Surface displacement, tessellation,and subdivision

Ikrima Elhassan

Overview

The Reyes image rendering architecture", Cook et al.,SIGGRAPH 1987

Curved PN triangles", Vlachos, Peters, Boyd, and Mitchell,Symposium on Interactive 3D Graphics, 2001

Reyes Architecture: Support Goals

Speed (render high quality film in less than a year)

Shading/Model Complexity & Diversity Minimal Raytracing Image Quality Flexibility

Design Goals

Natural Coordinates Vectorization Common underlying representation Locality Linearity Large Models Back door

Geometric Locality & Sampling

Raytracing can cause model and texture paging to dominate rendering time as model complexity increases

Uses stochastic sampling called jittering

MicroPolygons

½ pixel in length for Nyquist limit Dice primitives along natural boundaries Done in eyespace Results in a grid with shared vertices

Micropolygons: Adv vs. Disadvantages Vectorizable Texture locality &

filtering Subdivision

coherence Ease of Clipping &

Displacement maps No perspective

Shading occurs on nonvisible micropolygons

Rendering time becomes tied to depth complexity

Texture Locality

2 Classes of Textures: CATs & RATs

Sequential access with CATs

Can eliminate filtering

Description Algorithm

Bounded primitives (not necessarily tight) Primitives must be able to break down into

diceable primitives Must be able to split primitives Diceability test – returns “diceable” or “not

diceable”

Algorithm Description (Continued)

Does not require clipping Use ε plane to avoid invalid perspective

calculation Primitives with 0<z < ε are split until no

primitives span the ε plane

Extensions

Constructive Solid Geometry Transparency Depth of field Motion Blur

Implementation

Bucket Rendering Each primitive is diced or split and put into

corresponding bucket Only one bucket is needed at a time Lowers memory requirements

Final Thoughts on Reyes

No inverse calculations No clipping calculations Very vectorized No texture thrashing and

can eliminate run time filtering

Sampling occurs after shading

Difficult to handle metaballs

Hard to bound primitives such as particle systems for bucket sort

Polygons don’t have natural coordinate system

N-Patches

Issues with new geometric primitives Must be compatible with work already in

progress Must be backward compatible Fit existing hardware designs

N-patches: Advantages

Curved surfaces Improved visual quality (smooth

silhouettes and better vertex shading) Do not require developers to store

geometry differently (triangles) Minimize change to API’s Minimize bandwidth

Goals

Isolation (cannot access mesh neighbors) Fast Evaluation (including normal) Modeling range (smoother contours and

better shading)

Interpolation Use barycentric coordinates for

triangular domain Consider a set of points P0, P1,…, Pn,

and consider the set of all affine combinations taken from these points. That is all points that can be written as

                                                      for some

                                                    This set of points forms an affine

space, and the coordinates                                              

are called the barycentric coordinates of the points of the space.

Recall that a point within a triangle Δp0p1p2, can be described as p(u,v) = p0 + u(p1-p0) + v(p2-p0) = (1-u-v)p0 + up1 + vp2, where (u,v) are the barycentric coordinates

Bicubic interpolation results in C2 surfaces

Given a tabulated function yi = y(xi), i = 1...N , focus attention on one particular interval, between xj and xj+1. Linear interpolation in that interval gives the interpolation formulay = Ayj + By(j+1)

If we have yi”, we can add to the right-hand side of equation a cubic polynomial whose second derivative varies linearly from a value y j on the left to a value y (j+1) on the right.

Geometry: cubic B´ezier

Bijk = control points = coefficients

Makes up the “control net”

Cubic interpolation

Normal: quadratic B´ezier

Linear or Quadratic Interpolation

Algorithm

LOD = # vertices -2 on an edge

Tangent coefficients determined by planer projection

Algorithm (Continued)

Quadratic interpolation allows for inflection between vertices

Examples of N-patches

Sharp Edges

Proven that you cant have creases with purely local information

More than distinct normal per vertex causes holes or cracks

Not really discussed in detail, solution is to add more triangles

Hardware Performance

Operations are dot products, addition of two vectors, scaling, and per-component multiply of two vectors

Uses 6.8 to 11.6 vector operations per generated vertex Fill rate is not a bottleneck, since screen area is

unchanged Key limiting factor, most of time, is bandwidth Overhead in additional transformation of vertices Reduces calculation for key-frame interpolation and

collision detection Might be able to shift pixel shading to vertex shading

Advantages

Generated on-chip Saves bandwidth and

memory Curved surfaces and

better shading

Cant control curvature No sharp edges