Patent Publication Number: US-7589725-B2

Title: Soft shadows in dynamic scenes

Description:
BACKGROUND 
     Computer-generated scenes continue to become increasingly more realistic. One technique that greatly increases the realism of the scenes is the use of soft shadows. Soft shadows arise from area light sources for which the lighting&#39;s low-frequency directional dependence predominates over the effect of its high frequencies. Conversely, hard shadows arise from point or directional (i.e. high frequency) light sources. Several techniques exist to generate hard shadows in real-time such as shadow buffers. Hard shadow techniques can be extended to render soft shadows by numerically integrating over many light directions constituting the area light source. Unfortunately, these techniques do not generate soft shadows in real-time because integrating over a large light source area requires too many directional samples and thus too many rendering passes. 
     One recent real-time approach computes soft shadows using shadow and illumination fields. Shadow fields describe the shadowing effects of an individual scene entity at sampled points in its surrounding volumetric space. The illumination field for a local light source is referred to as a source radiance field (SRF). The SRF consists of cube maps that record incoming light from the illuminant at sample points in a surrounding volumetric space. An infinitely-distant environment map is a special case that can then be represented as a single SRF cube map. Each object in the scene is represented by an object occlusion field (OOF). The OOF records the occlusion of radiance by the object as viewed from sample points around the object. The soft shadows are then computed at runtime by rotating each blocker visibility function into the local coordinate frame and computing the spherical harmonic product over all of the blockers. While this technique improves the generation of soft shadows in dynamic scenes, the technique is still too computationally complex (e.g., spherical harmonics rotation and products) to allows the soft shadows to be efficiently generated when there are several objects moving in the scene. 
     SUMMARY 
     The present soft shadowing technique pre-computes visibility of blockers using a log of a spherical harmonic visibility function. These logs can then be accumulated and exponentiated in real-time to yield the product visibility vector over all the blockers. The product visibility vector is combined with the light intensity and surface reflectance (bi-directional reflection distribution functions (BRDF)) to determine shading at a receiver point in a computer-generated scene. Diffuse surfaces are a special case for which only the surface normal, rather than a general BRDF, is required at the receiver point. 
     This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an illustrative system that may be used to implement the soft shadowing technique described herein in accordance with one embodiment. 
         FIG. 2  is a graphical representation of a scene having a large area light source, local light sources, and objects which block the light from a receiver point. 
         FIG. 3  is a flow diagram illustrating an exemplary soft shadowing process performed at each receiver point of an object. 
         FIG. 4  is a flow diagram illustrating a process for determining the log visibility vectors which are pre-computed for use during the soft shadowing process of  FIG. 3 . 
         FIG. 5  illustrates a computer-generated object and a corresponding blocker geometric representation of the computer-generated object. 
         FIG. 6  is a flow diagram illustrating an exemplary process for obtaining a blocker geometric representation of a computer-generated object. 
         FIG. 7  is a graphic depiction of blocker geometric shapes and their impact on self-shading at a receiver point. 
         FIG. 8  is a set of exemplary spherical harmonics exponentiation calculations suitable for use in the soft shadowing process of  FIG. 3 . 
         FIG. 9  is a block diagram illustrating exemplary components for implementing the present soft shadowing technique. 
     
    
    
