Patent Grant
7400331
This invention relates generally to graphical rendering. More particularly, in certain embodiments, the invention relates to texture mapping techniques for the graphical rendering of a virtual object.
Certain computer graphics applications graphically render the surface of a three-dimensional virtual object. The surface of a three-dimensional virtual object is generally represented as a mesh of polygonal surface elements, for example, triangles. Attributes, such as color intensity, are assigned to surface elements of the virtual object, which are then displayed to the user.
The resolution of the surface mesh is typically chosen according to the three-dimensional complexity of the object. For example, rendering a virtual object having a complex geometry may require a tight mesh of many small surface elements, whereas rendering a virtual object of simpler shape may be performed with a looser mesh of fewer, larger surface elements.
It is typically not possible to achieve sufficient realism by assigning color values to vertices of each surface element, particularly if the surface elements are large. This problem may be overcome by representing three-dimensional surface properties of the object using a texture in two-dimensional space. Texture mapping permits improved resolution of surface properties, such as color values, within each surface element, and allows certain operations to be performed on the three-dimensional object as if it were flat.
Mapping methods are used to relate points on the surface of a three-dimensional virtual object to corresponding points of a two-dimensional surface. For example, mapping methods can be used to relate points on the surface of a three-dimensional sphere, such as the Earth, to corresponding points of a two-dimensional map. Certain graphics applications perform texture mapping for a three-dimensional virtual object by establishing a global parameterization that links every surface element in the three-dimensional object space to an element in two-dimensional texture space. The mapping process may not be automatic and may require manual input from the user. A goal of these mapping schemes is to produce a coherent two-dimensional texture whose elements are arranged in such a way that the distortion of the object surface is acceptably low. For many complex virtual objects, this poses a problem that is either computationally difficult or intractable.
Furthermore, current graphics applications require a user to paint the surface of a virtual object in two-dimensional texture space before applying the texture on the virtual object in three-dimensional object space. This results in a less interactive, less intuitive experience for the user.
Moreover, current methods do not allow a user to modify the shape of the object after its surface has been painted without losing surface data from unmodified portions of the object. This is due, in part, to the need to re-mesh and re-parameterize the entire model following any modification of the underlying three-dimensional model.
A method of texture painting has recently been introduced to allow texture mapping without global parameterization. M. Foskey et al., “ArtNova: Touch-Enabled 3D Model Design,” IEEE Virtual Reality 2002, pp. 119-126, (March 2002).
However, there remains a need for mapping methods that allow a user to modify the shape of an object after its surface has been painted, without losing the surface data from unmodified portions of the object. There also exists a need for a more efficient graphical rendering method that is able to support the rendering of complex virtual objects, while still allowing a user to interactively paint directly onto the object in object space. Furthermore, there is a need for a method of blending textures while maintaining the interactivity of painting, without creating graphical artifacts.
The invention provides improved methods of graphically rendering virtual objects. More specifically, the invention provides improved texture allocation and texture rendering techniques for graphically representing the surface of a three-dimensional virtual object.
The improved methods allow a user to paint directly onto the surface of a three-dimensional virtual object. The methods also allow a user to interactively modify the shape of a painted, three-dimensional object without losing surface data from unmodified portions of the model. For example, the improved methods permit a user to paint an object, then carve portions of the object without losing the painted surface data from unmodified portions of the model. The improved methods also permit a user to easily switch back and forth between a painting mode and a sculpting mode, without having to reallocate texture for unmodified portions of the object.
Additionally, the invention provides techniques for blending a brush stroke into a surface texture without creating artifacts in the overlapping segments of the stroke, and without diminishing the interactivity of the user's painting experience.
Furthermore, the invention features improved methods for the combined graphical and haptic rendering of a virtual object, thereby permitting a user to experience force feedback via a haptic interface device during the painting of the object in object space. In this way, a user can “feel” the virtual object as the user paints its surface while, at the same time, the user observes a display of the object in object space.
