Patent Application: US-8839293-A

Abstract:
a method for rendering a three - dimensional image on a computer graphics display device involving the steps of providing a data base defining at least the geometry and reflectivity of light emitters and object in the scene and approximating surfaces of each object in the scene as patches in three dimensional space . each patch is approximated as one or more elements defined by vertices . a source of irradiated light in the scene is selected and defined as a light source . then , for each vertex , a form factor representing the fraction of light energy that arrives at the vertex from the source is determined by ray tracing from the vertex to the source . radiosity is then determined at each vertex based on the form factors determined for each vertex . this process is repeated a selected number of times using a different light source each time . in this manner a plurality of radiosities are determined that collectively indicate the global illumination of the scene . the scene is then displayed based upon these radiosities .

Description:
the following definitions are employed herein for convenience , though their meaning and significance is well known in the art . surface : a bounded area with a constant or continuous surface normal , for example , polygons and curved surfaces . surfaces are not limited by complexity , and may be concave polygons with holes . patch : a whole or part of a surface which has been divided through some number of binary subdivisions . it is the basic geometric unit which acts as an illuminator of the environment , whether direct ( light ) or indirect ( reflector ). reflectivity and emission characteristics are assumed constant over the patch . global illumination is solved in terms of patch radiosities . element : whole or part of a patch which has been divided through some number of binary subdivisions . it is the smallest geometric unit for which the environment intensities are computed . it acts as a receiver of light for patches and as a discrete unit for numerical integration of the patch to patch form factors . an element is usually a polygon , but may assume other shapes . vertices : points having x , y and z coordinates defining the elements . according to the present invention , the aforementioned problems with the prior art method of progressive radiosity employing the hemi - cube approach may be eliminated by determining the illumination directly at the vertices . as before , at each solution step , the surface with the most radiated and / or emitted energy to contribute to the environment is treated as the source surface . instead of performing a hemi - cube at the source , each element vertex in the scene is then visited and a form factor is computed form the source surface to the vertex by casting rays ( i . e ., by ray tracing ) from the vertex to sample points on the source . this process is sometimes referred to as &# 34 ; shooting rays .&# 34 ; this process eliminates sampling problems inherent in the hemi - cube approach since illumination is guaranteed to be computed at every vertex . this is schematically illustrated in fig3 wherein rays from a source s are cast to each of the vertices defining the surface elements or polygons of a patch p . as illustrated in fig3 a plurality of sample points sp 1 , sp 2 and sp 3 , are defined on the area of the source and are uniformly distributed thereon . the number of sample points on the source s may vary from one vertex to the next , thus allowing area sources to be approximated as accurately as desired . since the form factors are computed independently at each vertex , each form factor may be computed to any desired accuracy . the true surface normal at the vertex may be used , thus solving the problem of continuous shading of independent surface facets . fig4 conceptually illustrates the concept of a form factor from a differential area da 1 to a differential area da 2 , wherein the differential areas are separated by a distance r . as illustrated in fig4 the ray cast between the two differential areas subtends an angle θ 1 to the surface normal of differential area da 1 and subtends an angle θ 2 to the surface normal of the differential area da 2 . the form factor from differential area da 1 to differential area da 2 is : ## equ2 ## for a finite area a 2 , this equation must be integrated over that area and can be solved analytically for simple configurations , but , for computer usage , it must be evaluated numerically . numerical integration may be accomplished in straightforward manner by approximately the integral as the sum of form factors computed for smaller regions of areas &# 34 ; delta a 2 &# 34 ; ( each corresponding to a patch i ). this is conceptually illustrated in fig5 where the subscript &# 34 ; i &# 34 ; has been added to the terms r , θ and θ 2 to indicate that form factors to a number &# 34 ; n &# 34 ; of patches i are to be determined . with reference to fig5 the form factor from the differential area da 1 to the finite area a 2 can be shown to be : ## equ3 ## a problem with the foregoing equation is that it breaks down if the size of delta a 2 is large relative to the distance r ; thus as r becomes less than unity , the size of delta a 2 must be shrunk correspondingly , or the resulting form factor grows without bound . to limit the subdivision of the source area a 2 , and thus the computational cost of computing form factors while avoiding the difficulty of the preceding equation , the delta areas ( delta a 2 or patches i ) are explicitly treated as finite areas . an equation for the form factor from finite delta areas ( patches i ) can be obtained by approximating each delta area by a simple finite geometry for which an analytical solution is available . for simplicity , a disk has been chosen as the finite area upon which to base the approximation . with reference to fig6 the form factor df 21 , from a source area a 2 to a directly opposing , parallel differential area da 1 is provided by the following relationship : fig7 illustrates the case where the source area a 2 and the differential area da 1 are not in parallel planes . for this case , the form factor from the area a 2 ( source surface ) to the differential area da 1 ( receiving area ) is approximated