A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids

Transcription

1 A Two-Step Approach for Interactive Pre-Integrated Voume Rendering of Unstructured Grids Stefan Roettger and Thomas Ert Visuaization and Interactive Systems Group University of Stuttgart Abstract In the area of voume visuaization the ce projection technique is used widey to dispay unstructured tetrahedra grids. In this fied, the pre-integration of the ray integra has proven to be a powerfu method for the dispay of tetrahedra grids with high quaity and at interactive frame rates. Besides the actua performance of the projected tetrahedra agorithm of Shirey and Tuchman, the degree of interactivity is aso imited by the update rate of the pre-integrated ray integra and the performance of the graphics card at high screen resoutions. In order to widen those two bottenecks, we propose a hardware-acceerated pre-integration approach and a rendering method which appies standard 2D texture mapping instead of the previousy used 3D texturing. As a resut, the update rate of the transfer function is increased by amost a factor of hundred and the rasterization of the tetrahedra is sped up by up to a factor of three without degrading accuracy. As a side effect, pre-integrated unstructured voume rendering can be utiized on a broader range of graphics patforms, since the support of 2D texture mapping is much more common. In summary, our two-step approach greaty increases the interactivity of unstructured pre-integrated voume rendering aowing for a much wider area of appication. 1 Introduction and Reated Work In order to dispay unstructured tetrahedra grids, the ce projection technique is used very widey in the area of voume visuaization. In the iterature many other methods are known for voume rendering [Drebin et a. 1988] of unstructured tetrahedra grids such as ray tracing [Kajiya 1984], scan ine agorithms [Westermann and Ert 1997; Farias et a. 2000] or hardware acceerated architectures [King et a. 2001]. In this paper, however, we wi concentrate soey on enhancing the projected tetrahedra (PT) agorithm of Shirey and Tuchman [Shirey and Tuchman 1990; Wihems and van Geder 1991; Wiiams et a. 1998], which athough pubished more than ten years back sti has great potentia for improvement. The pre-integration technique [Max et a. 1990; Roettger et a. 2000] has contributed significanty to the PT agorithm. By means of pre-integrating the three-dimensiona ray integra high quaity visuaizations of unstructured tetrahedra grids can be achieved. Previous voume rendering approaches based on the PT agorithm were ony abe to support inear transfer functions [Stein et a. 1994] without introducing severe artifacts. Actua pre-integrated voume rendering approaches can cope with arbitrary transfer functions. For this purpose, the pre-integrated ray integra is stored in a 3D texture which can be mapped efficienty onto the tetrahedra using graphics hardware [Roettger et a. 2000] (see aso [Enge et a. 2001] as an exampe of pre-integrated voume rendering on reguar meshes). However, the numerica pre-integration of the ray integra consumes a fair amount of preprocessing time. Thus, interactive updates of the transfer function are not yet possibe in order to expore unstructured tetrahedra data sets comfortaby. As a first step towards more interactive voume rendering of tetrahedra grids, we present a new technique which is abe to recacuate the preintegrated ray integra at interactive rates. In comparison to standard software pre-integration, a significant speed up is achieved by cacuating a the coors and opacities of one 3D texture sice in parae using the graphics hardware. Therefore, the approach is caed hardware-acceerated pre-integration. As a second step towards improved interactivity, we show how to speed up the the rasterization of the projected tetrahedra, which becomes the main imiting factor at high screen resoutions. The previousy utiized 3D texture mapping approach [Roettger et a. 2000] suffers from a reduced rasterization performance, since most graphics hardware is optimized for 2D texture mapping ony. In order to account for this, we transform the three-dimensiona ray integra into a cyindrica coordinate system. In this system the rasterized pixes of each singe tetrahedron ie on a pane going through the main cyinder axis. These panes can be easiy arranged into a set of 2D textures by effectivey resamping the pre-integrated ray integra in the cyinder coordinate system. Without oss of accuracy, the tetrahedra can now be rendered using the much faster 2D texture mapping. As a side effect, the method is compatibe with amost every graphics acceerator avaiabe today. Thus, a much broader range of graphics patforms is supported than with 3D texture mapping. 2 Ce Projection In this section we give a brief summary of the ce projection technique which is the basis of our new proposed methods. Basicay, an unstructured tetrahedra mesh is visuaized by sorting each tetrahedra ce with respect to visibiity. After that, the ces are composed (bended) in a back to front fashion. Efficient visibiity sorting agorithms have been studied in depth in the iterature. As a starting point, we refer the reader to [Wiiams 1992] and [Comba et a. 1999]. For the remainder of this paper, it is supposed that the mesh is aready sorted with respect to visibiity. Then each ce is pro-