Texture mapping methods of the invention include texture allocation methods and texture rendering methods. Texture allocation refers to the process of allocating elements in texture space to corresponding surface elements in object space. Texture rendering refers to the process by which data in the texture space is sent to a graphics card and/or graphics application for display. Texture rendering also involves sending information to the graphics application that allows the application to correctly relate the texture data to corresponding portions of the surface of the virtual object. Texture rendering methods of the invention are able to interface with any commercially-available standard for three-dimensional rendering and/or three-dimensional hardware acceleration, including cross-platform standards.
In one embodiment, the invention provides an improved method for graphically rendering a virtual object by using information produced during mesh generation. In the process of generating a surface mesh for a voxel-based virtual object, an index is produced for each resulting jack, where a jack is a surface-containing voxel, and a voxel is a volumetric element in object space. Each jack contains a certain configuration of surface elements. There are a finite number of possible configurations of surface elements within each jack; and the number of possible configurations depends on the mesh generation scheme employed. The index for each jack corresponds to one of the finite number of surface element configurations. A first lookup table is created by determining a local configuration of texture elements for each of the known set of surface element configurations. The first lookup table is preferably computed prior to object rendering. Thus, graphical rendering of a given virtual object proceeds by using the indices for the jacks of the virtual object to generate coordinates of texture elements in texture space corresponding to the surface elements of the virtual object.
Thus, in one embodiment, the invention provides a method of graphically rendering a virtual object including the steps of: (1) using an index corresponding to each of a plurality of jacks of a voxel-based virtual object to identify texture elements to which surface elements of the virtual object are mapped; and (2) generating texture coordinates in texture space for each of the identified texture elements. The index may be, for example, a marching cubes index, derived from an implementation of a marching cubes algorithm that is used to generate a surface mesh for the virtual object. Alternatively, the index may be, for example, a marching tetrahedra index, derived from an implementation of a marching tetrahedra algorithm used to generate a surface mesh for the virtual object.
It is found that dividing the surface of a virtual object into texture regions may overcome hardware limitations in rendering complex virtual objects. Thus, in one embodiment, the texture space in the graphical rendering method summarized above includes a plurality of texture regions. Moreover, in addition to steps (1) and (2) above, the rendering method may further include the steps of: (3) binding to a graphics application a blended texture corresponding to one of the plurality of texture regions; and (4) transmitting to the graphics application the texture coordinates for each of a plurality of surface elements associated with the blended texture. The blended texture may be a contiguous texture formed by blending together a plurality of texture layers, as discussed in more detail hereinbelow.
The index produced for each jack as a result of the mesh generation procedure can be used to create any number of lookup tables. In one embodiment, the graphical rendering method includes creating two lookup tables, using a first lookup table to determine to which of a plurality of texture elements a given surface element is mapped, and/or using a second lookup table to determine to which of a plurality of positions within a texture element a given surface element is mapped. For example, where either zero, one, or two triangular surface elements in object space are mapped to a quadrilateral texture element in texture space, a first lookup table is used to determine to which quadrilateral a given triangle is mapped, and the second lookup table is used to determine to which of two positions within the quadrilateral the triangle is mapped. More lookup tables may be used in more complex embodiments, for example, where all of the surface elements of a given jack are mapped to a single texture element.
The invention also includes methods of dynamic texture allocation and deallocation, allowing a user to interactively modify the shape of a painted, three-dimensional model, without losing texture data associated with unmodified portions of the object. Thus, the invention provides a method of mapping texture onto a virtual object that includes: allocating texture for at least one newly-created jack of a virtual object following an object modification; and/or de-allocating texture in the texture space for at least one newly-eliminated jack of the virtual object following an object modification.