as : as in the case of fig5 to evaluate the form factor from a general source a 2 to a vertex at the receiving area da 1 , the source a 2 is divided into delta areas or patches . the form factor due to each patch is computed utilizing the immediately preceding equation . occlusion between the source surface and the vertex by an object in the scene is tested by applying ray tracing methods , i . e ., by &# 34 ; shooting &# 34 ; or &# 34 ; casting &# 34 ; a single ray from the receiving vertex to the center of the patch on the source . this ray is a simple shadow ray , in other words , it provides only a yes or no answer to the question of whether anything is intersected . for a yes answer , the magnitude of the form factor is zero . for a no answer , the magnitude of the form factor is computed according to the form factor equation . in the remaining discussion herein , this process of &# 34 ; shooting &# 34 ; rays to determined whether anything is intersected will be referred to as &# 34 ; sampling the source &# 34 ; and rays will be said to be &# 34 ; shot &# 34 ; from the vertex to sample points on the source . if sample points are distributed uniformly on the source , then the total form factor is simply the sum of the form factors computed for each sample point i according to the following relationship : ## equ4 ## where n = number of sample points on source δ i = 1 if sample point is visible to vertex , 0 if occluded . in the immediately preceding equation , the presence of the area term , a 2 , in the denominator prevents the result from growing without bound for small values of r . this allows approximate form factors to be computed using large delta areas or patches , and hence a low number of sample points . thus , a fast , approximate radiosity solution can be obtained using a very small number of rays ( as few as one ray per vertex per source ). the accuracy of the result increases as the number of sample points on the source is increased . this can be seen by observing that as the value of n grows larger , the term a 2 / n tends towards zero and each term of the summation approaches the equation for a form factor from a differential area to a differential , as in the case of the equation for fig4 . thus , a smooth continuum of results is available depending upon the computational time that one is willing to expend on the executing computer system . it can be shown that the radiosity of patch i due to energy received from another patch j ( i . e ., patch j being a source ) can be determined according to the following relationship : if the radiosities are computed at vertices , instead of at the center of a patch , it can be shown that , by combining the form factor equation with the above radiosity equation , the equation for radiosity at a vertex 1 due to illumination by a source surface 2 is : ## equ5 ## where : b 1 = radiosity at the vertex 1 ; n = the number of sample points defined on the source surface ( 2 ); i represents one of the n sample points on the source surface ( 2 ) ( for i = 1 to n ); δ i = 1 if the sample point i is not occluded at vertex 1 and zero if occluded ; θ 1i = is the angle between a surface normal of the source surface 2 and the ray cast from the sample point i to the vertex 1 ; θ 2i = is the angle between the surface normal of the patch at the vertex 1 and the ray cast from the sample point i to the vertex 1 ; at each step of the progressive solution , the contribution to the radiosity of each element vertex in the environment due to the current source is computed using this relationship . it was previously mentioned that a uniform distribution of sample points on the source is used . however , it has been found that this can produce a form of aliasing , particularly at shadow edges . there are a number of solutions to this problem . the vertex radiosities can be filtered using a weighted averaged of neighboring vertex radiosities . alternatively , the accuracy of form factors can be increased by shooting a variable number of rays for each vertex - source pair . the form factor for a given vertex - source pair can be computed using successively greater numbers of rays , until the variation and the resulting form factors drops below a certain criteria . thus , at vertices lying on shadow boundaries , many rays may be shot to achieve an accurate form factor , while vertices completely inside or outside the shadow will require a much smaller number . this approach also increases the accuracy of form factors for sources that are very close to the vertex , in which case a large number of sample points may also be required . form factors obtained using ray tracing from a vertex to an area source are compared to analytical results for several configurations in fig8 a , 8b and 9a , 9b . as can be seen , the actual and analytical results are quite close for a square source , as illustrated in fig9 a and 9b . accuracy decreases as the source becomes more oblong , as illustrated in fig8 a and 8b . the results also becomes less accurate as the source and vertex move off axis from one another , as illustrated in fig9 a and 9b . however , since the magnitude of the form factors decreases at the same time , relative error is greatest at angles where incoming energy will make the smallest contribution . in each case , increasing the number of sample points brings the approximate result closer to the correct result . the use of ray tracing to compute form factors in radiosity allows the geometry used for shadow testing to be different from that used to represent objects for shading and rendering . when shooting a ray , it is not necessary to test for intersection against all the elements into which the environment has been subdivided . the original surface geometries can be used instead . however , for shadow testing during the form factor computation , the ray tracer intersects the original , true surface geometries . the ability to use different geometric representations for shadow testing and for shading and rendering provides an important source of efficiency for radiosity . adequately representing complex shadows can require subdividing surfaces into a very large number of small elements . by ray tracing against the original geometries , surfaces can be represented by as many facets as desired for shading and display , without increasing the cost of each shadow test . the following is a pseudo - code listing setting forth the method of the present invention . the listing includes self - evident explanatory comments . at a given step of the radiosity solution , a single surface patch is selected to act as the source surface . the contribution of the energy reflected or emitted by that patch to the radiosity of every element vertex in the environment is determined based on the form - factor from the source to the vertex . the form - factor for a source patch to a vertex is determined as follows : ______________________________________current . sub .