2 jected into screen space coordinates resuting in either three or four trianges depending on the orientation of the tetrahedron (see Figure 1). These projected trianges are paced around a thick vertex in the center. They can be rendered and composed efficienty using standard graphics hardware, which is the main advantage of the agorithm. = 0 Sf = For a constant chromaticity vector and a inear optica density the ray integra simpifies to the soution found by Stein et a. [Stein et a. 1994]. As proposed by Roettger et a. [Roettger et a. 2000], arbitrary transfer functions can be appied by first setting up a 3D texture which contains the integrated chromaticity C 3D and the integrated transparency α 3D. Secondy, the three parameters S f, S b, and serve as texture coordinates s, t, and r assigned to each vertex of the projected trianges. At the sihouette of a projected tetrahedron, for exampe, the ength of each ray segment is zero, thus the texture coordinate r is set to zero at the sihouette vertices. Within each tetrahedron the scaar vaues are interpoated ineary. For obvious geometrica reasons the same is true for the ength of the ray segment. Since the texture coordinates are interpoated ineary within each tetrahedron, it is straight forward to see that each pixe is rasterized with the correct coors and opacities obtained from the pre-integrated 3D texture map. For each compositing step the coor I of a pixe is cacuated from the previous coor I in the frame buffer using the foowing bend operation with back to front sorting: I : C 3D 1 α 3D I (4) >0 Figure 1: Decomposition of a projected tetrahedron into either three or four trianges paced around a thick vertex in the center (back dot). The meaning of the parameters S f, S b, and is iustrated in Figure 2. Using the voume density optica mode of Wiiams et a. [Wiiams and Max 1992] (see aso [Max 1995]) the coor and opacity of each pixe of the projected tetrahedra is a function of the three parameters S f, S b, and, which are the scaar vaues at the entry and the exit point of each ray segment and the ength of the ray segment, respectivey (see Figure 2). For a given scaar density function f f x y z, the chromaticity vector κ κ f x y z and the scaar optica density ρ ρ f x y z are the transfer functions of this mode. The three parameters S f, S b, and define the domain of the three-dimensiona ray integra which is given as the integrated chromaticity C 3D and the integrated transparency α 3D : S u C 3D S f S b α 3D S f S b Sf S f 0 e 1 e u S b S f (1) v 0 ρ S u du κ S v dv (2) 0 ρ S u du Figure 2: Intersection of the viewing ray with a tetrahedron: S f and S b are the scaar vaues of the front and back face, respectivey. The ength of the ray segment is denoted by. (3) 3 First Step: Hardware-Acceerated Pre- Integration As a first step towards more interactive pre-integrated voume rendering, we discuss how to speed up pre-integration in this section. Prior to rendering, the three-dimensiona ray integra has to be computed in a preprocessing step. Anaytica soutions are known for simpe transfer functions such as a inear optica density. For the usua definition of the transfer function as a ookup tabe consisting of m entries, a time consuming numerica integration is necessary. The time spent by the numerica integration depends by an order of four on the resoution of the 3D textures as shown beow. Therefore, the computation of a pre-integrated 3D texture with, for exampe, voxes requires far too many operations to achieve interactive update rates with standard software integration. As a soution to this probem, we propose to utiize graphics hardware in order to speed up the pre-integration of the ray integra. The three-dimensiona ray integra is integrated numericay by iterating over the three parameters S f, S b and in the range max. For each evauation of the integra we sampe the transfer function n times. The number of necessary sampes shoud be greater or equa to the number of touched entries in the ookup tabe. Therefore, the tota number of integration steps depends by an order of four on the resoution of the 3D texture. For each numerica integration step we use the approximation of Stein et a. [Stein et a. 1994] which assumes the chromaticity vector to be constant. Since this assumption does not hod for a ineary interpoated ookup tabe we oversampe by a factor of o to achieve more accurate resuts. From empirica studies an oversamping factor of o 2 is regarded to be sufficient for the described type of transfer function. Thus, the minimum number of sampes per integration is n 2m S f S b. The emission and absorption of the ight on its way aong each ray segment is integrated numericay by repeatedy evauating the foowing formua n times. The fina resuting RGBA coor vector is stored in a 3D texture map, which is denoted by T 3D S f S b C 3Dn 1 α 3Dn 1 with θ 1 α: i w S b n 1 S f S b! i 0 " " " n 1 (5) C 3Di# C 1 3Di e ρ w n κ w $ n C 3D (6) θ 3Di# θ 1 3Di e ρ w n θ 3D0 1 (7)