In one embodiment, the method proceeds by stepping through contiguous blocks of jacks of the model in render order to determine whether texture allocation is needed therein, and by keeping texture corresponding to a given jack block together within the same texture region. This dynamic allocation method provides improved efficiency, since new texture is only created for the modified jacks, and since there is no global parameterization required. Advantages are achieved by allocating texture in render order and by keeping texture corresponding to a given jack block together within the same texture region. This promotes efficient binding and results in a more efficient texture rendering process. In one embodiment, texture is allocated so as to minimize the number of texture binds required during rendering.
The invention also provides for the compact storage of texture information for a virtual object. Since many texture elements may contain a uniform value, it is efficient to store this information as a single value, instead of storing values for all texels of every texture element. It is not necessary to store a full contiguous texture for a given texture region, since the user does not need to see the texture space at any time, as all edits may be performed by the user in object space. Accordingly, the invention provides a method of creating a blended texture for use in the rendering of a virtual object, the method including the step of blending two or more texture layers, where at least one of the texture layers is a grid of pointers indicating at least two of: (i) uniform texture elements, (ii) nonuniform texture elements, and (iii) the location of the nearest free texture element in the grid.
A blended texture may be created by combining scratch textures, stencil textures, and/or paint textures associated with one or more brush strokes. A scratch texture indicates one or more intensity values of pixels associated with each brush stroke, and a paint texture indicates one or more color values associated with each brush stroke.
The methods of blending texture presented herein are compatible with a method of stenciling in which a user may specify a maximum amount by which selected portions of an image are allowed to change over the course of a series of brush strokes. A first type of stencil texture indicates a level of protection to apply to a given region of an image, and a second type of stencil texture contains brush strokes that are accumulated and are modified according to the protection levels of the first type of stencil texture prior to blending into the image.
Methods of the invention include blending texture layers of at least two of the three types listed above—scratch texture layers, paint texture layers, and stencil texture layers—in the order in which the brush strokes are performed. Enhanced interactivity is achieved, since texture data is blended into the image pixel-by-pixel in discrete time slices, without interrupting the user's application of brush strokes. Additional model complexity can be handled, since texture layers are blended into composite textures for only those texture regions affected by the brush strokes. Furthermore, one embodiment stores the texture layers in a “collapsed” form that includes both uniform and nonuniform texture elements, and then combines them to form a contiguous blended texture suitable for binding to a graphics application.
By combining the various texture allocation, texture rendering, and texture blending methods disclosed herein, the invention provides a complete graphics application feature that allows a user to paint directly on the surface of a virtual object in object space. Accordingly, the invention provides a method of interactively representing application by a user of at least one brush stroke directly onto a virtual object in object space, the method comprising the steps of: (a) allocating a plurality of texture elements in two-dimensional texture space for a plurality of jacks of a virtual object; (b) graphically rendering the allocated texture in real time as a user applies at least one brush stroke onto the virtual object, wherein rendering includes creating at least one blended texture for binding to a graphics application; and (c) updating the blended textures according to the at least one brush stroke applied by the user, wherein the method involves one or more of the following: (i) using an index corresponding to each of a plurality of jacks of the virtual object to identify texture elements to which surface elements of the virtual object are mapped; (ii) dynamically allocating texture in the texture space; and (iii) blending a set of texture layers corresponding to a first of a plurality of texture regions in the texture space and binding the blended texture to the graphics application during rendering of the first texture region.
Finally, the invention features an architecture for combined graphical rendering and haptic rendering of a virtual object, allowing a user to experience force feedback during the painting of the object in object space.
The objects and features of the invention can be better understood with reference to the drawings described below, and the claims. The drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the drawings, like numerals are used to indicate like parts throughout the various views. The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the U.S. Patent and Trademark Office upon request and payment of the necessary fee.
The invention provides improved methods of graphically rendering the surface of a three-dimensional virtual object. The invention relates to a method of texture mapping that allows a user to switch between a paint mode and a sculpt mode without losing data. The method also provides for accurate and efficient rendering without requiring global parameterization.
