Classifications

• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
  • G06T15/00—3D [Three Dimensional] image rendering
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
  • G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
  • G06T17/20—Finite element generation, e.g. wire-frame surface description, tesselation
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
  • G06T15/00—3D [Three Dimensional] image rendering
  • G06T15/005—General purpose rendering architectures
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
  • G06T9/00—Image coding

Definitions

• the invention relates to a method and apparatus for rendering computer generated images, in which the images include at least one closed polygonal contour.
• the invention relates to rendering of images using image space calculations and standard graphics hardware and in which the polygons in the images include 'arbitrary' shaped polygons, where 'arbitrary' permits the existence concavities, self-intersections, and even multiple 'contours'.
• Figures 1a to 1e show some examples of 'arbitrary' polygons (including concave polygons, polygons with self-intersections and multiple contour polygons).
• Figures 1a and 1e each show an example of a concave polygon, that is to say, a polygon in which at least one of the internal angles is greater than 180°.
• Figure 1b shows an example of a polygon with self intersections, that is to say, a polygon in which not every part of each line segment between two vertices remains inside or on the boundary of the polygon.
• Figure 1c shows an example of a polygon with multiple contours, that is to say, a polygon with a hole requiring an external and an internal contour to define the overall shape.
• Figure 1d shows an example of a polygon including all these features.
• SVG Scalable Vector Graphics
• OpenVG Open Vector Graphics
• SVG Scalable Vector Graphics
• OpenVG Open Vector Graphics
• API application programming interface
• the first family performs calculations in model space and are generally referred to as triangulation algorithms. These take a polygon outline and produce a set of non-overlapping triangles that exactly cover the filled area of the original polygon. To avoid confusion with other uses of "triangulation” in this document, we will refer to such algorithms as "true triangulation”.
• An example of the possible results of such a process, as applied to the polygon of Figure 1a is shown in Figure 2.
• the original shape can thus be constructed from the triangles: ⁇ [v2,v3,v5], [v3,v4,v5], [v5,v6,v2], [v6,v2,v7], [v7,v2,v1] ⁇ . Assuming a simple polygon with N-sides and that no extra vertices are added (note some algorithms do introduce additional vertices), we will obtain N-2 triangles. Once these triangles are generated, they can easily be rendered on any commodity graphics hardware.
• Meister's method removes one vertex at a time from a (simple) polygon in such a way that the reduced polygon remains simple. It repeatedly 'clips an ear', formed by a triple of consecutive vertices, from the polygon. This algorithm runs in O(n 3 ) time and, although it has been subsequently improved to be O(n 2 ), it is not particularly attractive except for polygons with relatively few vertices.
• Seidel's method runs in O( ⁇ l lOQ fl) time for simple polygons
• log (n) is defined as ...
• the application's inventor has implemented a version of Seidel's algorithm that has been enhanced to support completely arbitrary polygons. This still achieves virtually linear performance (assuming the implicit vertices created by self-intersections are included in 'n'). However, on a -2GHz CPU, the process still takes an average of 1 ⁇ 2 ⁇ s per polygon vertex. For polygons that will be drawn multiple times over numerous frames, the triangulation results can be cached, and so the pre-processing cost is amortised by the rendering process. For situations, however, where a polygon is only drawn a few times or is being dynamically altered on a frame-by-frame basis - which forces re-triangulation - the true triangulation process can be a very significant penalty. (Note that applying linear transformations to the model does not require re-triangulation.)
• the second family of methods with the capability of rendering arbitrary polygons uses image space calculations.
• the rendering/sampling process itself is adapted to determine which pixels fall inside the arbitrary polygon and which are outside.
• this can be done with scan line rendering algorithms, we are primarily interested in those that break the arbitrary polygon into smaller polygons (usually triangles) for which the hardware has direct rendering support, render those smaller polygons and make use of the hardware stencil buffer to determine which of the rendered pixels are inside the original arbitrary polygon.
• OpenGL programming guide the official guide to learning OpenGL, version 1.4
• Shreiner et al ISBN 0321173481
• Shreiner et al or http://en. wikipedia. or ⁇ /wiki/Trian ⁇ le fan ) is created by simply submitting the vertices in order, i.e. [V p V 21 V 3 ,../ ⁇ ] which implicitly creates the set of ⁇ /-2 triangles with vertices ⁇ [v, , v 2 , v 3 J [V 1 , v 3 , v 4 ⁇ [v, , v 4 , v 5 J ..[v, , v w _, ,v N ] ⁇ .
