Illumination dependent refinement of ... - Semantic Scholar
Illumination dependent refinement of multiresolution meshes Reinhard Klein, Andreas Schilling and Wolfgang Straßer Wilhelm-Schickard-Institut, GRIS, Universit¨at T¨ubingen, Germany
Abstract State of the art multiresolution modeling allows to selectively refine a coarse mesh of an object on the visually important parts. In this way it is possible to render the geometry of a given object accurately with a minimum number of triangles. There are also approaches that use cones of normal vectors to perform illumination dependent selective refinement. This includes the correct rendering of highlights and transitions between lit and unlit areas of an object. The drawback of these approaches is that using only the normal cone for the detection gives only a very poor criteria to detect the problematic cases. Therefore, too much refinement has to be performed which in most cases prevents real time update rates between successive frames. In this paper we describe a new approach for better estimation of normal deviations between different levels of detail. This allows for accurate lighting with a minimum number of triangles.
the necessary information. Selective refinement becomes possible if the sequence defined by the simplification process is changed. This is only possible under certain circumstances which guarantee that the parts of the mesh on which
1 Introduction and previous work Recently multiresolution modeling (MRM) for simplicial geometric models (any dimension, non-orientable, nonmanifold, non-regular) gain massive interest. Multiresolution models provide the basis to handle, visualize and transmit over the network and edit very large datasets [14, 2, 9, 1, 4, 5, 12, 13, 15]. The generation of a MRM of an object generally involves a sequence of local simplification operations like vertex removal, edge collapse, triangle collapse and vertex clustering. The sequence of local simplification operations defines a sequence of coarser and coarser approximations of the original model, the MRM. In order to be able to refine a coarse approximation of the model, the inverse local simplification operations are needed and the MRM must contain 1 Universit¨ at
T¨ubingen, Auf der Morgenstelle 10 / C9, 72076 T¨ubingen, Germany. E-mail: andreas reinhard @gris.uni-tuebingen.de http:==www.gris.uni-tuebingen.de
f
j
g
Figure 1. Example of a hierarchy of modification operations. The left side shows different level of detail triangulations. On the right side the corresponding directed acyclic graph. Operation 2 can only be performed after operation 1. Operation 4 requires operations 1 and 2 to be performed first but is independent of operation 3.
the operation is performed are present in the current mesh. This requirement automatically defines a hierarchy of modification operations and the corresponding triangles. This hierarchy can be considered as a directed acyclic graph of modification operations or the associated triangles. Figure 1 shows an example. For illumination depended selective refinement the surface normals are essential. For the decision if the mesh at a certain location has to be refined further, the normals contained in the actual coarse approximation cannot be used. Information about the normals in the corresponding area of the original model (which can be totally different) is required. Therefore, in previous work the variation of the normals in a certain area is represented by a cone of normals represented by a vector together with an angle. The normal cone conservatively bounds all the normals of all the triangles in the corresponding area. This corresponding area is defined by several authors [16, 5, 10] as all the triangles in the original mesh which are descendents of the area being considered in the simplification hierarchy. However, the dependencies introduced by the simplification hierarchy are not related to the geometry of the relevant area. Therefore, in general the area considered for the calculation of the normal cone comprises in many cases more than twice the actual needed area. This leads to much larger normal cones than required and therefore to unnecessary refinements. Therefore, we suggested in [6] not to use the hierarchy to determine the corresponding area but instead to use a geometric correspondence. Despite of this improvement there still remains a general problem when a normal cone is used: The cone of normals does not tell anything about the distribution of the normals in the relevant area. Figure 2 shows the result of lighting dependent simplification using normal cones. Note that in the area of the highlight the car bonnet is refined down to triangles of pixel size although this would be unnecessary especially if Phong-interpolation is used. If we knew that the normals over the area of a triangle can be sufficiently well approximated by a linear interpolation of the vertex normal (which occurs very often) the refinement could be avoided in the following cases:
always, if Phong interpolation is used,
2 The simplification algorithm The mesh simplification algorithm always starts with the finest triangulation in 3D space approximating the original model surface. In the following this starting triangulation is denoted with M , indicating that it contains M vertices. The simplification algorithm successively simplifies the starting triangulation by removing vertices from the current triangulation and retriangulating the resulting hole or by collapsing edges or triangles. This is done until no further vertices can be removed from the simplified triangulation without exceeding a predefined distance between the original triangulation and the simplified one. If after a single simplification step the simplified triangulation contains m vertices it is denoted by m : The order in which the simplification steps are performed is determined by a priority queue. For every single vertex vi of the simplified mesh an error value i is computed and stored. This value describes the potential error, that is the modified one-sided Hausdorff distance that would occur between the simplified and the original mesh if the simplification step was performed. At the beginning of the algorithm this error value is computed for every vertex, edge or triangle in the original mesh. The vertices, edges or triangles of the original mesh are then sorted into a priority queue according to their potential error. Afterwards, in each simplification step we successively eliminate the vertex, edge or triangle with the smallest potential error on top of the priority queue. If a simplification step is actually performed the potential errors of the neighbor elements are recomputed and the priority queue is updated.