Texture mapping is a process by which points on the surface of a three-dimensional virtual object are related to points on a two-dimensional surface. Texture mapping allows certain operations to be performed on the three-dimensional object as if it were flat. The three-dimensional virtual object is generally represented in object space, for example, as a system of elements and/or points in a Cartesian (x,y,z) coordinate system. The surface elements of the object are mapped to texture elements in two-dimensional texture space. The texture is then rendered by relating texels, or pixels in texture space, to corresponding points in object space, and displaying them to a user. Additional effects, such as lighting, shadowing, manipulation of orientation and/or size, animation, and other effects, may be performed once the surface of the object is initially rendered.
Because embodiments of the invention allow a user to paint directly on the object in object space, the texture space need not be coherent. There is no need to display the texture space to the user. Accordingly, mapping methods of the invention do not require global UV parameterization.
Texture mapping includes texture allocation and texture rendering. Texture allocation refers to the process of allocating elements in texture space to corresponding elements in object space.
The method of
Texture allocation proceeds such that a plurality of texture regions are defined for the virtual object being rendered. The organization of allocated texture into texture regions is important in the rendering methods described herein, since it allows binding and rendering texture from one texture region at the time, providing faster, more efficient rendering of complex virtual objects.
Furthermore, jacks may be organized into jack blocks. For example, a jack block may be an 8×8×8 contiguous block of voxels, although any other number or configuration of voxels may be used. The voxels within a given jack block may or may not include surface-containing voxels (jacks).
The allocation method produces a list, for example, the “Texture Mapper” depicted in the system architecture of
The texture allocation method of
If there is enough space in the current texture region for the current jack, step 310 of
Step 312 of
Texture allocation methods of the invention support dynamic texture allocation and deallocation—that is, the creation and/or elimination of texture elements corresponding to edited portions of the object, without affecting texture previously allocated for texture elements corresponding to unedited portions of the object. In this way, the invention supports a quick transition from a sculpting mode, or other object editing mode, to a painting mode, or other surface editing mode, and vice versa. The allocation method of
In addition to texture allocation, texture mapping involves the process of texture rendering. While texture allocation need only be performed if texture has not yet been allocated for a given virtual object or when there has been a change that affects the amount of surface of the virtual object, texture rendering is performed once for every frame that is graphically rendered. For example, texture rendering may take place up to about 30 times per second, or more, depending on the complexity of the virtual object being rendered and the,capability of the computer system and graphics application used. Texture rendering involves sending data representing the contents of each texture element to a graphics rendering interface, such as a graphics card and/or graphics application. Thus, the texture rendering methods of the invention may optionally include using a commercially-available platform for graphical rendering and hardware acceleration. Moreover, the texture rendering methods of the invention may optionally include the use of a commercially-available graphics application programming interface. Along with sending data representing the contents of each texture element, texture rendering methods of the invention send texture coordinates to the graphics rendering interface/application in order to correctly relate the texture data to corresponding portions of the surface of the virtual object.
Steps 708, 710, 712, and 714 of
In another embodiment, the method of texture rendering proceeds jack block by jack block, irrespective of texture regions. Here, unlike the method shown in
The method illustrated in
The rendering method of
The results of the method of
More sophisticated surface-element-to-texture-element mapping schemes are possible. For example, in one embodiment, where there are a finite number of surface element configurations within a jack that are created during mesh generation, texture elements indicative of each possible surface element configuration within a jack can be pre-computed so that a single texture element is mapped to all the surfaces within a given jack.
When the user paints a virtual object in object space, values are assigned to corresponding texels in texture space. The user “paints” texture elements, in essence, via the link between surface elements of the virtual object in object space and corresponding texture elements in texture space. This process is invisible to the user, who only sees the effects of the painting operation in object space. However, the process of blending brush strokes, occurs in texture space using two-dimensional texture compositing techniques. Graphical user input fills texture layers corresponding to a given texture region; then, the texture layers are combined into a blended composite texture for that texture region. The blended composite texture for each texture region is bound one-at-a-time to a graphics application during texture rendering, as described in more detail elsewhere herein.
