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Ray-Tracing a Polygon Mesh
Keywords: ray-tracing, polygon mesh, tessellation, triangulation, triangle, barycentric coordinates, interpolation, normals, vertex attribute, texture coordinates.

Introduction

We learned how to render triangles in the previous lesson which leads us to the next step: rendering more complex objects. Usually the simplest form of object we can find right after simple primitives such as spheres and triangles, are polygonal objects (which are also called polygon meshes).

Polygon meshes are not used much in film (where a more sophisticated type of primitive, the subdivision surface is preferred) but they stay very popular in games and real time applications in general. These objects are made out of polygons which we also call faces. A polygon or a face is a flat surface (it should be but if the vertices making up the face are not coplanar it won't be) which is defined by a set of vertices connected to each other to form a closed shape. Two vertices connected to each other form an edge. Therefore, the number of vertices making up a face has to be greater or equal to three. Three vertices define a triangle, and four define a quad. We can also combine polygons together to create complex polyhedral objects.

Polygon meshes are simple in principle but have they own set of problems: for instance how do we smooth a low poly mesh (a mesh that has a small number of polygons may appear blocky), or how do we efficiently encode the vertex connectivity data of a mesh? These techniques will be studied in the advanced section. The goal of this basic lesson is to learn how to define a polygon mesh in our program (chapter 2) and render this mesh using ray tracing (chapter 3).

Mesh Definition

Figure 1: a simple mesh with 2 faces and 6 vertices.

A polygon mesh is made out of vertices which are connected to each other to form faces. The first thing we expect to define a mesh is therefore a list of vertices. Let's consider an example of two quads sharing a common edge (figure 1). In this example we would have 6 vertices v0, v1, v2, v3, v4 and v5. The other information we need is the total number of faces the mesh contains. In our example this number is 2. We also need an array of integers which indicates how many vertices each face is made of (the face index array). The first and the second face from our mesh have 4 vertices therefore this array would contain the following data: [4 4]. The size of this array is equal to the number of faces in the mesh (2). Finally for each face, we need to know how the vertices are connected to other to create the faces, their indexes in the vertex array list, and the order in which they should be connected. The vertices v0, v1, v2 and v3 are making up the first face, while connecting vertices v0, v3, v4 and v5 in this order gives the second face. We can store this information in the form of an array of integer values which size if equal to the sum of the number of vertices each face is made of. The mesh contains 2 faces which are each connected to 4 vertices, therefore this array should be of size 8 and would contain the following data: [0 1 2 3 0 3 4 5]. Here is what we would get in pseudocode:

// mesh vertices Vec3f P[6] = {Vec3f (-5, -5, 5), Vec3f ( 5, -5, 5), Vec3f ( 5, -5, -5), Vec3f (-5, -5, -5), Vec3f (-5, 5, -5), Vec3f (-5, 5, 5)}; uint32_t npolys = 2; // number of faces uint32 faceIndex[2] = {4, 4}; // face index array uint32 vertexIndex[8] = {0, 1, 2, 3, 0, 3, 4, 5}; // vertex index array
Remember that if the current orientation is left-handed, then a polygon whose vertices are specified in clockwise order (from the point of view of the camera) will be a front-facing polygon (that is, will have a normal vector which points toward the camera). If the current orientation is right-handed, then polygons whose vertices were specified in counterclockwise order will be front-facing.

Names of the variables in the pseudocode follow the conventions defined in the RenderMan Interface Specification for the RiPointsPolygons method. The RenderMan Interface "is a standard interface between modeling programs and rendering programs capable of producing photorealistic quality images" proposed and maintained by Pixar. This document can easily be found on the web and reading it is recommended as it contains a great deal of very interesting and useful information (even though Pixar doesn't maintain it anymore).

