The Visual Hull Concept for Silhouette-Based Image Understanding

Aldo Laurentini

article1994TPAMI1,772 citations

Introduces the foundational concept of the visual hull to establish the exact theoretical limits and computational algorithms for reconstructing and recognizing 3D objects from 2D silhouette intersections.

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Silhouette-based image processing is widely used in automated inspection, navigation, and robotic manipulation because extracting two-dimensional silhouettes from camera images is computationally simple and robust to degraded conditions. However, standard systems using multi-view volume intersection cannot fully identify or reconstruct complex, non-convex objects because concavities and hidden recesses do not appear in silhouette projections. This fundamental limitation creates uncertainty regarding which surface features can actually be reconstructed or distinguished from silhouette data alone.

The article introduces the formal geometric concept of the "visual hull" to mathematically define and compute the theoretical limits of silhouette-based object recognition and three-dimensional shape reconstruction. It establishes exact geometric conditions under which non-convex surfaces can be recovered or differentiated using multiple camera viewpoints.

To solve this, the author develops computational geometric frameworks that categorize viewing zones and surface visibility across two-dimensional and three-dimensional spaces. The visual hull is defined as the closest possible volumetric approximation of an object that can be obtained using volume intersection, or equivalently, the maximal object that produces identical silhouettes from all allowable viewpoints within a specified region. The article evaluates two primary viewing domains: the external visual hull, where viewpoints remain outside the object's convex hull, and the internal visual hull, where viewpoints are constrained only by the object's surface.

The findings establish that an object can only be distinguished or reconstructed on its "silhouette-active" surfaces, which lie directly on the visual hull's boundary. For standard external viewing, there is a single, unique external visual hull that never exceeds the object's convex hull. The analysis demonstrates that the visual hull of a three-dimensional planar-faced polyhedron is not strictly planar; its boundaries consist of planar patches and curved ruled quadric surfaces formed by alignments between vertices and edges. Computationally, two-dimensional visual hulls can be computed efficiently in polynomial time, allowing the active surfaces of three-dimensional polyhedra to be determined by slicing planes along each polyhedral face.

These results demonstrate that investing in additional silhouette cameras or finer image intersections cannot overcome geometric concavity limits; any surface feature residing in an inactive concavity is physically unrecoverable from external silhouette data alone. In terms of engineering and algorithm design, teams must recognize that silhouette-based reconstruction may return an object larger than the actual part or introduce spurious unconnected components, which directly affects quality inspection, collision-free path planning, and automated tolerance verification.

Organizations developing automated optical inspection or robotic systems should use the visual hull algorithm as a design-time evaluation tool to determine whether silhouette methods are sufficient for a specific component's geometry before deploying physical sensor hardware. If critical features fall within silhouette-inactive areas, vision systems must be augmented with alternative cues such as stereo depth, structured light, or photometric sensors. Current algorithms for full three-dimensional polyhedra rely on high-complexity brute-force methods, meaning computational pipelines should focus on the more efficient face-by-face active surface algorithms. Future work is required to develop optimized three-dimensional implementations and expand theoretical formulations to objects with smooth, curved surfaces.

Politecnico di Torino facial images image understanding nonconvex visual hull concept silhouette-based image understanding object reconstruction

Laurentini (1994).pdf

Paper: Marching cubes: A high resolution 3D surface construction algorithm, William E. Lorensen et al. (1987). Introduces the standard volumetric surface extraction technique that underpins volumetric intersection and 3D geometric mesh representations.
Paper: Surface reconstruction from unorganized points, Hugues Hoppe et al. (1992). Presents foundational algorithms for inferring 3D surface geometry and topology from unorganized geometric point observations.
Paper: Comparing Images Using the Hausdorff Distance, Daniel P. Huttenlocher et al. (1993). Establishes core geometric distance and set-comparison measures used in silhouette matching and 2D/3D shape alignment.

