A volumetric method for building complex models from range images

B. CurlessM. Levoy

article1996SIGGRAPH3,881 citationsTest of Time Award

Introduces a foundational volumetric method that integrates multiple noisy range scans into a cumulative weighted signed distance function, using space carving to produce complete, watertight 3D surface models with least-squares optimality.

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A new volumetric algorithm integrates multiple aligned range images into high-resolution surface models that are seamless, watertight, and free of the catastrophic failures seen in earlier methods. Range scanners now produce dense depth data for applications such as reverse engineering and digital archiving, yet single scans rarely cover an entire object, and prior merging techniques either discard directional uncertainty, require global reprocessing, or leave holes that break downstream uses such as rapid prototyping.

The work set out to demonstrate a single framework that simultaneously represents sensor uncertainty along lines of sight, updates the model incrementally and in any order, fills unobserved gaps plausibly, and remains robust to outliers while scaling to dozens of input scans. The approach stores a weighted signed-distance field on a run-length-encoded voxel grid. Each new range image is scan-converted into local distance and weight ramps aligned with the sensor’s viewing rays, then combined with existing values through simple additive rules. Space carving marks voxels behind observed surfaces as empty, allowing an isosurface algorithm to extract both measured geometry and hole-filling patches at the empty-to-unseen boundaries. The final surface is shown under stated assumptions to be the least-squares minimizer of distances measured along the original lines of sight.

In practice the method merged as many as 70 scans containing roughly 12 million input vertices into models of 1.8–2.6 million triangles. Root-mean-square deviation between the original range points and the reconstructed surface remained approximately 0.1 mm, comparable to the scanner’s own accuracy. On a thin drill bit the volumetric surface stayed topologically correct where a leading polygon-zippering technique produced intersecting geometry and holes. Adding a backdrop behind the object enabled more aggressive carving and produced clean, filtered hole-fill surfaces on a 20 cm Buddha statuette without eroding fine surface detail.

These results imply that complex objects can now be digitized to manufacturing-grade fidelity in a few hours of computation, removing a major barrier to automated model creation for rendering, simulation, and physical reproduction. The same data structure also supplies a ready-made occupancy map useful for robot navigation or further geometric analysis.

Further gains are possible by parallelizing the scanline updates, extending carving to rays that return no valid range, and adapting the weighting scheme to additional sensor technologies. Limitations remain for surfaces thinner than the maximum uncertainty interval and for sharp corners when no single scan spans both faces; these cases can still produce local thickening or rounding. The reported experiments on real scanned objects nevertheless give high confidence that the core algorithm delivers the claimed combination of robustness, efficiency, and completeness for the great majority of objects that can be captured by optical triangulation.

Stanford University high resolution image reconstruction signed distance function Volumetric Integration Signed Distance Function Run‑Length Encoding

Curless et al (1996).pdf

Paper: Marching cubes: A high resolution 3D surface construction algorithm, W. Lorensen et al. (1987). Reading Marching Cubes first provides essential background on extracting discrete polygonal surfaces from volumetric scalar fields, a foundational technique that the source paper adapts and builds upon.

Paper: DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation, Jeong Joon Park et al. (2019). DeepSDF extends the volumetric signed distance function representation introduced in the source paper into continuous, neural network-parameterized implicit fields.

Cover for A volumetric method for building complex models from range images

Abstract

A number of techniques have been developed for reconstructing surfaces by integrating groups of aligned range images. A desirable set of properties for such algorithms includes: incremental updating, representation of directional uncertainty, the ability to fill gaps in the reconstruction, and robustness in the presence of outliers. Prior algorithms possess subsets of these properties. In this paper, we present a volumetric method for integrating range images that possesses all of these properties.

Our volumetric representation consists of a cumulative weighted signed distance function. Working with one range image at a time, we first scan-convert it to a distance function, then combine this with the data already acquired using a simple additive scheme. To achieve space efficiency, we employ a run-length encoding of the volume. To achieve time efficiency, we resample the range image to align with the voxel grid and traverse the range and voxel scanlines synchronously. We generate the final manifold by extracting an isosurface from the volumetric grid. We show that under certain assumptions, this isosurface is optimal in the least squares sense. To fill gaps in the model, we tessellate over the boundaries between regions seen to be empty and regions never observed.

