DOT3 Normal Mapping on the PS2 1 Introduction
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1. nx scale int iY int ny scale clamp to 127 128 iX iX lt 127 127 iX gt 128 7128 iX iY CiY lt 127 127 iY gt 128 128 iY Choose iZ so length N lt 128 int iZ int sqrt 128 128 iX iX iY iY assert iX iX iY iYt iZ iZ lt 128 128 assert iX iXt iY iY iZ 1 iZ 1 gt 128 128 positive side const int R_pos Max iX 0 const int G_pos Max iY 0 const int B_pos Max iZ 0 negative side const int R_neg Max iX 0 const int G_neg Max iY 0 const int B_neg Max iZ 0 B_neg should always be zero const int delta R_neg R_pos G_neg G_pos B_neg B_pos int alpha 3 128 delta 1 3 alpha alpha lt 0 0 iX gt 255 7255 alpha 232. Figure 7 Shown to the left a and c is the front and the back of the head rendered using the two pass DOT3 solution To the right b and d shows the same shots using the four pass method b is the same picture as seen in 6 c The edge highlighting is especially noticable on c 18 5 Conclusion We have shown that it is possible to do DOT3 normal mapping very similar in quality to PC style normal mapping Two methods have been presented A cheap two pass solution without per pixel normalization and a more expen sive four pass alternative with per pixel normalization Both take advantage of the hardware available on the PS2 more specifically the GS and the VU1 This paper has also described how it is possible to integrate the normal mapping with ordinary shading of non normal mapped objects without any additional effect on performance 6 Acknowledgements The author would like to thank Kasper H y Nielsen for his help restruc turing this paper and for his many suggetions that improved its readabil ity Thanks also to Mircea Marghidanu Steven Osman Lionel Lemarie and Trine Mikkelsen for additional proof reading and their insightful comments Finally thanks to IO Interactive and to Eidos for letting me publish this paper References Blinn78 Blinn J F Simulation of wrinkled surfaces Proceedings of the 5th annual conference on Computer graphics and interactive techniques ACM Press pp 286 292 1978 Breugel
3. g b Since the result is used as a palette look up we can simply compensate by reordering the palette so the new palette is a simple permutation of the original palette Leaving out the subtraction by 128 in 9 just offsets the entries of the ILP by 128 The main principle of fetching any channel in a 24 32 bit frame buffer on the PS2 is using the buffer as an 8 bit texture twice the width and twice the height Looking at the tables in section 8 3 of the GS users manual Sony02 makes it clear that real time swizzling is needed to get this to work Note that all calculations are made using modulo by 256 on the GS with color clamping disabled The good news is that this can be done using a selection of pretesselated sprites and the region repeat mode The details of how to fetch channels in a 24 32 bit buffer on the PS2 are available on the playstation2 linux website Breugelmans01 Unlike Breugelmans01 which keeps the GS primitives in a DMA chain this paper s implementation uses a VU1 program which tessellates the prim itives required to do the operation The same primitives are then reused for every page This is done by delivering the settings of the FRAME TEXO and ZBUF registers for every page The unpacking is done using difference mode and STCYCL 3 1 on the VIF1 For every sequence the ROW register is set to the value of the first register and then only the VRAM pointer dif ferences are unpacked to keep the chain sma
4. pipeline For a single light source pre filling the z buffer step 1 is not necessary Alternatively 2 to 3 lights can be used by storing their dot product layers in unused alpha pixels in the VRAM Then use the frame buffer as the LAB Once the LAB is done copy the red green and blue to the free alphas and render the unlit diffuse buffer Afterwards at step 4 apply the LAB by using these alphas Note this is not necessary if more than one draw buffer is available 3 2 Achieving signed multiplication with unsigned in put without the per vertex clamping problem Assume we have two signed values a and b and we wish to calculate the product a b implicit signed shift right by 7 Since the input is 8 bit we assume a 127 126 128 and b 128 127 127 so that a b 128 127 127 We create two intermediate values az and b by the equations 1 and 2 dg 128 a 1 bo b 128 2 a b 128 ag b 128 128b azb 128a2 128 3 We can rearrange a little and take advantage of the fact that every mul tiplication has an implicit shift to the right by 7 a b 128 ag b2 ag 128 4 Now 4 is quite close to an equation that can be computed using the GS blend mode function Furthermore a gt and bz are both 8 bit unsigned inputs Alternatively the signed product can be expressed as 5 a b Maz a 0 bo Max a 0 b2 Maz a 0 Ma