2.1 Generating information about the normals 2.1.1 The normals in a simplified triangulation During the simplification process the normals at the vertices must be recalculated to avoid alias problems due to undersampling, see Figure 3. After a simplification step the normals at the vertices vi incident to the modified area are given by
ni = P 1A
X
j j j
if Gouraud interpolation is used and a linear brightness distribution can be assumed.
In the remainder of the paper, we will introduce a suitable measure to be stored within the MRM, that allows us to detect these cases during the refinement process at rendering time and to obtain substantially smaller approximating meshes for the same high image quality.
nj Aj ;
where j enumerates all triangles incident to vertex vi :. 1 Note that as a consequence of this filtering procedure, the normals at the vertices of a given triangle may change during the simplification process. 1 In order to model sharp edges several normals at a vertex are necessary. These cases can be handled in a similar way taking into account only the respective incident triangles.
Figure 2. The upper two pictures show the illumination of a car bonnet without and with adaptive triangulation. The lower two pictures show the corresponding triangulations. Data courtesy of Volkswagen AG.
ni nij1
nij2
vk1
Vk2
v’i v'ij1 v'ij2
Figure 3. Alias problems due to sampling of normals.
vi
vk3
Figure 4. Determination of deviation angles by projecting original vertices on plane of triangle.
2.1.3 Deviation of normals due to normal filtering 2.1.2 Determination of deviation angles
To get information about the deviation of the normals, the correspondence map between the vertices and triangles of the original triangulation and the triangles of the simplified triangulation used for the distance calculations [9] is exploited. For a given triangle (vk1 ; vk2 ; vk3 ) of the simplified triangulation this map delivers all vertices vi 2 M of the original triangulation that are nearer to than to any other triangle of the simplified triangulation. Each of these vertices corresponding to is projected onto the triangle : In addition to these vertices we determine all vertices which are connected to one of the vertices vi via an edge and project them onto the plane defined by : The projected vertices are denoted with v 0 : They are connected by the same edges as their counterparts in the original triangulation. For each projected triangle ij of which at least one vertex does not belong to ; the two intersection points vij0 1 and vij0 2 of their projected edges with the edges of triangle are determined and corresponding normal vectors nij1 and nij2 of these points in the original mesh are interpolated, see Figure 4. In the next step for each of these 0 the barycentric covertices vi0 ; and intersection points vij k ordinates with respect to are computed, and using these coordinates, linearly interpolated normals n i based on the current normals at the vertices vk1 ; vk2 ; vk3 are computed. In a last step the deviation angles i between the original normals ni and corresponding interpolated normals n i in the vertices and intersection points are computed. After computing all these angles a linear deviation function f is computed, such that i f (vi0 ): The function f is defined by its values 1 ; 2 ; 3 at the vertices vk1 ; vk2 ; vk3 :
As described above, normal filtering is performed by recalculating the normals at the vertices of a simplified triangulation based on the adjacent triangles. This filtering does not only occur during the simplification process but also during the extraction of selectively refined meshes from the MRM. During the recalculation of these normals, different constellations of adjacent triangles must be taken into account. The deviation between the normal vector used to determine the deviation angles (see previous paragraph)and an actual normal vector varies. To get a conservative estimation for this deviation after the simplification process all possible normals have to be computed. This can be done with a simple recursive algorithm operating on the directed graph of the hierarchy.