class PolygonMesh : public Object { public: PolygonMesh(uint32_t nfaces, int *fi, int *vi, Vec3f *p) : numFaces(nf), faceIndex(NULL), vertexIndex(NULL), P(NULL) { // compute vertArraySize and maxVertexIndex uint32_t vertArraySize = 0; uint32_t maxVertexIndex = 0, index = 0; for (uint32_t i = 0; i < numFaces; ++i) { vertArraySize += nv[i]; for (uint32_t j = 0; j < fi[i]; ++j) if (vi[index + j] > maxVertexIndex) maxVertexIndex = vi[index + j]; index += fi[i]; } maxVertexIndex += 1; pts = std::unique_ptr<Vec3f []>(new point[maxVertexIndex]); for (uint32_t i = 0; i < maxVertexIndex; ++i) P[i] = p[i]; vertexIndex = std::unique_ptr<uint32_t []>(new int[maxVertexIndex]); for (uint32_t i = 0; i < maxVertexIndex; ++i) vertexIndex [i] = vi[i]; faceIndex = std::unique_ptr<uint32_t>(new int[numFaces]); for (uint32_t i = 0; i < numFaces; ++i) faceIndex[i] = fi[i]; }; ~PolygonMesh() { /* release memory */ ... } bool intersect(...) const { ... } void getSurfaceData(...) const { ... } uint32_t numFaces; // number of faces std::unique_ptr<uint32_t []> faceIndex; // face index std::unique_ptr<uint32_t []> vertexIndex; // vertex index std::unique_ptr<Vec3f []> P; }; int main(...) { ... uint32_t numFaces = 2; uint32_t faceIndex[2] = {4, 4}; uint32_t vertexIndex[8] = {0, 1, 2, 3, 0, 3, 4, 5}; Vec3f P[6] = { Vec3f (-5, -5, 5), Vec3f ( 5, -5, 5), Vec3f ( 5, -5, -5), Vec3f (-5, -5, -5), Vec3f (-5, 5, -5), Vec3f (-5, 5, 5), }; PolygonMesh *mesh = new PolygonMesh(numFaces, faceIndex, vertexIndex, P); ... }

The code is quite simple (a functional C++ version of this code is available in the last chapter of this lesson). The point list, face and vertex index array are passed to the constructor of the MeshPolygon class as well as the number of faces. However we don't know how many points are in the point array. To find out, we look for the vertex with the maximum index value in the vertex index array (lines 13-14). The first element of an array in C++ start at 0, therefore the total number of vertices in the point list is the maximum index value plus 1 (line 17).

As you can see in the code above, we added the two methods intersect() and getSurfaceData() but we left them empty for now. We will fill them in the next chapter.

Convert Polygons to Triangles

Figure 2: example of a convex (left) and concave (right) polygon.

Unfortunately it is not easy to directly ray-trace a polygon mesh. For the sake of simplicity and generality it is usually much better to convert the polygon mesh into a triangle mesh and ray-trace each individual triangle rather than developing an algorithm to ray trace polygons. This partially explains why we put so much effort into ray-tracing triangles efficiently (as explained in the previous lesson). It is much easier to have one efficient highly optimized ray-primitive intersection and convert every type of geometry to this primitive rather than implementing an efficient ray-geometry intersection routine every time a new type of geometry is added to the system. In addition, converting polygons to triangles is actually really simple. However our technique will only works for convex polygons. Concave polygons require a different method which we won't look at in this lesson. The technique we will use for this lesson is simple to understand and implement but other methods can be used to produce a better result (there is a lot of research on the topic of meshing) as we already mentioned it only works for convex polygons. If you are unsure about whether or not your mesh contains concave polygons, the best is to triangulate it first in a program such as Maya before exporting it to the renderer.

Figure 3: triangulating a convex face.

