Splatting

1
Splatting

• Jian Huang, CS 594, Spring 2002
• This set of slides reference slides made by Ohio
  State University alumuni over the past several
  years.

2
Volumetric Ray Integration
color
opacity
object (color, opacity)
3
Splatting

• Lee Westover - Vis 1989 SIGGRAPH 1990
• Object order method
• Front-To-Back or Back-To-Front
• Original method - fast, poor quality
• Many many improvements since then!
• Crawfis93 textured splats
• Swan96, Mueller97 anti-aliasing
• Mueller98 image-aligned sheet-based splatting
• Mueller99 post-classified splatting
• Huang00 new splat primitive FastSplats

4
Splatting

• Volume field of 3D interpolation kernel
• One kernel at each grid voxel
• Each kernel leaves a 2D footprint on screen
• Voxel contribution footprint (C, opacity)
• Weighted footprints accumulate into image

screen footprints splats
voxel kernels
screen
5
Splatting

• Volume field of 3D interpolation kernel
• One kernel at each grid voxel
• Each kernel leaves a 2D footprint on screen
• Voxel contribution footprint (C, opacity)
• Weighted footprints accumulate into image

screen footprints splats
voxel kernels
screen
6
Splatting

• Volume field of 3D interpolation kernel
• One kernel at each grid voxel
• Each kernel leaves a 2D footprint on screen
• Voxel contribution footprint (C, opacity)
• Weighted footprints accumulate into image

screen footprints splats
voxel kernels
screen
7
Splatting

• Volume field of 3D interpolation kernel
• One kernel at each grid voxel
• Each kernel leaves a 2D footprint on screen
• Voxel contribution footprint (C, opacity)
• Weighted footprints accumulate into image

screen footprints splats
voxel kernels
screen
8
Ray-casting - revisited
Interpolationkernel
volumetric compositing
color c c s ?s(1 - ?) c
opacity ? ? s (1 - ?) ?
object (color, opacity)
9
Ray-casting - revisited

• (ideally) we would reconstruct the continuous
  volume (cloud) using the interpolation kernel h
• the we would compute the analytic integral along
  a ray r
• this can only be approximated by discretization

(hey! Which optical model is this equation??)
10
Splatting - principal idea

• This last equation
• can be rewritten in the following way

Splatting Kernel or Splat

• Which can be computed analytically known as
  footprint

11
Footprint Extent
Approximate the 3D kernel (h(x,y,z))extent by a
sphere
12
Footprint Table
A popular kernel is a three-dimensional Gaussian
(radially symmetric) As 1D integration of 3D
Gaussian is still a 2D Gaussian we can just
skip the Z integration and evaluate the
Gaussian function on 2D image space after voxel
projection
Generic footprint table
preprocessing
13
View-dependent footprint
It is possible to transform a sphere kernel into
A ellipsoid

• The projection of an
• ellipsoid is an ellipse
• We need to transform the
• generic footprint table
• to the ellipse

14
View-dependent footprint (2)
15
Example Footprint at Different Resolutions
16
Footprint - principal idea

• Draw each voxel as a cloud of points (footprint)
  that spreads the voxel contribution across
  multiple pixels.
• Larger footprint -gt larger spatial kernel extent
  -gt lower frequency components -gt more blurring
• Large pixel/voxel ratio

17
Rendering a Splat

• Use texture mapping hardware to resample
  footprint table (either single density channel or
  separate classified r,g,b,a channels)
• Or, use FastSplats to render each splat as a
  graphics primitive of itself

18
Splatting - efficiency

• footprint - splatted (integrated) kernel
• if interpolation kernel is isotropic (spherical)
  then its footprint is independent of the view
  point (for orthographic viewing)
• for perspective - footprint can be approximated
  with an ellipse
• Hence, for common cases, we can pre-integrate it
  (efficient!)
• for perspective projection, to approximate, we
  have to compute the orientation of the ellipse

