Abstract:
The present invention relates to a method and apparatus for obtaining a bounding sphere for the ellipsoid that results when an affine transformation matrix transforms a bounding volume. The present invention accomplishes this by using Gershgorin intervals to obtain a new radius for the bounding volume. The invention operates on a scene graph, which is comprised of a number of nodes arranged in a hierarchical organization. At least one of these nodes is an affine transformation matrix, which is used for operations such as rotations, scaling, and translations. Associated with the transformation matrix is a bounding sphere. The present invention calculates the radius of a new bounding sphere that encircles the ellipsoid formed when the affine matrix transforms the old bounding volume. The use of Gershgorin intervals is fast and yields accurate results. Thus, the present invention provides for a method and apparatus that is computationally fast and produces a well-behaved affine transformation of bounding spheres.

Description:
FIELD OF INVENTION 
     The present invention relates to the field of processing scene graphs used in the display of computer-generated images. More specifically, the present invention relates to a method and apparatus for quickly and accurately creating well-behaved affine transformations of bounding spheres. 
     BACKGROUND OF THE INVENTION 
     Computer graphics is used in a wide variety of applications, such as in business, science, animation, simulation, computer-aided design, process control, electronic publication, gaming, medical diagnosis, etc. In an effort to portray a more realistic real-world representation, three-dimensional objects are transformed into models having the illusion of depth and displayed onto a two dimensional computer screen. Conventionally, polygons are used to construct these three-dimensional objects. Next, a scan conversion process is used to determine which pixels of a computer display fall within each of the specified polygons. Thereupon, texture is selectively applied to those pixels residing within specified polygons. In addition, hidden or obscured surfaces are eliminated from view. Finally, lighting shading, shadowing, translucency, and blending effects are applied. 
     For a high-resolution display, having over a million pixels, displaying a three-dimensional scene on a computer system is mathematically intensive and requires tremendous processor power. Furthermore, the computer system must be extremely fast to handle dynamic computer graphics, for example, displaying a three-dimensional object in motion. Even more processor power is required for interactive computer graphics, whereby three-dimensional images change in response to user input. Still more processor power is required to achieve a scene with more intricate details. 
     Given the tremendous demands that rendering scene graphs places on the processor. the number of complex mathematical computations performed should be kept to a minimum. One conventional alternative to performing complex mathematical operations is using mathematical shortcuts to estimate parameters. However, these shortcuts often fail to yield accurate results. 
     An example of this problem is calculating which portion of the object or scene graph needs to be displayed on the computer monitor. Conventionally, a bounding volume, which is stored in the scene graph, is compared with a viewer frustum or viewpoint. If they intersect. the computer displays the portion of the scene graph which corresponds to the bounding volume. If they do not intersect, there is no need to process the details of the un-displayed portion of the scene graph. During conventional image processing, the size and shape of the bounding volume is altered by a transform matrix. This matrix rotates, translates, squashes, or scales the object, as well as the bounding volume. It is computationally very difficult to arrive at the size and shape of the altered bounding volume. Conventionally, the exact size of a bounding sphere around the altered bounding volume may be calculated. However, this can require complex mathematical operations, such as solving a cubic polynomial. Another conventional method is to estimate the size of the bounding sphere. However, estimation is sometimes mathematically intensive, and mathematical shortcuts are often inaccurate. For example, some conventional methods produce accurate results for only a limited class of transform matrices. 
     Therefore, a need exists for a method and apparatus for obtaining a bounding sphere for the ellipsoid that results when an affine transformation matrix operates on a bounding volume. The method and apparatus needs to be computationally fast, as well as accurate for any type of affine transform matrix. 
     SUMMARY OF THE INVENTION 
     The present invention relates to a method and apparatus for obtaining a bounding sphere for the ellipsoid that results when an affine transformation matrix transforms a bounding volume. The present invention accomplishes this by using Gershgorin intervals to obtain a new radius for the bounding volume. The invention operates on a scene graph, which is comprised of a number of nodes arranged in a hierarchical organization. At least one of these nodes is an affine transformation matrix, which is used for operations such as rotations, scaling, and translations. Associated with the transformation matrix is a bounding sphere. The present invention calculates the radius of a new bounding sphere that encircles the ellipsoid formed when the affine matrix transforms the old bounding volume. The use of Gershgorin intervals is fast and yields accurate results. Thus, the present invention provides for a method and apparatus that is computationally fast and produces a well-behaved affine transformation of bounding spheres. 
     In another embodiment, the present invention compares the transformed bounding sphere with a viewer frustum to decide whether to display the corresponding portion of the scene graph and whether to continue traversing down the scene graph. Thus, the present invention provides a method and apparatus for fast and accurate rendering of scene graphs. 
    
    
     These and other objects and advantages of the present invention will no doubt become obvious to those of ordinary skill in the art after having read the following detailed description of the preferred embodiments, which are illustrated in the various drawing figures. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The operation of this invention can best be visualized by reference to the drawings. 