     DETAILED DESCRIPTION 
     The present soft shadowing technique determines the shading that is applied at a point within a scene based on shadows that are cast from several objects (as approximated by a set of blockers), the lighting within the scene, and the surface BRDF (surface normal for diffuse surfaces) at the point. The shadows of the blockers are each represented by a visibility function that determines the amount of light the blocker blocks from reaching the point being shaded. In contrast with techniques that numerically integrate over a large number of lighting directions and test blocker visibility in each direction, the present technique accumulates the visibility in log space and computes the spherical harmonic exponential in real-time to arrive at the product visibility vector over all the blockers. This decreases the per-blocker computation which allows the present technique the ability to handle more blockers and the ability to map the computation to a graphics processing unit in a single shading pass. As will be described, the present technique is applicable to any type of object, including objects that have a dynamic geometry, such as deforming characters whose motion may not be known in advance. These and other aspects of the present soft shadowing technique are now described in detail. 
       FIG. 1  is an illustrative system that may be used to implement the soft shadowing technique described herein in accordance with one embodiment. The system includes a computing device, such as computing device  100 . Computing device  100  represents any type of computing device such as a personal computer, a laptop, a server, a game console, a handheld or mobile device (e.g., a cellular phone, digital assistant), and the like. In a very basic configuration, computing device  100  typically includes at least one processing unit  102  and system memory  104 . Depending on the exact configuration and type of computing device, system memory  104  may be volatile (such as RAM), non-volatile (such as ROM, flash memory, etc.) or some combination of the two. System memory  104  typically includes an operating system  106 , one or more program modules  108 , and may include program data  110 . For the present soft shadowing technique, the program modules  108  may include one or more components  140  for implementing the soft shadowing technique. In addition, program modules  108  may include a graphics application  142  that utilizes the soft shadowing technique implemented within components  140 . Alternatively, the operating system  106  may include one or more components for implementing the soft shadowing technique. Program data  110  may include a tabulation of visibility logs  150 . This basic configuration is illustrated in  FIG. 1  by those components within dashed line  112 . 
     Computing device  100  may have additional features or functionality. For example, computing device  100  may also include additional data storage devices (removable and/or non-removable) such as, for example, magnetic disks, optical disks, or tape. Such additional storage is illustrated in  FIG. 1  by removable storage  120  and non-removable storage  122 . Computer storage media may include volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data. System memory  104 , removable storage  120  and non-removable storage  122  are all examples of computer storage media. Thus, computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by computing device  100 . Any such computer storage media may be part of device  100 . 
     Computing device  100  may also have input device(s)  124  such as keyboard, mouse, pen, voice input device, touch input device, etc. Computing device  100  may also contain communication connections  128  that allow the device to communicate with other computing devices  130 , such as over a network. Communication connection(s)  128  is one example of communication media. Communication media may typically be embodied by computer readable instructions, data structures, program modules, or other data. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Computer readable media can be any available media that can be accessed by a computer. By way of example, and not limitation, computer readable media may comprise “computer storage media” and “communications media.” 
     Various modules and techniques may be described herein in the general context of computer-executable instructions, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. for performing particular tasks or implement particular abstract data types. These program modules and the like may be executed as native code or may be downloaded and executed, such as in a virtual machine or other just-in-time compilation execution environment. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments. An implementation of these modules and techniques may be stored on or transmitted across some form of computer readable media. 
       FIG. 2  is a graphical representation of a scene  200  having a large area light source  202  and objects  204 - 216 . In  FIG. 2 , object  210  is shown with a receiver point p at which soft shadowing is applied. In overview, soft shadowing depends on lighting, reflectance, and visibility. The present soft shadowing technique focuses on calculating the visibility efficiently so that soft shadowing may be applied to scene that has numerous moving objects which block the light received at a point p. For example, in dynamic scenes, objects  204 - 216  may each move within the scene and may deform and be articulated. Therefore, the visibility may rapidly change and vary significantly among different scene points. 
     In overview, the visibility of the light source  202  from the receiver point p needs to be computed with respect to the occluding objects (i.e., objects  204 - 216 ) in the scene  200 . While this conventionally requires tracing rays between the illuminant and the receiver point, new techniques have efficiently calculated the visibility using simple operations on pre-computed data. For example, one technique uses pre-computed shadow fields. While using pre-computed shadow fields is more efficient than ray tracing, the technique is still limited to roughly six rigidly-moving blocker objects to maintain real-time performance. The limitation is due mainly because it requires the shadow fields to be aligned to the scene positions using coordinate transformations and multiplications of each shadow field to obtain the aggregate visibility of all the blockers. In contrast, as will be described below, the present soft shadowing technique does not require expensive rotation operations and products. 
     The present technique was formulated after recognizing that the visibility functions could be represented as logs which could then be added in real-time to give a good approximation of the aggregate visibility. Thus, the present technique accelerates the process of soft shadowing and allows soft shadowing to be implemented in real-time using the graphics processing unit. 
     Before describing the present technique in further detail, the following is an overview of some the concepts and terminology used though-out the following discussion. Spherical harmonics are used to represent low-frequency spherical functions, such as radiance incident at a point and blocker visibility functions which modulate distant radiance. Given a real spherical function f(s), the spherical function may be projected to determine a vector f that represents its low-frequency behavior using the following equation:
 
 f=∫   S   f ( s ) y ( s ) ds   (1)
 
where y(s) is the vector of the spherical harmonic basis function. The spherical harmonic basis functions are orthogonal polynomials in s(x,y,z) restricted to the sphere sεS. An order n spherical harmonic projection has n 2  vector coefficients. Conversely, given a spherical harmonic vector f a continuous spherical function {tilde over (f)}(s) can be reconstructed using the following equation:
 
 {tilde over (f)} ( s )=Σ i=0   n     2     −1   f   i   y   i ( s )= f·y ( s )  (2)
 
     For computing combined shadowing effects of multiple blockers directly in the spherical harmonic basis, without resorting to numerical integration over directions or performing complicated geometric clipping operations, spherical harmonic products and a triple product tensor may be used. The spherical harmonic product, denoted f·g, represents the order-n projected result of multiplying the reconstruction of two order-n vectors, f times g, as follows: 
                     f   *   g     =           ∫   S     ⁢       f   ⁡     (   s   )       ⁢     g   ⁡     (   s   )       ⁢     y   ⁡     (   s   )       ⁢     ⅆ   s         ⇒       (     f   *   g     )     i       =       ∑   jk     ⁢       Γ   ijk     ⁢     f   j     ⁢     g   k                   (   3   )               
The spherical harmonic triple product tensor, Γ ijk , is defined using the following equations:
 Γ ijk =∫ S   y   i ( s ) y   j ( s ) y   k ( s ) ds.   (4) 
The spherical harmonic triple product tensor is a symmetric, sparse, order-3 tensor. Equation (4) above incurs truncation error because the product of two order-n vectors is actually order 2n−1. Spherical harmonic products are expensive operations, even at low orders.
 