Paper: The lumigraph, Steven J. Gortler et al. (1996). Directly employs visual hull geometry extracted from silhouettes to bound and rebin plenoptic sampling rays for novel view synthesis.
Paper: A volumetric method for building complex models from range images, Brian Curless et al. (1996). Extends volume intersection and space-carving principles to fuse multiple line-of-sight range images into seamless signed-distance volumetric meshes.
Paper: A Comparison and Evaluation of Multi-View Stereo Reconstruction Algorithms, Steven M. Seitz et al. (2006). Surveys and benchmarks advanced multi-view 3D reconstruction frameworks, many of which use visual hulls for bounding volume initialization.
Paper: NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction, Peng Wang et al. (2021). Modernizes multi-view silhouette and appearance reconstruction by learning neural implicit signed distance functions via volume rendering.
Paper: 3D-R2N2: A Unified Approach for Single and Multi-view 3D Object Reconstruction, Christopher B. Choy et al. (2016). Applies deep recurrent neural networks to sequentially reconstruct 3D volumetric object shapes from multi-view 2D images.

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Abstract

Abstract—Many algorithms for both identifying and reconstructing a 3-D object are based on the 2-D silhouettes of the object. In general, identifying a nonconvex object using a silhouette-based approach implies neglecting some features of its surface as identification clues. The same features cannot be reconstructed by volume intersection techniques using multiple silhouettes of the object. This paper addresses the problem of finding which parts of a nonconvex object are relevant for silhouette-based image understanding. For this purpose, the geometric concept of visual hull of a 3-D object S is introduced. The visual hull of a 3-D object S is the closest approximation of S that can be obtained with the volume intersection approach. An equivalent statement, relative to object identification, is that the visual hull of S is the maximal object silhouette-equivalent to S, i.e., which can be substituted for S without affecting any silhouette. Only the parts of the surface of S that also lie on the surface of the visual hull can be reconstructed or identified using silhouette-based algorithms. The visual hull of an object depends not only on the object itself but also on the region allowed to the viewpoint. Two main viewing regions can be considered, resulting in the external and internal visual hull. In the former case the viewing region is related to the convex hull of S, in the latter it is bounded by S itself. The internal visual hull also admits an interpretation not related to silhouettes: the features of the surface of S that is not coincident with the surface of the internal visual hull cannot be observed from any viewpoint lying outside the convex hull. After a general discussion of the visual hull and its properties, algorithms for computing the internal and external visual hulls of 2-D objects and 3-D planar face objects are presented and their complexity analyzed. In general, the visual hull of a 3-D planar face object turns out to be bounded by planar and curved patches. A precise statement of the concept of visual hull appears to be novel, as is the problem of its computation.

Topics

facial images image understanding nonconvex viewpoint interaction techniques multi-view stereo silhouette images visual informations

Keywords

visual hull concept silhouette-based image understanding object reconstruction visual cone aspect graph polyhedral objects convex hull quadric surfaces silhouette-active surface VH algorithm

I. INTRODUCTION
II. THE VISUAL HULL: DEFINITIONS AND GENERAL PROPERTIES
A. The Visual Hull of an Object Relative to a Viewing Region
B. The Main Case: The External Visual Hull
C. The Internal Visual Hull
III. THE COMPUTATION OF VH(S) AND OF IVH(S) IN 2-D
A. An Efficient Algorithm for Computing the 2-D Visual Hull
B. An Algorithm for Computing the 2-D Internal Visual Hull
IV. COMPUTING THE SILHOUETTE-ACTIVE SURFACES OF POLYHEDRAL OBJECTS
V. COMPUTING VH AND IVH FOR POLYHEDRAL OBJECTS
A. The 3-D Visual Number
B. The Active Surfaces
C. Examples of Visual Hulls of Simple Polyhedra
D. The Visual Hull and the Aspect Graphs
E. A Brute-Force Algorithm for Computing the Visual Hull
F. The Computation of the Internal Visual Hull
VI. CONCLUSION
REFERENCES

Knowls

Knowl 1 — Visual Hull Relative to a Viewing Region
definition
Let $S \subset \mathbb{E}^3$ be a 3-D object, and let $R \subseteq \mathbb{E}^3$ be a viewing region containing all allowable viewpoints from which $S$ may be observed.