3 The extraction algorithm The goal of the multiresolution model is to minimize the number of polygons that are sent to the rendering pipeline, without affecting the quality of the resulting image. To achieve this, the multiresolution model allows the extraction of selectively refined models either by coarsening or refining the current triangulation. The decision whether a given triangle may remain in the triangulation or if its surrounding area must be refined further or if a coarser triangulation of its surrounding area is possible, is the most important step in this process. For an implementation this can be reduced to the single question of whether a given triangle fulfills the quality requirements or not [8]. To answer this question we have to investigate certain characteristics of that part of the original surface that corresponds to the triangle in question. In the following we call this part of the original surface corresponding surface for simplicity. The flow diagram of the
algorithm we use for this decision is shown in figure 5.
3.1 Handling backfacing surface parts Backside?
The first question in the algorithm is to decide if a triangle (vi1 ; vi2 ; vi3 ) and its corresponding part on the original surface is on the backside with respect to the current viewing position. It can easily be answered in a conservative way using the normals ni1 ; ni2 ; ni3 at the vertices of the triangle together with the corresponding deviation angles i1 ; i2 ; i3 . Let ei1 ; ei2 ; ei3 be the vectors pointing from the vertices to the observer then we get the following criteria. If
Do not refine
No
Projected Triangle Smaller than pixel?
Yes
Do not refine
No
6 (,eij ; nij ) < =2 , ij ; j = 1; 2; 3; then the surface is backfacing and the triangle need not be refined any more. In the other cases we proceed with checking the geometric and shading errors.
Yes
Refine
No
Geometry ok?
Yes
3.2 Controlling the geometric error Facing away from light source
To control the geometric error the distance (Hausdorffdistance) between original and simplified triangulation [9] must be converted to a screen space error. Several approaches to this problem have been published and can be used [7, 16, 5, 10]. A further significant reduction can be achieved by avoiding the refinement outside of the view frustum. However, refinement outside the view frustum is inevitable to support fast changes of the viewing direction in VR-applications.
No
Lin. approx. of normals sufficient?
Yes
Do not refine
Yes
Linear Phong Interpolation sufficient?
3.3 Controlling the shading errors In the following we assume that the Blinn-Phong shading model is used. For one light source it is given by:
8< L k + L k (n l) a a d d +Ls ks (n h)m ; L=: La ka
Refine
No
Yes
if (n l) > 0; else,
where ka ; kd and ks are the ambient, diffuse and specular material coefficients, La ; Ld ; Ls the corresponding luminaces, l is the direction from the surface point to the light e source, and h = jjll+ +ejj the so called halfway vector between the vector to the light source l and the vector to the observer e.
α*dL/dα small enough?
Phong interpolation used?
Yes
Do not refine
No
Refine
No
Brightness sufficiently linear?
Yes
Do not refine
3.3.1 Only ambient light contributions The first test determines for a given triangle if the corresponding surface faces away from the light source and therefore, gets only (constant) ambient lighting. To perform this test we determine if the normal cones at the vertices defined by the normals ni1 ; ni2 ; ni3 at the vertices and
Figure 5. Flow diagram of the extraction algorithm.
