from images to voxels
DESCRIPTION
SIGGRAPH 2000 Course on 3D Photography. From Images to Voxels. Steve Seitz Carnegie Mellon University University of Washington http://www.cs.cmu.edu/~seitz. 3D Reconstruction from Calibrated Images. Scene Volume V. Input Images (Calibrated). - PowerPoint PPT PresentationTRANSCRIPT
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From Images to VoxelsFrom Images to Voxels
Steve SeitzSteve Seitz
Carnegie Mellon UniversityCarnegie Mellon University
University of WashingtonUniversity of Washingtonhttp://www.cs.cmu.edu/~seitzhttp://www.cs.cmu.edu/~seitz
SIGGRAPH 2000 Course onSIGGRAPH 2000 Course on3D Photography3D Photography
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3D Reconstruction from Calibrated Images3D Reconstruction from Calibrated Images
Scene VolumeScene Volume
VV
Input ImagesInput Images
(Calibrated)(Calibrated)
Goal: Goal: Determine transparency, radiance of points in VDetermine transparency, radiance of points in V
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Discrete Formulation: Voxel ColoringDiscrete Formulation: Voxel Coloring
Discretized Discretized
Scene VolumeScene Volume
Input ImagesInput Images
(Calibrated)(Calibrated)
Goal: Goal: Assign RGBA values to voxels in VAssign RGBA values to voxels in Vphoto-consistentphoto-consistent with images with images
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Complexity and ComputabilityComplexity and Computability
Discretized Discretized
Scene VolumeScene Volume
N voxelsN voxels
C colorsC colors
33
All Scenes (CN3)Photo-Consistent
Scenes
TrueScene
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IssuesIssues
Theoretical QuestionsTheoretical Questions• Identify class of Identify class of allall photo-consistent scenes photo-consistent scenes
Practical QuestionsPractical Questions• How do we compute photo-consistent models?How do we compute photo-consistent models?
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1. C=2 (silhouettes)1. C=2 (silhouettes)• Volume intersection [Martin 81, Szeliski 93]Volume intersection [Martin 81, Szeliski 93]
2. C unconstrained, viewpoint constraints2. C unconstrained, viewpoint constraints• Voxel coloring algorithm [Seitz & Dyer 97]Voxel coloring algorithm [Seitz & Dyer 97]
3. General Case3. General Case• Space carving [Kutulakos & Seitz 98]Space carving [Kutulakos & Seitz 98]
Voxel Coloring SolutionsVoxel Coloring Solutions
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Reconstruction from Silhouettes (C = 2)Reconstruction from Silhouettes (C = 2)
BinaryBinary ImagesImages
Approach: Approach: • BackprojectBackproject each silhouette each silhouette
• Intersect backprojected volumesIntersect backprojected volumes
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Volume IntersectionVolume Intersection
Reconstruction Contains the True SceneReconstruction Contains the True Scene• But is generally not the sameBut is generally not the same
• No concavitiesNo concavities
• In the limit get In the limit get visual hullvisual hull
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Voxel Algorithm for Volume IntersectionVoxel Algorithm for Volume Intersection
Color voxel black if on silhouette in every imageColor voxel black if on silhouette in every image• O(MNO(MN33), for M images, N), for M images, N33 voxels voxels
• Don’t have to search 2Don’t have to search 2NN33 possible scenes! possible scenes!
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Properties of Volume IntersectionProperties of Volume Intersection
ProsPros• Easy to implement, fastEasy to implement, fast
• Accelerated via octrees [Szeliski 1993]Accelerated via octrees [Szeliski 1993]
ConsCons• No concavitiesNo concavities
• Reconstruction is not photo-consistentReconstruction is not photo-consistent
• Requires identification of silhouettesRequires identification of silhouettes
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1. C=2 (silhouettes)1. C=2 (silhouettes)• Volume intersection [Martin 81, Szeliski 93]Volume intersection [Martin 81, Szeliski 93]
2. C unconstrained, viewpoint constraints2. C unconstrained, viewpoint constraints• Voxel coloring algorithm [Seitz & Dyer 97]Voxel coloring algorithm [Seitz & Dyer 97]
3. General Case3. General Case• Space carving [Kutulakos & Seitz 98]Space carving [Kutulakos & Seitz 98]
Voxel Coloring SolutionsVoxel Coloring Solutions
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1. Choose voxel1. Choose voxel2. Project and correlate2. Project and correlate3. Color if consistent3. Color if consistent
Voxel Coloring ApproachVoxel Coloring Approach
Visibility Problem: Visibility Problem: in which images is each voxel visible?in which images is each voxel visible?
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The Global Visibility ProblemThe Global Visibility Problem
Inverse VisibilityInverse Visibilityknown imagesknown images
Unknown SceneUnknown Scene
Which Points are Visible in Which Images?Which Points are Visible in Which Images?
Known SceneKnown Scene
Forward VisibilityForward Visibilityknown sceneknown scene
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LayersLayers
Depth Ordering: Visit Occluders First!Depth Ordering: Visit Occluders First!