Triangulating a convex polygon is very simple. We simply take the first vertex in the face, and draw triangles by connecting the vertices v[0], v[n + 1] and v[n + 2] to create a triangle, where $n$ is the current number of triangles formed minus one. When $n = 0$, the first triangle is formed by the vertices v[0] v[1] and v[2], when $n = 1$ the second triangle is formed by the vertices v[0], v[2] and v[3], and for $n = 2$, the third triangle is formed by v[0], v[3], v[4], and so on. Figure 3 illustrates this process. The number of triangles created for a polygon is always equal to the number of vertices in the polygon minus two (create some polygons yourself to verify this observation).

numTrisInFace = numVertsInFace - 2;

Our program will need to triangulate the original mesh before it gets rendered, which practically means that we don't want to store the original face and vertex index arrays. We will create new ones from the data passed to the mesh constructor class. We first want to compute how many triangles the original mesh will generate (which we can easily do since we know that each face containing $N$ vertices results in $N - 2$ triangles). We also know that each face of a triangulated mesh contains three vertices therefore we don't need to store the face index array for triangulated mesh. However we will need to regenerate ourselves a vertex index array to store which vertices each triangle in the mesh is connected to. The size of this array is equal to the number of triangles multiplied by 3. The pseudocode should make this process easier to understand:

TriangleMesh( const uint32_t nfaces, const std::unique_ptr<uint32_t []> &faceIndex, const std::unique_ptr<uint32_t []> &vertsIndex, const std::unique_ptr<Vec3f []> &verts, const std::unique_ptr<Vec3f []> &normals = nullptr, const std::unique_ptr<Vec2f []> &st = nullptr) : numTris(0) { uint32_t k = 0, maxVertIndex = 0; // find out how many triangles we need to create for this mesh for (uint32_t i = 0; i < nfaces; ++i) { numTris += faceIndex[i] - 2; for (uint32_t j = 0; j < faceIndex[i]; ++j) if (vertsIndex[k + j] > maxVertIndex) maxVertIndex = vertsIndex[k + j]; k += faceIndex[i]; } maxVertIndex += 1; // allocate memory to store the position of the mesh vertices P = std::unique_ptr<Vec3f []>(new Vec3f[maxVertIndex]); for (uint32_t i = 0; i < maxVertIndex; ++i) { P[i] = verts[i]; } // allocate memory to store triangle indices trisIndex = std::unique_ptr<uint32_t []>(new uint32_t [numTris * 3]); uint32_t l = 0; // generate the triangle index array for (uint32_t i = 0, k = 0; i < nfaces; ++i) { // for each face for (uint32_t j = 0; j < faceIndex[i] - 2; ++j) { // for each triangle in the face trisIndex[l] = vertsIndex[k]; trisIndex[l + 1] = vertsIndex[k + j + 1]; trisIndex[l + 2] = vertsIndex[k + j + 2]; l += 3; } k += faceIndex[i]; } // store normals and st coordinates ... }

The computation of the triangle vertex indexes is straightforward. We create an array which size is equal to the number of triangles multiplied by 3 (since each triangle requires three indexes into the point array - line xx). We loop over each face then over each triangle in this face and compute the indexes for the current triangle from the input vertex index array which we store in the triangle vertex index array (line xx to xx). At this stage we have a renderable polygon mesh in the form of a triangulated polygon mesh.

Normals

Figure 4: face normals (left) and vertex normals (right). Face normal are used for flat shading and vertex normals are used to fake smooth shading.

The normal of a triangle can be computed from the cross product of vectors AB and AC (figure 4). This normal is called a face normal, because it corresponds to a vector perpendicular to the plane defined by the three vertices of the triangle. Therefore the normal vector is also perpendicular to the face (or triangle in that case. Remember than polygons should be coplanar) hence the term face normal. However we can also compute what is called a vertex normal which as its name suggests, is a normal defined at the position of each vertex in the mesh. This normal is useful for smooth shading (as shown in figure 4) and is usually computed from an average of all the face normals the vertex is connected to. Vertex normals are required for shading and they are usually computed by the modeling program and passed on to the renderer as part of the mesh description. When they are not defined, the renderer needs to compute them itself. Computing vertex normals will be addressed in a future lesson. For now we will assume that they are always defined in the object geometry file.

Texture Coordinates

Unless the geometry is generated on the fly in the program, st coordinates are generally generated in the modeling program and exported to the file in which the object is stored. Similarly to normals, we will expect texture coordinates to be defined in the object geometry file.

What's Next?