     A spherical harmonic product matrix, M f , given a spherical harmonic vector f, may be defined. The product matrix is a symmetric matrix which encapsulates spherical harmonic product with f. In other words, f·g=M f g for an arbitrary vector g. M f  is defined by the following equation: 
     
       
         
           
             
               
                 
                   
                     
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                         Γ 
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                           f 
                           k 
                         
                         . 
                       
                     
                   
                 
               
               
                 
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     Based on these concepts, the present soft shadowing technique was formulated.  FIG. 3  is a flow diagram illustrating an exemplary soft shadowing process  300  performed at each receiver point of an object. Prior to soft shadowing, the shape of each object is represented by a set of blockers which may take any arbitrary shape. One embodiment for representing the shape of the objects using a set of blockers is illustrated later in conjunction with  FIG. 5 . As will be described, in that embodiment, the arbitrary shape is a sphere. Process  300  begins at block  302  where the blockers that impact the soft shadow at receiver point p are determined. Briefly, determining the blockers that impact the soft shadow, described later in detail in conjunction with  FIG. 7 , distinguishes blockers that correctly self-shadow from blockers that incorrectly self-shadow at receiver point p. Processing continues at block  304 . 
     At block  304 , a pre-computed visibility log for each blocker that affects the shading at the receiver point is obtained. One embodiment for pre-computing the visibility log for blockers is described later in conjunction with FIG.  4 . In overview, in one embodiment, at run-time, receiver point p is assumed to be shadowed by one or more blockers each having a sphere of radius r centered at P. The pre-computed log visibility vector, f(p,P,r) for one single sphere blocker is determined with the following equation:
 
θ( p,P,r )=sin −1 ( r/∥P−p ∥)
 
 s ( p,P )=( P−P )/∥ P−p∥ 
 
 f ( p,P,r )= F (θ( p,P,r )) y ( s ( p,P ))  (6)
 
where F(θ) is the diagonal matrix corresponding to vector f*(θ). Vector f*(θ) is determined based on the visibility function g(θ) for a canonical circle centered at the z axis. The spherical harmonic visibility function for circles of angular radius θ centered around the z-axis is given by the following equation:
 
                     g   ⁡     (     s   ,   θ     )       =     {           0   ,             if   ⁢           ⁢     s   ·     (     0   ,   0   ,   1     )         ≥     cos   ⁡     (   θ   )                   1   ,           otherwise   .                     (   7   )               
Projecting g yields many zero components because of its circular symmetry around z. The visibility function g may then be represented using doubly-indexed spherical harmonic notation, g lm , where l represents the band index (l=0, 1, . . . , n−1) and m indexes the 2l+1 component of band l. This visibility function is circularly symmetric around z (i.e., g l0  components are non-zero). Therefore, the visibility function g can be represented with n rather than n 2  components via vector g 1 (θ). This defines the g* vector as a function of θ. The logs of the circle visibility vector is then the f* vector defined as f*=log(g*). Thus, as will be described, the logs of circle visibility vectors f* are tabulated rather than their direct projections. These pre-computed visibility logs are tabulated and stored. The tabulated data is then queried in real-time to obtain the visibility log for each blocker that affects the shading at the receiver point. Processing continues at block  306 .
 
     At block  306 , each visibility log that is obtained is rotated based on the direction of the receiver point. When the blocker geometry is represented using a set a spheres, the visibility function may be rotated from z to an arbitrary axis z′ via a rotation rule as follows:
 
 g   z* (θ)= G (θ) y ( z ′)=diag( g   0 *(θ), g   1 *(θ), g   1 *(θ), g   1 *(θ), . . . ) y ( z′ ).  (8)
 
This allows the visibility for circles of any angular radius and around any axis to be defined using a one dimensional table of n projection coefficients, g(θ), and a two dimensional table of n 2  spherical harmonic basis functions, y(s). In another embodiment, the visibility is evaluated without being tabulated. Processing continues at block  308 .
 