The visual hull of $S$ relative to viewing region $R$ , denoted $VH(S, R)$ , is defined as: $VH(S, R) = \{ P \in \mathbb{E}^3 \mid \forall V \in R, \text{ the half-line starting at } V \text{ and passing through } P \text{ intersects } S \text{ at least once} \}$

Key properties of $VH(S, R)$ include:
1. $S \subseteq VH(S, R)$ .
2. $VH(S, R)$ is the maximal object that is silhouette-equivalent to $S$ with respect to $R$ (it yields the identical silhouette as $S$ from every viewpoint $V \in R$ ).
3. $VH(S, R)$ is the closest volume approximation of $S$ obtainable by intersecting visual cones (the circumscribed cones with apex at $V$ and tangent to $S$ ) from all viewpoints $V \in R$ .
4. If $R \supset R'$ , then $VH(S, R) \subseteq VH(S, R')$ .
5. An object $S$ can be reconstructed exactly from silhouettes observed from $R$ if and only if $S = VH(S, R)$ .
6. Two objects $S$ and $S'$ can be distinguished from their silhouettes observed from $R$ if and only if $VH(S, R) \neq VH(S', R)$ .
The surface $s(S)$ of $S$ is partitioned relative to $R$ into:
- Silhouette-active surface $sa(S, R)$ : the subset of $s(S)$ that lies on the boundary $vh(S, R)$ of $VH(S, R)$ , i.e., $sa(S, R) = s(S) \cap vh(S, R)$ . This surface uniquely determines $VH(S, R)$ .
- Silhouette-inactive surface $si(S, R) = s(S) \setminus sa(S, R)$ : points on $s(S)$ not on $vh(S, R)$ , which can be reshaped arbitrarily within $VH(S, R) \setminus S$ without altering any silhouette observed from $R$ .
Knowl 2 — External Visual Hull and Line Intersection Characterization
theoretical result
Let $S \subset \mathbb{E}^3$ be a bounded 3-D object, and let $CH(S)$ denote its convex hull. The standard viewing region outside the convex hull of $S$ is $R_e = \mathbb{E}^3 \setminus CH(S)$ .

The external visual hull (or simply the visual hull) of $S$ , denoted $VH(S)$ , is defined as $VH(S) = VH(S, R_e)$ .

Key theoretical properties established for $VH(S)$ include:
1. Convex Hull Containment: For any viewing region $R' \subseteq R_e$ that completely encloses $S$ , $VH(S, R') \subseteq CH(S)$ .
2. Viewing Region Invariance: For any two viewing regions $R'$ and $R''$ that enclose $S$ and lie entirely outside $CH(S)$ , $VH(S, R') = VH(S, R'') = VH(S)$
3. Line Intersection Characterization: A point $Q \in \mathbb{E}^3$ belongs to $VH(S)$ if and only if every straight line passing through $Q$ contains at least one point of $S$ : $Q \in VH(S) \iff \forall \text{ lines } L \text{ passing through } Q, \; L \cap S \neq \emptyset$
The silhouette-active surface of $S$ relative to $R_e$ is denoted $sa(S) = s(S) \cap vh(S)$ , where $vh(S)$ is the boundary of $VH(S)$ .
Knowl 3 — Internal Visual Hull and Half-Line Characterization
theoretical result
Let $S \subset \mathbb{E}^3$ be a 3-D object. When viewpoints are permitted everywhere outside $S$ itself (including within concavities inside the convex hull $CH(S)$ ), the viewing region is $R_i = \mathbb{E}^3 \setminus S$ .

The internal visual hull of $S$ , denoted $IVH(S)$ , is defined as $IVH(S) = VH(S, R_i)$ .

Theoretical properties of $IVH(S)$ include:
1. Half-Line Intersection Characterization: A point $Q \in \mathbb{E}^3$ belongs to $IVH(S)$ if and only if every half-line starting at $Q$ intersects $S$ at least once: $Q \in IVH(S) \iff \forall \text{ half-lines } H \text{ starting at } Q, \; H \cap S \neq \emptyset$
2. Containment: Because $R_e \subset R_i$ , the internal visual hull is always contained within the external visual hull: $IVH(S) \subseteq VH(S)$
3. Unobservability of Internal Silhouette-Inactive Surface: The boundary surface of $S$ is partitioned into the internal silhouette-active surface $isa(S) = s(S) \cap ivh(S)$ and the internal silhouette-inactive surface $isi(S) = s(S) \setminus isa(S)$ , where $ivh(S)$ is the boundary of $IVH(S)$ . Any feature lying on $isi(S)$ cannot be observed from any viewpoint $V \notin CH(S)$ .
Knowl 4 — Geometric Boundary Structure of 3-D Polyhedral Visual Hulls
theoretical result
For a general 3-D polyhedron $S$ with planar faces and straight edges, the boundary of its visual hull $VH(S)$ is not planar in general; it consists of planar patches and ruled quadric surface patches.