19
Splatting - Highlights

• Footprints can be pre-integrated
• fast voxel projection
• Advantages over ray-casting
• Fast voxel interpolation is in 2D on screen
• More accurate integration (analytic for X-ray)
• More accurate reconstruction (afford better
  kernels)
• Only relevant voxels must be projected

20
Early Implementation Axis Aligned Splatting

• Voxel kernels are added within axis-aligned
  sheets
• Sheets are composited front-to-back
• Sheets volume slices most perpendicular to the
  image plane

volume slices
volume slices
image plane at 70
image plane at 30
21
Early Implementation Axis Aligned Splatting

• Volume

volume slices
sheet buffer
image plane
compositing buffer
22
Early Implementation Axis Aligned Splatting

• Add voxel kernels within first sheet

volume slices
sheet buffer
image plane
compositing buffer
23
Early Implementation Axis Aligned Splatting

• Transfer to compositing buffer

volume slices
sheet buffer
image plane
compositing buffer
24
Early Implementation Axis Aligned Splatting

• Add voxel kernels within second sheet

volume slices
sheet buffer
image plane
compositing buffer
25
Early Implementation Axis Aligned Splatting

• Composite sheet with compositing buffer

volume slices
sheet buffer
image plane
compositing buffer
26
Early Implementation Axis Aligned Splatting

• Add voxel kernels within third sheet

volume slices
sheet buffer
image plane
compositing buffer
27
Early Implementation Axis Aligned Splatting

• Composite sheet with compositing buffer

volume slices
sheet buffer
image plane
compositing buffer
28
What Doesnt Work?

• Mathematically, the early splatting methods only
  work for X-ray type of rendering, where voxel
  ordering is not important
• Bad approximation for other types of optical
  models
• Object ordering is important in volume rendering,
  front objects hide back objects
• need to composite splats in proper order, else we
  get bleeding of background objects into the image
  (color bleeding!)
• Axis- aligned approach add all splats that fall
  within a volume slice most parallel to the image
  plane, composite these sheets in front- to- back
  order
• Incorrect accumulating on axis-aligned face cause
  popping
• A better approximation with Riemann sum is to use
  the image-aligned sheet-based approach

29
Problems Early Implementation Axis Aligned
Splatting

• In-accurate compositing, result in color bleeding
  and popping artifacts (Demo)!

Part of this voxel
gets composited beforepart of this voxel
30
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
31
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
32
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
33
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
34
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
35
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
36
Image-Aligned Sheet-Buffer

• Slicing slab cuts kernels into sections
• Kernel sections are added into sheet-buffer
• Sheet-buffers are composited

sheet buffer
image plane
compositing buffer
37
Image-Aligned Splatting

• Note We need an array of footprint tables now. A
  separate footprint table for each slice of the 3D
  reconstruction kernel.

38
Volume Rendering Pipeline Two Variations
39
Volume Rendering Pipeline Two Variations
40
IASB Splatting

• No popping or color bleeding
• Sharp, noise-free images

41
Occlusion Culling

• A voxel is only visible if the volume material in
  front is not opaque

screen
occluded voxel does not pass visibility test
wall of occluding voxels
occlusion map opacity image
42
Visibility Test Based on SAT of Occlusion Buffer

• Compute occlusion map after each sheet
• Cull both individual voxel and voxel sets with a
  summed area table of occlusion map

Do not project
Project
opacity ? threshold
opacity lt threshold
occlusion map
opacity 0
43
Occlusion Culling

• Build a summed area table (SAT) from the opacity
  buffer
• To test whether a rectangular region is opaque or
  not, check the four corners
• Can cull voxel sets directly

44
Anti-aliasing

• Needed to preserve small features
• Needed for the diverging rays in perspective
• In splatting, resize the footprint according to
  depth

Aliased anti-aliased
45
Motion Blur

• Stretch the reconstruction kernel in the
  direction of movement.
• Stretch the splat footprint in the direction of
  the projected movement (2D).