In this chapter we have explained what information we needed to store and compute polygon meshes. In the next chapter we will explain how these meshes can be ray traced and provide a functional C++ example.


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5 and mentally projecting the sequential ROC curve to the right, it seems fairly safe to assume it would still fall below the simultaneous ROC. This empirical measure of discriminability is not based on any theoretical assumptions about memory. c If, for some reason, policymakers preferred a FAR of approximately .06 because of the higher HR that could be achieved, the fact that pAUCSIM > pAUCSEQ over the tested FAR range (0 to FAR z m Such research is often focused on testing theories that may have applied significance. Huntington: Robert E. Krieger Publishing Co. Gronlund, S. D., Carlson, C. A., Neuschatz, J. S., Goodsell, C. A., Wetmore, S. A., Wooten, A., & Graham, M. (2012). Target Under those assumptions, d' can be safely compared for Condition A vs. The same receiver operating characteristic (ROC) data as in Fig. And what would the policy implications be in a case like that?

) is equal for the three procedures and that the difference in pAUC arises because the simultaneous procedure is less susceptible to the deleterious effects of criterion variability than showups and sequential lineups. In both cases, it is the structural constraints of the testing procedure itself, not a difference in underlying latent variables, that results in a difference in the empirical ROC curves. 5 except that the smooth curves generated by a theoretical (signal detection) model are drawn through the ROC data points.

Journal of Applied Psychology, 70, 556–564. volume 3, Article number: 9 (2018) A photo lineup consists of a picture of one suspect (the person who the police believe may have committed the crime) plus several additional photos of physically similar foils (i.e., fillers) who are known to be innocent. and variance σ

In other words, the probability that a suspect identified from a sequential procedure is guilty is .746, whereas the probability that a suspect identified from a simultaneous procedure is guilty is .576. max

is the same in both cases.

Journal of Experimental Psychology: Applied, 19, 345–357. Finally, criterion variance (σ The same basic model applies to a lineup, but the way it works is somewhat different. (2016).

5, it is visually obvious that pAUC for the simultaneous procedure is greater than the pAUC for the sequential procedure over the FAR range of 0 to .038 (the maximum FAR for the sequential procedure). Z Instead, empirical discriminability refers to the degree to which participants correctly sort target and foil stimuli into their true categories.

2.

through μ Figure 4 presents the ROC data computed from the values shown in Table 1. would be estimated to be about 1.4. = 2.0, empirical discriminability for the showup procedure is impaired to a greater extent than empirical discriminability for the simultaneous lineup procedure. If, instead of using confidence ratings, instructions were used to induce conservative responding from the outset such that only IDs made with high confidence were obtained in the first place, the police would lose the potentially useful investigatory information that a suspect ID made with low or medium confidence might provide. SAE International's core competencies are life-long learning and voluntary consensus standards development. (2015). Viewed in this light, the “controversy” over ROC analysis of lineup performance actually consists of a normal scientific debate about which theory of underlying latent variables better accounts for the empirical data. Empirical is the information you received and found out, and theoretical the information that is set. Google Scholar. Then compute d’, not the diagnosticity ratio. However, unlike a filler, an innocent suspect is not known to be innocent and will be imperiled (and perhaps wrongfully convicted) if mistakenly identified. To appreciate why the two measures can go in opposite directions without contradiction, it is important to consider how d'

Wells, G. L., Yang, Y., & Smalarz, L. (2015). Their simulations showed that, in the absence of criterion variability and with d' m

However, because this “hand waving” analysis of the effect of sequential lineups on the HR and FAR is clearly insufficient, a quantitative assessment of some kind is needed. Psychology, Public Policy, and Law, 17, 99–139. For example, if the prior odds of guilt are even (i.e., half target-present lineups, half target-absent lineups), one can ask about the posterior odds of guilt for the subset of lineups that resulted in a filler ID or No ID. Computer software is needed to precisely measure the size of the shaded area, and the tutorial videos associated with Gronlund, Wixted, and Mickes (2014) explain how to use one such R program, called pROC (Robin et al., 2011), to do that. (2012) actually estimated pAUC – not d' The data in Table 1 allow one to compute not only the overall HR and FAR but also a HR and FAR separately for varying degrees of response bias specified by the different confidence ratings. Foil