the corresponding deviation angles i1 ; i2 ; i3 are facing away from the direction to the light source. Let li1 ; li2 ; li3 be the vectors pointing from the vertices to the light source then we get the following criteria. If
6 (,lij ; nik ) < =2 , ij ; j; k = 1; 2; 3; only the constant ambient term has to be considered and the triangle is not refined further 2 . 3.3.2 Is linear Phong interpolation sufficient? In the next test we determine if linear Phong interpolation on the given triangle adequately describes the illumination of the corresponding surface. As a criteria we assume that it can be tolerated if the correct color in a pixel is replaced by a color that would be correct on a point, that is located in a one pixel-wide neighborhood of the pixel itself. In other words, the deviation angle , introduced by the varying normals on the corresponding surface must be less than the minimal deviation between Phong interpolated normals of two points that are one (screen-) pixels apart3 . Let nx ; ny respectively, be the partial derivatives of n in the directions of (screen-) x and y and cos( ) nx +sin( ) ny the derivative of n in the direction of (cos( ); sin( )). Now, we have to guarantee that a deviation of the normal by an angle in an arbitrary direction is always less than the minimum absolute deviation value d determined by nx and ny . To get d (and with that ) we could solve the corresponding eigenvalue problem, see Figure 6. To avoid this computation we use the simple conservative bound for d (and therefore for tan( )) based on a projection in the normal vector space, see Figure 6. If nx ny 0 we get the following formula
tan( ) jjnx ,(nx (ny ,nx)) ((n , nny), (nnx , n )) jj: y x y x Otherwise, nx is replaced by ,nx : For sake of simplicity, for nx and ny in this calculation the Phong increments can
be used. Up to this point the angular deviations are controlled. Therefore, the resulting meshes cannot only be used for Phong-illumination but also in ray-tracing applications, or with environment mapping which is of special interest in the CAD-industry to visualize the quality and smoothness of free-form surfaces. If linear interpolation of the normals is not sufficient, there still may be cases where refinement may be unnecessary, since the gradient of the intensity may be so small that
2 Although cases can be constructed where these nine tests are required, they can normally be reduced to three tests ( lij ; nij ) < =2 ij ; j = 1; 2; 3 with only marginal effects on the resulting illumination 3 To be correct the deviation of the halfway vector should also be considered. However, we neglect the deviation of this vector in a one pixelwide neighbourhood.
,
6
,
n
n+∂n/∂x
α
Figure 6. Finding the smallest deviation in directional deviation space.
in larger neighborhoods of the pixel or even on the whole triangle the differences of intensities cannot be recognized by an observer. This cases are detected in a subsequent test. For this purpose let us assume that we use a Phong model. Than the original normals deviate from the interpolated normals at most by an angle of : Now we have to find an upper bound for the changes in the brightness that might be caused by a variation of the normals by an angle of in arbitrary direction. To end up this we consider the derivatives @Ld of the diffuse term and @L of the specular term, where @ @ cos() = (n l) and cos = (n h): Then the changes that can be caused by an angle are bounded by
@L () + max @L ()) ; (max @ @ where min ,
Abstract State of the art multiresolution modeling allows to selectively refine a coarse mesh of an object on the visually important parts. In this way it is possible to render the geometry of a given object accurately with a minimum number of triangles. There are also approaches that use cones of normal vectors to perform illumination dependent selective refinement. This includes the correct rendering of highlights and transitions between lit and unlit areas of an object. The drawback of these approaches is that using only the normal cone for the detection gives only a very poor criteria to detect the problematic cases. Therefore, too much refinement has to be performed which in most cases prevents real time update rates between successive frames. In this paper we describe a new approach for better estimation of normal deviations between different levels of detail. This allows for accurate lighting with a minimum number of triangles.
the necessary information. Selective refinement becomes possible if the sequence defined by the simplification process is changed. This is only possible under certain circumstances which guarantee that the parts of the mesh on which
1 Introduction and previous work Recently multiresolution modeling (MRM) for simplicial geometric models (any dimension, non-orientable, nonmanifold, non-regular) gain massive interest. Multiresolution models provide the basis to handle, visualize and transmit over the network and edit very large datasets [14, 2, 9, 1, 4, 5, 12, 13, 15]. The generation of a MRM of an object generally involves a sequence of local simplification operations like vertex removal, edge collapse, triangle collapse and vertex clustering. The sequence of local simplification operations defines a sequence of coarser and coarser approximations of the original model, the MRM. In order to be able to refine a coarse approximation of the model, the inverse local simplification operations are needed and the MRM must contain 1 Universit¨ at
T¨ubingen, Auf der Morgenstelle 10 / C9, 72076 T¨ubingen, Germany. E-mail: andreas reinhard @gris.uni-tuebingen.de http:==www.gris.uni-tuebingen.de
f
j
g
Figure 1. Example of a hierarchy of modification operations. The left side shows different level of detail triangulations. On the right side the corresponding directed acyclic graph. Operation 2 can only be performed after operation 1. Operation 4 requires operations 1 and 2 to be performed first but is independent of operation 3.