     At block  308 , the visibility log vectors are summed. Accumulating the log involves vector sums which are independent of the blocker ordering and much cheaper than spherical harmonic products. Therefore, the present soft shadowing technique significantly reduces the per-blocker computational costs. Processing continues at block  310 . 
     At block  310 , a spherical harmonic exponentiation is performed on the sum to obtain the total product visibility vector over all the blockers. While the exponentiation calculation is an expensive operation, it only needs to be performed once for each receiver point. For order-n spherical harmonic vectors, this reduces per-blocker computation from O(n 5 ) to O(n 2 ). Thus, for order-4 spherical harmonic vectors, this results in less than 1/20 th  the per-blocker cost. This reduction allows the present soft shadowing technique the ability to handle complicated scenes with many more blockers than prior techniques could handle. Processing continues at block  310 . 
     At block  310 , a triple product of lighting, reflectance, and visibility is performed to obtain the shading at the receiver point. For diffuse surfaces, shading is determined by computing (H(N),L,g) where L is the light vector, g is the total product blocker visibility vector from block  310 , and H(N) is the irradiance weighting function given the surface normal N as determined using the following conventional equation: 
                     H   ⁡     (   N   )       =       1   Π     ⁢       ∫   S     ⁢       max   ⁡     (       s   ·   N     ,   0     )       ⁢     y   ⁡     (   s   )       ⁢       ⅆ   s     .                   (   9   )               
For diffuse surfaces in lighting environments, L H (N)=L*H(N) may be tabulated. At runtime, L H  may then be indexed at the receiver normal N p  to obtain the cosine-weighted incident radiance at point p. This result is dotted with the exponentiated blocker vector g to produce the shadowed result L H (N p )·g. However, if the lighting changes every frame, tabulating L H  may be difficult. Therefore, in a further refinement, shading may instead by calculated by forming the light&#39;s product matrix M L  and computing (M L g)·H(N p ). Processing continues at block  314 .
 