3 s=1-i/(n-1) ceared frame buffer rendered quads for i=0,...,n-1 Sf s=1 s=0 s=i/(n-1) fina sice with =const Figure 3: Hardware-acceerated pre-integration: First, the transfer function is stored in a 1D texture map. Then n rectanges are drawn and bended with the texture coordinate s assigned in the depicted fashion. In order to speed up the numerica cacuation of the ray integra, we rearrange the nesting of the four oops, such that the two innermost oops iterate over S f and S b. Now, texture mapping hardware can be used to substitute the two inner oops. This eads to an approach which effectivey integrates the coors and opacities of one 3D texture sice in parae. Since a voxes of a sice are integrated simutaneousy we use the upper bound n o " m as the number of sampes. For each sice with const, we need two rendering passes to cacuate both the RGB and the apha component of the voxes. For the RGB component of the voxes, the frame buffer is initiay set to and the transfer function is stored in a 1D texture: T 1D s % & n κ s! 1 e n ρ s (8) Then n rectanges with a size equa to the resoution of the 3D texture sices are rendered and bended into the frame buffer. The texture coordinate s is set according to Figure 3 ensuring that the RGBA coor of each rendered pixe corresponds to the coor at position s w of the transfer function. For the apha component of the voxes, we proceed as described above, except that we use the primary coor to moduate the 1D texture. After initiaizing the frame buffer to the transparencies θ are cacuated in a three RGB channes. If the graphics hardware supports bending with a constant coor then the two passes can be combined into a singe one by using the constant bending coor (1,1,1,0) in the source path, which resuts in the cacuation of the transparencies in the α-channe of the frame buffer. As the fina step, the contents of the frame buffer are transferred back into main memory or copied directy into a 3D texture map residing in dedicated graphics memory. In order to iustrate the progress of the hardware-acceerated pre-integration (of the RGB component), four intermediate steps from a numerica integration with a tota of n 128 samping steps are shown in Figure 4. The fina resut can be seen on the eft side of Figure 5. The appied transfer function consisted of 64 entries, which can be seen besides the images in Figure 4. The number of integration steps and the bit depth of the frame buffer imit the accuracy of the integration. Whie SGI workstations support 12 bits per channe and achieve good integration quaity for transfer functions of up to 256 entries (see aso Tabe 1), current PC graphics hardware supports a maximum of 8 bits per channe. Using the pixe shaders of the NVIDIA GeForce3 or the ATI Radeon 8500 graphics acceerator severa integration steps can be combined Figure 4: The RGB contents of the frame buffer after 16, 48, 80, and 112 integration steps out of a tota of 128 steps. The appied transfer functions κ S and ρ S are depicted besides the images. taking advantage of the increased interna accuracy of the graphics acceerators. In the case of the GeForce3, the three avaiabe genera combiners are utiized to compute three steps of the integration with 12 bits of interna accuracy. The fina combiner is configured to perform another integration step. The precision gain is therefore equivaent to reducing the number of integration steps by a factor of 4. Besides the improved accuracy, this muti-texturing approach is approximatey twice as fast as singe-texturing. The RMS (root mean square) deviation of our hardwareacceerated approach from standard numerica integration is given in Tabe 1. Frame buffer depths of 12 bits per channe or mutipe integration steps at once aow us to use transfer functions of up to 256 or 128 entries, respectivey. Beyond that imit the quaity of the hardware-acceerated integration is not sufficient, because quantization artifacts become visibe. Transfer functions with more entries require a frame buffer depth of 16 bits per channe, which currenty is supported ony on SGI Octane2 and Onyx patforms. Since the quaity gain achieved by oversamping is sma compared to the quaity oss of twice the number of samping steps we choose o 1 for hardware-acceerated pre-integration. As an exampe of the quantization artifacts which arise with insufficient frame buffer depths, the right image in Figure 5 shows the resut of a hardware-acceerated pre-integration using 1024 integration steps and 12 bits per channe. 