Using this method, we are able to integrate a large number of range images (as many as 70) yielding seamless, high-detail models of up to 2.6 million triangles.

Topics

high resolution image reconstruction signed distance function computational efficiency

Keywords

Volumetric Integration Signed Distance Function Run‑Length Encoding Space Carving Hole Filling Incremental Updating Directional Uncertainty Isosurface Extraction Optical Triangulation Scanner Marching Cubes

A Volumetric Method for Building Complex Models from Range Images
1 Introduction
2 Previous work
3 Volumetric integration
4 Hole filling
5 Implementation
5.1 Hardware
5.2 Software
5.2.1 Run-length encoding
5.2.2 Fast volume traversal
5.2.3 Fast surface extraction
6 Results
7 Discussion and future work
Acknowledgments
References
A Isosurface as least squares minimizer

Knowls

Knowl 1 — Cumulative Weighted Signed Distance Function for Range Image Integration
model/method
Multiple range images are integrated into a single volumetric representation represented as a discrete voxel grid. Each voxel $\mathbf{x} \in \mathbb{R}^3$ stores a cumulative signed distance $D(\mathbf{x})$ and a cumulative weight $W(\mathbf{x})$ . For $n$ range images, where the $i$ -th range image supplies a signed distance $d_i(\mathbf{x})$ (measuring distance along the sensor line of sight to the nearest surface point) and a certainty weight $w_i(\mathbf{x})$ , the cumulative fields are defined by:

$D(\mathbf{x}) = \frac{\sum_{i=1}^n w_i(\mathbf{x}) d_i(\mathbf{x})}{\sum_{i=1}^n w_i(\mathbf{x})}$

$W(\mathbf{x}) = \sum_{i=1}^n w_i(\mathbf{x})$

These fields can be updated incrementally as each new range image $i+1$ arrives:

$D_{i+1}(\mathbf{x}) = \frac{W_i(\mathbf{x}) D_i(\mathbf{x}) + w_{i+1}(\mathbf{x}) d_{i+1}(\mathbf{x})}{W_i(\mathbf{x}) + w_{i+1}(\mathbf{x})}$

$W_{i+1}(\mathbf{x}) = W_i(\mathbf{x}) + w_{i+1}(\mathbf{x})$

where $D_i(\mathbf{x})$ and $W_i(\mathbf{x})$ denote the state after integrating the first $i$ range images.

The per-image weight $w_i(\mathbf{x})$ incorporates directional sensor uncertainty by scaling with the dot product between the surface normal and the viewing direction (down-weighting grazing angles) and falling off near mesh boundaries. To prevent surfaces on opposite sides of an object from corrupting one another and to minimize compute costs, $d_i(\mathbf{x})$ and $w_i(\mathbf{x})$ are truncated such that the weights taper off behind the surface at a distance equal to half the maximum sensor uncertainty interval, and do not extend far in front of the surface. The reconstructed continuous surface is extracted as the zero-crossing isosurface $D(\mathbf{x}) = 0$ .
Knowl 2 — Least Squares Optimality of Weighted Signed Distance Isosurface
theoretical result
Under the assumptions that the range sensor is orthographic and range measurement errors are independently distributed along sensor lines of sight $\mathbf{v}_i$ , extracting the zero-crossing isosurface of the cumulative weighted signed distance function is equivalent to finding the surface that minimizes the continuous least squares error of squared distances along the sensing directions.

Let the reconstructed surface be parameterized over a canonical parameter domain $A$ as $z = f(x, y)$ . The weighted total squared error across $n$ range observations is:

$E(z) = \iint_A \sum_{i=1}^n w_i(x, y, z) d_i(x, y, z)^2 \left| \mathbf{v}_i \cdot \left( \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}, -1 \right) \right| dx dy$

where $d_i(x, y, z)$ is the signed distance to the $i$ -th observed range surface along sensing direction $\mathbf{v}_i$ , and $w_i(x, y, z)$ is its corresponding weight. Applying the calculus of variations to minimize $E(z)$ yields the stationarity condition:

$\sum_{i=1}^n \partial_{\mathbf{v}_i} \left[ w_i(x, y, z) d_i(x, y, z)^2 \right] = 0$

where $\partial_{\mathbf{v}_i}$ is the directional derivative along $\mathbf{v}_i$ . Because the weight $w_i$ does not vary along the line of sight ( $\partial_{\mathbf{v}_i} w_i = 0$ ) and the signed distance derivative along the line of sight is unity ( $\partial_{\mathbf{v}_i} d_i = 1$ ), this simplifies to:

$\sum_{i=1}^n w_i(x, y, z) d_i(x, y, z) = 0$

Because the cumulative divisor $\sum_{i=1}^n w_i(x, y, z)$ is strictly positive wherever observations exist, this condition is identically satisfied at the zero-crossing isosurface $D(\mathbf{x}) = 0$ of the cumulative weighted signed distance function.
Knowl 3 — Volumetric Three-State Classification and Space Carving for Watertight Hole Filling
model/method
To automatically fill unobserved regions (holes) with plausible watertight geometry, all voxels in the volumetric grid are classified into one of three states using fixed-point signed distance bounds $D_{\min} < 0 < D_{\max}$ :
1. Unseen: Voxels that have not been probed by any sensor ray, represented by $D(\mathbf{x}) = D_{\max}$ and $W(\mathbf{x}) = 0$ .
2. Empty: Voxels that lie in free space along the line of sight between the sensor and an observed surface point, marked during space carving, represented by $D(\mathbf{x}) = D_{\min}$ and $W(\mathbf{x}) = 0$ .
3. Near surface: Voxels located within the truncation band around observed range surfaces, represented by continuous values $D_{\min} < D(\mathbf{x}) < D_{\max}$ and cumulative weight $W(\mathbf{x}) > 0$ .
Holes in the object correspond to the frontiers between unseen regions ( $D_{\max}$ ) and empty regions ( $D_{\min}$ ). By running an isosurface extraction algorithm (such as Marching Cubes) across the entire grid without restricting evaluation to $W(\mathbf{x}) > 0$ voxels, the extraction simultaneously recovers the real zero-crossings where $W(\mathbf{x}) > 0$ and creates bridging zero-crossing triangles across the discontinuous $D_{\max} \to D_{\min}$ boundaries. The hole-filling triangles connect seamlessly to the observed surface triangles because both meet where the near-surface distance ramp transitions into the empty and unseen states.
Knowl 4 — Synchronous Scanline Streaming and Range Image Resampling for Volumetric Updates
algorithm
Updating a 3D volumetric distance grid from range images is accelerated by resampling range images to align with voxel grid scanlines and using hierarchical depth tests to skip empty or unseen regions.
Input: Set of $N$ calibrated range image meshes, voxel grid bounding volume with fixed-point bounds $[D_{\min}, D_{\max}]$
Output: Updated volumetric grid storing cumulative signed distance $D(\mathbf{x})$ and weight $W(\mathbf{x})$
Initialize all voxels $\mathbf{x}$ in volume to unseen: $D(\mathbf{x}) \leftarrow D_{\max}$ , $W(\mathbf{x}) \leftarrow 0$
for each range image $i = 1$ to $N$ do
    Render/resample range surface into an image plane oriented parallel to voxel grid scanlines
    Compute vertex weights $w$ from sensor line-of-sight angles and boundary proximity, interpolating them across pixels
    Build 1D min/max depth binary trees for adjacent pairs of resampled range scanlines
    for each voxel scanline in the volume do
        Query binary tree with scanline bounds to locate voxels within distance truncation band
        for each voxel $\mathbf{x}$ in the near-surface segment do
            Compute distance $d_i(\mathbf{x})$ along line of sight to range surface
            Compute interpolated weight $w_i(\mathbf{x})$
             $D_{new}(\mathbf{x}) \leftarrow \frac{W(\mathbf{x}) D(\mathbf{x}) + w_i(\mathbf{x}) d_i(\mathbf{x})}{W(\mathbf{x}) + w_i(\mathbf{x})}$
             $W_{new}(\mathbf{x}) \leftarrow W(\mathbf{x}) + w_i(\mathbf{x})$
            Write updated voxel into double-buffered run-length encoded volume
        end for
        if space carving is enabled then
            for each voxel $\mathbf{x}$ along sensor ray between sensor and surface do
                if $W(\mathbf{x}) == 0$ and $D(\mathbf{x}) == D_{\max}$ then
                     $D(\mathbf{x}) \leftarrow D_{\min}$
                end if
            end for
        end if
    end for
end for
Extract isosurface at $D(\mathbf{x}) = 0$ via Marching Cubes over varying and empty/unseen boundary voxels
Knowl 5 — Run-Length Encoded Representation for Volumetric Distance Grids
model/method
To handle large volumetric grids (up to 160 million voxels) within limited system memory, the volume is stored as a run-length encoded (RLE) data structure containing three distinct run types:
1. Empty runs: Continuous sequences of voxels verified to be free space ( $D(\mathbf{x}) = D_{\min}, W(\mathbf{x}) = 0$ ).
2. Unseen runs: Continuous sequences of unobserved voxels ( $D(\mathbf{x}) = D_{\max}, W(\mathbf{x}) = 0$ ).
3. Varying runs: Sequences of voxels near observed surfaces containing non-constant, continuously varying signed distances $D(\mathbf{x})$ and weights $W(\mathbf{x})$ , stored as an uncompressed stream of values.
This encoding achieves compression ratios between 10:1 and 20:1 compared to uncompressed 3D grids. The resulting voxel representation typically requires less memory than the final extracted triangle mesh (vertices and connectivity indices). To maintain linear memory layout without pointer overhead during incremental integration, the volume is double-buffered: scanlines are streamed from a source RLE buffer and written to a destination RLE buffer.
Knowl 6 — Localized Mesh Post-Filtering for Hole-Fill Boundaries
model/method
Isosurfaces extracted across the step transition between unseen ( $D_{\max}$ ) and empty ( $D_{\min}$ ) voxels exhibit aliasing and stair-step artifacts because the underlying field change is a discontinuous step rather than a smooth distance ramp.