the operation is performed are present in the current mesh. This requirement automatically defines a hierarchy of modification operations and the corresponding triangles. This hierarchy can be considered as a directed acyclic graph of modification operations or the associated triangles. Figure 1 shows an example. For illumination depended selective refinement the surface normals are essential. For the decision if the mesh at a certain location has to be refined further, the normals contained in the actual coarse approximation cannot be used. Information about the normals in the corresponding area of the original model (which can be totally different) is required. Therefore, in previous work the variation of the normals in a certain area is represented by a cone of normals represented by a vector together with an angle. The normal cone conservatively bounds all the normals of all the triangles in the corresponding area. This corresponding area is defined by several authors [16, 5, 10] as all the triangles in the original mesh which are descendents of the area being considered in the simplification hierarchy. However, the dependencies introduced by the simplification hierarchy are not related to the geometry of the relevant area. Therefore, in general the area considered for the calculation of the normal cone comprises in many cases more than twice the actual needed area. This leads to much larger normal cones than required and therefore to unnecessary refinements. Therefore, we suggested in [6] not to use the hierarchy to determine the corresponding area but instead to use a geometric correspondence. Despite of this improvement there still remains a general problem when a normal cone is used: The cone of normals does not tell anything about the distribution of the normals in the relevant area. Figure 2 shows the result of lighting dependent simplification using normal cones. Note that in the area of the highlight the car bonnet is refined down to triangles of pixel size although this would be unnecessary especially if Phong-interpolation is used. If we knew that the normals over the area of a triangle can be sufficiently well approximated by a linear interpolation of the vertex normal (which occurs very often) the refinement could be avoided in the following cases:
always, if Phong interpolation is used,
2 The simplification algorithm The mesh simplification algorithm always starts with the finest triangulation in 3D space approximating the original model surface. In the following this starting triangulation is denoted with M , indicating that it contains M vertices. The simplification algorithm successively simplifies the starting triangulation by removing vertices from the current triangulation and retriangulating the resulting hole or by collapsing edges or triangles. This is done until no further vertices can be removed from the simplified triangulation without exceeding a predefined distance between the original triangulation and the simplified one. If after a single simplification step the simplified triangulation contains m vertices it is denoted by m : The order in which the simplification steps are performed is determined by a priority queue. For every single vertex vi of the simplified mesh an error value i is computed and stored. This value describes the potential error, that is the modified one-sided Hausdorff distance that would occur between the simplified and the original mesh if the simplification step was performed. At the beginning of the algorithm this error value is computed for every vertex, edge or triangle in the original mesh. The vertices, edges or triangles of the original mesh are then sorted into a priority queue according to their potential error. Afterwards, in each simplification step we successively eliminate the vertex, edge or triangle with the smallest potential error on top of the priority queue. If a simplification step is actually performed the potential errors of the neighbor elements are recomputed and the priority queue is updated.
2.1 Generating information about the normals 2.1.1 The normals in a simplified triangulation During the simplification process the normals at the vertices must be recalculated to avoid alias problems due to undersampling, see Figure 3. After a simplification step the normals at the vertices vi incident to the modified area are given by
ni = P 1A
X
j j j
if Gouraud interpolation is used and a linear brightness distribution can be assumed.
In the remainder of the paper, we will introduce a suitable measure to be stored within the MRM, that allows us to detect these cases during the refinement process at rendering time and to obtain substantially smaller approximating meshes for the same high image quality.
nj Aj ;
where j enumerates all triangles incident to vertex vi :. 1 Note that as a consequence of this filtering procedure, the normals at the vertices of a given triangle may change during the simplification process. 1 In order to model sharp edges several normals at a vertex are necessary. These cases can be handled in a similar way taking into account only the respective incident triangles.
Figure 2. The upper two pictures show the illumination of a car bonnet without and with adaptive triangulation. The lower two pictures show the corresponding triangulations. Data courtesy of Volkswagen AG.
ni nij1
nij2
vk1
Vk2
v’i v'ij1 v'ij2
Figure 3. Alias problems due to sampling of normals.
vi
vk3
Figure 4. Determination of deviation angles by projecting original vertices on plane of triangle.