5. type can be put in the same position One is rendered using Gouraud shading and the other using the fancy per pixel lighting If one is rendering at say 70 then the other renders at 30 and so on Once 0 intensity on the expensive version of the light is reached it is removed from the rendering process Models using 4 8 bit normal maps can switch from four pass to two pass e g based on distance using the same normal map but with a modified palette This is due to the fact that every index represents a unique normal so only the palette will be different when switching between the two methods 15 4 Results The methods have been implemented and tested on the PS2 The results for a low resolution model rendered using the four pass method running in real time on the PS2 are shown in figure 6 The model was lit by two point light sources a green and a blue The frame buffer resolution was 512 x 448 and the normal map figure 5 is 256 x 256 in 8 bit A comparison of the two methods is shown in figure 7 Figure 5 The normal map 16 a Wireframe b Gouraud shading c DOTS diffuse d DOT3 diffuse specular Figure 6 a Wireframe model 412 triangles to give an idea of the amount of actual detail in the model b Traditional Gouraud shading c The model with normal mapping applied d Normal mapping with specular highlights 17 a Two pass front Four pass front c Two pass back d Four pass back
6. with the sphere map applied into the DPB e Pass 2 Render the triangles again but use palette for X set mask to affect red only and use the blend mode above use the normal map as a texture e Pass 3 Same as pass 2 but use palette for Y and affect green only e Pass 4 Same as pass 2 but use palette for Z and affect blue only 12 a Spheremap Pass 1 Apply sphere map Pass 2 Pass 3 Pass 4 Figure 4 a The sphere map normalization table texture b A low resolu tion model rendered with the sphere map applied first pass c e The 2nd 3rd and 4th pass Each pass updates a single channel in the framebuffer red green and blue respectively Thus e shows the final signed multiplication offset by 128 Adding these yields the final dot products see section 3 3 13 The sphere map forms tolight vectors which are packed according to equa tion 1 and hence are in the set 0 1 255 see the code in appendix 1 As mentioned in section 3 a traditional normal map is used centered at 128 and also in the set 0 1 255 equation 2 In general it is common to fade off the results of the normal mapping near the silhouette of the models as seen from the light source since the illusion fails there On PS2 this attenuation must be applied per vertex in the following way fade_att clamp 6 n 1 0 1 The term n is the dot product between the unit length vertex normal and the unit length direction towar
7. A so we have two roots lz_1 B 2 B 2 which is 1 B usable lz_2 B 2 B 2 which is 1 not usable so this means lz 1 B once we have the lz component we can calculate lx and ly aswell 24bit sphere map for int y 0 y lt iHeight y for int x 0 x lt iWidth x reversed GS remapping section 3 4 8 const float s xt 0 5f float iWidth const float t y 0 5f float iHeight const float s2 2 s 1 const float t2 2 t 1 21 float Lz 1 s2 s2 t2 t2 2 zero vector int r 128 int g 128 int b 128 if Lz gt 1 const float m 2 sqrt 2 Lz 2 float Lx float Ly s 0 5 m t 0 5 m normalize for accuracy const float div sqrt Lx Lx Ly Ly Lz Lz Lx div Ly div Lz div must remember to subtract the vector from 128 r 128 nearest Lx 127 g 128 nearest Ly 127 b 128 nearest Lz 127 assert r gt 0 amp amp r lt 256 assert g gt 0 amp amp g lt 256 assert b gt 0 amp amp b lt 256 write const int vect int b lt lt 16 1 g lt lt 8 1 r lt lt 0 Cint mem y iWidth x vect 22 Appendix 2 Packing normals for two pass packing the normals for two pass solution IN nz ny nz is the unit length normal in tangent space OUT R_pos G_pos B_pos R_neg G_neg B_neg alpha const float scale 127 9f scale to range int iX int