The bounding surface patches arise from two classes of active surfaces generated by straight lines tangent to $S$ :
1. $VE$ Surfaces (Planar Patches): Formed by lines that pass through a vertex $V$ of $S$ and are tangent to an edge $E_k$ of $S$ (or part of it) without intersecting the interior of $S$ . These surfaces are planar sheets.
2. $EEE$ Surfaces (Ruled Quadric Patches): Formed by lines tangent to three mutually skew straight edges $E_i, E_j, E_k$ $E_{i}, E_{j}, E_{k}$ of $S$ $S$ without intersecting the interior of $S$ $S$ . These lines sweep out a ruled quadric surface:
  - A hyperboloid of one sheet when the three edges are mutually skew and not all parallel to a single plane.
  - A hyperbolic paraboloid when the three edges are parallel to a common plane.
Consequently, the boundary $vh(S)$ of the visual hull of a polyhedron is a piecewise smooth surface composed of planar and quadric patches.
Knowl 5 — Polyhedral Silhouette-Active Surface Characterization via 2-D Cross Sections
theoretical result
Let $S \subset \mathbb{E}^3$ be a polyhedron with planar faces $\{F_i\}$ . For each face $F_i$ , let $p_i$ be the plane supporting $F_i$ , and let $PS_i = (p_i \cap S) \setminus F_i$ be the 2-D polygonal cross-section of $S$ in plane $p_i$ , excluding face $F_i$ itself.

A point $Q$ on face $F_i$ belongs to the silhouette-active surface $sa(S)$ if and only if $Q$ does not belong to the 2-D visual hull of $PS_i$ within plane $p_i$ : $Q \in F_i \cap sa(S) \iff Q \notin VH(PS_i)$