A scratch texture indicates one or more intensity values of pixels (and/or texels) associated with each brush stroke. A scratch texture may be thought of as a “template” describing the brush characteristics; for example, the scratch texture may contain intensity values that reflect brush size and/or edge effects. The brush may be assigned a certain width, a varying width along the length of a stroke, a certain shape characteristic of the end of the brush, and/or a “falloff” or transition region along a portion of the brush, such as the edges of the brush. The intensity values in the scratch texture may be used to determine the intensity of a selected color at a given pixel within the brush stroke. For example, a scratch texture comprising values representing a given brush stroke may represent a “falloff” region of the brush stroke by assigning intensity values from 1 (full intensity) at the center, or near to the center, of the stroke, down to 0 (no intensity) at points at the edges of the stroke. Thus, a blended composite of the scratch texture with a paint texture containing a single monochromatic color value would portray a paint stroke with the most intense color at the center of the stroke, falling off to no color at the very edges of the stroke.
As used herein, a brush stroke is an operation performed on a collection of pixels in a texture and/or image. Brush strokes include, for example, paint strokes, erase strokes, pen strokes, lines, characters, and text. The manner in which a brush stroke operates on a collection of pixels may be determined interactively by a user, as in the performance of a paint or erase stroke, using a graphical interface device, or non-interactively—for example, in batch—without any user input. Thus, brush strokes include, for example, batch deletions, batch pastes, and flood fills. The pixels/texels on which a brush stroke operates may be either contiguous or non-contiguous.
The texture layers 1102, 1104, 1106, 1108, 1110 portrayed in
Texture layers may be stored in “collapsed” form, as shown in
A representative contiguous texture 1200 is shown in
The collapsible texture 1202 in
Texture layers are combined in discrete time slices to form the blended composite image in a pixel-by pixel manner, without interrupting application of additional brush strokes by the user. Each brush stroke within a given texture region may be represented using a scratch texture. An individual brush stroke is divided into a series of stroke segments that are separately blended into the scratch texture. The invention provides a method that prevents double-blending of overlapping portions of segments of the individual brush stroke, thereby avoiding undesired artifacts at the segment ends.
The blending method uses a scratch texture to describe the size and shape of the brush. By assigning a new pixel value to the scratch texture only if it exceeds the existing value at that pixel, the double-blending artifact is avoided. The blending method includes the steps of: (1) receiving brush stroke data from a graphical user interface; (2) for each of a plurality of pixels of a scratch texture, comparing a received pixel value to a value previously written at the corresponding pixel of the scratch texture and assigning the received value to the corresponding pixel of the scratch texture only if it exceeds the existing value; and (3) blending the scratch texture into the target image. Here, the scratch texture is a texture having one or more intensity values of pixels (and/or texels) associated with each brush stroke. The blending step may be performed substantially upon completion of the performance of the paint stroke by the user. Each scratch texture is emptied or deleted after its contents are blended, so that each new brush stroke fills an empty scratch texture.
In order to enhance interactivity, texture layers are blended in the order in which the brush strokes they represent are performed. Texture data is blended pixel-by-pixel in discrete time slices, without interrupting the user's application of additional brush strokes. Thus, multiple scratch textures layers may be blended together where each scratch texture represents an individual brush strokes within a given texture region. This technique is demonstrated in
The texture layers that are combined to form a blended composite texture for a given texture region may include stencil textures. Stencil textures are used to protect a selected region of an image from subsequent editing. The methods of blending texture are compatible with stenciling methods that allow a user to more accurately control the editing of an image by specifying a maximum amount by which selected portions of the image are allowed to change over the course of a series of brush strokes.
The stenciling methods involve protecting an image using a first texture and a second texture, rather than a single texture alone.
