# Splatting Jian Huang, CS 594, Spring 2002 This

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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 object ( color , opacity )1.

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! – 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: Fast. Splats

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

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

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

Ray-casting — revisited color c = c s s (1 — ) + c opacity = s (1 — ) + 1. 0 object ( color , opacity )volumetric compositing. Interpolation kernel

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 k kkrdrvfvrphdrrpfp. I)()( k kkrvfvvhvf)()()( (hey! Which optical model is this equation? ? )

Splatting — principal idea • This last equation • can be rewritten in the following way: k kkdrvfvrphp. I)( k kkdrvrphvfp. I)( Splatting Kernel or “Splat” Which can be computed analytically : known as footprint dzzyxhyx , , ), (Splat

Footprint Extent Approximate the 3 D kernel (h(x, y, z))extent by a sphere

Footprint Table A popular kernel is a three-dimensional Gaussian (radially symmetric) As 1 D integration of 3 D Gaussian is still a 2 D Gaussian – we can just skip the Z integration and evaluate the Gaussian function on 2 D 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 Fast. Splats 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: – Fast: voxel interpolation is in 2 D 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 at 70° image plane at 30° volume slices xyz volume slices

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Volume

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Add voxel kernels within first sheet

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Transfer to compositing buffer

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Add voxel kernels within second sheet

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Composite sheet with compositing buffer

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Add voxel kernels within third sheet

Early Implementation – Axis Aligned Splatting sheet buffer compositing buffer volume slices image plane • Composite sheet with compositing buffer

What Doesn’t 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

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

Image-Aligned Sheet-Buffer sheet buffer compositing buffer • Slicing slab cuts kernels into sections • Kernel sections are added into sheet-buffer • Sheet-buffers are composited image plane

Image-Aligned Sheet-Buffer sheet buffer compositing buffer • Slicing slab cuts kernels into sections • Kernel sections are added into sheet-buffer • Sheet-buffers are composited image plane

Image-Aligned Splatting • Note: We need an array of footprint tables now. A separate footprint table for each slice of the 3 D 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 occluded voxel: does not pass visibility test wall of occluding voxels occlusion map = opacity image screen

Visibility Test Based on SAT of Occlusion Buffer opacity threshold occlusion map. Do not project Project opacity < threshold opacity = 0 • Compute occlusion map after each sheet • Cull both individual voxel and voxel sets with a summed area table of occlusion map

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 (2 D).

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 2 D 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 Fast. Splat • We desire – fast scan conversion – minimum or controllable errors – compact storage – simple integer operation

Fast. Splats 1 D Linear 1 D Squared 2 D 1 D with Radius Look Up Table (RLUT)

1 D Linear, 1 D Squared Fast. Splats • On the radial line 1 D Linear Fast. Splat, indexed by r x, y 1 D Squared Fast. Splat, indexed by r 2 x, y ( x, y ) ( x +1 , y )r x, y ( x o , y o )

1 D Squared Fast. Splat (Elliptical) • For elliptical kernels, if we define a canonical radius: • The incremental scan-conversion still works at the same low cost

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

2 D Fast. Splat (Bit. Blt, Vox. Blt) • No run-time computation Pre-rasterized footprint with center at ( x o , y o ), radius r Pre-rasterized footprints for all possible center positions , radius r Snap splat center to a k by k sub-pixel grid

2 D Fast. Splats • 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

2 D Fast. Splats (2) • The storage need: • When storage is limited, the quality is limited too • [Mora’ 00], similar

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

1 D RLUT Fast. Splat • For hardware, we need finer parallelism than scan-line

1 D Fast. Splat with RLUT

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • At a k by k subpixel precision

1 D Fast. Splat with RLUT • k by k subpixel precision

1 D Fast. Splat with RLUT • Due to symmetry, the RLUT set for x component is the same as the RLUT set for the y component • one RLUT set – k 1 D tables – each of splat_extent length x or y offset k splat_extent k k

Comparisons Among the Fast. Splats • 1 D Linear: very accurate, compact, slow • 1 D Squared: accurate, compact, fast • 1 D RLUT: accurate, compact, intended for hardware • 2 D Fast. Splat: can be very fast, accurate and compact under constrained conditions