csl 859: advanced computer graphicssubodh/courses/csl859/pdfslides/... · 2010-11-15 · storing...

CSL 859: Advanced Computer Graphics

Dept of Computer Sc. & Engg.IIT Delhi

Monday 15 November 2010

Point Based Representation

Point sampling of Surface Mesh construction, or Mesh-less

Often come from laser scanning Or even natural light

How do you render How do you peform other processing

Visibility, Collision etc.

Related concepts: image based representation, particle systems


Laser Range Scanning

Laser scanners sample points on a 3D shape Often on a grid

Large number of samples Most surfaces have a high frequency

Michaelangelo Project


Point-Based Rendering


Surfels

[Pfister et al., SIGGRAPH 2000]


Sampling Known Objects

3D “Rasterization” Layered Depth Images Cast rays through the

object along each axis Advance ray by regular

increments Store pre-filtered texture

colors at the points Project the surfel into texture

space and filter to get a single color

Disk radius >= maximum inter sample distance

rk = r0 2k


Filter shape (e.g., Gaussian) is usually circular/spherical We want that shape on screen => ellipse in space EWA Filter = low-pass filter warped reconstruction filter

Low-Pass Filter

Volume

Elliptical Weighted Average



W

Low-Pass Filter

Volume




W

Low-Pass Filter

Volume

Projection




W

Low-Pass Filter

Volume

Projection

Convolution



Storing surfels

3 Layered depth images (LDI), one per axis LDI = Images with multiple <depth,color> per pixel Makes a Layered Depth Cube (LDC)

Pixel spacing h0 related to expected screen resolution

Octree with node = bxb block of pixels Bottom-up construction Make pixel spacing = h0 2i

Filter down each block

Grossman and Dally [12] also use view-independent texture fil-tering and store one texture sample per surfel. Since we use a mod-ified z-buffer algorithm to resolve visibility (Section 7.3), not allsurfels may be available for image reconstruction, which leads totexture aliasing artifacts. Consequently, we store several (typicallythree or four) prefiltered texture samples per surfel. Tangent diskswith dyadically larger radii are mapped to texturespace and used to compute the prefiltered colors. Because of itssimilarity to mipmapping [13], we call this a surfel mipmap. Fig-ure 4b shows the elliptical footprints in texture space of consecu-tively larger tangent disks.

6 Data StructureWe use the LDC tree, an efficient hierarchical data structure, tostore the LDCs acquired during sampling. It allows us to quicklyestimate the number of projected surfels per pixel and to trade ren-dering speed for higher image quality.

6.1 The LDC TreeChang et al. [4] use several reference depth images of a scene toconstruct the LDI tree. The depth image pixels are resampled ontomultiple LDI tree nodes using splatting [29]. We avoid these inter-polation steps by storing LDCs at each node in the octree that aresubsampled versions of the highest resolution LDC.The octree is recursively constructed bottom up, and its height is

selected by the user. The highest resolution LDC — acquired dur-ing geometry sampling — is stored at the lowest level . If thehighest resolution LDC has a pixel spacing of , then the LDC atlevel has a pixel spacing of . The LDC is subdividedinto blocks with user-specified dimension , i.e., the LDIs in a blockhave layered depth pixels. is the same for all levels of the tree.Figure 5a shows two levels of an LDC tree with using a 2Ddrawing. In the figure, neighboring blocks are differently shaded,

b)a)

Figure 5: Two levels of the LDC tree (shown in 2D).

and empty blocks are white. Blocks on higher levels of the octreeare constructed by subsampling their children by a factor of two.Figure 5b shows level of the LDC tree. Note that surfels athigher levels of the octree reference surfels in the LDC of level 0,i.e., surfels that appear in several blocks of the hierarchy are storedonly once and shared between blocks.Empty blocks (shown as white squares in the figure) are not

stored. Consequently, the block dimension is not related to thedimension of the highest resolution LDC and can be selected ar-bitrarily. Choosing makes the LDC tree a fully volumetricoctree representation. For a comparison between LDCs and vol-umes see [19].

6.2 3-to-1 ReductionTo reduce storage and rendering time it is often useful to optionallyreduce the LDCs to one LDI on a block-by-block basis. Becausethis typically corresponds to a three-fold increase in warping speed,we call this step 3-to-1 reduction. First, surfels are resampled tointeger grid locations of ray intersections as shown in Figure 6.Currently we use nearest neighbor interpolation, although a more

resampled surfelson grid locations

LDI 1 surfelsLDI 2 surfels

Figure 6: 3-to-1 reduction example.

sophisticated filter, e.g., splatting as in [4], could easily be imple-mented. The resampled surfels of the block are then stored in asingle LDI.The reduction and resampling process degrades the quality of

the surfel representation, both for shape and for shade. Resampledsurfels from the same surface may have very different texture col-ors and normals. This may cause color and shading artifacts thatare worsened during object motion. In practice, however, we didnot encounter severe artifacts due to 3-to-1 reduction. Because ourrendering pipeline handles LDCs and LDIs the same way, we couldstore blocks with thin structures as LDCs, while all other blockscould be reduced to single LDIs.As in Section 5.2, we can determine bounds on the surfel density

on the surface after 3-to-1 reduction. Given a sampling LDI withpixel spacing , the maximum distance between adjacent surfelson the object surface is , as in the original LDC tree.The minimum distance between surfels increases todue to the elimination of redundant surfels, making the imaginaryDelaunay triangulation on the surface more uniform.