     At block  314 , the shading is applied at the receiver point. The total product blocker visibility vector may also be used to shade other types of bi-directional reflection distribution functions (BRDFs) or textural detail. 
       FIG. 4  is a flow diagram illustrating a process for determining the log visibility vectors which are pre-computed for use during the soft shadowing process of  FIG. 3 . Process  400  begins at block  402  where a blocker geometry is obtained. The blocker geometry may take any arbitrary shape. For example, the blocker geometry may utilize conventional blocker geometry, such as octree and medial axis. In another embodiment, illustrated in  FIG. 5 , the blocker geometry utilizes a set of varying-sized spheres to represent the object. The arbitrary shapes of the blocker geometry are referred to as blockers because they block the light from reaching the receiver point p. Once the blocker geometry is obtained, processing continues at block  404 . 
     At block  404 , the log visibility vector for each blocker in a scene is determined. Briefly, the visibility vector represents the fraction of light that a blocker blocks from reaching receiver point p. These vectors represent low-frequency visibility of blockers in the spherical harmonic basis. Processing continues at block  406 . 
     At block  406 , the log visibility vector for each blocker may be stored based on properties of the blocker. In the embodiment in which the blocker geometry utilizes spheres, the properties that are stored may include the sphere&#39;s center position and radius. This allows a single look-up table to be used to obtain the log visibility vectors. The log visibility vector for each blocker may then be computed based on the center location of the sphere and the radius of the sphere. This allows a compact representation for the visibility vector for each blocker. The coordinates for the center of each blocker that affects the shading at the receiver point can then be easily queried at run-time. Self-shadowing can also be efficiently handled as will be described in conjunction with  FIG. 6 . 
       FIG. 5  illustrates a computer-generated object  500  and a corresponding blocker geometric representation  510  of the computer-generated object. The computer-generated object  500  has a geometry T. The blocker geometric representation  510  is formed using spheres (e.g., spheres  512  and  514 ) to approximate the geometry T of the computer-generated object  500 . In generating the geometric representation  510 , variational shape approximation is applied. In addition, a set of n S  spheres S i  each having a center P i  and radius r i  are determined that bound the geometry T but have minimal outside volume E. The volume within T&#39;s interior may be neglected regardless of how many spheres overlap because shadowing does not need to take into account the number of times light is blocked by a solid object. In addition, the sphere set&#39;s bounding property eliminates gaps that would otherwise let light leak though solid objects. 
       FIG. 6  is a flow diagram illustrating an exemplary process  600  for obtaining a blocker geometric representation of a computer-generated object. Process  600  begins at block  602  where the geometry T is divided into a set of points which include points on the surface (i.e., triangle midpoints) and points in the interior (i.e., grid corners from a mesh voxelization). Processing continues at block  604 . 
     At block  604 , initial blockers representing geometry T are initialized with the randomly picked set of points from block  602 . Thus, the blockers, which in one embodiment are spheres (S i ), are initialized with one of the randomly picked center points P i . Processing continues at block  606 . 
     At block  606 , the radius r i  of each sphere S i  is then initialized with the value of 0. The spheres then undergo an iterative process as described below in blocks  608 - 616 , to obtain their optimal shape. Processing continues at block  608 . 
     At block  608 , point clustering is applied. Point clustering is performed using a “flood fill” order away from the sphere centers. The flood fill order may be stack-based. The next point is repeatedly popped from the stack and is inserted into the cluster having minimal error. Then, each of the neighbors of the point are inserted onto the stack if the neighbor is not already on the stack. An outside volume error is calculated when adding a point by extending the cluster sphere&#39;s radius r i  so as to include the new point. The outside volume of the new sphere is then measured with respect to geometry T. Block  608  is performed until each of the points have been clustered. The set of cluster spheres then may be set to “bound” the set of points. Processing continues at block  610 . 
     At block  610 , each cluster sphere center P i  is independently updated. The update attempts to minimize outside volume V(S i −T) while constraining r i  to continue bounding the cluster&#39;s points. The minimal P i  may be found using conjugate gradient. Outside volume of a sphere S, V(S−T), may be computed by summing over the triangles of geometry T of its signed outside volume with respect to S. Processing continues at decision block  612 . 
     At decision block  612 , a determination is made whether the last iteration of point clustering and updating significantly reduced the outside volume error or not. If the volume error was significantly reduced, processing loops back up to block  608  to perform another iteration. Otherwise, processing continues at block  614 . 
     At block  614 , cluster teleportation is applied. Cluster teleportation is applied to attempt to subdivide the cluster of maximal error into two by finding a pair of points that are farther away from each other in comparison with pairs of other points. The cluster of maximal overlap, defined as the volume the cluster shares with other cluster spheres divided by its own volume, is chosen for deletion. In one embodiment (not shown), after teleporting the cluster center, another iteration of clustering (block  608 ) and updating (block  610 ) may be performed. If the error is reduced, the teleported perturbation is accepted. Otherwise, the previous state of the clustered spheres is kept. Processing is then complete. 
     As geometry T moves between scenes, the corresponding bounding spheres are updated. Model may be animated using “skinning” which applies a weighted combination of tranformations, attached to bones in an articulated skeleton, the mesh vertices. Mean value coordinate (MVCs) for each sphere center P i  with respect to the “rest pose” of the mesh are found. This expresses P i  as a weighted combination of the vertices. Applying the same weights to vertices in a deformed pose yields the corresponding deformed sphere center P i ′ The original sphere radius r i  is kept which remains a bound, assuming typical articulated motion. In a further refinement, the vector or vertex MVCs may be pre-multiplied by a matrix of bone weights per vertex to speed up weighting calculations. The sphere center transformation is then expressed as a weighted combination over a few bones rather than over many vertices. 
       FIG. 7  is a graphic depiction of blocker geometric shapes and their impact on self-shadowing at a receiver point p. As shown, there a number of blocker spheres (e.g., spheres  702 - 708 ) and a low-frequency light source  720 . A bounding sphere set implies that every point on an object is completely self-shadowed. Therefore, blockers that represent the same local geometry as the receiver (e.g., sphere  700 ) need to be distinguished from blockers that represent non-local geometry (e.g., sphere  708 ). Blockers that represent the same local geometry as the receiver cast incorrect self-shadows. At a receiver point p on an object  700 , there is an outward-facing normal N p . Together, the receiver point p and the outward facing normal N p  define a tangent plane T p . If a blocker sphere S contains point p, the relationship of S&#39;s center to T p  is used to determine whether the sphere is local or non-local. If the center of S is behind T p , S is eliminated (e.g., sphere  202  is eliminated). If the center of S is in front of T p , the radius of sphere S is reduced until it becomes tangent to T p  (see reduction of sphere  708  to sphere  710 ). The term “behind” means that the center of blocker sphere S is on the opposite side of the tangent plane as the normal for receiver point p. The term “in front” means that the center of the blocker sphere S is on the same side of the tangent plane as the normal for receiver point p. If the receiver point p is outside the blocker sphere S, a simple method accumulates the blocker spheres regardless of its position relative to T p . However, this method produces objectionable banding because different shadows are obtained depending on whether the receiver is inside one or more local blockers. However, entirely eliminating local spheres fails to capture important local self-shadowing effects. 
     In a further refinement, a blocking sphere S that is outside receiver point p is removed only if it is entirely behind T p  (e.g., blocking sphere  706  is removed). If the blocking sphere partially passes through the tangent T p , the blocking sphere is replaced with a blocking sphere S′ that is tangent to and in front of T p  (e.g., block sphere  704  is replaced with blocking sphere  712 ). S′ is determined by a point of maximal distance of S in front of T p , q 0 , as well as the projection of q 0  onto T p , q 1 . q 0  and q 1  form a diameter of S′ and their midpoint form its center. Using this replacement rule, a spatial discontinuity occurs as the receiver point p moves from inside S to outside. To minimize this spatial discontinuity, the radius of S′ is gradually scaled up as a function of p&#39;s distance to S along the tangent plane, using the scale factor α as follows:
 
α=max(1,(∥ p−q   1   ∥−d )/ d ),  (10)
 
where d=√{square root over (r 2 −(r−∥q 1 −q 0 ∥) 2 )}. Radius r is the radius of S and d represents the distance of q 1  to the outside of S along the tangent plane.
 