46
Camera Depth-of-Field

• Two possible approaches
• Low-pass filter the splats
• Low-pass filter the sheets

Plain with Depth-of-Field
47
Procedural Textures

• Easily done with pre-coloring
• Per-pixel

48
Bump-Mapping

• If calculating the normal per-pixel, we can
  modulate it to achieve bump mapping.

49
Per-pixel Classification

• Per-pixel classification can be based on gray
  scale, position, gradient, or ...

7.25 sec
9.41 sec (procedural)
7.99 sec
50
Scan-Converting A Splat

• Scan-convert an arbitrary-size radially symmetric
  2D function centered at arbitrary position
• circular or elliptical
• Texture mapping hardware is not the solution
• We want a hardware accelerated splat or point
  primitive

51
Fast Splats FastSplat

• We desire
• fast scan conversion
• minimum or controllable errors
• compact storage
• simple integer operation

52
FastSplats

• 1D Linear
• 1D Squared
• 2D
• 1D with Radius Look Up Table (RLUT)

53
1D Linear, 1D Squared FastSplats

• On the radial line

1D Linear FastSplat, indexed by rx,y
rx,y
1D Squared FastSplat, indexed by r2x,y
(x,y)
(xo,yo)
(x1,y)
54
1D Squared FastSplat (Elliptical)

• For elliptical kernels, if we define a canonical
  radius
• The incremental scan-conversion still works at
  the same low cost

55
FastSplats

• 1D Linear
• 1D Squared
• 2D
• 1D with Radius Look Up Table (RLUT)

56
2D FastSplat (BitBlt,VoxBlt)

• No run-time computation

Pre-rasterized footprint with center at (xo,yo),
radius r
Pre-rasterized footprints for all possible center
positions, radius r
Snap splat center to a k by k sub-pixel grid
57
2D FastSplats

• No run-time computation

Pre-rasterized footprints for all possible center
positions, for all possible radii
Snap splat center to a k by k sub-pixel grid
Allow for a set of radius values
58
2D FastSplats (2)

• The storage need
• When storage is limited, the quality is limited
  too
• Mora00, similar

59
FastSplats

• 1D Linear
• 1D Squared
• 2D
• 1D with Radius Look Up Table (RLUT)

60
1D RLUT FastSplat

• For hardware, we need finer parallelism than
  scan-line

61
1D FastSplat with RLUT
RLUT
62
1D FastSplat with RLUT

• At a k by k subpixel precision

63
1D FastSplat with RLUT

• At a k by k subpixel precision

64
1D FastSplat with RLUT

• At a k by k subpixel precision

65
1D FastSplat with RLUT

• At a k by k subpixel precision

66
1D FastSplat with RLUT

• At a k by k subpixel precision

67
1D FastSplat with RLUT

• At a k by k subpixel precision

68
1D FastSplat with RLUT

• At a k by k subpixel precision

69
1D FastSplat with RLUT

• At a k by k subpixel precision

70
1D FastSplat with RLUT

• At a k by k subpixel precision

71
1D FastSplat with RLUT

• At a k by k subpixel precision

72
1D FastSplat with RLUT

• At a k by k subpixel precision

73
1D FastSplat with RLUT

• At a k by k subpixel precision

74
1D FastSplat with RLUT

• At a k by k subpixel precision

75
1D FastSplat with RLUT

• At a k by k subpixel precision

76
1D FastSplat with RLUT

• At a k by k subpixel precision

77
1D FastSplat with RLUT

• At a k by k subpixel precision

78
1D FastSplat with RLUT

• k by k subpixel precision

79
1D FastSplat with RLUT
x or y offset

• Due to symmetry, the RLUT set for x component is
  the same as the RLUT set for the y component
• one RLUT set
• k 1D tables
• each of splat_extent length

k
splat_extent
80
Comparisons Among the FastSplats

• 1D Linear very accurate, compact, slow
• 1D Squared accurate, compact, fast
• 1D RLUT accurate, compact, intended for hardware
• 2D FastSplat can be very fast, accurate and
  compact under constrained conditions