     FIG. 1 shows a computer system upon which the present invention may be practiced. 
     FIG. 2 shows an exemplary scene graph. 
     FIG. 3 shows a viewer frustum for determining which portion of a scene graph should be displayed. 
     FIG. 4 shows a viewer frustum and bounding volumes for determining which portion of a scene graph should be displayed. 
     FIG. 5 shows a scene graph with an affine transformation matrix node and bounding sphere nodes. 
     FIG. 6 shows a bounding sphere and the ellipsoid that is formed when the sphere is transformed by an affine matrix. 
     FIG. 7 is a flowchart describing the steps in calculating the new radius for the bounding sphere. 
     FIG. 8 provides details for calculating the Gershgorin intervals. 
    
    
     DETAILED DESCRIPTION 
     A method and apparatus for a computationally fast and well-behaved affine transformation of bounding spheres is described. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be obvious, however, to those skilled in the art that the present invention may be practiced without these specific details, in other instances, well-known structures and devices are shown in block diagram form in order to avoid obscuring the present invention. 
     FIG. 1 shows a computer system upon which the present invention may be practiced. Initially, an original three-dimensional scene is created and described in file format (e.g., VRML) by a programmer. The programmer describes both components of the scene (e.g., geometry, materials, lights, images, movies, and sounds) as well as the relationships among those components. These relationships may be static (e.g., a transformation hierarchy) or dynamic (e.g., the values of transformations in the hierarchy). Changes in one element of the scene graph may be made to effect changes in others. For instance, a change in position of a light-bulb shaped geometry can be made to cause a change in position of light source. In addition, information about the locations and characteristics of viewpoints may be included in the scene. Once created, the files are stored in the storage device  104  (e.g., a hard disk drive) of the computer system  110 . 
     In practice, applications running on a processor  101  will be used to build a scene graph in order to have a graphics subsystem  108  draw it onto a display device  105 . Optionally, the user may choose to edit the three-dimensional scene by inputting specific commands via the input device  115  (e.g., a keyboard, mouse, joystick, lightpen, etc.). The user may also interact with the three-dimensional scene (e.g., flight simulation, game playing, etc.) through user input device  115 . 
     FIG. 2 shows a pictorial representation of a scene graph. The graph is comprised of nodes, which are abstract representations of objects. In FIG. 2, the overall person is represented by node  201 . The person node  201  is comprised of an upper body node  203  and a lower body node  205 . Each of these nodes has its own set of nodes as well (i.e., child nodes). For example, the upper body is comprised of a left arm node  207  and a right arm node  209 . Likewise, the lower body is comprised of a left leg node  211  and a right leg node  213 . In turn, the right leg node  213  is comprised of a right thigh node  221 , a right calf node  223 , and a right foot node  225 . The scene in FIG. 2 contains little detail because only a few nodes are shown. However, almost limitless detail can be achieved by adding more levels of nodes. 
     Rendering a hierarchical scene graph for display requires traversing it to decide which portions should be displayed. FIG. 3 shows a view frustum  300  from point of view  301  for determining what should be displayed on computer screen  305 . Anything falling within the view frustum  300  should be displayed. In this example, since left arm  307  falls within the view frustum  300 , it should be displayed. However, the right arm  309  falls outside the view frustum  300  and should not be displayed. 
     To assist in the rendering process, the preferred embodiment of the present invention stores bounding volumes at each node of the scene graph. FIG. 4 shows bounding volumes for the whole person  402 , lower body  406 , right leg  414 , right thigh  422 , right calf  424 , and right foot  426 . In one embodiment, when rendering a scene with respect to a certain viewpoint  401 , a bounding volume is compared with the frustum  400 . If they do not intersect, there is no need to traverse further down the scene graph. For example, because the lower body bounding sphere  406  does not intersect with the view frustum  400 , not only is the lower body not displayed, but also there is no need to traverse further down the scene graph. In other words, it is not necessary to compare the bounding volumes of the lower body&#39;s children nodes (i.e., the leg nodes) with the view frustum. This is because the bounding volume of the lower body node  406  contains the union of the bounding volumes of all of the nodes in its subgraph (i.e., its children nodes, grandchildren nodes, and so on down the hierarchy). 
     The preferred embodiment also stores affine transform matrix nodes in the scene graph. FIG. 5 shows such a matrix node  508 . These affine matrices transform a portion of the scene graph in some manner, such as scaling rotating, translating, or squashing. FIG. 5 also shows bounding volumes for the right leg  514 , right thigh  522 , right calf  524 , and right foot  526 . Each node is the scene graph has such a volume, although for simplicity others are not shown. As an example, the affine matrix  508  might perform right hip rotation, although it will be understood by those in the art that many other operations are possible. Furthermore, the matrix transforms all the nodes below it. For example, the affine matrix transforms nodes  513 ,  514 , and nodes  521 - 526 . Thus it transforms not just the object nodes (e.g., legs) but also transforms the bounding spheres. 