• FIG. 3 Borrowing the example (figure 1 (a)) from Shreiner et al, part of this process is shown in figure 3.
• the seven sided figure is rendered as a fan of five triangles. Assuming the odd- even fill rule, the pixels of the screen which are covered by an odd number of triangles will be deemed interior while those covered by an even number will be exterior. For example, the area bounded by v1 , v3, and location 20, is covered by triangles [v,,v 2 ,v 3 ] and [v,, V 3 , v 4 ] .
• FIG 15(b) Another method, as shown in figure 15(b), - in this example using a set of triangles generated using the invention's method (refer figure 7) - sends more geometry than the bounding box method but generally results in less redundant pixel filling.
• much of the shape is filled with a single 'layer/pass' of pixels, 450, but there are regions where pixels are filled multiple times, 460. This typically becomes worse with polygons with greater numbers of regions of, or total area of, concavity.
• the method also works unaltered for self-intersecting and multiple contour polygons - Shreiner et al also provide an example of the latter. In effect they just concatenate all the vertices of all the contours and treat the result as larger triangle fan.
• Each edge contour is drawn as a fan of triangles about an arbitrary center point, which we choose to be the 3D centroid of the contour vertices.
• Sander et al suggest a further improvement: "To further reduce the eccentricity of the fan triangles, we break up each large contour into a set of smaller loops. More precisely, we pick two vertices on the contour, add to the data structure two opposing directed edges between these vertices, and proceed as before on the smaller loops thus formed.”
• each triangle is defined as three integer indices, each say of 16 or 32 bits, which select vertices from a vertex array.
• this format offers the opportunity to exceed the 1 :1 barrier of strips and fans, though this is unlikely for 2D shapes.
• graphics hardware now frequently employs a vertex caching technique such as that described by Hoppe ("Optimization of mesh locality for transparent vertex caching”. Computer Graphics (SIGGRAPH '99 Proceedings) pp 269-276).
• Hoppe Optimization of mesh locality for transparent vertex caching. Computer Graphics (SIGGRAPH '99 Proceedings) pp 269-276).
• the hardware maintains a cache of the last K vertices used in past triangles.
• a FIFO replacement policy is generally employed rather than, say, a Least Recently Used (LRU) scheme as the former is not only simpler but, more importantly, generally results in a higher cache hit rate for 3D models.
• LRU Least Recently Used
• the supplied primitives e.g. triangles
• a vertex on a polygon more typically forms a convex, rather than concave, angle.
• Vertices with numerically local indices tend to also be spatially local. This is typically more apparent with polygons with larger numbers of vertices such as those illustrated in Figure 6a, 6b, and 6c. .
• a 'safe' vertex is identified and then removed to reduce an N sided shape to an N-1 shape.
• the clipped vertex then forms a triangle with its two original neighbours.
• the identification of a 'safe' vertex, i.e. where the produced triangle is entirely inside the original polygon is expensive.
• the inventor has appreciated that if a simple ear clipping algorithm is applied to every second vertex of an original (single) contour of an N-sided polygon, the ⁇ _ N / 2 ⁇ triangles thus formed will typically be more equilateral in shape than those produced by the fan or strip method. Furthermore, these triangles are less likely to have large regions that either lie outside of the final filled shape or overlap other triangles, than those typically generated by the fan or strip methods. After this process, one will be left with a 'residual' polygon with ⁇ % ⁇ (ie ceiling(N/2)) vertices. The same process may then be reapplied to produce another set of triangles and another residual polygon with even fewer vertices.
• the rendering system supports triangle primitives.
• Some rendering systems also support the rendering of quads (either convex or arbitrary) or even M-sided 'arbitrary' polygons with say, M ⁇ 6.
• the preferred embodiments detailed below demonstrate the invention for triangle primitives, the invention is not restricted to the output of triangle primitives and may output M-sided primitive units, or output primitive units of more than one type, eg triangles and quads, if used with more flexible rendering systems.
• one type of primitive e.g. triangles, will be outputted to cover the majority of the contour.
• Figures 1a to 1e show five examples of 'arbitrary' polygons
• Figure 2 shows the possible results of applying a 'true triangulation' process to the polygon of Figure 1a
• Figure 3 shows triangles generated by the most well known, prior art process for rendering in conjunction with a stencil buffer as applied to the polygon of Figure 1a;
• Figure 4 shows an additional point and triangles generated using Sander et al's method as applied to the polygon of Figure 1a;
• Figure 5 shows Sander et al's method as applied to the polygon of Figure 1e;
• Figures 6a to 6e show five examples of 'arbitrary' polygons with more significant numbers of vertices.