) was set to 0 for the simultaneous lineup and to 0.75 for the sequential lineup, which is why the sequential lineup, despite its higher d', yields a lower ROC than the simultaneous lineup. Moreover, no model of memory would be needed to reinforce the decision as to which of the two procedures is diagnostically superior. A stickler might contend that a minimum FAR greater than 0 should also be specified, one that is equal to the FAR associated with the leftmost ROC point from the condition with the larger minimum FAR (e.g., FAR Foil Nevertheless, to be sure about that, one would have to actually perform the experiment because it is at least theoretically possible that the ROC curves would cross and the sequential procedure would become superior in that higher FAR range. In addition, a slight variation of an earlier model [1] is presented to test the sensitivity of empirical models. Hypothetical receiver operating characteristics (ROC) curve for a lineup procedure in which a 5-point confidence scale was used. Live lineups were once the norm, but nowadays, the police almost always administer photo lineups after they identify a suspect in the days or weeks following a crime.

The model shown in Fig. Both were referring to what we have here denoted d' C To compare the two procedures with respect to pAUC, it is essential to use the same FAR Wetmore, S., Neuschatz, J. S., Gronlund, S. D., Wooten, A., Goodsell, C. A., & Carlson, C. A. Wixted, J.T., Mickes, L. Theoretical vs. empirical discriminability: the application of ROC methods to eyewitness identification. The diagnostic feature-detection theory attributes the difference to a d' However, unlike Fig. Vision Research, 40, 1227–1268. However, in the presence of criterion variability (equated across the two procedures), simultaneous lineups yielded higher empirical discriminability (measured by pAUC) than showups. 1 Cameron, E. L., Tai, J. C., Eckstein, M. P., & Carrasco, M. (2004). Simulated receiver operating characteristic (ROC) data generated by a simultaneous lineup using the MAX decision rule, a sequential lineup using the “first-above-criterion” decision rule, and a showup. The absolute/relative distinction was originally advanced as a theory of response bias, with a relative judgment strategy corresponding to increased pressure to choose someone from the lineup. For example, in the latest critique of ROC analysis, Smith et al. = 1.4 and with σ In essence, that kind of comparison is what ROC analysis is all about, and it illustrates why ROC analysis is needed to unambiguously determine the diagnostically superior procedure. = μ Thus, the DR (i.e., the likelihood ratio) is equal to the correct ID rate divided by the false ID rate (HR/FAR).

(i.e., over all possible memory-strength values for a filler) is given by: Again, this is the likelihood of observing a filler ID from a target-absent lineup. Other studies have reported no significant difference between the two procedures, but with a trend still favoring the simultaneous procedure (e.g., Andersen, Carlson, Carlson, & Gronlund, 2014; Exp. (2017). advantage enjoyed by simultaneous lineups compared to the other two procedures. Here, we describe how to write the likelihood function for that probability and then describe the similar approach used to write the likelihood functions for the probability of observing a filler ID from a target-present and then from a target-absent lineup. Alternatively, as noted earlier, confidence in No IDs could be collected in such a way as to allow one to project the ROC further to the right (i.e., by collecting a confidence rating in connection with the face that the witness believes is most likely to be the perpetrator). (1975). 1 $$,$$ \mathrm{f}=@\left(\mathrm{x}\right)\ \mathrm{normpdf}\left(\mathrm{x},\mathrm{mu}\_\mathrm{t},\mathrm{sigma}\_\mathrm{t}\right).\ast \mathrm{normcdf}\left(\mathrm{x},\mathrm{mu}\_\mathrm{d},\mathrm{sigma}\_\mathrm{d}\right).\hat{\mkern6mu} \left(\mathrm{k}-1\right).\ast \Big(1-\mathrm{normcdf}\left(\mathrm{c},\mathrm{x},\mathrm{sigma}\_\mathrm{c}\right), $$,$$ \mathrm{p}\_\mathrm{t}=\mathrm{integral}\left(@\left(\mathrm{x}\right)\ \mathrm{f}\left(\mathrm{x}\right),-15,15\right).