SceneScene
TraversalTraversal
Condition: Condition: depth order is depth order is view-independentview-independent
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Panoramic Depth OrderingPanoramic Depth Ordering
• Cameras oriented in many different directionsCameras oriented in many different directions
• Planar depth ordering does not applyPlanar depth ordering does not apply
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Panoramic Depth OrderingPanoramic Depth Ordering
Layers radiate outwards from camerasLayers radiate outwards from cameras
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Panoramic LayeringPanoramic Layering
Layers radiate outwards from camerasLayers radiate outwards from cameras
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Panoramic LayeringPanoramic Layering
Layers radiate outwards from camerasLayers radiate outwards from cameras
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Compatible Camera ConfigurationsCompatible Camera Configurations
Depth-Order ConstraintDepth-Order Constraint• Scene outside convex hull of camera centersScene outside convex hull of camera centers
Outward-LookingOutward-Lookingcameras inside scenecameras inside scene
Inward-LookingInward-Lookingcameras above scenecameras above scene
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Calibrated Image AcquisitionCalibrated Image Acquisition
Calibrated TurntableCalibrated Turntable360° rotation (21 images)360° rotation (21 images)
Selected Dinosaur ImagesSelected Dinosaur Images
Selected Flower ImagesSelected Flower Images
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Voxel Coloring Results (Video)Voxel Coloring Results (Video)
Dinosaur ReconstructionDinosaur Reconstruction72 K 72 K voxels coloredvoxels colored7.6 M 7.6 M voxels testedvoxels tested7 min. 7 min. to compute to compute on a 250MHz SGIon a 250MHz SGI
Flower ReconstructionFlower Reconstruction70 K 70 K voxels coloredvoxels colored7.6 M 7.6 M voxels testedvoxels tested7 min. 7 min. to compute to compute on a 250MHz SGIon a 250MHz SGI
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Limitations of Depth OrderingLimitations of Depth Ordering
A View-Independent Depth Order May Not ExistA View-Independent Depth Order May Not Exist
p q
Need More Powerful General-Case AlgorithmsNeed More Powerful General-Case Algorithms• Unconstrained camera positionsUnconstrained camera positions
• Unconstrained scene geometry/topologyUnconstrained scene geometry/topology
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1. C=2 (silhouettes)1. C=2 (silhouettes)• Volume intersection [Martin 81, Szeliski 93]Volume intersection [Martin 81, Szeliski 93]
2. C unconstrained, viewpoint constraints2. C unconstrained, viewpoint constraints• Voxel coloring algorithm [Seitz & Dyer 97]Voxel coloring algorithm [Seitz & Dyer 97]
3. General Case3. General Case• Space carving [Kutulakos & Seitz 98]Space carving [Kutulakos & Seitz 98]
Voxel Coloring SolutionsVoxel Coloring Solutions
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Space Carving AlgorithmSpace Carving Algorithm
Space Carving AlgorithmSpace Carving Algorithm
Image 1 Image N
…...
• Initialize to a volume V containing the true sceneInitialize to a volume V containing the true scene
• Repeat until convergenceRepeat until convergence
• Choose a voxel on the current surfaceChoose a voxel on the current surface
• Carve if not photo-consistentCarve if not photo-consistent
• Project to visible input imagesProject to visible input images
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ConvergenceConvergence
Consistency PropertyConsistency Property• The resulting shape is photo-consistentThe resulting shape is photo-consistent
> all inconsistent voxels are removedall inconsistent voxels are removed
Convergence PropertyConvergence Property• Carving converges to a non-empty shapeCarving converges to a non-empty shape
> a point on the true scene is a point on the true scene is nevernever removed removed
V’
V
p
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Output of Space Carving: Photo HullOutput of Space Carving: Photo Hull
The The Photo HullPhoto Hull is the UNION of all photo-consistent scenes in V is the UNION of all photo-consistent scenes in V
• Tightest possible bound on the true sceneTightest possible bound on the true scene
True SceneTrue Scene
VV
Photo HullPhoto Hull
VV
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Space Carving AlgorithmSpace Carving Algorithm
The Basic Algorithm is UnwieldyThe Basic Algorithm is Unwieldy• Complex update procedureComplex update procedure
Alternative: Multi-Pass Plane SweepAlternative: Multi-Pass Plane Sweep• Efficient, can use texture-mapping hardwareEfficient, can use texture-mapping hardware
• Converges quickly in practiceConverges quickly in practice
• Easy to implementEasy to implement
Results Related Methods
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