     FIG. 6 shows how the affine transform matrix will generally transform the original bounding sphere  601 , having a first radius  611 , into an ellipsoid  602 . The preferred embodiment of the present invention teaches a way to calculate a radius  612  for a bounding sphere  603 , such that the bounding sphere  603  completely surrounds the ellipsoid  602 . Sphere  605  is of the ideal size, as it completely encloses the ellipsoid  602  and has the smallest possible radius  607 . According to the present invention, the second sphere  603  will always be large enough to fully contain the ellipsoid  602 , although it could be somewhat larger than absolutely necessary. It can be shown that the new radius  612  can never be more than the sqrt(3) times as large as the ideal size radius  607 , given that a 3×3 affine transform matrix is used. Thus, the present invention provides a method and apparatus for an accurate calculation of the radius  612  of transformed bounding spheres. 
     More specifically, the present invention calculates how much the radius  611  for the sphere  601  should be expanded or contracted to form a new sphere  603 , as described. The most accurate answer could be found by multiplying radius  611  by the singular value of the affine transform matrix. (Which, in this case, is equal to the square root of the largest eigenvalue of the matrix times its transform.) Unfortunately, this is a complex calculation. However, the eigenvalues of any real valued 3×3 symmetric matrix always lie within the respective Gershgorin intervals. The present invention calculates the upper bounds of the three Gershgorin intervals for the affine matrix times its transpose. Thus, an upper bound on the singular value is found. Finally, the old sphere radius  611  is multiplied by the upper bound on the singular value. This method is much simpler mathematically than conventional methods. Thus, the present invention provides for a method and apparatus for fast calculation of the radius of transformed bounding spheres. 
     FIG. 7 shows a flowchart for calculating the new radius of the transformed sphere, according to the present invention. In step  701 , the six unique entries of the affine matrix times its transpose are calculated. (Because there will always be only six and not nine unique entries for the 3×3 matrix, the present invention saves even more processing time.) In step  703 , the upper bounds of the three Gershgorin intervals are found. In step  705 , the largest of the upper bounds is selected. In step  707 , the square root is taken. Finally, in step  709 , the radius of the old bounding sphere is multiplied by the square root just calculated. This provides a radius for the transformed bounding sphere that is at least as large as necessary, but may never be more than sqrt(3) times as large as necessary, given that a 3×3 affine matrix is used. 
     FIG. 8 shows more details for the calculation of the upper bounds of the Gershgorin intervals. In FIG. 8, a 3×3 matrix  801  and its transpose  803  are shown. Matrix  805  is the symmetric matrix that results from multiplying the matrix by its transform. The three Gershgorin intervals  807  are shown as well. 
     The following is a mathematical proof that the Gershgorin upper bound on the maximum singular value of a 3×3 matrix M is at most sqrt(3) times the actual singular value of M. (And thus the radius calculation can never be more than sqrt(3) times as large as necessary.) 
     Let M′ denote transpose (M). 
     We know that (the maximum singular value of M)&gt;=(maximum column length in M), since M scales the length of the corresponding unit basis vector by that amount. 
     Squaring both sides, we get: (maximum eigenvalue of M′M)&gt;=(maximum diagonal entry of M′M). 
     But note that (maximum diagonal entry of M′M) is actually the maximum of the absolute values of all the entries of M′M (since the entries of M′M are the pairwise dot products of the columns of M, and the maximum such dot product is achieved as the dot product of the longest column vector with itself). 
     So (maximum eigenvalue of M′M)&gt;=the absolute value of each entry of M′M. 
     So the Gershgorin upper bound on the maximum eigenvalue of M′M, which is of the form: |a|+|b|+|c| for some entries a,b,c of M′M,&lt;=3*(maximum eigenvalue of M′M). 
     Taking the square roots of both sides, this says that the Gershgorin upper bound on the maximum singular value of M is&lt;=sqrt(3) times the actual maximum singular value of M. 
     Another embodiment of the present invention multiples the transpose of the affine matrix times the affine matrix instead of the reverse. As the actual eigenvalues are the same regardless of the order of multiplication, the result is acceptable. However, reversing the multiplication may result in a faster calculation. This is because in certain cases the matrix columns are more likely to be pairwise orthogonal than the rows. Thus, this embodiment of the present invention provides for a method and apparatus that is computationally fast, as well as accurate. 
     The foregoing descriptions of specific embodiments of the present invention have been presented for purposes of illustration and description. They are not intended to be exhaustive or to limit the invention to the precise forms disclosed, and obviously many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application, to thereby enable others skilled in the art to best utilize the invention and various embodiments with various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims appended hereto and their equivalents.