-- ff . sub .-- estimate = 0 . 0 ( ff = form factor ) do ( previous . sub .-- ff . sub .-- estimate = current . sub .-- ff . sub .-- estimate for several different points on the light source ( determine direction from vertex to point on light source if vertex and light source face each other for this point ( shoot ray from vertex to point if ray does not intersect anything between vertex and light ( label point on light as visible ) ) ) current . sub .-- ff . sub .-- estimate = 0 . 0 for each point on light ( if point was labeled as visible ( current . sub .-- ff . sub .-- estimate = ## str1 ## )) until absolute value of current . sub .-- ff . sub .-- estimate minus previous . sub .-- ff . sub .-- estimate is less than some accuracy criterionform . sub .-- factor = current . sub .-- ff . sub .-- estimatethe following terms used above are defined here : area = area of source patch theta . sub .-- v = angle between vertex normal and direction to point on light theta . sub .-- s = angle between source normal and direction to the vertex num . sub .-- points = total number of points to which rays have been shot on the light source patch for this vertexpoint lights and directional lights use the same basic algorithm , except that the equation for the form - factor is different , and only one ray needs to be shot ( since the lightsource has no area ) determine direction from vertex to light if vertex is within cone of influence of the light ( shoot ray from vertex to light source ( for directional light , shoot ray in direction of lightsource ) if ray does not intersect anything between vertex and light ( ## str2 ## ) ) the equation given here for the form . sub .-- factoris a general equation that encompasses the most widely used &# 34 ; non - physical &# 34 ; light source types . these areexplained in more detail below . cone lights are takeninto account by the step in which the vertex is checked forinclusion in the &# 34 ; cone of influence &# 34 ; of the light . a1 , a2 and a3 are the attenuation coefficients . n is a spot light power . ______________________________________ ** means to the power of , e . g . distance ** 2 means distance squared . fig1 a , b illustrate , in flow chart form , the method of the present invention described above and set forth in the pseudo code listing . the present invention may be employed in connection with any suitable three dimensional graphics software , such as the commercially available starbase graphics software package and hpux hardware manufactured and sold by hewlett - packard company , palo alto , calif . attached hereto as appendix a is a source code listing , written in the programming language &# 34 ; c &# 34 ;, implementing the method of the present invention on such system . the implementation is for a three dimensional rendering of an object floating in the space of a room . as pointed out in the pseudo code listing above , an important feature of the present invention is the ability to employ so called &# 34 ; non - physical &# 34 ; light sources in radiosity . a &# 34 ; non - physical &# 34 ; light source is one which has no physical counterpart outside of the computer . the ability to employ such light sources can be helpful in creating certain shading effects that help to define special relationships , surface geometries , etc . of the same scene . according to the invention , different light sources including &# 34 ; non - physical &# 34 ; light sources are modeled by varying the form factor equation . this follows since light energy received at a point from a source is determined according to the selected form factor equation . thus , by varying the form factor equation from point to point , variations in light characteristics , such as attenuation , distribution , etc ., can be simulated . as set forth in the pseudo code listing and with reference to fig1 , the basic equation for defining characteristics of light source according to the present invention is : ## equ6 ## where : f = form factor ; θ v , θ s and r are as show in fig1 ; a 1 , a 2 , and a 3 = constants representing attenuation behavior of emitted light , i . e ., the attenuation coefficients . thus , light having arbitrary quadratic attenuation can be simulated by arbitrarily selecting values of the constants a 1 , a 2 and a 3 . alternatively , a 1 can be set to unity and a 2 and a 3 can be set to zero , in which case the preceding equation becomes : fig1 illustrates the case of a &# 34 ; point light &# 34 ; pl . by definition , a &# 34 ; point light &# 34 ; has no area and therefore the term ( cos θ s ) n becomes unity , and the form factor from the point light pl to any vertex on a surface becomes : fig1 illustrates the case of a &# 34 ; directional light &# 34 ; dl . in the case of a directional light , all emitted rays are parallel and the term cos θ s is ignored ( treated as unity ) and thus the form factor employed in connection with a directional light becomes : where θ v gives the angle between the surface normal at the vertex and the light source direction . &# 34 ; positional lights &# 34 ; e . g ., spot lights , cone lights , and the like can be simulated by varying the value of n in the equation f = cos θ v ( cos θ s ) n . distribution patterns approximating those of spot lights , cone lights and other types of positional lights can also be simulated . the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof and , accordingly , reference should be made to the appended claims rather than to the foregoing specification as indicating the scope of the invention . ## 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