Path Tracing. Mikael Persson mpersson December 3, 2001

Transcription

1 Path Tracing Mikael Persson mpersson December 3, 2001 This report is available online for better viewing:

2 Project Report 1. Topics The rendering equation Monte Carlo integration for direct illumination Bidirectional reflectance distribution function Thin-lens camera model 2. Statement The raytracer previously implemented can produce very nice pictures. But it is still a long way from photo-realistic rendering. This project attempts to take the ray tracer one step closer to that goal. The first problem that will be addressed is the simulation of light transport. The theory of the rendering equation [4], and the bidirectional reflectance distribution function will be used. To approximate the rendering equation Monte Carlo integration will be used for direct illumination and path tracing for indirect illumination. Combined with a more accurate light model [3] and other features such Beer s law, Fresnel effects [6] and depth of field [5]. The latter feature will address the simulation of the camera used to capture the scene. 3. Technical issues The path-tracing algorithm is formulated as a possible solution strategy for the rendering equation [4]. This technical outline will describe how this project has applied the path-tracing algorithm for that purpose. The details of the algorithm are addressed below, the implementation issues section covers the general description of the algorithm. If good references exist, the technical issue will only be discussed briefly, if not, then a more detail description will be given. 3.1 The rendering equation The rendering equation [4] that has to be solved for each surface point is stated below. L( x, v) = L ( x, v) + ρ ( x, v, i) L ( x, i)( n i di e ) i Where L(x,v) is the radiance leaving the point x in direction v. L e ( x, v) is the amount of radiance emitted by the surface from point x in direction v. The BRDF ρ( x, v, i) computes a weight factor for the amount of radiance incoming from i at point x reflected in direction v. L i ( x, i) is the radiance incoming from direction i at point x, n is the surface normal at the point x. The expression is integrated over all possible incomming directions i, usually the visible hemisphere. 3.2 Approximation of the rendering equation using path tracing The challenge of approximating the rendering equation will be attempted with a technique called path-tracing, fully described in [2]. Rays will be traced from the camera (eye) and into the scene. At each ray surface intersection two steps need to be performed, first a Monte Carlo integration of direct illumination, fully described in [1] and second a path-tracing resulting in simulation of indirect illumination. The

3 procedure of sampling possible paths from the light source to the eye, although the paths are created in the opposite order, composes the path-tracing algorithm. Direct lighting using Monte Carlo integration Indirect lighting using pathtracing Figure 3.1. Illustration of the path-tracing method with direct and indirect illumination. A great number of sample paths (in the order of ) are gathered for each pixel, and thus the rendering equation is approximated by using the Monte Carlo integration, as described below. 3.3 Monte Carlo integration Approximating an integral using Monte Carlo methods can be performed as stated below. Where p(x) is an arbitrary probability density function and x 1...xM are samples achieved using the pdf p. I = f ( x) dx = f ( x) p( x) f ( x) M i dx = Ε p( x) p( x) M i= 1 p( xi ) 1 f ( x ) 3.4 Direct illumination and area light sources At each ray-object intersection, the direct illumination is computed forming one of the subpaths along the way. As can be seen in Figure 3.1, a sample path is actually composed of n subpaths from lightsource to the pixel, n being the total path length. The direct illumination is computed by integrating illumination contributions over the area of the light source. The following expression: L( x, v) = L e ( x, v) + ' s( x, x )cosθ cosθ M ' ' i i ρ ( x, v, xi xi ) Ae Le i= 1 xi where L is the radiance leaving the surface point, x, in direction v, results from the application of the Monte Carlo integration to the rendering equation for a surface point, while one must take into account that direct illumination is composed of deterministic rays, i.e. the shadow ray. ' 2 i x

4 The light source is sampled M times, s(x,y) is a shadow function being 0 if an object is in between point x and y, else being 1. The extra cosine factor, of the angle between the shadow ray and the light source normal, and the distance squared arises from the fact that the shadow ray is deterministic. A e is the area of the light source, the reciprocal of the probability of each shadow ray, and L e is the radiance leaving the lightsource. Figure 3.2, below shows an example of soft shadows produced using Monte Carlo integration. Figure 3.2. The cube casts soft shadows created using Monte-Carlo integration. 3.5 The Bidirectional Reflectance Distribution Function The BRDF, ρ ( x, v, i), mentioned above must compute a weight factor stating how much radiance is reflected from direction i in direction v at a surface point x. The BRDF describes the properties of the surface being rendered. The function must also supply a probability density function relating incoming and reflection directions, i.e. computing the probability of a certain pair of incoming and reflection directions. The pdf will be used for importance sampling. The materials described below are implemented in the path tracer Diffuse materials Diffuse materials have a uniform distribution i.e. incoming radiance is evenly distributed in all directions. This results in a constant BRDF, being the diffuse colour of the material Shirley materials For metals and other more exotic materials the BRDF described in [3] is used. The BRDF has a number of attributes under user control, R d, the diffuse reflectance, R s, the specular reflectance, nu and nv being the exponents, i.e. controlling the specular behaviour of the material. In Figure 3.3 below, a couple of example materials produced using the above BRDF are shown.