12 bits per chan. RMS error max. error steps= % % steps= % % steps= % % steps= % % steps= % % steps= % % 8 bits per chan. RMS error max. error steps= % % steps= % % steps= % % steps= % % Tabe 1: Integration accuracy of our hardware-acceerated preintegration approach: The RMS error is given in comparison to standard software pre-integration. The deviations were cacuated on a SGI Octane MXI with 12 bits per channe and on a PC with NVIDIA GeForce3 and 8 bits per channe for the transfer function depicted in Figure 4. In comparison to standard numerica pre-integration our proposed hardware-acceerated pre-integration technique achieves a speedup of approximatey , depending on the size of the cacuated 3D texture (see Tabe 2). For a 3D texture with voxes the update rate is 5.5 Hertz on a SGI Octane MXI with 250 MHz MIPS R10K processor. Two passes for the coors and opacities are required on this patform, since the SGI Octane MXI ony supports an RGB but no RGBA visua with 12 bits per channe. This imitation is not present on SGI Onyx systems.

4 Figure 5: Left: The resut of a pre-integration with 128 integration steps using 12 bits per channe on a SGI Octane MXI. Right: Quantization artifacts become visibe at 1024 integration steps time steps speedup software int s 1-64 hardware int s time steps speed up software int s hardware int s time steps speed up software int s hardware int s Figure 6: The rasterized pixes (dark grey) of a projected tetrahedron ie on a pane (ight grey) defined by the front diagona and the parameters S f, S b, and assigned to the thick vertex (back dot). cyinder axis Sf Tabe 2: Performance comparison of the standard software and the hardware pre-integration approach on a SGI Octane MXI with 250 MHz MIPS R10K processor. For comparabiity, the number of integration steps was chosen such that each method performs the same number of steps on the average. t s 4 Second Step: Cyindrica Mapping of the Ray Integra As a second step towards more interactive pre-integrated voume rendering of unstructured grids, we propose to speed up rasterization of the tetrahedra by using a cyindrica mapping of the ray integra. For high screen resoutions the rasterization performance of the graphics card becomes a imiting factor for unstructured voume rendering. Since the approach of Roettger et a. [Roettger et a. 2000] reies on 3D textures which contain the pre-integrated ray integra, actua impementations of the agorithm are restricted to graphics hardware which supports 3D textures efficienty. However, actua graphics hardware performs much sower when using 3D textures instead of standard 2D textures. This is due to additiona texture memory accesses and a reduced texture cache coherency. In order to overcome this restriction, we seek to find a soution which utiizes 2D textures rather than the sower 3D textures. We attain this goa by first considering the foowing observation: Observation: At the sihouette of each projected tetrahedron the ength of the ray segments is zero and the scaar vaues at the front and the back face are equa (compare Figure 1). For this reason, the rasterized pixes of the sihouette ie on the front diagona of the pre-integrated 3D texture (see Figure 6). For the same reason, a rasterized pixes of the tetrahedron ie on a pane defined by the front diagona and the parameters S f, S b, and ' 0 assigned to the thick vertex of the projected tetrahedron. For convenience, we shear the texture coordinate system such that the front diagona becomes the vertica axis of our new coordinate system defined by the base vector S f S b S b (see Figure 7). Next, the new coordinate system is transformed into cyinder coordinates with the front diagona being the cyinder axis and α ( 0)* 180)+ being the rotationa parameter. In this coordinate system α Sf- Figure 7: Sheared cyindrica representation of the ray integra: A rasterized pixes of a tetrahedron ie on a cyinder pane (grey triange) with α const. the rasterized pixes of a tetrahedron ie on a pane with α const. For each projected tetrahedron the three-dimensiona ray integra is reduced to a two dimensiona integra in sheared cyinder coordinates. Instead of computing a two-dimensiona tabe for each projected tetrahedron, we resampe the ray integra at equidistant vaues of α and reconstruct the exact coors and opacities of each cyinder pane by inear interpoation of the two nearest resamped panes. Each of these resamped panes is stored as a 2D texture map in dedicated graphics memory. The inear interpoation of the resamped panes