     Once the blockers have been determined, the visibility for each of the blockers is pre-computed so that the product of the blockers can be determined in realtime. Obtaining the product of a collection m blockers, g[1], g[2], . . . , g[m] in log space may be denoted as follows:
 
 g =exp( f )=exp( f[ 1]+ f[ 2]+ . . . + f[m ]).  (11)
 
Each f[i]=log(g[i]), where g[i] is the spherical harmonic projection of the corresponding blocker visibility function as follows:
 
                       g   ⁡     [   i   ]       ⁢   s     =     {           0   ,               ⁢       if   ⁢           ⁢   object   ⁢           ⁢   i   ⁢           ⁢   blocks   ⁢           ⁢   in   ⁢           ⁢   direction   ⁢           ⁢   s     ;                 1   ,               ⁢     otherwise   .                       (   12   )               
Because the present soft shadowing technique utilizes equation (11) above, the accumulation of the logs involve vector sums which are independent of the blocker ordering and much cheaper than spherical harmonic products.
 
     The spherical harmonic log used in the present soft shadowing technique is derived based on observations for calculating the spherical harmonic exponential. It was determined that the spherical harmonic exponential could be evaluated using the following equation: 
                       g   =       exp   ·     (   f   )       =     1   +       R   f   T     ⁢     q   ⁡     (     D   f     )       ⁢     R   f     ⁢   f           ,   where     ⁢     
     ⁢       q   ⁡     (   x   )       =       1   +     x     2   !       +       x   2       3   !       +   …     =           exp   ⁡     (   x   )       -   1     x     .                 (   13   )               
R f   T  is a rotation matrix and D f  is a diagonal matrix. Using eigenanalysis of the product matrix g, Mg=R g   T q′(D g )R g , yields the following approximations:
 log( g )= R   g   T   q ′( D   g ) R   g ( g− 1)   q′ ( x )=1 /q (log( x ))=log( x )/( x− 1)  (14) 
where the function q′ is applied to each diagonal component.
 
     In a further refinement, the eignenvalues of M g  are clipped to avoid applying logs to values that are negative or close to 0 using the following equation:
 
 {tilde over (D)}   g =max( D   g ,ε), {tilde over (M)}   g   =R   g   T   {tilde over (D)}   g   R   g   (15)
 
Equation 15 then uses {tilde over (D)} g  rather than D g . Eigenvalue clipping yields smaller error ∥M g −{tilde over (M)} g ∥ compared with clipping values of g(s) over the sphere. In practice, it has been found that setting the threshold ε to 0.02 times the largest eigenvalue works well for low-order spherical harmonic vectors.
 
       FIG. 8  is a set of exemplary spherical harmonics exponentiation calculations  800  suitable for use in the soft shadowing process of  FIG. 3 . This set of exemplary spherical harmonics exponentiation calculations utilize several exponentiation techniques, such as product series approximation, scalar/matrix exponentials using scaling/squaring, and the like. Exponential technique  802 , referred to as PS−p, is a product series evaluation of degree p. Exponential technique  802  was formulated after recognizing that the result of numerical integration or high-order tensors could be approximated by substituting repeated spherical harmonics products for true spherical harmonic powers. Even though this resulted in approximation error, the approximation was typically accurate, especially for vectors representing bandlimited visibility functions. A spherical harmonic product series is based on these repeated spherical harmonic products and is more practical for real-time evaluation. The Volterra series was applied using a Taylor expansion for h(x)=exp(x) to obtain exponential technique  802 . 
     Exponential technique  810 , referred to as PS*−p combines a DC isolation technique  812  and scaling/squaring  814  applied to a factored degrees product series  816 . Exponential technique  810  analytically computes the exponential of the DC component which reduces the magnitude of the residual vector {circumflex over (f)}, where the {circumflex over (f)}=(0,f 1 ,f 2 , . . . f n     2     −1 ). The scaling/squaring  814  applies where p is a positive integer. The input is divided by a power of 2, the exponential of this scaled input is computed, and then the result is repeatedly squared p times. Scaling/squaring  814  approximates the product series in exponential technique  802 . However, it typically reduces errors relative to a spherical harmonic power series h(f)=(h 0 1+h 1 1f 1 +h 2 f 2 +h 3 f 3 + . . . ), where 1=(√{square root over (4π)},0,0, . . . ,0) and 
               f   p     =       ∫         f   p     ⁡     (   s   )       ⁢     y   ⁡     (   s   )       ⁢     ⅆ   s         =     ∫         (       ∑   i     ⁢       f   i     ⁢       y   i     ⁡     (   s   )           )     p     ⁢     y   ⁡     (   s   )       ⁢       ⅆ   s     .                 
Because exponential squares are more efficient then general spherical harmonic products, this approximation is more useful. Parameter p is chosen as a function of ∥f∥ using p=max(0,└log 2 ∥f∥+3┘). Results show that typically at most p=3 squaring are needed for low-order (n≦6) spherical harmonic vectors. Exponential technique  802  may be evaluated by accumulating successively higher powers of f via f p+1 =f p *f. Thus, p−1 spherical harmonic products are needed for a degree p expansion. The number of spherical harmonic products can be reduced by segregating even and odd powers as shown in factored degrees product series  816 . This results in fewer products which results in smaller truncation error. Thus, factored degree-p product series  816  provides a better approximation than the spherical harmonic power series discussed above. Powers of f can be computed to minimize the number of products in each term: f 2* =f*f, f 4* =f 2* *f 2* , f 6* =f 4* *f 2* , and so on. Other conventional factoring techniques may be applied for series degree p&gt;12.
 