8. DOTS3 Normal Mapping on the PS2 Morten Mikkelsen IO Interactive mmQioi dk November 4 2004 Abstract This paper describes a method for doing PC style normal mapping on the Playstation 2 by taking advantage of the GS and VU1 units Two variations are described A cheap two pass solution without per pixel normalization and a per pixel normalized alternative which requires four passes 1 Introduction Bump mapping is a technique originally developed by Blinn in 1978 Blinn78 where the surface normal is perturbed by information stored in a two dimen sional bump map While bump mapping perturbs the existing normal of a model normal mapping Cohen98 replaces the normal entirely by doing a look up into a normal map which usually is a texture with tangent space normals stored as RGB Both are inexpensive ways to fake geometric shading detail on low resolution models While normal mapping today is standard on X Box and PC hardware platforms Kilgard00 it has not yet been done satisfactorily on the PS2 The aim of this paper is to show how to achieve normal mapping on the PS2 with a visual quality comparable to the PC and X Box The following section describes a related method and discusses its limita tions Section 3 describes the proposed approach which is split in two parts One without per pixel normalization see section 3 5 and one with per pixel normalization see section 3 6 Section 4 discusses the results Finally sec tion 5
9. azx n 0 L 128 Max ny 0 Maz ny 0 128 Max n 0 Maz nz 0 8 The vector Ly Ly L is the barycentrically weighted result of the sur rounding per vertex packed tolight vectors The final dot product is com puted as in equation 9 without subtraction by 128 which yields mul muly mul ie 0 Ly Max n z 0 Ly ax ny 0 Ly Max ny 0 i 0 L Max n 0 128 Max n 0 Maz nz 0 128 Max n 0 Max ny 0 128 Max n 0 Mazx n z 0 L 4 L Ber This means that we can precompute the three last terms and store the result in alpha of the normal map Na 128 Maz n 0 Max nz 0 128 Max n 0 Max n 0 128 Max n z 0 Mazx nz 0 Since na is potentially larger than 255 visible wrapping errors are likely to occur i e the bilinearly interpolated values will be wrong To solve this we accept to lose a bit of precision and round n to the nearest multiple of three and divide by three n During the post filter pass step 2 4 we can then add n to red green and blue of the DPB To completely avoid the precision loss one could also do a three pass solution by adding the terms as a separate additive pass but it is probably not worth it To summarize the procedure is as follows e Pass 1 Render triangles using the positive palette with alpha blending disabled and the te
10. draws the conclusion The reader is expected to be familiar with the PS2 architecture and the general concepts of normal mapping as known from the PC 2 Previous work In 2002 Mark Breugelmans Breugelmans02 suggested a method for achiev ing bump mapping on the PS2 The clever part of this method is how it computes the sum R G B on the GS The weakness is how the signed multiplications are handled Mark splits the normals into a positive and a negative side by using two palettes The components in either palette are clamped to zero if they are of the opposite sign The vectors towards the light source transformed into tangent space are delivered in the vertex colors T L They are also split into a positive and a negative side With NM being the normal map the equation to obtain the signed multiplication becomes four passes N Mpos T Lpos N Mneg T Lneg N Mneg T Lpos N Mpos T Lneg One problem is that TL is clamped to zero per vertex and not per pixel which may result in incorrect dot products Figure 1 illustrates the interpo lation problem Figure 1 The blue line represents the interpolation from signed to unsigned values yl t q 1 t p The thick red line is what we would like to have which is the blue line clamped to zero per pixel y2 Max t q 1 t p 0 The green dotted line is what we actually get using per vertex clamping y3 t Maz q 0 1 t Maz p 0 This approximation results in artifacts
11. ds the light in object space This is actually fortunate for this four pass method since it fixes a well known sphere mapping artifact i e that projecting onto a sphere map per vertex and not per pixel goes wrong once the look ups reach the far back of the sphere Applying this factor will make sure triangles facing away from the light remain unlit The question is how the attenuation factor should be applied now that the light vectors are in a sphere map and not in the vertex colors Applying any kind of attenuation factor can be done by scaling down the vectors in the sphere map during the first pass The attenuation cannot be put directly into the vertex colors of this pass since the sphere map is stored as vectors subtracted from 128 A solution is to use any of the HIGHLIGHT texture functions to get the correct result by undoing equation 1 applying the scale and applying equation 1 again 128 95 128 45 sphmap 9 scale sphmap g scale 1 scale 128 9p So by setting red green and blue of the vertex color to scale 128 and then 1 scale 128 in the vertex color alpha and by using a HIGHLIGHT texture function during first pass per vertex attenuation is possible For a spot light it is possible to apply a projective texture of intensities to the DPB after the first pass and have the attenuation applied per pixel 14 This is done by using the GS blend mode to apply the attenuation in the same way a