To eliminate these artifacts without degrading fine geometric details on observed surfaces, a smoothing filter based on weighted averages of nearest vertex neighbors is applied exclusively to the hole-fill mesh regions. Hole-fill vertices are identified during isosurface extraction as those generated from voxel edges interpolating between zero-weight voxels ( $W=0$ ). The filter weights extend slightly beyond the hole boundaries and taper off into the neighboring observed surface mesh, producing a smooth geometric blend between hole fills and real geometry while keeping observed high-frequency features intact.
Knowl 7 — Space Carving with External Backdrops
model/method
When scanning objects with hollow interiors, through-holes, or deep gaps, sensor rays may pass entirely through a void without intercepting any object geometry, leaving internal cavities classified as unseen rather than empty.

To force space carving through such voids, a physical backdrop surface is placed behind the target object during scanning. The backdrop is positioned strictly outside the bounding box of the volumetric voxel grid. When sensor rays strike the backdrop, the algorithm follows the line of sight back toward the sensor and carves all traversed voxels within the grid as empty ( $D=D_{\min}, W=0$ ). Because the backdrop coordinates reside outside the volume grid, no geometry from the backdrop itself is incorporated into the reconstructed model, while the void is successfully emptied to allow correct hole-filler placement.

Knowl 8 — Reconstruction Statistics for Dragon and Happy Buddha Datasets

data/table

The performance of volumetric range integration was evaluated on physical statuettes scanned from dozens of viewpoints using optical triangulation. The table summarizes the scan counts, input/output geometric complexity, voxel resolutions, execution times (measured on a 250 MHz MIPS R4400 processor), and resulting hole counts with and without space carving and hole filling:

Model	Scans	Input triangles	Voxel size (mm)	Volume dimensions	Exec. time (min)	Output triangles	Holes
Dragon	61	15 M	0.35	$712 \times 501 \times 322$	56	1.7 M	324
Dragon + fill	71	24 M	0.35	$712 \times 501 \times 322$	257	1.8 M	0
Buddha	48	5 M	0.25	$407 \times 957 \times 407$	47	2.4 M	670
Buddha + fill	58	9 M	0.25	$407 \times 957 \times 407$	197	2.6 M	0

Without space carving, the models retain hundreds of topological boundary holes (324 for Dragon, 670 for Buddha). Adding space carving and backdrop scans eliminates all holes (0 holes), yielding completely watertight manifolds. The root-mean-square (RMS) distance between original range image points and points on the extracted surfaces is approximately 0.1 mm for both models, matching the intrinsic accuracy limit of the optical triangulation scanner.