     Exponential technique  820 , referred to as OL (Optimal Linear Approximation) applies an optimal linear method. For spherical harmonics order-4 or lower, an extension of a simple two-term series exp·(f)≈1+f from exponential technique  802  provides good accuracy without the need for even a single spherical harmonic product. Given an input vector f to be exponentiated, DC isolation  812  may be applied to obtain {circumflex over (f)} and compute the magnitude ∥{circumflex over (f)}∥. Exponential technique  820  can then be used to provide an optimal linear approximation. The coefficients a and b may be pre-determined by generating a set of spherical harmonic vector pairs representing circles of increasingly angular radius. One vector in the pair is the visibility function g and the other it is corresponding log vector f. The DC component of f is zeroed out to account for DC isolation, resulting in {circumflex over (f)}. Visibility function g is correspondingly scaled to obtain 
               g   ^     =       exp   ⁡     (     -       f   0         4   ⁢   π           )       ⁢     g   .             
The least-squares best projection of ĝ onto the orthogonal vectors  1  and {circumflex over (f)} is found via
 
             a   =           g   ^     ·   1       1   ·   1       =             g   ^     0         4   ⁢   π         ⁢           ⁢   and   ⁢           ⁢   b     =           g   ^     ·     f   ^           f   ^     ·     f   ^         .               
This results in the minimum error being ∥ĝ−(a1+b{circumflex over (f)})∥. The least-squares projection may be performed for each circle of a different angular radius. The resulting a and b coefficients may be tabulated as a function of ∥{circumflex over (f)}∥, which increases with angular radius.
 
