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                    Graphics II 91.547


                  Image Based Rendering
                          Session 11
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          A Rendering Taxonomy
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          The Plenoptic Function




                                 p  P(, ,  ,Vx ,Vy ,Vz , t )

                    “… the pencil of rays visible from any point in space,
                    at any time, and over any range of wavelengths”


                  Given a set of discrete samples (complete or incomplete)
                  from the plenoptic function, the goal of image-based
                  rendering is to generate a continuous representation of
                  that function.
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          Movie Map
          (Lippman 1980)

                                         Image




                                      Find Nearest
                  Vx ,Vy ,Vz , ,       Sample       Movie
                                                     Storage
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          Taxonomy of “Virtual Camera” Movement
          (Chen et al. 1995)
                  0 Camera Rotation
                     - Camera fixed at a particular location
                     - Three rotational degrees of freedom
                         =Pitch (up and down)
                         =Yaw (about vertical axis)
                         =Roll (about camera axis)
                  0 Object Rotation
                     - Camera always pointing at center of object
                     - Viewpoint constrained to move over surface of sphere
                     - Three angular degrees of freedom
                  0 Camera movement
                     - Viewpoint unconstrained
                     - Viewing direction unconstrained
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          Environment Maps
          Map Geometries




                  Cube       Sphere   Cylinder
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          Quick Time VR    TM


          (Chen 1995)
                                     2500 Pixels




                                              768 Pixels
                  2500 x 768 = 1.9 G Pixels x 3 B/pixel = 5.8 GB
                  10:1 compression       500 MB/panorama
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          Image Distortion from
          Cylindrical Environment Map

                  Projection Plane


                                        Pre-warped
                                        Projection onto
                                        Plane




                  Cylindrical
                  Environment
                  Map
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          Quick Time VR
          Image Warping for Correct Perspective View
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          Quick Time VR
          Panoramic Display Process
                              Compressed Tiles




                                                                CD ROM or
                                                                Hard Disk
                                            Visible Tiles


                               Compressed
                                  Tiles                         Main Memory
                                 Cache

                                      Decompress


                  Offscreen                      Warp       Display
                                Visible
                  Buffer                                    Window
                                Region
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          Quick Time VR
          Accomplishing (Limited) Camera Motion
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          Accomplishing Camera Motion
          Greene&Kass (1993 Apple Tech Doc.)
                  0 Regular 3-D lattice of cubic environment maps
                  0 Each environment map is a z-buffered rendering from a
                      discrete viewpoint
                  0   Image from a new viewpoint is generated by re-sampling the
                      environment map
                  0   Re-sampling involves rendering the pixels in the environment
                      maps as 3-D polygons from the new viewpoint
                  0   Rendering time proportional to the environment map
                      resolution but independent of scene complexity
                  0   Not suitable for real-time walkthrough performance on typical
                      desktop computers (especially in 1993!)
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          Alternative approach:
          Work entirely in Image Space
                  0 Sequence of images from closely spaced viewpoints is highly
                    coherent
                  0 Depends upon the ability to establish a pixel-by-pixel
                    correspondence between adjacent images
                      - Can be computed if range data and camera parameters
                        are known (true for rendered images)
                      - For natural images, there are several techniques including
                        manual user intervention
                  0 Pairwise correspondence between two images can be stored
                    as a pair of morph maps
                      - Bi-directional maps required because of possible many to
                        one and one to many pixel correspondences
                  0 Can be represented by graph data structure where nodes are
                    images and arcs are bi-directional morph maps
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          N-Dimensional Graph Data Structure
                              Bi-directional Morph Maps



                  Image    Image      Image        Image    Image




                  Image    Image       Image        Image   Image




                   Image    Image       Image       Image    Image
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          Simple View Interpolation

                   Reference Image 1   Reference Image 2

                                                           Corresponding
                                                           Pixels




                                                            Morph maps
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          Image Overlap or
          Image Folding
                                 P1
                                      P2




