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Hierarchical Volume Rendering. …. 8x8x8 4x4x4 2x2x2 …. Big Data Problem …. One way to handle the big data problem is to use hierarchical data structures (hierarchical volumes). Less amount of data are required. Issues.

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Presentation Transcript
big data problem

8x8x8 4x4x4 2x2x2 …

Big Data Problem …
  • One way to handle the big data problem is to use hierarchical data structures (hierarchical volumes)

Less amount of data are required

issues
Issues
  • Creation of hierarchical data structures
  • Utilization of hierarchical data structures
hierarchical volume

Entire volume

Half in each dim

Hierarchical Volume
  • A hierarchical volume can be constructed using the octree
octrees
Octrees
  • An octree can be full or non-full
    • Full octree – each parent node has exact 8 children
    • Condition: all the dimensions are the same power of 2
    • A full octree example: 16 x 16 x 16
      • Level 1: (8) 8 x 8 x 8; level 2: (8) 4 x 4 x 4

Level 3: (8) 2 x 2 x 2; level 4: (8) 1 x1 x 1

octrees 2
Octrees (2)
  • For a volume with resolution s in each dimension, the total number of tree nodes is:

8^0 + 8 ^ 1 + … + 8 ^ (log S –1) = (s ^ 3 –1 )/7

  • Node to data ratio: (S^3 –1 ) / 7*S^3 = 0.1428

this is the optimal ratio

  • If a volume can not be represented by a full octree, the node to data ratio will increase -> overhead increases

2

octrees 3
Octrees (3)
  • Other benefit of full octree - Can be stored in a linear array without using pointers
    • Each parent has exact 8 children – their positions in an array are totally predictable
    • For a parent stored at T[k], then its eight children are stored at T[8k+1], T[8k+2], …, T[8k+8].
    • All the nodes at the same level are stored in a contiguous memory space
octrees 4
Octrees (4)
  • Power of 2 volume is not a norm
  • Full-octree thus is not always possible
    • A non-full octree example: 16 x 8 x 4
      • Level 1: (8) 8 x 4 x 2; level 2: (4) 4 x 2 x 1

Level 3: (4) 2 x 1 x 1; level 4: (2) 1 x1 x 1

  • Total number of nodes: 1 + 8 +8* 4 + 8*4*4 = 161
  • The node to data ratio = 161 / 512 = 0.314 > optimal ( 0.1428)
  • Also , we need to store the pointers from parents to children
octrees 5
Octrees (5)
  • In general, there are two ways to construct an octree for a volume
    • Top-down:

1 -> 8 -> 8x8 -> 8x8x8 -> 8x8x8x4 -> ….

-> n (n = total number of data)

    • Buttom-up:

n -> n / 8 -> n / 8x8 -> n/8x8x8 -> n/8x8x8x4

-> … -> 1

  • Which one has a better node/data ratio?
branch on need bon octrees
Branch-on-Need (BON) Octrees
  • A top-down implementation of bottom-up octrees
  • Overlay the volume with a conceptual all power of 2 volume
  • Subdivide the volume based on the concpetual volume
bono construction

(1)

(4)

(9)

(2) (2) (1) (4)(4)(4)(4) (2) (2)

BONO Construction

10

11

00

01

bono v s top down
BONO v.s top-down
  • Put more 8-way branches toward to the leaf nodes
  • In general, BONO requires a smaller number of tree nodes
octrees and visualization
Octrees and visualization
  • Isosurface Extraction
  • Each BONO node stores the min/max values of the corresponding subvolume
  • The empty node can be skipped rapidly
octrees for volume rendering
Octrees for Volume Rendering
  • Each tree node stores the mean value of the encompassed voxels
  • Alternatively, each tree node stores a subvolume of reduced resolution
  • Each tree node also stores an error measure: e.g. standard deviation of the voxels (root mean square to the approximated value)
octrees for volume rendering1
Octrees for Volume Rendering
  • At run time, the error measures stored in the octree nodes are compared with a threshold
  • The traversal is stopped at the tree nodes that pass the threshold

… …

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