• Figure 7 shows the triangles generated by one embodiment of the present invention as applied to the polygon of Figure 1a;
• Figure 8 shows the first step of processing of an embodiment of the invention, in which a multi-contoured arbitrary polygon is logically split into its contours;
• Figures 9a to 9f each show a subset of the triangles, each set corresponding to a certain hierarchical "size" of triangle, as generated by an embodiment of the invention as applied to the polygon of Figure 6a;
• Figure 10 shows a subset (every 10 th triangle) of the output of a modified fan method applied to the polygon of Figure 6a;
• Figures 11a and 11b illustrate areas of overdraw that occur with the triangles generated by an embodiment of the invention, Figure 11a as applied to the polygon of Figure 6a, and Figure 11b the overdraw caused by a modified fan algorithm;
• Figure 12 shows an overview of an exemplary embodiment of the apparatus of the invention
• Figures 13a and 13b show a pathological polygon case for an embodiment of the invention
• Figures 14a and 14b show an alternate configuration of the case of Figures 13a and 13b, that is not pathological
• Figure 15 shows two methods of filling the polygon's pixels once the stencil buffer values have been determined.
• Method (A) uses a rectangular bounding box determined from the extremities of the polygon while method (B) resends the triangulation used to create the stencil; and
• Figure 16 shows the result of filling the polygon's pixels once the stencil buffer has been determined for a polygon with a consistent winding order, using the triangulation generated to create the stencil but where triangles with the opposite winding order have been eliminated.
• a contour is converted to a set of primitives, in this case triangles, with the method described below, and the resulting primitives e.g. triangles are concatenated with those sets generated by any earlier contours of the polygon.
• the combined sets of primitives e.g. triangles are rendered using the stencil buffer.
• PtB MIN(J+ StepSize, N-1 );
• PtC MIN(J + 2 * StepSize, N-1 );
• StepSize StepSize * 2; ⁇
• Figure 6 shows five examples of arbitrary polygons with more significant numbers of vertices.
• the map of Australia in figure 6a contains approximately 1500 vertices with 26 contours.
• the 'Hubert curve' shown in figure 6b contains 1000 vertices.
• the "Th" text in figure 6c has 807 vertices, while both the random walk, shown in figure 6d, and star, shown in figure 6e, both have100 source vertices.
• Any of these arbitrary polygons may be rendered using the invention.
• figure 9 shows certain subsets of the output triangles of the method, as applied to the polygon of figure 6a, wherein the six images show the results of capturing and rendering the triangle output when StepSize is 1 , 4, 16, 64, 256, and 512 respectively.
• figure 10 shows every 10 th triangle generated by a modified prior-art fan algorithm (the modification being to treat each contour separately rather than the less efficient merging as suggested by Shreiner et al). Comparing figure 9 and figure 10 one can see that triangles thus produced by the invention are, for this source model, significantly better shaped than those generated by the fan algorithm.
• Figure 11 (A) shows the pixels that are covered by more than one triangle. For simplicity, this figure does not distinguish between areas covered by two triangles and those covered by three or more triangles even though the latter clearly is increasingly costly. Nevertheless, if we just count the number of pixels touched by the overdraw region relative to the pixels of the final intended result, we find that overdraw has an area that is approximately 15% of the area of the intended final result.
• figure 11 (B) shows the overdraw area using the modified fan algorithm. In this situation, the overdrawn pixels represent a region that, surprisingly, has an area that is approximately 39% of the final result. Note that the unmodified Shreiner fan algorithm would be significantly worse.
• This first embodiment of the invention described above does, on average, address the two most important issues, i.e. that of improved triangle shape and reduced overdraw. Though of less importance, the first embodiment does not achieve the triangle data efficiency of fans and strips, whereby an ⁇ /-sided contour, requiring N-2 triangles, can be transmitted with just N vertices.
• Another potential drawback of the method of the first embodiment is that, with the possible exception of the final larger values of StepSize, generated triangles that are 'chronologically' local, are often only spatially local in the vicinity of a single shared vertex.
• This second embodiment incorporates the additional feature of a very small stack. Assuming the maximum number of vertices in a contour that will ever be encountered is N Max , then this stack needs at most[log 2 (N M ⁇ )]+l (ie ceiling(log 2 ( ⁇ max )) + 1)entries. For example, if in a given system a contour could have at most 2 16 vertices, then the stack would have no more than 17 entries.
• This embodiment provides interleaving of primitive levels to maximise vertex temporal locality and produces at least one second level primitive before all the first level primitives are produced.