Overlapping portions may result from multiple overlapping brush strokes and/or from a single brush stroke that overlaps itself. Despite the presence of any overlapping portion(s), and despite the number of brush strokes applied following activation of the stencil, methods of the invention can prevent the opacity of an initial version, or initial layer, of the selected region(s) from decreasing below a specified minimum in any subsequent composite.
The stenciling method illustrated in
The interactivity of the user's image editing experience may be preserved by use of a display texture 1510. In one embodiment, texels of the protected texture 1504 are copied directly into a display texture 1510, while texels of the second texture 1508 are modified according to the first texture 1502, then blended into the display texture 1510.
User interactivity is preserved by modifying texels of the second texture and blending the modified texels into the display image on a texel-by-texel basis. In this way, the user sees the resulting image emerge, subject to the user-specified protection, as the user continues to apply brush strokes.
The display texture can reflect real-time user brush strokes, as well as preserve a minimum opacity of the original texture layer within a protected region, regardless of the number of brush strokes that follow. This is because, in a preferred embodiment, each update of the display texture is performed by: (1) re-copying texel values of the original protected texture layer into the display texture texels, and then (2) compositing the modified second texture texels with the display texture texels. The display texture 1510 may be updated at a rate of up to about 30 times per second, or more, depending on the complexity of the image and the computational speed for graphical rendering. The update rate may be less for more complex images or for slower machines—for example, from about 10 times per second to about 20 times per second.
The use of a display texture 1510 is optional. Whether or not a display texture 1510 is used, methods of the invention can preserve a minimum opacity of the original image layer within a protected region by accumulating graphical input in the second texture 1508, modifying the second texture 1508 using the first texture 1502, and, subsequently, blending the modified second texture 1508 into the protected texture 1504.
The user may provide one or more signals indicating the beginning and/or ending of the period of time in which brush strokes are accumulated in the second texture. The user provides a first signal, such as a button click, to indicate the beginning of the application of a stencil. Alternatively, the stencil may self-activate once the user defines the first texture. The step of defining the first texture may include indicating: (1) one or more regions of the surface of a virtual object to be protected; and/or (2) one or more levels of protection to apply within the one or more regions. The surface elements within the region indicated by the user in object space are mapped to corresponding texture elements in texture space.
Once the stencil is active, graphical input representing brush strokes by the user are directed into the second texture 1508. When the user has completed one or more brush strokes, the user may provide a second signal to deactivate the stencil. In one embodiment, once the stencil is deactivated, the second texture 1508, modified by the first texture, is blended into the protected texture 1504.
Real-time display of erase strokes requires modification of paint strokes applied both before and after activation of the stencil.
In one embodiment, a minimum opacity of an original texture layer may be preserved despite repeated erase strokes within the protected region, where overlapping erase strokes result in the attenuation of a texel value of the protected texture down to a minimum value, but no further. The minimum value is determined from the protection level of that texel as indicated in the first texture 1502. In cases where texels are represented by RGBA quadruples, the first texture 1502 may be used to derive a minimum alpha channel value below which the texel values of the protected texture 1504 are not allowed to fall. Pixel values and/or texel values include values of any bit depth, where bit depth generally ranges from 1 to 64 bits per pixel. Pixel values include grayspace values, RGB colorspace values, RGBA colorspace values, or any other colorspace values, such as HSL, HSV, CMY, CMYK, CIE Lab, and R-Y/B-Y. Preferred methods of the invention are performed using 24-bit, RGBA colorspace pixels, although any bit depth and/or colorspace format may be used.
The display methods discussed herein work where a user performs paint strokes, erase strokes, or both paint and erase strokes, while the stencil is active. Additionally, these display methods preserve the interactivity experienced by the user during the editing process, while maintaining the protection offered by the stenciling methods described herein.
Methods of the invention involve the modification of paint colors of images and/or textures, as well as the blending of paint colors between two or more images and/or textures. For example, in one embodiment, pixels of the second texture 1508 are modified using the first texture 1502, where the first texture 1502 is a map of values indicating protection levels for the protected texture 1504 and the second texture 1508 contains accumulated graphical input. A texel of the second texture 1508 may be modified by attenuating its color value according to the protection level in the first texture 1502 corresponding to that texel.