True Scene Reconstruction
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Multi-Pass Plane SweepMulti-Pass Plane Sweep• Sweep plane in each of 6 principle directionsSweep plane in each of 6 principle directions
• Consider cameras on only one side of planeConsider cameras on only one side of plane
• Repeat until convergenceRepeat until convergence
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Other ApproachesOther Approaches
Level-Set Methods Level-Set Methods [Faugeras & Keriven 1998][Faugeras & Keriven 1998]• Evolve implicit function by solving PDE’sEvolve implicit function by solving PDE’s
Transparency and Matting Transparency and Matting [Szeliski & Golland 1998][Szeliski & Golland 1998]• Compute voxels with alpha-channelCompute voxels with alpha-channel
Max Flow/Min Cut Max Flow/Min Cut [Roy & Cox 1998][Roy & Cox 1998]• Graph theoretic formulationGraph theoretic formulation
Mesh-Based Stereo Mesh-Based Stereo [Fua & Leclerc 95][Fua & Leclerc 95]• Mesh-based but similar consistency formulationMesh-based but similar consistency formulation
Virtualized Reality Virtualized Reality [Narayan, Rander, Kanade 1998][Narayan, Rander, Kanade 1998]• Perform stereo 3 images at a time, merge resultsPerform stereo 3 images at a time, merge results
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Advantages of VoxelsAdvantages of Voxels• Non-parametricNon-parametric
> can model arbitrary geometrycan model arbitrary geometry
> can model arbitrary topologycan model arbitrary topology
• Good reconstruction algorithmsGood reconstruction algorithms
• Good rendering algorithms (splatting, LDI)Good rendering algorithms (splatting, LDI)
DisadvantagesDisadvantages• Expensive to process hi-res voxel gridsExpensive to process hi-res voxel grids
• Large number of parametersLarge number of parameters
> Simple scenes (e.g., planes) require lots of voxelsSimple scenes (e.g., planes) require lots of voxels
ConclusionsConclusions
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Volume IntersectionVolume Intersection• Martin & Aggarwal, “Volumetric description of objects from multiple views”, Martin & Aggarwal, “Volumetric description of objects from multiple views”,
Trans. Pattern Analysis and Machine Intelligence, 5(2), 1991, pp. 150-158.Trans. Pattern Analysis and Machine Intelligence, 5(2), 1991, pp. 150-158.
• Szeliski, “Rapid Octree Construction from Image Sequences”, Computer Vision, Szeliski, “Rapid Octree Construction from Image Sequences”, Computer Vision, Graphics, and Image Processing: Image Understanding, 58(1), 1993, pp. 23-32.Graphics, and Image Processing: Image Understanding, 58(1), 1993, pp. 23-32.
Voxel Coloring and Space CarvingVoxel Coloring and Space Carving• Seitz & Dyer, “Photorealistic Scene Reconstruction by Voxel Coloring”, Proc. Seitz & Dyer, “Photorealistic Scene Reconstruction by Voxel Coloring”, Proc.
Computer Vision and Pattern Recognition (CVPR), 1997, pp. 1067-1073.Computer Vision and Pattern Recognition (CVPR), 1997, pp. 1067-1073.
• Seitz & Kutulakos, “Plenoptic Image Editing”, Proc. Int. Conf. on Computer Seitz & Kutulakos, “Plenoptic Image Editing”, Proc. Int. Conf. on Computer Vision (ICCV), 1998, pp. 17-24.Vision (ICCV), 1998, pp. 17-24.
• Kutulakos & Seitz, “A Theory of Shape by Space Carving”, Proc. ICCV, 1998, pp. Kutulakos & Seitz, “A Theory of Shape by Space Carving”, Proc. ICCV, 1998, pp. 307-314.307-314.
BibliographyBibliography
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Related ReferencesRelated References• Faugeras & Keriven, “Variational principles, surface evolution, PDE's, level set Faugeras & Keriven, “Variational principles, surface evolution, PDE's, level set
methods and the stereo problem", IEEE Trans. on Image Processing, 7(3), 1998, methods and the stereo problem", IEEE Trans. on Image Processing, 7(3), 1998, pp. 336-344.pp. 336-344.
• Szeliski & Golland, “Stereo Matching with Transparency and Matting”, Proc. Int. Szeliski & Golland, “Stereo Matching with Transparency and Matting”, Proc. Int. Conf. on Computer Vision (ICCV), 1998, 517-524.Conf. on Computer Vision (ICCV), 1998, 517-524.
• Roy & Cox, “A Maximum-Flow Formulation of the N-camera Stereo Roy & Cox, “A Maximum-Flow Formulation of the N-camera Stereo Correspondence Problem”, Proc. ICCV, 1998, pp. 492-499.Correspondence Problem”, Proc. ICCV, 1998, pp. 492-499.
• Fua & Leclerc, “Object-centered surface reconstruction: Combining multi-Fua & Leclerc, “Object-centered surface reconstruction: Combining multi-image stereo and shading", Int. Journal of Computer Vision, 16, 1995, pp. 35-56.image stereo and shading", Int. Journal of Computer Vision, 16, 1995, pp. 35-56.
• Narayanan, Rander, & Kanade, “Constructing Virtual Worlds Using Dense Narayanan, Rander, & Kanade, “Constructing Virtual Worlds Using Dense Stereo”, Proc. ICCV, 1998, pp. 3-10.Stereo”, Proc. ICCV, 1998, pp. 3-10.
BibliographyBibliography