12. for a wide range of 3D models since it is assumed that every triangle has the same tangent space assigned 2 to all three vertices and that the light is a directional light This means it usually works well for flat models but fails for smooth shaded models 3 The Approach This paper proposes a different way to achieve signed multiplications on the GS which improves the speed and quality compared to Breugelmans02 Two methods are presented One that does the signed multiplication in two passes without the per vertex clamping problem And an alternative method which does it in four passes and also normalizes the tolight vectors per pixel using a sphere map look up The first method uses a positive negative side strategy The second method uses a traditional normal map as known from the PC centered at 128 but with a reorganized palette Both methods work with directional spot and point light sources 3 1 Overview The two normal mapping methods share the same general rendering steps The only difference between the two is how they achieve signed multiplica tions In the general case two buffers are needed A 32 bit light accumulation buffer LAB and a 32 bit dot product buffer DPB The rendering steps 1 4 for rendering normal mapped objects are as follows 1 Render all the visible normal mapped objects to fill the z buffer and to set the ambient color in the LAB 2 For every light hitting the objects 2 1 Clear the DPB to 0
13. ight source Ts Ty Tz Through trial and error good results i e without wrapping errors have been achieved using the factor s 122 The code for packing the normals is shown in appendix 2 It is possible that larger factors may be used for s depending on how rounding was performed during normal map creation 10 3 6 Four pass and per pixel normalization solution When performing a linear interpolation of the tolight vectors across a triangle in the two pass solution the result is not a unit vector since it is not renor malized This causes lighting to decrease in intensity across the triangle and results in edge highlighting mach band artifacts The solution is of course to renormalize the tolight vector per pixel Again we only consider step 2 3 One way to achieve renormalization on the GS is to do look up into a sphere map containing packed unit vectors Since the sphere map is a texture and since reading from another texture the normal map is required during blending the tolight vectors will have to be delivered to the DPB during the first pass Thus the first pass simply renders geometry with the sphere map applied i e the interpolated tangent space tolight vectors are used as look ups into the sphere map using the regular sphere map look up equations m 2 4 T T T 1 9 8 To Ts s Apg m T t 05 m So using this method the tolight vectors are delivered via the texture coordinates and not the ver
14. ll This is done double buffered and 40 page settings are delivered per source buffer In addition the mask of the FRAME register is set in the COLUMN register so the mask may be set independently from the rest of the chain 3 4 Clearing the DPB As mentioned in step 2 1 of the overview it is necessary to initialize the DPB to 0x00808080 before rendering the signed multiplications The reason is that pixels that are not rendered to should have an intensity level of zero after look up into the ILP When using multiple lights one trick is to clear the DPB by setting the z buffer 24 bit to point to the DPB when adding the contribution of the current light to the LAB step 2 6 The depth is set to 0x00808080 and the TEST register is set to all pixels pass This involves reading from the DPB as an 8 bit texture and writing to it as a z buffer There are two ways to make this work without getting texels overwritten before they are read e Set the XYZ2s of the sprites according to the Z24 Z32 layout by setting them so the four 32x16 regions in the pages a page is 64x32 are swapped along the diagonals e Alternatively render to the LAB in PSMZ32 They both give the same result It works because the z buffer is forced to write to pixels inside the 32x16 region that is currently being read as a texture and is already cached This eliminates the deleting of pixels in 32x16 regions that have not yet been read Of course this also means the con