This equivalence holds because $Q \in sa(S)$ if and only if there exists a visual line passing outside $S$ at an infinitesimal distance from $Q$ . In the limit, this visual line must lie in the supporting plane $p_i$ and not intersect $PS_i$ , which is precisely the condition that $Q \notin VH(PS_i)$ .
Knowl 6 — Algorithm for 2-D Visual Hull Computation
algorithm
The 2-D visual hull algorithm computes $VH(SP)$ for a set of planar polygons $SP$ with $n$ total vertices. A point $Q \in \mathbb{E}^2$ belongs to $VH(SP)$ if and only if its 2-D visual number $VN(Q, SP) = 0$ , defined as the number of disconnected families of visual lines passing through $Q$ that do not intersect $SP$ .
Input: Set of planar polygons $SP$ with $n$ vertices
Output: 2-D visual hull $VH(SP)$
1. Find all active lines:
   a. Identify candidate bitangent lines passing through pairs of vertices $(V_i, V_j)$ that are tangent to $SP$ at two points and do not cross $SP$ locally.
   b. Prune lines that cross $SP$ at either vertex or lie entirely outside the convex hull $CH(SP)$ .
   c. For remaining lines, test intersection against the $O(n)$ edges of $SP$ to extract active segments (segments where crossing changes $VN$ by $+1$ or $-1$ ).
2. Construct the planar arrangement / partition of $\mathbb{E}^2$ induced by the active segments and the edges of $SP$ using a plane sweep algorithm.
3. Traverse the planar partition via a dual-graph depth-first search:
   a. Start at a known region inside $SP$ where $VN = 0$ .
   b. Cross between adjacent regions across active segments, incrementing or decrementing $VN$ according to the segment crossing orientation.
4. Merge all adjacent regions having visual number $VN = 0$ to obtain $VH(SP)$ .
Complexity:
- Finding active segments takes $O(n^3)$ time ( $O(n^2)$ vertex pairs tested against $O(n)$ polygon edges) and yields $O(n^2)$ active segments.
- Constructing the planar partition takes $O(m \log m)$ time, where $m$ is the number of vertices in the partition ( $m = O(n^4)$ in the worst case).
- Partition traversal and merging take $O(m)$ time.
- Total time complexity: $O(n^3 + m \log m)$ .
Knowl 7 — Algorithm for 2-D Internal Visual Hull Computation
algorithm
The 2-D internal visual hull algorithm computes $IVH(SP)$ for a set of planar polygons $SP$ with $n$ total vertices. A point $Q \in \mathbb{E}^2$ belongs to $IVH(SP)$ if and only if its internal visual number $IVN(Q, SP) = 0$ , defined as the number of disconnected families of visual half-lines starting at $Q$ that do not intersect $SP$ .
Input: Set of planar polygons $SP$ with $n$ vertices
Output: 2-D internal visual hull $IVH(SP)$
1. Compute active segments with respect to $IVH$ :
   a. Identify candidate lines tangent to $SP$ at two points or tangent to a vertex and an edge.
   b. If all active lines belong to configurations that cannot bound $IVH$ , terminate and output $IVH(SP) = SP$ .
   c. Otherwise, extract the active segments that separate regions whose $IVN$ differs by 1.
2. Construct the planar partition generated by the active segments and all edges of $SP$ using a plane sweep algorithm.
3. Compute the starting $IVN$ for an initial region containing a point $P$ :
   a. Connect $P$ to all $n$ vertices $V_i$ of $SP$ with straight lines.
   b. Count the number of visual lines passing through $P$ tangent to $SP$ at $V_i$ that bound families of internal visual lines, computing the starting $IVN$ in $O(n^2)$ time.
4. Traverse the dual graph of the planar partition:
   a. Propagate $IVN$ values to adjacent regions by adding or subtracting 1 according to the crossing orientation of each active segment.
   b. Do not cross boundary edges of $SP$ during traversal (evaluating separate connected components of $\mathbb{E}^2 \setminus SP$ independently).
5. Merge all regions with $IVN = 0$ to form $IVH(SP)$ .
Complexity:
- Active segment computation takes $O(n^3)$ time.
- Planar sweep and partition construction takes $O(m \log m)$ time, where $m$ is the number of vertices in the partition.
- Traversal and merging take $O(m)$ time.
- Total time complexity: $O(n^3 + m \log m)$ .
Knowl 8 — Algorithm for Computing Silhouette-Active Surfaces of a Polyhedron
algorithm
The silhouette-active surface $sa(S)$ of a 3-D polyhedron $S$ with $K$ planar faces $\{F_1, \dots, F_K\}$ and $n$ total vertices is computed face-by-face using 2-D visual hull operations.
Input: Polyhedron $S$ with planar faces $F_1, \dots, F_K$ and $n$ total vertices
Output: Silhouette-active surface $sa(S) \subseteq \partial S$
1. Compute the convex hull $CH(S)$ in $O(n \log n)$ time. Any face $F_i$ belonging to $\partial CH(S)$ is immediately added to $sa(S)$ .
2. For each remaining face $F_i$ :
   a. Compute the plane $p_i$ supporting $F_i$ .
   b. Compute the 2-D polygonal cross-section $PS_i = (p_i \cap S) \setminus F_i$ in $O(n)$ time.
   c. Apply the 2-D visual hull algorithm to compute $VH(PS_i)$ within plane $p_i$ , restricting active segments and sweep partition to the region enclosed by $F_i$ .
   d. Set $sa(S) \cap F_i = F_i \setminus VH(PS_i)$ .
3. Output the union of all computed face active regions: $\bigcup_{i=1}^K (sa(S) \cap F_i)$ .
Complexity:
- For each face $F_i$ , the 2-D visual hull computation takes $O(n^3 + k_i \log k_i)$ time, where $k_i$ is the size of the planar partition on face $F_i$ .
- Summing over all $O(n)$ faces, the overall time complexity is $O(n^4 + n p)$ , where $p$ is the average value of $k_i \log k_i$ across all faces.
Knowl 9 — Algorithm for 3-D Polyhedral Visual Hull and Internal Visual Hull Computation
algorithm
The 3-D visual hull $VH(S)$ and internal visual hull $IVH(S)$ of a general polyhedron $S$ with $n$ edges and vertices are computed by constructing the 3-D spatial arrangement induced by active surfaces and evaluating visual numbers in each cell.