2.1.3 Deviation of normals due to normal filtering 2.1.2 Determination of deviation angles
To get information about the deviation of the normals, the correspondence map between the vertices and triangles of the original triangulation and the triangles of the simplified triangulation used for the distance calculations [9] is exploited. For a given triangle (vk1 ; vk2 ; vk3 ) of the simplified triangulation this map delivers all vertices vi 2 M of the original triangulation that are nearer to than to any other triangle of the simplified triangulation. Each of these vertices corresponding to is projected onto the triangle : In addition to these vertices we determine all vertices which are connected to one of the vertices vi via an edge and project them onto the plane defined by : The projected vertices are denoted with v 0 : They are connected by the same edges as their counterparts in the original triangulation. For each projected triangle ij of which at least one vertex does not belong to ; the two intersection points vij0 1 and vij0 2 of their projected edges with the edges of triangle are determined and corresponding normal vectors nij1 and nij2 of these points in the original mesh are interpolated, see Figure 4. In the next step for each of these 0 the barycentric covertices vi0 ; and intersection points vij k ordinates with respect to are computed, and using these coordinates, linearly interpolated normals n i based on the current normals at the vertices vk1 ; vk2 ; vk3 are computed. In a last step the deviation angles i between the original normals ni and corresponding interpolated normals n i in the vertices and intersection points are computed. After computing all these angles a linear deviation function f is computed, such that i f (vi0 ): The function f is defined by its values 1 ; 2 ; 3 at the vertices vk1 ; vk2 ; vk3 :
As described above, normal filtering is performed by recalculating the normals at the vertices of a simplified triangulation based on the adjacent triangles. This filtering does not only occur during the simplification process but also during the extraction of selectively refined meshes from the MRM. During the recalculation of these normals, different constellations of adjacent triangles must be taken into account. The deviation between the normal vector used to determine the deviation angles (see previous paragraph)and an actual normal vector varies. To get a conservative estimation for this deviation after the simplification process all possible normals have to be computed. This can be done with a simple recursive algorithm operating on the directed graph of the hierarchy.
3 The extraction algorithm The goal of the multiresolution model is to minimize the number of polygons that are sent to the rendering pipeline, without affecting the quality of the resulting image. To achieve this, the multiresolution model allows the extraction of selectively refined models either by coarsening or refining the current triangulation. The decision whether a given triangle may remain in the triangulation or if its surrounding area must be refined further or if a coarser triangulation of its surrounding area is possible, is the most important step in this process. For an implementation this can be reduced to the single question of whether a given triangle fulfills the quality requirements or not [8]. To answer this question we have to investigate certain characteristics of that part of the original surface that corresponds to the triangle in question. In the following we call this part of the original surface corresponding surface for simplicity. The flow diagram of the
algorithm we use for this decision is shown in figure 5.
3.1 Handling backfacing surface parts Backside?
The first question in the algorithm is to decide if a triangle (vi1 ; vi2 ; vi3 ) and its corresponding part on the original surface is on the backside with respect to the current viewing position. It can easily be answered in a conservative way using the normals ni1 ; ni2 ; ni3 at the vertices of the triangle together with the corresponding deviation angles i1 ; i2 ; i3 . Let ei1 ; ei2 ; ei3 be the vectors pointing from the vertices to the observer then we get the following criteria. If
Do not refine
No
Projected Triangle Smaller than pixel?
Yes
Do not refine
No
6 (,eij ; nij ) < =2 , ij ; j = 1; 2; 3; then the surface is backfacing and the triangle need not be refined any more. In the other cases we proceed with checking the geometric and shading errors.
Yes
Refine
No
Geometry ok?
Yes
3.2 Controlling the geometric error Facing away from light source
To control the geometric error the distance (Hausdorffdistance) between original and simplified triangulation [9] must be converted to a screen space error. Several approaches to this problem have been published and can be used [7, 16, 5, 10]. A further significant reduction can be achieved by avoiding the refinement outside of the view frustum. However, refinement outside the view frustum is inevitable to support fast changes of the viewing direction in VR-applications.
No
Lin. approx. of normals sufficient?