15. mans01 Breugelmans M 32bit Colour Channel Shifting us ing 8bit and 16bit texture formats Sony Computer Entertainment Europe www playstation2 linux com files p21sd 32bit_colour_ channel_shifting pdf October 2001 Breugelmans02 Breugelmans M PS2 Bump mapping Sony Computer Entertainment Europe Published on PlayStation 2 professional devel oper s resources January 2002 19 Cohen98 Cohen J Olano M Manocha D Appearance Preserving Sim plification Computer Graphics SIGGRAPH Procedings July 1998 Kilgard00 Kilgard M J A practical and robust bump mapping tech nique for today s GPU s GDC 2000 Advanced OpenGL Game Devel opment 2000 Sony02 Playstation 2 GS User s Manual 6th Edition Sony Computer Entertainment Inc 2002 20 Appendix 1 Code to generate the sphere map texture int nearest const float x if x lt 0 return nearest x else return int x 0 5f I use 64264 in 24bit myself void GenerateSphereMap void mem int iWidth int iHeight if l 2 ly 2 lz 2 1 m 2xsqrt lr 2 ly 2 lz 1 2 2xsqrt 2 lz 2 s la m 0 5 t ly m 0 5 we setup a second degree polynomial for lz by using lg 2 ly 2 lz 2 1 0 We then find the roots A 1 B 2 s 1 2 2xt 1 2 2 C B 1 det B 2 44C lz Bt sqrt det 2A this can be reduced since det 2 B 2 lz B 2 B 2
16. s when using the HIGHLIGHT texture function Alternatively this can also be applied after all four passes are complete 3 7 Normal Mapped Specular Highlights By using half vectors instead of directions towards the light it is possible to do normal mapped specular highlights A half vector is computed by adding the direction from the vertex to the eye and the direction from the vertex to the light source The resulting half vector must still be transformed into tangent space and normalized In addition the ILP should have its entries raised by an appropriate power term to match the properties of the material only for specular not diffuse To apply any form of attenuation such as distance attenuation it is neces sary to compensate for the power term otherwise the result will be att dot and not att dot as it should be The obvious way to fix it is by using att dot There are different ways to take a fast Nth root on the VU1 of values between 0 and 1 which are covered on the pro news groups and the playstation2 linux developers forum 3 8 Using Level of Detail As it was noted in step 5 of the overview it is still possible to render non normal mapped geometry in a single pass They are just rendered into the frame buffer using standard Gouraud shading An additional nice property is that it is possible to fade at distance continuously to the Gouraud shaded lighting Since attenuation is possible two lights of the same
17. tex colors After the first pass the DPB contents is rasterized triangles with their normalized tolight vectors To execute the signed dot product multiplications on the GS we take advantage of equation 8 So for the subsequent 3 passes the following GS blendmode is used normalMmaprg framebuf fer g normalmapa framebuf ferrgb This blend mode implies that X Y and Z of the normal map must be passed through via alpha so 3 palettes have to be used One for each X Y Code for precomputing the sphere map is shown in Appendix 1 11 255 r g b X256 Y 256 Z256 a Ordinary palette 0 1 cieee e b Reordered Palette Figure 3 a The structure of a palette of an 8 bit normal map b The reordered palette used to represent X during the signed multiplication pass Similar ones are made for Y and Z and Z of the normal map A quantized 8 bit normal map can be used with three palettes or alternatively three 8 bit normal maps which will give the same quality as using a standard 24 bit normal map The RGB of the 3 palettes should be set to 128 see figure 3 b so tolight will be subtracted from 128 during blending the first part of equation 8 The resulting signed multiplications do not have rounding errors and are identical with an ordinary char multiplication a b with a signed shift right by 7 offset by 128 To summarize the procedure is as follows e Pass 1 Render triangles
18. tribution gets added to the LAB in PSMZ32 layout This can be fixed by rendering the signed multiplications in the DPB in PSMZ32 layout as well which will take us back to PSMCT32 layout in the LAB Alternatively one could also just unswizzle the LAB on the GS at the end once all lights have been processed before step 3 3 5 Two pass DOTS solution on the GS Step 2 3 can be achieved in two passes In order to do so the tolight vector must be packed and passed per vertex stored as vertex colors The packed tolight vector l ly l is the normalized direction towards the light source Ts T T transformed into tangent space scaled and then decentralized using equation 2 lo ly lz 128 128 128 char s T char s T3 char s T The value s is an empirical scale factor and is given later in this section The surface normal nz ny nz of length 128 in tangent space is packed by splitting it into a postive and a negative side This is done similar to Breugelmans02 but in addition an alpha term is computed Two palettes are used one for each side An additional difference is that the tolight vectors are not divided into positive negative sides but simply offset by 128 In order to do the per pixel shading we need to compute the following according to equation 10 muls Maz ns 0 Ls Mas ns 0 Ls 128 Maa nz 0 Maa ng 0 muly Maz ny 0 Ly Max ny 0 Ly mul Mazr n 0 L M
