Splatting
Jian Huang, CS 594, Spring 2002
This set of slides reference slides made by Ohio State
University alumuni over the past several years.
Volumetric Ray Integration
color
opacity
1.0
object (color, opacity)
Splatting
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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!
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Crawfis’93: textured splats
Swan’96, Mueller’97: anti-aliasing
Mueller’98: image-aligned sheet-based splatting
Mueller’99: post-classified splatting
Huang’00: new splat primitive: FastSplats
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
voxel kernels
screen footprints =
splats
screen
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
voxel kernels
screen footprints =
splats
screen
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
voxel kernels
screen footprints =
splats
screen
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
voxel kernels
screen footprints =
splats
screen
Ray-casting - revisited
Interpolation
kernel
volumetric compositing
color c = c s s(1 - ) + c
opacity  =  s (1 - ) + 
1.0
object (color, opacity)
Ray-casting - revisited
• (ideally) we would reconstruct the continuous
volume (cloud) using the interpolation kernel h:
f r (v )   h(v  vk ) f (vk )
k
• the we would compute the analytic integral along
a ray r: (hey! Which optical model is this equation??)
I ( p)   f r ( p  r )dr    h p  r  vk  f vk dr
k
• this can only be approximated by discretization
Splatting - principal idea
• This last equation
I ( p)    h p  r  vk  f vk dr
k
• can be rewritten in the following way:
I ( p)   f vk  h p  r  vk dr
k
Splatting Kernel or “Splat”
 Which can be computed analytically: known as footprint
Splat ( x, y )   hx, y, z dz
Footprint Extent
Approximate the 3D kernel (h(x,y,z))extent by
a sphere
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
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
View-dependent footprint (2)
Example Footprint at Different
Resolutions
Footprint - principal idea
• Draw each voxel as a cloud of
points (footprint) that spreads
the voxel contribution across
multiple pixels.
• Larger footprint -> larger spatial
kernel extent -> lower frequency
components -> more blurring
– Large pixel/voxel ratio
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
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
Splatting - Highlights
• Footprints can be pre-integrated
– fast voxel projection
• Advantages over ray-casting:
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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
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
z
y
x
image plane at 30°
image plane at 70°
Early Implementation – Axis
Aligned Splatting
• Volume
volume slices
sheet buffer
image plane
compositing buffer
Early Implementation – Axis
Aligned Splatting
• Add voxel kernels within first sheet
volume slices
sheet buffer
image plane
compositing buffer
Early Implementation – Axis
Aligned Splatting
• Transfer to compositing buffer
volume slices
sheet buffer
image plane
compositing buffer
Early Implementation – Axis
Aligned Splatting
• Add voxel kernels within second sheet
volume slices
sheet buffer
image plane
compositing buffer
Early Implementation – Axis
Aligned Splatting
• Composite sheet with compositing buffer
volume slices
sheet buffer
image plane
compositing buffer
Early Implementation – Axis
Aligned Splatting
• Add voxel kernels within third sheet
volume slices
sheet buffer
image plane
compositing buffer
Early Implementation – Axis
Aligned Splatting
• Composite sheet with compositing buffer
volume slices
sheet buffer
image plane
compositing buffer
What Doesn’t Work?
• Mathematically, the early splatting methods only work for Xray 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 imagealigned sheet-based approach
Problems Early Implementation –
Axis Aligned Splatting
• In-accurate compositing, result in color bleeding
and popping artifacts (Demo)!
Part of this voxel
gets composited before
part of this voxel
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
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
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
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
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
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
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
Image-Aligned Splatting
• Note: We need an array of footprint tables now. A
separate footprint table for each slice of the 3D
reconstruction kernel.
Volume Rendering Pipeline: Two Variations
Volume Rendering Pipeline: Two Variations
IASB Splatting
• No popping or color bleeding
• Sharp, noise-free images
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
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
occlusion map
opacity < threshold
opacity = 0
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
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
Motion Blur
• Stretch the reconstruction kernel in the direction
of movement.
• Stretch the splat footprint in the direction of the
projected movement (2D).
Camera Depth-of-Field
• Two possible approaches:
– Low-pass filter the splats
– Low-pass filter the sheets
Plain
with Depth-of-Field
Procedural Textures
• Easily done with pre-coloring
• Per-pixel
Bump-Mapping
• If calculating the normal per-pixel, we can
modulate it to achieve bump mapping.
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
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
Fast Splats
FastSplat
• We desire
– fast scan conversion
– minimum or controllable errors
– compact storage
– simple integer operation
FastSplats
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1D Linear
1D Squared
2D
1D with Radius Look Up Table (RLUT)
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) (x+1,y)
(xo,yo)
1D Squared FastSplat (Elliptical)
• For elliptical kernels, if we define a
canonical radius:
• The incremental scan-conversion still works
at the same low cost
FastSplats
 1D Linear
 1D Squared
 2D
• 1D with Radius Look Up Table (RLUT)
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 subpixel grid
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 subpixel grid
Allow for a set of radius values
2D FastSplats (2)
• The storage need:
• When storage is limited, the
quality is limited too
• [Mora’00], similar
FastSplats
 1D Linear
 1D Squared
 2D
• 1D with Radius Look Up Table (RLUT)
1D RLUT FastSplat
• For hardware, we need finer parallelism
than scan-line
1D FastSplat with RLUT
RLUT
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• At a k by k subpixel precision
1D FastSplat with RLUT
• k by k subpixel precision
1D FastSplat with RLUT
x or y offset
• Due to symmetry, the RLUT
set for x component is the
k
same as the RLUT set for the y
component
• one RLUT set
splat_extent
– k 1D tables
– each of splat_extent length
k
k
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