can be performed efficienty by the graphics hardware using twopass muti-texturing. Since the 2D textures have different size and shape for different vaues of α, a ot of texture memory woud be wasted by scaing up to the next power of two. Aso, costy trigonometric evauations woud be required for the mapping to the cyinder coordinate system. Instead, we choose a simiar cyindrica mapping, which produces equay sized textures by orthogonay projecting the cyinder panes onto the nearest pane defined Assuming normaized texture coordi- by α 0), 90), or 180). nates (α s t S f S b ( 0 1 ), the pre-integrated ray integra is resamped into a set of square 2D textures with the same resoution as the 3D texture using the foowing mapping:

5 α ( α ( f : α s t %-. S f S b (9) α ( 0 0/ 25 : t s t 4αs 0/ 25 0/ 5 : t 4 0/ 5 α s t s 0/ 5 0/ 75 : t 4 α 0/ 5 s t s α ( 0/ 75 1 : t s t 4 1 α s Figure 9: Eight sices with increasing ray segment ength from a pre-integrated 3D texture with 64 3 voxes. The transfer function is shown besides the images. Whie rendering the tetrahedra, the parameters S f, S b and have to be transformed into cyinder coordinates in order to choose the appropriate two textures next to α and to set the 2D texture coordinates s and t at the projected vertices. At the sihouette vertices the texture coordinates are just 0 S b. The additiona overhead for the thick vertex is at most one division, one mutipication, and two subtractions: f 1 : S f S b -. α s t (10) S f S b ' S f S b : 4 S f S b S f S b S b S S f S b '3 01 ' S f S b : 0/ 5 f S b 4 S b S S f S b 2 01 '3 S b S f : 0/ 5 b S f 4 S b S f S b S b S f : 1 4 S b S f S b S f S b In order to reconstruct the origina pre-integrated 3D tabe exacty, the number of cyinder panes must be four times the resoution of the 3D texture. In most cases, however, the number of panes can be reduced by a factor of two or four without introducing visibe artifacts. In the atter case the memory consumption of the 2D textures is the same as the memory consumption of the origina 3D texture. By combining two of the trianguar shapes into a singe quadriatera the texture memory consumption coud be reduced further. 3dfx 3D texture 2D muti-texture speedup Voodoo fps NVIDIA 3D texture 2D muti-texture speedup GeForce2 MX 33.8 fps GeForce fps 79.0 fps 66% ATI 3D texture 2D muti-texture speed up Rage 128 M3 5.5 fps Radeon fps 30.6 fps 235% Radeon fps 38.2 fps 13% SGI 3D texture 2D texture speedup Octane MXI 8.5 fps 12.7 fps 49% Octane2 V fps 32.2 fps 282% Onyx fps 24.1 fps 213% Figure 8: Performance measurements: 3D texturing versus twopass 2D muti-texturing (singe 2D texture with nearest neighbour interpoation on SGI patforms). For each rendered tetrahedron the α-parameter is usuay distinct. Switching to the appropriate textures for each tetrahedron causes significant overhead for texture state management. A soution to minimize this overhead is to arrange the resamped textures into one arge tied texture. In Figure 9 eight sices of a preintegrated 3D texture are dispayed for an exampe transfer function which consisted of four intereaved orange and bue regions with constant opacity. The corresponding tied 2D texture is given in Figure 10. Figure 10: Tied cyindrica representation of the pre-integrated 3D texture shown in Figure 9 using 64 cyinder panes with a tie size of 64 2 pixes. The tie in the eft bottom corner corresponds to α 0) and the tie in the right top corner to α 180). By means of the described two-pass muti-texturing approach pre-integrated unstructured voume rendering can be utiized on a much broader range of graphics patforms. For exampe, it can be utiized on an Appe ibook with Rage 128 Mobiity M3 graphics chip which has no support for 3D texturing (see Figure 12 which shows a bonsai [Roettger and Thomand 2000] rendered on an Appe ibook). On patforms that support 3D texturing the rasterization performance is improved by up to an factor of three. The experimenta measurements in Tabe 8 have been conducted by rendering a spherica distance voume (depicted in Figure 11) with both the 3D texture from Figure 9 and the two-dimensiona anaogue from Figure 10. The distance voume consisted of ony 2560 tetrahedra so that the rasterization of the tetrahedra was the main imiting factor. This was true even for the comparaby sma window size of pixes. 5 Concusion The described enhancements widen the two main bottenecks of pre-integrated voume rendering of unstructured grids, namey the sow update rate of the pre-integrated ray integra and the poor rendering performance at high screen resoutions. As a consequence of the proposed hardware-acceerated numerica pre-integration, the transfer function can be manipuated interactivey to expore unstructured data sets comfortaby. In comparison to previous approaches the update rate was sped up by a factor of amost 100.