Knowl 9 — Robustness Against Triangulation Scanner Edge Distortions Compared to Polygonal Zippering
empirical result
Traditional laser triangulation scanners produce systematic false edge extensions (flaring artifacts) along depth discontinuities and silhouette boundaries. When reconstructed using surface-based polygonal methods such as zippering (which erodes overlapping boundaries and stitches mesh edges), these false extensions cause catastrophic failures, including self-intersecting surfaces, spurious geometry, and open holes (as demonstrated on a 1.6 mm drill bit scanned from 12 rotational views at 30° spacing).

In contrast, the volumetric integration method combines redundant signed distance ramps from multiple viewpoints according to their directional lines of sight. False edge extensions from one view are intersected and overridden by positive distance ramps and space carving from other views. As a result, the volumetric approach averages out sensor noise and artifacts to yield a clean, watertight model of complex, thin geometry.
Knowl 10 — Geometric Interference on Thin Features and Sharp Corners
limitation
Because signed distance ramps and non-zero weight functions must extend behind observed surfaces by a finite distance (equal to half the maximum sensor uncertainty interval) to guarantee proper zero-crossing convergence, opposing surfaces that are closer together than this truncation distance will interfere with each other.

When scanning opposite sides of thin geometric features or acute sharp corners, the negative distance values behind one surface overlap and conflict with the positive distance values in front of the opposite surface. This interference imposes a fundamental lower limit on the reconstructible thickness of an object, causing unmodeled thickening of thin plates or rounding/blunting of sharp apexes and corners unless surface normal directions are explicitly factored into the volumetric distance evaluation.

Coverage note — None was omitted; all primary contributions, including the cumulative signed distance function, least squares optimality formulation, space carving and hole filling methods, scanline-streaming algorithm, RLE data structure, localized mesh filtering, backdrop carving, empirical benchmarks, and geometric limitations, have been captured.

References

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Citation

MLA

Curless, B., and M. Levoy. “A Volumetric Method for Building Complex Models from Range Images”. Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, 1996, pp. 303–12, https://doi.org/10.1145/237170.237269.

APA

Curless, B., & Levoy, M. (1996). A volumetric method for building complex models from range images. Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, 303–312. https://doi.org/10.1145/237170.237269

Chicago

Curless, B., and M. Levoy. 1996. “A Volumetric Method for Building Complex Models from Range Images”. Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, 303–12. https://doi.org/10.1145/237170.237269.

Harvard

Curless, B. and Levoy, M. (1996) “A volumetric method for building complex models from range images”, Proceedings of the 23rd annual conference on Computer graphics and interactive techniques. ACM, pp. 303–312. Available at: https://doi.org/10.1145/237170.237269.

Vancouver

1. Curless B, Levoy M (1996) A volumetric method for building complex models from range images. In: Proceedings of the 23rd annual conference on Computer graphics and interactive techniques. ACM, pp 303–312

BibTeX

@inproceedings{Curless_1996, series={SIGGRAPH96}, title={A volumetric method for building complex models from range images}, url={http://dx.doi.org/10.1145/237170.237269}, DOI={10.1145/237170.237269}, booktitle={Proceedings of the 23rd annual conference on Computer graphics and interactive techniques}, publisher={ACM}, author={Curless, Brian and Levoy, Marc}, year={1996}, month=Aug, pages={303–312}, collection={SIGGRAPH96} }

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A volumetric method for building complex models from range images

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Cumulative Weighted Signed Distance Function for Range Image Integration

Knowl 2 — Least Squares Optimality of Weighted Signed Distance Isosurface

Knowl 3 — Volumetric Three-State Classification and Space Carving for Watertight Hole Filling

Knowl 4 — Synchronous Scanline Streaming and Range Image Resampling for Volumetric Updates

Knowl 5 — Run-Length Encoded Representation for Volumetric Distance Grids

Knowl 6 — Localized Mesh Post-Filtering for Hole-Fill Boundaries

Knowl 7 — Space Carving with External Backdrops

Knowl 8 — Reconstruction Statistics for Dragon and Happy Buddha Datasets

Knowl 9 — Robustness Against Triangulation Scanner Edge Distortions Compared to Polygonal Zippering

Knowl 10 — Geometric Interference on Thin Features and Sharp Corners

References

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