Yes
Do not refine
Yes
Linear Phong Interpolation sufficient?
3.3 Controlling the shading errors In the following we assume that the Blinn-Phong shading model is used. For one light source it is given by:
8< L k + L k (n l) a a d d +Ls ks (n h)m ; L=: La ka
Refine
No
Yes
if (n l) > 0; else,
where ka ; kd and ks are the ambient, diffuse and specular material coefficients, La ; Ld ; Ls the corresponding luminaces, l is the direction from the surface point to the light e source, and h = jjll+ +ejj the so called halfway vector between the vector to the light source l and the vector to the observer e.
α*dL/dα small enough?
Phong interpolation used?
Yes
Do not refine
No
Refine
No
Brightness sufficiently linear?
Yes
Do not refine
3.3.1 Only ambient light contributions The first test determines for a given triangle if the corresponding surface faces away from the light source and therefore, gets only (constant) ambient lighting. To perform this test we determine if the normal cones at the vertices defined by the normals ni1 ; ni2 ; ni3 at the vertices and
Figure 5. Flow diagram of the extraction algorithm.
the corresponding deviation angles i1 ; i2 ; i3 are facing away from the direction to the light source. Let li1 ; li2 ; li3 be the vectors pointing from the vertices to the light source then we get the following criteria. If
6 (,lij ; nik ) < =2 , ij ; j; k = 1; 2; 3; only the constant ambient term has to be considered and the triangle is not refined further 2 . 3.3.2 Is linear Phong interpolation sufficient? In the next test we determine if linear Phong interpolation on the given triangle adequately describes the illumination of the corresponding surface. As a criteria we assume that it can be tolerated if the correct color in a pixel is replaced by a color that would be correct on a point, that is located in a one pixel-wide neighborhood of the pixel itself. In other words, the deviation angle , introduced by the varying normals on the corresponding surface must be less than the minimal deviation between Phong interpolated normals of two points that are one (screen-) pixels apart3 . Let nx ; ny respectively, be the partial derivatives of n in the directions of (screen-) x and y and cos( ) nx +sin( ) ny the derivative of n in the direction of (cos( ); sin( )). Now, we have to guarantee that a deviation of the normal by an angle in an arbitrary direction is always less than the minimum absolute deviation value d determined by nx and ny . To get d (and with that ) we could solve the corresponding eigenvalue problem, see Figure 6. To avoid this computation we use the simple conservative bound for d (and therefore for tan( )) based on a projection in the normal vector space, see Figure 6. If nx ny 0 we get the following formula
tan( ) jjnx ,(nx (ny ,nx)) ((n , nny), (nnx , n )) jj: y x y x Otherwise, nx is replaced by ,nx : For sake of simplicity, for nx and ny in this calculation the Phong increments can
be used. Up to this point the angular deviations are controlled. Therefore, the resulting meshes cannot only be used for Phong-illumination but also in ray-tracing applications, or with environment mapping which is of special interest in the CAD-industry to visualize the quality and smoothness of free-form surfaces. If linear interpolation of the normals is not sufficient, there still may be cases where refinement may be unnecessary, since the gradient of the intensity may be so small that
2 Although cases can be constructed where these nine tests are required, they can normally be reduced to three tests ( lij ; nij ) < =2 ij ; j = 1; 2; 3 with only marginal effects on the resulting illumination 3 To be correct the deviation of the halfway vector should also be considered. However, we neglect the deviation of this vector in a one pixelwide neighbourhood.
,
6
,
n
n+∂n/∂x
α
Figure 6. Finding the smallest deviation in directional deviation space.
in larger neighborhoods of the pixel or even on the whole triangle the differences of intensities cannot be recognized by an observer. This cases are detected in a subsequent test. For this purpose let us assume that we use a Phong model. Than the original normals deviate from the interpolated normals at most by an angle of : Now we have to find an upper bound for the changes in the brightness that might be caused by a variation of the normals by an angle of in arbitrary direction. To end up this we consider the derivatives @Ld of the diffuse term and @L of the specular term, where @ @ cos() = (n l) and cos = (n h): Then the changes that can be caused by an angle are bounded by
@L () + max @L ()) ; (max @ @ where min ,