6 Figure 11: Spherica distance voume rendered with an intereaved orange and bue transfer function which is shown besides. The rasterization performance, which is the main botteneck at high screen resoutions, was increased significanty by utiizing 2D muti-texturing instead of the previousy appied 3D texturing. As a wecome side effect, a much broader range of graphics patforms is supported by our 2D muti-texturing approach. This is an important improvement for the practica appication and the interactive usabiity of pre-integrated unstructured voume rendering. References COMBA, J., KLOSOWSKI, J. T., MAX, N. L., MITCHELL, J. S. B., SILVA, C. T., AND WILLIAMS, P. L Fast Poyhedra Ce Sorting for Interactive Rendering of Unstructured Grids. Computer Graphics Forum (Proc. Eurographics 99) 18, 3, DREBIN, R. A., CARPENTER, L., AND HANRAHAN, P Voume Rendering. Computer Graphics 22, 4, ENGEL, K., KRAUS, M., AND ERTL, T High-Quaity Pre-Integrated Voume Rendering Using Hardware-Acceerated Pixe Shading. In Eurographics Workshop on Graphics Hardware 01, ACM SIGGRAPH, FARIAS, R., MITCHELL, J., AND SILVA, C An Efficient and Exact Projection Agorithm for Unstructured Voume Rendering. In Proc. Voume Visuaization Symposium 00, ACM Press, KAJIYA, J. T Ray Tracing Voume Densities. In Proc. SIGGRAPH 84, ACM, KING, D., WITTENBRINK, C., AND WOLTERS, H An Architecture For Interactive Tetrahedra Voume Rendering. Proc. Internationa Workshop on Voume Graphics 01, MAX, N. L., HANRAHAN, P., AND CRAWFIS, R Area and Voume Coherence for Efficient Visuaization of 3D Scaar Functions. Computer Graphics (San Diego Workshop on Voume Visuaization) 24, 5, Figure 12: Hierarchica unstructured bonsai data set with approximatey 1,200,000 tetrahedra rendered on an Appe ibook with our proposed muti-texturing technique. The transfer functions are depicted besides. MAX, N. L Optica Modes for Direct Voume Rendering. IEEE Transactions on Visuaization and Computer Graphics 1, 2, ROETTGER, S., AND THOMANDL, B Three Litte Bonsais. roettger/data/bonsai. ROETTGER, S., KRAUS, M., AND ERTL, T Hardware- Acceerated Voume and Isosurface Rendering Based On Ce- Projection. In Proc. Visuaization 00, IEEE, SHIRLEY, P., AND TUCHMAN, A A Poygona Approximation to Direct Scaar Voume Rendering. ACM Computer Graphics (San Diego Workshop on Voume Visuaization) 24, 5, STEIN, C. M., BECKER, B. G., AND MAX, N. L Sorting and Hardware Assisted Rendering for Voume Visuaization. In Symposium on Voume Visuaization 94, IEEE, WESTERMANN, R., AND ERTL, T The Vuffer: Visibiity Ordering of Unstructured Voume Primitives by Poygon Drawing. In Proc. Visuaization 97, IEEE, WILHELMS, J., AND VAN GELDER, A A Coherent Projection Approach for Direct Voume Rendering. Computer Graphics 25, 4, WILLIAMS, P. L., AND MAX, N. L A Voume Density Optica Mode. In Computer Graphics (Workshop on Voume Visuaization 92), ACM, WILLIAMS, P. L., MAX, N. L., AND STEIN, C. M A High Accuracy Voume Renderer for Unstructured Data. Transactions on Visuaization and Computer Graphics 4, 1, WILLIAMS, P. L Visibiity Ordering Meshed Poyhedra. ACM Transactions on Graphics 11, 2,