GPU-Accelerated Iterated Function Systems. Simon Green, NVIDIA Corporation

Transcription

1 GPU-Accelerated Iterated Function Sstems Simon Green NVIDIA Corporation

2 Iterated Function Sstems Fractal Conceived b John Hutchinson 1981 Popularized b Michael Barnsle Fractals Everwhere 1998 Consists of a set of functions Functions map points from one point in space to another Traditionall functions are affine transformations Should be contractive move points closer together

3 IFS Mathematics Image is set of points that are the solution to recursive set equation: S = n 1 U i= 0 f i S where S R f i : R R fi = ai + bi + c i di + ei + f i

4 Generating Images of an IFS Commonl solved using the chaos game Pick random point in space Repeat { Pick function at random Appl function to point Plot the point } Converges to correct solution in ~0 steps

5 Eample: Sierpinski Triangle Eample: Sierpinski Triangle = = = F F F

6 Eample: Sierpinski Triangle Eample: Sierpinski Triangle = = = F F F

7 Eample: Sierpinski Triangle Eample: Sierpinski Triangle = = = F F F

8 Eample: Sierpinski Triangle Eample: Sierpinski Triangle = = = F F F

9 Eample: Sierpinski Triangle Eample: Sierpinski Triangle = = = F F F

10 Fractal Flames Invented b Scott Draves Used in Electric Sheep screen saver Modification of traditional IFS: Non-linear functions as well as affine transforms Log-densit displa Color b structure Animates transformations and function weights

11 Fractal Flames

12 GPU Particle Sstems Render-to-verte arra made particle sstems possible on the GPU Lutz Latta Building a Million Particle Sstem GDC 004 Kipfer et al Uberflow Graphics Hardware 004 Positions / velocities of particles stored in tetures Simulation of particles computed using fragment programs Results rendered directl as point-sprites No read-back to CPU required Enables millions of particles at interactive speeds

13 GPU Particle Sstem Demo

14 GPUflame Uses the GPU to accelerate computation and displa of IFS fractals Etends functions to 3 dimensions Uses floating point blending for HDR displa

15 GPUflame Computation Uses standard GPGPU techniques Point positions stored in floating point teture Single quad drawn per iteration Fragment program applies IFS functions Uses Cg language to describe functions Random numbers come from pre-calculated teture

16 GPUflame Colouring Uses color inde value Tracks number of function applied at each step Stored in alpha channel of position teture Value is used in final point shader to inde into 1D palette teture

17 Colour Palettes

18 Colour Palettes

19 Colour Palettes

20 IFS Cg Code // interface for variations interface Variation { float3 funcfloat3 p float r float theta; }; float3 linearfloat3 p float r float theta { return p; } float3 sphericalfloat3 p float r float theta { return normalizep; } float3 swirlfloat3 p float r float theta { return float3r*costheta+r r*sintheta+r p.z; }

21 IFS Cg Code #include "variations.cg" float4 mainin float uv : TEXCOORD0 uniform samplerrect pos_te uniform samplerrect rand_te uniform float time uniform float44 matri[] uniform Variation variation : COLOR { // get position and color float4 te = terectpos_te uv; float3 p = te.z; float c = te.w; // get random number float4 rand = tedrand_te uv + floattime 0;

22 IFS Cg Code // appl transformation int ri = floorrand. * matri.length; forint i=0; i<matri.length; i++ { if ri==i { p = mulmatri[i] float4p 1.0; ci = i / float matri.length-1; } } // appl function float r = lengthp; float theta = atanp. p.; p = variation.funcp r theta; } c = c + ci /.0; // mi old and new color return float4p c;

23 GPUflame Rendering Points rendered using render-to-verte arra VBO/PBO OpenGL etensions Use of point sprites possible Rendered to 16-bit floating point buffer Using additive blending

24 Tone Mapping Simple tone mapping pass Converts floating point values to displaable 8-bit values Eposure and gamma controls Glow effect Perform Gaussian blur on image Mi original and blurred image Helps communicate ver bright regions

25 Without Tone Mapping

26 With Tone Mapping

27 Without Glow

28 With Glow

29 Performance 1 million 45 frames/sec GeForce 6800 Go 3 transforms 1 function High qualit 10 frames/sec 56k points 10 passes Ever function must be evaluated for each point

30 GPUflame Demo

31 Future Work Analtical point anti-aliasing Tiled rendering for high resolution prints More comple colouring schemes New tone mapping algorithms New IFS functions

32 Conclusion GPU hardware allows interactive eploration of a comple mathematical space Questions?