m Retrieved 29 Mar 2016, from http://www.policeforum.org/. Only recently, however, has signal detection theory been brought to bear on this issue. Their current model policy states: “This policy recognizes that the sequential and simultaneous approaches are both valid methods of conducting an identification procedure and does not recommend one over the other.” (International Association of Chiefs of Police, 2016, p. 1). According to this model, different eyewitness identification procedures are differentially susceptible to the deleterious effects of criterion variability. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. If these data were fit by a model to estimate d'

is usually set to 0 because no specific theory is relied upon to justify the seemingly safe assumption that if responding were infinitely conservative, both the HR and the FAR would be 0. 6, even though underlying theoretical discriminability (d' This function corresponds to the probability of observing a target ID from a target-present lineup made with a particular level of confidence associated with criterion, c. If there are five confidence criteria for making a positive ID (as in Fig. 4 (integrated from − ∞ to + ∞): Again, this is the likelihood of observing a filler ID from a target-present lineup. Finally, the area beneath the curve was estimated from a FAR of 0 to FAR In summary, according to this theory, pAUCSIM > pAUCSEQ and pAUCSIM > pAUCSHOWUP because d'm-SIM > d'm-SEQ and d'm-SIM > d'm-SHOWUP, respectively. An understanding of that distinction is important for both theoreticians and policymakers because the two measures need not agree.

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zbMATH — the first resource for mathematics

Quantization of Chern-Simons gauge theory with complex gauge group. (English) Zbl 0717.53074
The author studies the canonical quantization of Chern-Simons gauge theory in $$2+1$$ dimensions considering the case in which the gauge group is a complex Lie group. The Lagrangian is defined on a three-dimensional manifold $$M=\Sigma *{\mathbb{R}}$$, with $${\mathbb{R}}'$$ being the time axis and $$\Sigma$$ an oriented closed surface. The special case in which $$\Sigma$$ is a genus one surface is also analyzed. The quantization is described in close analogy with the compact gauge group case and the gravitational interpretation of the SL(2,$${\mathbb{C}})$$ theory is also considered.
Reviewer: V.Silveira

MSC:
 53C80 Applications of global differential geometry to the sciences 81T13 Yang-Mills and other gauge theories in quantum field theory 81T70 Quantization in field theory; cohomological methods
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On Some Fully Invariant Subgroups of Summable Groups
Annales Mathématiques Blaise Pascal, Volume 15 (2008) no. 2, p. 147-152

We show the inheritance of summable property for certain fully invariant subgroups by the whole group and vice versa. The results are somewhat parallel to these due to Linton (Mich. Math. J., 1975) and Linton-Megibben (Proc. Amer. Math. Soc., 1977). They also generalize recent assertions of ours in (Alg. Colloq., 2009) and (Bull. Allah. Math. Soc., 2008)

DOI : https://doi.org/10.5802/ambp.244
Classification:  20K10
Keywords: fully invariant subgroups, $\lambda$-large subgroups, summable groups, free valuated vector spaces
@article{AMBP_2008__15_2_147_0,
author = {Danchev, Peter},
title = {On Some Fully Invariant Subgroups of Summable Groups},
journal = {Annales Math\'ematiques Blaise Pascal},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {15},
number = {2},
year = {2008},
pages = {147-152},
doi = {10.5802/ambp.244},
mrnumber = {2468040},
zbl = {1155.20053},
language = {en},
url = {https://ambp.centre-mersenne.org/item/AMBP_2008__15_2_147_0}
}

Danchev, Peter. On Some Fully Invariant Subgroups of Summable Groups. Annales Mathématiques Blaise Pascal, Volume 15 (2008) no. 2, pp. 147-152. doi : 10.5802/ambp.244. https://ambp.centre-mersenne.org/item/AMBP_2008__15_2_147_0/

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