• Primitive polygon level is reflected in the separation of the vertices of the primitive polygon relative to the vertices of the original arbitrary closed polygonal contour.
• a first level primitive has vertices which are contiguous vertices in the original closed polygonal contour.
• Higher level primitives have vertices which are offset from each other in the original closed polygonal contour. The amount of offset is related to the primitive level and the number of vertices in the primitive.
• An (i+1 ) th level (i ⁇ ) primitive is formed of end points of consecutive i th level primitives. Considering the set of i th level primitives, and associated vertices, each vertex will belong to exactly one member of the set except for the special case of end points, an end point being shared with at most one other i th level primitive.
• the vertices in the i th level primitive correspond to vertices offset from each other by (Q-1 ) ⁇ (i-1 ) in the original arbitrary closed polygonal contour .
• first level primitives have vertices [v1 , v2, v3], [v3, v4, v5]...
• the second level primitives will have vertices [v1 , v3, v5], [v5,v7,v9]...
• the second embodiment uses the same steps 1 and 3 as the first variant, but replaces step 2, i.e. the triangulation of a single contour, with the following method, as expressed in pseudo code.
• step 2 i.e. the triangulation of a single contour
• Vstack[0] : 0;
• the triangles are produced in an order which is far more 'vertex cache friendly 1 . Specifically, the triangles produced are...
• V 9l J V l» V 2» V 3l [ V 3» V 4» V 5l [ V l» V 3» V sl[ V 5» V 6» V 7l[ V 7» V 8» V 9l [ V 5» V 7» V 9l[ V I» V 5.
• StepSize [1,1,2,1,1,2,4,1,1,2,1,1,2,4,8,1...]
• the hit rate for, say, a 120 vertex contour is a respectable 62% which is nearly double that of the first embodiment and on- par with a fan or strip.
• An example of an apparatus implementing the invention is shown in figure 12.
• the parameters for a contour are supplied, 200, to an input buffer, 210. These are read by a state machine, 220, which implements the steps described in the pseudo code above.
• the state machine has access to a vertex stack.
• the vertex stack is preferably split into two portions, a first portion 230 containing P stack entries and a second portion 240.
• the second portion of the vertex stack 240 holds the remaining stack entries and would only require a single read/write port, and so could be implemented with a cheaper register file with an index to the element that is 4 th highest on the stack.
• the stack operations are done under the guidance of the state machine.
• the primitive output unit here a triangle output unit, 260, can be instructed by the state machine to select the top three stack elements, 230, and output the corresponding triangle.
• the "cycle" counts for filling of the stencil for the arbitrary polygons of figure 6, using a number of the described triangle generation techniques are compared against each other. Also, as a target benchmark, the rendering cycles (not including pre-processing costs) of the results of a true triangulation algorithm are supplied. For ease of interpretation, the scores are normalised so that the (modified) fan algorithm scores 1.0. Smaller figures are better.
• the "Sander” algorithm is the inventor's interpretation of Sander et al's document.
• the second preferred embodiment can further be adapted so that it is unnecessary to know, in advance, how many vertices are in a particular contour before starting to generate the triangle indices. Such an embodiment would be useful in an application which streams vertices that are generated 'on the fly'. Additionally, it can be modified to not require support for indexed triangles in the rendering hardware by also storing vertex coordinates in a wider stack.
• Some rendering systems also support the rendering of quads (either convex or arbitrary) or even M-sided 'arbitrary' polygons with, say, M ⁇ 16.
• Either of the presented preferred embodiments can be easily adapted, without departing from the scope of the invention, to output primitive units other than triangles to suit these more flexible rendering systems.
• FIG. 13 A very simple case is shown in figure 13 (A).
• the method effectively still has to fill the large pentagonal region, 310. As can be seen from the figure, all these triangles would form a considerable area of overdraw.
• the alternative embodiment uses a randomisation technique. For each contour, a random or pseudo random offset value can be supplied to or computed by the embodiment. This offset is then added, modulo N, to all the vertex indices in the contour to reduce the likelihood of recurring worst cases.
• the vertices themselves may be supplied as an array of arbitrary indices into a vertex array. It should be obvious one skilled in the art that the embodiments presented here can be extended to support such indirection.
• the stencil Once the stencil has been set up by drawing the triangulation by the methods above, they can be filled/shaded/textured by any of the known methods known in the art such as those described above, i.e. using the bounding box or re-use of the triangulation. Furthermore, if the winding order of the parent polygon is consistent, the additional enhancement, presented in this invention, may be used whereby the triangulation is resent to the rendering system but instructed to cull triangles with the opposite winding order.

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