7 The Rendering PipelineThe rendering pipeline takes the surfel LDC tree and renders it us-ing hierarchical visibility culling and forward warping of blocks.Hierarchical rendering also allows us to estimate the number of pro-jected surfels per output pixel. For maximum rendering efficiency,we project approximately one surfel per pixel and use the same res-olution for the z-buffer as in the output image. For maximum imagequality, we project multiple surfels per pixel, use a finer resolutionof the z-buffer, and high quality image reconstruction.

7.1 Block CullingWe traverse the LDC tree from top (the lowest resolution blocks)to bottom (the highest resolution blocks). For each block, we firstperform view-frustum culling using the block bounding box. Next,we use visibility cones, as described in [11], to perform the equiv-alent of backface culling of blocks. Using the surfel normals, weprecompute a visibility cone per block, which gives a fast, con-servative visibility test: no surfel in the block is visible from anyviewpoint within the cone. In contrast to [11], we perform all visi-bility tests hierarchically in the LDC tree, which makes them moreefficient.

7.2 Block WarpingDuring rendering, the LDC tree is traversed top to bottom [4]. Tochoose the octree level to be projected, we conservatively estimatefor each block the number of surfels per pixel. We can choose onesurfel per pixel for fast rendering or multiple surfels per pixel forsupersampling. For each block at tree level , the number of sur-fels per pixel is determined by , the maximum distance be-tween adjacent surfels in image space. We estimate by divid-ing the maximum length of the projected four major diagonals ofthe block bounding box by the block dimension . This is correctfor orthographic projection. However, the error introduced by usingperspective projection is small because a block typically projects toa small number of pixels.For each block, is compared to the radius of the de-

sired pixel reconstruction filter. is typically , where

338


Projection

View frustum culling Traverse blocks to find the right resolution

One surfel per pixel, or n for supersampling inmax = Length of projected block diagonals/b If inmax > filter footprint, traverse children

Warp blocks to screen space Fast incremental algorithms [GrossMan & Daly] Only a few operations per surfel

fewer than matrix multiplication due to regularity of samples


Filling

Use Z buffer Each surfel covers an area Project this area orthographically and scan convert

Approximate ellipse with a bounding parallelogram Depth varies linearly: surfel normal

May have holes (magnification or surfel orientation) Shading: Per-surfel Phong illumination

Also incorporates environment/normal maps Reconstruct image by filling holes between

projected surfels Apply a filter in screen space Super-sample to improve quality


Splatting and Reconstruction


QSplat

Primary goal is interactive rendering of very large point-data sets

Built for the Digital Michelangelo Project

[Rusinkiewicz and Levoy, SIGGRAPH 2000]


Sphere Trees

A hierarchy of spheres, with leaves containing single vertices

Each sphere stores center, radius, normal, normal cone width, and color (optional)

Tree built the same way one would build a KD-tree Median cut method

Rendered really large models for the day Focus on memory layout and data

quantization


Rendering Sphere Trees

Start at root Do a visibility test

View frustum Also back-face cull based on normal cone

Recurse or Draw Recurse based on projected area of sphere

with an adapted threshold To draw, use normals for lighting and z-buffer

to resolve occlusion


Splat Shape

Several options Square (OpenGL “point”) Circle (triangle fan or texture mapped square) Gaussian (have to do two-pass)

Can squash splats depending on viewing angle Sometimes causes holes at silhouettes, can

be fixed by bounding squash factor


Splat Shape


Splat Silhouettes


Few Splats


Many Splats


Surface Reconstruction: Cocone

• p+ ≡ pole of p = point in the Voronoi cell farthest from p

ε < 0.1 → the vector from p to p+ is

within π/8 of the true surface normal

The surface is nearly flat within the cell

Voronoi cell of p

p


Surface Reconstruction: Cocone

• p+ ≡ pole of p = point in the Voronoi cell farthest from p

ε < 0.1 → the vector from p to p+ is

within π/8 of the true surface normal

The surface is nearly flat within the cell

Voronoi cell of p

p+

p


Sample Reconstructed Surfaces

courtesy Tamal Dey, OSU


Moving Least Squares

For point set P, MLS(P) is a projection operator which maps each point to itself: MLS(P) = {x: (P, x) = x}

(P, x): Find a local reference plane at x: p(u,v)

Compute local polynomial approximation to surface on the plane: p(u,v)

ψψ


MLS

courtesy Luiz Velho


Reference domain The local plane:

is computed so as to minimize a local weighted sum of squared distances of the points pi to the plane


Light Field

Uniformly sample two planes plane of cameras

Many plane-pairs needed

Need compression Quantization


Rendering

For each desired ray: Compute intersection with (u,v) and

(s,t) planes Take closest ray Filter from the closest


Compression

Vector quantization Build a codebook of 4D tiles Each tile an index into the codebook

Example: 2x2x2x2 tiles, 16 bit index = 24:1 compression


csl 859: advanced computer graphicssubodh/courses/csl859/pdfslides/... · 2010-11-15 · storing...

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