Materiales utilizados para el recubrimiento pulpar directo.

In this section, we present surface distance maps and our algorithm to compute them efficiently using texture mapping hardware. We first introduce the notation used in the paper.

O

T

M

V

V

V

P

Figure 3.2:Affine map and distance computation: We compute the distance map at a point

qon trianglet (ofO1). The green vector shows the closest site ofO2 toq. The affine map

M1_{maps triangletto a triangle¯tin the 2D domainT} 1.

3.2.1

Notation

We denote piecewise linear 2-manifold objects or meshes in 3D asOi. Furthermore,Oiis decomposed into vertices, open edges and open faces, also known assites. A site is denoted aspi. Let_Ti ⊂R2 represent the 2D parametric domain for objectOi. We use an overbar to

represent the mapping of a 3D primitive to the 2D domain_T, for e.g. A pointqand triangle

tin 3D map toq¯and¯trespectively on_T.

The distance function of a site pi at a point q _{∈ R}3 is denotedd_{(q, pi). The distance}

pointq_∈ T topi. The closest vector from qtopi is known as the distance vector, denoted

~

d(q, pi).

Given two triangulated objectsO1 andO2, the surface distance mapD(O1)of an object

O1at a pointq∈O1 is the minimum value of the distance functions of all sitespk∈O2atq. We define an affine mappingM1

i :ti → T1 to transform the sampled points on the triangles

ti _∈ O1 into the 2D domainT1 ⊂ R2. For ease of notation, when the object idj is implicit (j = 1in this case), we shall drop the superscript fromMj_iand denote the affine map asMi.

3.2.2

Distance Fields: Background

Distance fields can be computed efficiently on discrete volumetric grids by rasterizing the distance function of each site to the points in the grid. Many algorithms compute the distance functions from each site to the points on the planes swept along the Z-axis of the grid [SOM04, SGGM06, SPG03]. These algorithms perform the distance field computation using one of these approaches:

1. Evaluate the distance functiond_{(q, pk)at each pointqin the plane directly by raster-}

izing the distance functions and use the depth-buffer hardware.

2. Compute the distance vector fromqto the site and use the magnitude of the distance vector to computed₍q, pk). This computation can be efficiently performed using the bilinear interpolation capabilities of the texture mapping hardware.

In order to accelerate the computations, prior algorithms construct a convex bounding poly- topeGto bound the Voronoi region of site. As a result, the distance function is only evaluated at the points inside. Details on the computation of these polytopes are given in Section 2.4. We use similar techniques to accelerate the computation of surface distance maps.

3.2.3

Planar Parameterization

Given a 3D meshO with trianglestk, k = 1, . . . , n, our algorithm transformstk intotk¯

by applying an affine mappingMk(see Fig. 3.2). Mkis represented as a matrix and ensures the following properties:

• There is a one-to-one mapping from a pointq_∈tkto the pointMkq∈tk¯.

• No two transformed trianglestk¯ =Mktkandtl¯ =Mltlshare a common interior point in the 2D domain_T.

These constraints are satisfied using piece-wise planar parameterizations of the surface in 3D space and the mapped triangles can be represented in a 2D texture atlas.

The affine transform for a triangletkwith verticesv0,v1,v2to a triangletk¯ with vertices

¯ v0,v¯1,v¯2in 2D domainTis given as M(x) = A(x₋v0) +v0 (3.1) where A= ¯ v1 −v¯0 v¯2−v¯0 v¯3−v¯0 v1−v0 v2−v0 v3−v0 −1 v3 = (v1−v0)×(v2−v0) ¯ v3 = (v¯1−v¯0)×(v¯2−v¯0)

SinceM is affine,Acan be written as a composition of a scale, shear and rotation ma- trices. Mathematically, A = AsAr where As represents a scale and shear matrix in the XY plane and Ar is a rotation matrix. We shall use this representation to perform an error analysis in Section .

3.2.4

Surface Distance Computation

Surface distance maps compute the distance-to-closest-primitive in the the scene to the sampled points on the surface of the mesh, excluding primitives on same mesh. We first compute the affine mappings, Mk for each triangle tk in the 3D mesh. These affine map defines a sampling on each triangletk in 3D space by sampling the projected triangletk¯ in the 2D domain _T. The surface distance map samples the domain _T uniformly using a 2D texture. Instead of computing distances using a volumetric grid, our algorithm computes the distance map on each triangletkusing affine transforms of distance functions to a 2D plane containingtk¯.

We present an algorithm to compute distance functions on a set of sampled points on the triangles of the 3D mesh. For each sitepi, we compute a convex bounding polytopeGi, which acts as a spatial bound on the Voronoi region ofpi. In other words, any point outside

Gi can not lie in the Voronoi region ofpi. We intersectGi with the triangle tk in 3D mesh. Let x1, . . . ,xl denote the vertices of Gi ∩tk. From the bilinear interpolation property of

distance vectors presented in Section 2.3, for a pointq _∈ tk the distance vector~d(q, pi)is a convex combination of the distance vectors at the verticesxj, j = 1, . . . , l. Since Mk is affine, the distance vector at a pointq¯ _∈ tk¯ is the convex combination of distance vectors at

¯

x_j =Mkxj, j = 1, . . . , l.

Thus the distance vector computation onGi∩tkcan be performed as follows:

1. Assign to each vertexxj a vector~d(xj, pi).

2. Each vertex xj, j = 1, . . . , lis transformed to a vertex x¯j in the 2D domainT, using

the affine mapMk.

3. The distance vector~d(¯q, pi)at a pointq¯ _∈tk¯ is computed as a convex combination of the vectors~d(xj, pi)associated with the verticesx¯j, j = 1, . . . , l.