                     Reference             Interpolated
                       View                    View
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          Image Holes or
          Image Stretching
                                       P2
                                  P1




                      Reference             Interpolated
                        View                    View
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          Example of Hole Region




                         Viewpoint 1   Viewpoint 2
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          Example of Hole Region
          Minimizing by Closely Spaced Viewpoints




                         Viewpoint 1   Viewpoint 2
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          Source Image Viewed from
          Camera Moved to the Right
                  Ref. View 1




                                      Ref. View 2
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          Offset Vectors for Camera Motion Morph Map
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          Locus of Morph Map for Motion
          Parallel to Image Plane and Floor
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          Distortion of Intermediate Images with
          Linear Warp


                              Linear path of one feature
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          Morphing Parallel Views


                       Reference image


                        V0
                                                 Interpolated image




                                                          Reference image
                                Ci

                                                            V1


                               Vi  (1  s) V0  sV1
                               p i  (1  s)p 0  sp1
                               p i  Vi P
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          View Interpolation:
          The Algorithm



                                I0         Is         I1

                                1            3               1


                                I0 '       Is '       I1 '
                                       2          2
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          Example 1 of calculated intermediate images


            Reference Image 1                  Reference Image 2




                                Intermediate
                                    Views
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          Example 2 of calculated intermediate images

                  Reference Image 1   Interpolated Image   Reference Image 2
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          Multiple-Center-of-Projection Images
          (Rademacher&Bishop 1998)
                  0 Information from a set of viewpoints stored in a single image
                  0 Features
                     - Greater connectivity information compared with
                       collections of standard images
                     - Greater flexibility in the acquisition of image-based
                       datasets, e.g. sampling different portions of the scene at
                       different resolutions
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          Multiple-Center-of-Projection Images
          Definition
                  0 A multiple-center-of-projection image consists of a two-
                   dimensional image and a parameterized set of cameras
                   meeting the following conditions:
                     - The cameras must lie on either a continuous curve or a
                       continuous surface
                     - Each pixel is acquired by a single camera
                     - Viewing rays vary continuously across neighboring pixels
                     - Two neighboring pixels must either correspond to the
                       same camera or to neighboring cameras
                     - Each pixel contains range information
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          MCOP Image
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          Strip Camera used for Capture of
          Real MCOP Images
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          Camera Path in Capturing
          MCOP Image of Castle
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          Image Plane for Camera Motion
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          Resulting 1000 x 500 MCOP Image
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          Reprojection

                      Camera model, stored per column:

                           Ci Center of projection

                           O i Vector from Ci to image plane origin

                           U i Horizontal axis of viewing plane

                           Vi Vertical axis of viewing plane

                           ij Disparity = distance from Ci to the image plane
                               divided by distance from Ci to the pixel’s world
                                 space point



                                   x        U ix   Vix   Oix   i  Cix 
                  Reprojection      y  1   U      Viy   Oiy   j   Ciy 
                  Formula:            ij    iy                 
                                   z
                                            U iz
                                                     Viz   Oiz  1 Ciz 
                                                                   
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          View of Castle
          Reconstructed from MCOP Image
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          AnotherView of Castle
          Reconstructed from MCOP Image
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          Lumigraphs
                  0 Lumigraph = a representation of the light resulting from a
                      scene
                  0   Limited data representation of the plenoptic function
                  0   Generated from multiple images and camera “poses”
                  0   Rendering: Image = Lumigraph + Camera Model
                  0   Special case of 4D Light Field (Levoy, Hanrahan)
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          What is a Lumigraph?

                   For all points on the surrounding surface,
                          For all directions,
                                 The color intensity of the ray.




                  Assumption: We are outside a convex hull containing
                       the objects
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          Parameterization of the Lumigraph




         Images from Steven Gortler, SIGGRAPH 1999
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          Building the Lumigraph
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          Approximating the Lumigraph
          With Discrete Samples
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          Views of a Light Field (Lumigraph)




      Levoy & Hanrahan, Light Field Rendering, Computer Graphics

				
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