19. x00808080 This pass is free for every light after the first one as will be explained in section 3 4 2 2 Disable color clamp 2 3 Render all objects that are hit by the current light so that signed multiplications are delivered to red green and blue 2 4 Apply a 2D post filter pass over the DPB to achieve R G B resulting in the final dot product 2 5 Enable color clamp 2 6 Add the lighting contribution of the DPB to the LAB To do this the DPB is read as an 8 bit texture with the dot products as texels We use an intensity look up palette ILP to add the dot product to all 3 channels of the LAB The ILP entries of the negative dot products are set to zero The look up is finally multiplied by the color of the light and added to the LAB 3 We are done with the DPB Clear it to zero and render all objects unlit with its diffuse texture 4 Multiply the buffer unsigned of the diffuse layer with the LAB redaifs Tedtight reeNaif f GTCCNight blueai sp blUetignt 5 Render all non normal mapped geometry as one generally would into the same buffer as the one containing the result of the multiplication in step 4 Even if there is not enough VRAM for two draw buffers it is still possible to use the techniques in this paper Instead use a single light and save the dot product layer in a free alpha channel like the one in the display buffer The intensities may be applied once it seems convenient during the rendering
20. xture function set to MODULATE e Pass 2 Render the triangles again but use the negative palette and enable alpha blending to do a subtraction of the source from the frame buffer Set vertex alpha to 128 so the texture alpha n may be passed to the frame buffer alpha set TCC RGBA and use MODULATE 9 a Pass 1 b Pass 2 c alpha added to rgb Figure 2 a A low resolution model rendered using MODULATE and the positive palette first pass b Second pass uses the negative palette and subtracts the source from the frame buffer Furthermore the alpha of the normal map is passed to alpha of the frame buffer c Third shot shows the result after adding the alpha This is not a part of step 2 3 but done at step 2 4 i e purely 2D without using the geometry of the model Adding r g b yields the final dot products see section 3 3 In the two pass case we have to modify the post filter of the DPB step 2 4 to get the final dot product result r g b a a a This is done by using one 2D pass reading the DPB in PSMT8H and adding the contents to red green and blue see figure 2 c The result at this point is not the signed multiplications since we have just added the last term na by adding an equally large slice one third of it to each channel The final dot products are obtained by completing the post filter adding red green and blue Per vertex attenuation of any kind may be achieved by scaling down the vectors towards the l
21. z a 0 5 The last term however is still signed In order to fix that we create another intermediate value cz and rewrite 5 to cg 128 Maz a 0 Maz a 0 6 a b Maz a 0 bg Mazx a 0 bg cp 128 7 On the GS clamping to 8 bit is performed once after the texture function operation and then clamping wrapping is performed at the end of the blend mode operation We can complete the dot product with full 8 bit precision by taking advantage of simple modulo tricks These are explained in section 3 3 3 3 Adding R G B Leaving out the subtraction by 128 in equation 4 yields a b 128 128 ay b2 a2 8 Applying 8 on the channels of the DPB to do the signed multiplications of the per pixel dot product the final dot product may be obtained by using equation 9 r g b 128 128 128 r g pb 128 9 dotprod Similar calculations can be made for 7 by again leaving out the sub traction by 128 a b 128 Maz a 0 bp Max a 0 b2 c2 10 Using 10 the final result can again be obtained by 9 As mentioned in step 2 6 an ILP is used to add the dot products to the LAB Assuming the pixels we look up in the DPB are the final dot products the ILP must contain 1 2 3 128 in red green and blue of the first 128 entries For free clamping we keep zero in the last 128 entries It is possible to simplify equation 9 to the sum of r
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