2025). Occupancy and fusion works use mIoU, RayIoU, NDS, and mAP (Wang et al., 6 Jul 2025, Wu et al., 6 May 2026). Detection-oriented BEV encodings use AP in BEV and 3D, while super-resolution studies emphasize EMD, Chamfer Distance, IoU, and F1 (Eskandar et al., 2022, Khoshkdahan et al., 12 May 2026).

Limitations are equally task-specific. Linear models are weak on curved surfaces, undulating terrain, and regions without explicit line structures (Zhang et al., 2020). ALICE-LRI assumes spinning LiDARs with repeated, uniform azimuth sampling and does not apply to non-repetitive solid-state LiDARs (Soutullo et al., 23 Oct 2025). The mining height grid cannot represent overhangs because it assumes a single height per $p=(x,y,z)$6 cell (Bhandari et al., 2024). HiPR’s max-height prior cannot represent multi-layer scenes such as bridges (Wu et al., 6 May 2026). UltraLiDAR notes that thin wires and very slender poles may be under-resolved at $p=(x,y,z)$7 vertical resolution or a $p=(x,y,z)$8 token footprint (Xiong et al., 2023). Forest-height products show underestimation or saturation in the tallest canopies, especially above about $p=(x,y,z)$9 or in regions with georegistration error, clouds, or difficult terrain (Wagner et al., 2023, Wagner et al., 17 Jan 2025, Kalinicheva et al., 12 Dec 2025).

A plausible implication is that height fidelity is best understood not as a single encoding recipe, but as a design constraint that recurs at several levels of the LiDAR stack: acquisition-time delay coding, projection-time beam assignment, representation-time vertical discretization, compression-time bit allocation, and task-time feature aggregation. The surveyed papers show that preserving vertical structure can mean exact beam-row recovery, tighter $$r=\sqrt{x^2+y^2+z^2},\qquad \alpha=\arctan2\!\big(z,\sqrt{x^2+y^2}\big),\qquad \phi=\operatorname{atan2}(y,x),$$0 quantization, percentile-preserving canopy statistics, continuous-space voxel coordinates, per-pillar height priors, or compact latent codes whose channels remain explicitly vertical.