     Experimental results show that models agree on their a and b curves over a substantial part of the domain, roughly ∥{circumflex over (f)}∥&lt;4.8, which corresponds to a blocker of angular radius less than 50°. For bigger ∥{circumflex over (f)}∥, the curves follow an initial baseline curve until they suddenly diverge. Thus, asymptotic behavior may be derived for a and b for certain geometry. 
     Exponential technique  820  may also apply one or more of the other techniques described above, such as scaling/squaring, to reduce the magnitude of the input vector and extend the domain over which accurate results are obtained. When scaling/squaring is applied to the optimal linear method, the resulting technique is referred to as HYB (hybrid of optimal linear method). 
       FIG. 9  is a block diagram illustrating exemplary components for implementing the present soft shadowing technique. As mentioned above, the exemplary components may be incorporated into a graphics application, implemented as other computer-readable instructions that perform graphics computations, or the like. In overview, the exemplary components include a blocker geometry module  902  and soft shadow module  904 . The blocker geometry module  902  is configured to represent each object in a scene using a set of shapes. These shapes are then made available to the soft shadow module  904  for pre-computing blocker visibility logs and for real-time computing of the soft shadow at each receiver point. Thus, soft shadow module includes a preprocessing where the blocker visibility logs  908  are determined and stored and real-time processing where the soft shadows are applied to the objects in the scene. Real-time processing queries the blocker visibility logs  908  to obtain the log for each blocker. As described above, the logs are summed and then exponentiated to obtain a product blocker visibility for all the blockers. The product is multiplied with illumination information  910  and reflectance information  912  to obtain the soft shadows. The lighting and the reflectance may be determined using conventional methods. Thus, as described above, the present soft shadowing technique is able to handle several blockers in real-time. 
     The present soft shadowing technique handles different types of light sources that may be static or moving. In addition, the technique handles receiver points on dynamic geometry. The following describe some of these variations. For static receiver points, local shadowing effects can be “baked in” by dotting with the precomputed vector H(N p )*g p , where g p  represents local visibility due to static occluders. The blocker accumulation (block  308 ) is then performed on blockers from the dynamic geometry. In a further refinement, if the lighting is static, L can be multiplied into g p . For receiver points on dynamic geometry, both static and dynamic blockers are accumulated every frame. 
     The present soft shadowing technique also handles circular/spherical light sources. In this embodiment, the light source is defined in terms of L(θ,d) where θ is the angular radius of the light circle and d is the central direction. Local light sources are handled by allowing θ and d to vary as a function of the receiver point p. This may be supported by tabulating L H (θ,φ)=H((0,0,1))*L(θ, ) where θ is the light&#39;s angular radius and φ is the angle the central light direction makes with the normal N. For this embodiment, the two dimensional table utilizes a canonical orientation aligning the normal N with z and the light direction in the xz plane, making an angle of φ with z. This canonical configuration is then rotated into its actual orientation at each receiver point before computing the dot product with g. This rotation may be accelerated by fitting a single-lobe ZH model to the 2-parameter family of vectors L H (θ,φ). 
     In another refinement, a cosine filter in frequency space may be used to window the lighting and visibility functions in order to remove ‘ringing’ artifacts caused by using spherical harmonics. The cosine filter scales spherical harmonic coefficients in band 1 by α 1 =cos (π/2(l/h). It was found that for order-n spherical harmonics, a window size h=2n worked well. It was also found that it may be beneficial to use a greater windowing (i.e., smaller h) for certain HDR lighting environments, depending on their frequency content. 
     The blocking geometry described above may be further approximated by applying sphere hierarchies. One will note that as a blocker gets closer to a receiver point, detailed knowledge about the blocker&#39;s shape is needed to accurately determine the blocker&#39;s impact on soft shadows at the receiver point. Likewise, as the blocker recedes further from the receiver point, the blocker&#39;s shape may be approximated more coarsely. In a further refinement, the blockers may be grouped in a hierarchical manner and clustered over receiver points. For each receiver point cluster, a cut or list of blocker spheres from an appropriate level of the hierarchy is assembled based on an angle the blocker sphere subtends over the cluster. The log visibility factor is then accumulated at each receiver vertex p in the cluster using a simple approximation that exploits spatial coherence and is based on detailed information computed a single point p* centered in the cluster. By doing so, artifacts from inconsistent blocker approximations used in different clusters may be reduced. 
     The sphere sets determined in  FIG. 6  form leaf notes in the blocker hierarchy. Then, hierarchy levels are constructed one at a time from the leaves up to the root, using a technique based on a conventional clustering technique. Each cluster stores its current bounding sphere. Clustering then iteratively assigns spheres to the closest cluster, based on the distance from the sphere center to the cluster&#39;s center. The cluster&#39;s bounding sphere is then updated. Initially, the cluster&#39;s center is an average center over all the spheres assigned to the cluster. This average is optimized by using the conjugate gradient method to minimize the bounding sphere&#39;s radius. After convergence, each cluster is made a parent node in the hierarchy. The spheres assigned to the cluster become the cluster&#39;s children. The number of clusters is chosen so that the average branching ratio in the hierarchy is around 4. During animation, the bounding spheres at each parent node are updated bottom-up in the hierarchy by applying a sphere pair bound to successively merge in each child node. Receiver point clusters may be computed using a conventional Lloyd clustering into a manually-specified number of clusters. Experiments have had successful results when a receiver cluster contains several hundred vertices. 
     These clusters may then be used to compute the log visibility function instead of using individual blockers as illustrated in  FIGS. 4-6 . First the bounding sphere nodes, S i , with centers P i  and radii r i  are determined that are appropriate for shadowing the cluster. This may be achieved by assuming that the receiver cluster is bounded by a sphere with center p R  and radius r R . Then the blocker sphere subtends an angle less than θ max  if r i &lt;sin (θ max )(∥P i −p R ). If S i =(P i , r i ) satisfies the above test, the sphere is inserted into the sphere blocker list. Otherwise, the node&#39;s children are checked. 
     For each S i  in the assembled list, two visibility vectors are computed at a central cluster point p*. A bounding log visibility vector f b [i](p*,P i ,r i ) applies equation (6) above to the bounding sphere S i . A detailed log visibility vector f d [i] sums log visibility over all leaf node spheres below S i . This computation may be accelerated by pruning detailed spheres using a minimum angular radius θ min . 
     A ratio vector, w[i] is then computed that represents the least-squares best per-band scaling of f b [i] to match f d [i] as follows: 
                       w   l     ⁡     [   i   ]       =       (       ∑     m   =     -   1         +   1       ⁢         (       f   b     ⁡     [   i   ]       )       l   ⁢           ⁢   m       ⁢       (       f   d     ⁡     [   i   ]       )       l   ⁢           ⁢   m           )     /     (       ∑     m   =     -   1         +   1       ⁢       (     ⁢         f   b     ⁡     [   i   ]         l   ⁢           ⁢   m     2         )               (   16   )               
Diagonal matrix W[i] is derived from w[i] by repeating its component 2l+1 times along the diagonal, as done for the diagonal matrix F in equation (6) above.
 
     The sphere hierarchy may also be used to compute a per-point log visibility vector. This is achieved by using information computed previously at the cluster center p* to accumulate a log visibility vector at p for each receiver point in the cluster. A modified version of equation (6) is then applied to each sphere i in the cluster&#39;s blocker list, which multiplies by the ratio vector W[i] as follows:
 
 f[i ]( p,P   i   ,r   i )= W[i]F (θ( p,P   i   ,r   i ))) y ( s ( p,P   i )).  (17)
 
The vectors are summed over i and the visibility vector is obtained by applying equation (11) above.
 
     As described, the present soft shadowing technique achieves significant savings in processing costs when calculating soft shadows for dynamic scenes in real-time. Although details of specific implementations and embodiments are described above, such details are intended to satisfy statutory disclosure obligations rather than to limit the scope of the following claims. Thus, the invention as defined by the claims is not limited to the specific features described above. Rather, the invention is claimed in any of its forms or modifications that fall within the proper scope of the appended claims, appropriately interpreted in accordance with the doctrine of equivalents.