For a point $Q \notin S$ :
- The 3-D visual number $VN3D(Q, S)$ is the number of faces of the 3-D visual cone of $Q$ relative to $S$ . A point $Q \in VH(S)$ if and only if $VN3D(Q, S) = 0$ .
- The internal 3-D visual number $IVN3D(Q, S)$ is the number of faces of the circumscribed visual half-cone from $Q$ relative to $S$ . A point $Q \in IVH(S)$ if and only if $IVN3D(Q, S) = 0$ .
Input: Polyhedron $S$ with $n$ vertices and edges
Output: 3-D visual hull $VH(S)$ (or $IVH(S)$ )
1. Compute all potentially active surfaces:
   a. Generate $O(n^2)$ vertex-edge ( $VE$ ) planar surfaces.
   b. Generate $O(n^3)$ edge-edge-edge ( $EEE$ ) ruled quadric surfaces.
   Total number of active surfaces is $N_s = O(n^3)$ .
2. Construct the 3-D spatial arrangement of $\mathbb{E}^3$ generated by the $O(n^3)$ algebraic surfaces using an incremental algorithm.
   - The partition contains $O(N_s^3) = O(n^9)$ cells, faces, edges, and vertices.
   - Partition construction runs in $O(n^9 \log n)$ time.
3. For each 3-D cell $C$ in the partition:
   a. Select an interior sample point $Q \in C$ .
   b. For each of the $O(n)$ potential visual angles of $S$ , compute the 2-D polygonal intersection $INT_i$ with $S$ in $O(n^2)$ time.
   c. Compute the 2-D visual number of $Q$ relative to $INT_i$ in $O(n^2)$ time.
   d. Sum the 2-D visual numbers over all visual angles to compute $VN3D(Q, S)$ (or $IVN3D(Q, S)$ ) in $O(n^3)$ time per cell.
4. Merge all cells in the 3-D arrangement where $VN3D(Q, S) = 0$ (or $IVN3D(Q, S) = 0$ ) to form $VH(S)$ (or $IVH(S)$ ).
Complexity: Evaluating $O(n^9)$ cells at $O(n^3)$ time per cell yields an overall worst-case computational complexity of $O(n^{12})$ for both $VH(S)$ and $IVH(S)$ .

Coverage note — The brief discussion linking active surfaces to visual event surfaces in polyhedral aspect graphs was omitted as it serves as a conceptual connection to external literature rather than a standalone technical contribution.

References

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Citation

MLA

Laurentini, A. “The Visual Hull Concept for Silhouette-based Image Understanding”. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 2, 1994, pp. 150–62, https://doi.org/10.1109/34.273735.

APA

Laurentini, A. (1994). The visual hull concept for silhouette-based image understanding. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(2), 150–162. https://doi.org/10.1109/34.273735

Chicago

Laurentini, A. 1994. “The Visual Hull Concept for Silhouette-based Image Understanding”. IEEE Transactions on Pattern Analysis and Machine Intelligence 16 (2): 150–62. https://doi.org/10.1109/34.273735.

Harvard

Laurentini, A. (1994) “The visual hull concept for silhouette-based image understanding”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(2), pp. 150–162. Available at: https://doi.org/10.1109/34.273735.

Vancouver

1. Laurentini A (1994) The visual hull concept for silhouette-based image understanding. IEEE Transactions on Pattern Analysis and Machine Intelligence 16:150–162

BibTeX

@article{Laurentini_1994, title={The visual hull concept for silhouette-based image understanding}, volume={16}, ISSN={0162-8828}, url={http://dx.doi.org/10.1109/34.273735}, DOI={10.1109/34.273735}, number={2}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Laurentini, A.}, year={1994}, pages={150–162} }

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The Visual Hull Concept for Silhouette-Based Image Understanding

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Visual Hull Relative to a Viewing Region

Knowl 2 — External Visual Hull and Line Intersection Characterization

Knowl 3 — Internal Visual Hull and Half-Line Characterization

Knowl 4 — Geometric Boundary Structure of 3-D Polyhedral Visual Hulls

Knowl 5 — Polyhedral Silhouette-Active Surface Characterization via 2-D Cross Sections

Knowl 6 — Algorithm for 2-D Visual Hull Computation

Knowl 7 — Algorithm for 2-D Internal Visual Hull Computation

Knowl 8 — Algorithm for Computing Silhouette-Active Surfaces of a Polyhedron

Knowl 9 — Algorithm for 3-D Polyhedral Visual Hull and Internal Visual Hull Computation

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