Blending of images and/or textures may be performed using one or more compositing operations. For example, in one embodiment, texels of the second texture 1508 are blended into the protected texture 1504. This may be done by performing a compositing operation, such as an “overlay” operation in RGBA colorspace, between the second texture 1508 and the protected texture 1504.
Because texture is dynamically allocated, a user may seamlessly switch between an object modification mode (for example, a sculpt mode) and a surface paint mode without losing paint information for unaffected surfaces of the virtual object.
Step 1702 of the painting method illustrated in
Step 1704 of
An embodiment of the painting method of
For each texture region, there is an interface, labeled “IComposite Color Texture Region” 2010 in the architecture 2000 of
The “Texture Region Component” 2014 in the architecture of
Any of the methods described herein for graphically rendering a virtual object may be combined with haptic rendering of the virtual object. For example, an embodiment of the invention haptically renders the virtual object in real time as the user applies a brush stroke onto the surface of the virtual object in object space. The haptic rendering process includes determining a force feedback corresponding to a position of a haptic interface device held by the user as the user applies a brush stroke using the device. The force is delivered to the user via the haptic interface device. A haptic rendering process is described, for example, in U.S. Pat. No. 6,552,722, by Shih et al., the text of which is hereby incorporated by reference in its entirety.
A combined haptic and graphical rendering method of the invention performs haptic rendering at a substantially faster rate than graphical rendering. This may be necessary to provide realistic force feedback to the user. In one embodiment, graphical rendering is performed within a range from about 5 Hz to about 150 Hz, while haptic rendering is performed within a range from about 700 Hz to about 1500 Hz. In another embodiment, haptic rendering is performed at about 1000 Hz, while graphical rendering is performed at up to about 40 Hz. Haptic and graphical rendering of a virtual object may be performed by different threads of an overall rendering process.
The sampling structure in a marching tetrahedra approach is a tetrahedron. Samples are at the vertices on a rectangular grid, where a mesh cell is made up of eight samples of the grid, and the mesh cell is split into six tetrahedra that tile the cell. FIG. 25 shows a mesh cell 2500 that is split into six tetrahedra 2502, 2504, 2506, 2508, 2510, 2512, where vertices of the tetrahedra coincide with vertices of the cube. The configuration of triangular surface elements through each tetrahedron is determined by a lookup table. The index into the lookup table for a given tetrahedron is determined from values at the four vertices of the tetrahedron, where each value indicates whether the vertex is interior or exterior to the isosurface of the virtual object being meshed. The sum of all triangular surface elements for all tetrahedra of a given cube tile the cube. The sum of all triangular surface elements for all the cubes tile the object space.
A computer hardware apparatus may be used in carrying out any of the methods described herein. The apparatus may include, for example, a general purpose computer, an embedded computer, a laptop or desktop computer, or any other type of computer that is capable of running software, issuing suitable control commands, receiving graphical user input, and recording information. The computer typically includes one or more central processing units for executing the instructions contained in software code that embraces one or more of the methods described herein. The software may include one or more modules recorded on machine-readable media, where the term machine-readable media encompasses software, hardwired logic, firmware, object code, and the like. Additionally, communication buses and I/O ports may be provided to link any or all of the hardware components together and permit communication with other computers and computer networks, including the internet, as desired.
While the invention has been particularly shown and described with reference to specific preferred embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.
This application is a divisional application of U.S. patent application Ser. No. 10/697,548, filed on Oct. 30, 2003, and issued on Aug. 22, 2006, as U.S. Pat. No. 7,095,418, which is related to commonly-owned U.S. patent application Ser. No. 10/697,174, filed on Oct. 30, 2003, each of which is hereby incorporated by reference in its entirety.
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