5 Figure 3.3. The image contains three spheres with different versions of the Shirley BRDF. The left sphere uses only specular settings, the middle sphere uses specular and diffuse settings and the right sphere is only diffuse Dielectric materials To simulate dielectric materials, such as water and glass, a combination of refractive path tracing, Beer s law and a phong model for direct illumination is used. The parameters of the lighting model are, Rs the specular reflectance used in the phong model for direct illumination, a r, ag, ab, the exponents of Beer s law attenuation controls the appeared colour of the material. The BRDF, Beer s law attenuation, as described below, is used for refractive paths, i.e. paths passing through the transmissive object. ρ x, k, k ) = e ( 1 2 at Rr Where R r is the recursive, i.e. transmitted, colour computed by recursive calling of the path-tracing algorithm. For direct illumination a phong like model is used, presented below. ρ ( x, k s k 10 1, k2 ) = R ( r 2 ) Where r is the reflected incoming, k 1, vector. Note that this material model is neither energy conserving nor reciprocal but gives reasonable result if used with caution. Figure 3.4, below shows a collection of different materials produced with the BRDF described above. Figure 3.4. Dielectric material samples. Left, sphere with reddish Beer s law attenuation. Middle sphere with bluish Beer s law attenuation. Right, sphere without Beer s law attenuation.

6 3.6 Sampling paths At each ray-object intersection a new direction for the path must be selected. This can be done in a number of different ways depending on the object material. At each surface intersection one of four different ray scattering methods are randomly chosen as described in the implementation section below Diffuse path sampling Diffuse paths are generated by uniform hemisphere sampling, i.e. a diffuse material scatter light uniformely in all directions of the hemisphere. The fact that the diffuse surfaces scatter light uniformly results in more noise in the rendering since more samples has to be made to cover the entire hemisphere in an feasible way. The details of the sampling used are described in [2]. incomming ray Figure 3.5. Illustrating the scattering of light on a diffuse surface Reflective path sampling Reflective paths are generated by computing and perturbing the specular reflection direction. The perturbation is uniform resulting in a constant probability density function. incomming ray normal reflected ray Figure 3.6. Illustrating the scattering of light on a reflective surface. The image below, Figure 3.7, shows a reflective material sampled using perturbed reflective sampling as described above. Note the blurry reflection as a consequence of the random perturbations. Figure 3.7. The sphere is rendered using the Shirley BRDF with reflective path sampling.

7 3.6.2 Refractive path sampling Refrective paths are computed by using Snell s law. A detailed description on refraction and its use in dielectric materials is presented in [6]. No perturbation is used to allow simulation of thin glass objects as often is the case. The image below, Figure 3.8, illustrates the effect of refraction on a sphere. Figure 3.8. A refractive sphere rendered using refractive path sampling. No perturbation is used Shirley path sampling When using shirley materials, diffuse, reflective and refractive sampling may not produce the desired results. If an accurate visual representation of the material is to be achieved one must sample the BRDF in the directions where it has the greatest weight. By implementing importance sampling as described in [3] this was achieved. The images below, Figure 3.9, shows a rendered sphere with and without importance sampling. Figure 3.9. Left, Shirley BRDF sphere with importance sampling. Right, Shirley BRDF sphere without importance sampling. 3.7 Depth of field The camera model, i.e. depth of field simulation, is achieved using a thin-lens model. The used specifies a lens aperature (radius) and a focal plane distance, i.e. the plane where everything is in perfect focus. First the intersection between the focal plane and the ray from the eye (center of the lens) through the current pixel is computed, all additional rays fired will pass through this point. For each sample ray used for pixel

8 sampling a random position on the lens is picked and used as origin for the ray resulting in a random ray within the field specified by Figure 3.10, below. Thin lens Image plane Focal plane Figure Illustrating the thin-lens camera model simulation. When the scene is path traced using the, according to the camera model computed, rays as primary rays objects in front of and behind the focal plane will become blurry, i.e. out of focus. 4. Software design The software design is straigh forward. A scene described by the extended scripting language is processed by the script shell module, grsh, an interface to the graphics database module, gr. As the script is executed, the scene database is built and finally the rendering command is executed. When rendering is initiated the render module starts rendering and queries the database for scene ray intersections. script Create scene Scene database enqueries Grsh Gr Render image Render command Figure 4.1. Illustrating the software design and basic communication scheme. Figure 4.1 above gives a simple view of the module communications described above. 4.1 Communication The communication of the path tracer is simple: Input will consist of a scene script written in the extended scene description language. Output will consist of the final rendered picture. The path tracer is executed using the following command line:./grsh scenefile.gr Where scenefile.gr is the extended scene description script.

9 4.2 Modules Mainly the same modules as used in the ray tracer assignment. Grsh, the scene description language interface. This module contains the TCL interface to the scene database in module gr. Gr, the scene database. Contains functions for modifying the scene database as well as performing ray tracing requests. It is divided into a number of parts each handling a specific scene graph node type. Each part has functions for node creation, data structure building and ray intersection where applicable. The parts of the module in order from the top of the file are: 1. Node managing functions 2. Transform functions 3. Uniform grid functions 4. Polyhedron functions 5. Instance functions 6. Sphere functions 7. Box functions 8. Material functions 9. Light source functions 10. Scene graph functions 11. Statistics functions Render, performs the actual rendering with support of the scene database request functions. This module also contains the lighting model and supporting functions for importance sampling. The module is divided into four parts, in order from the top of the file, BRDF functions, importance sampling functions, path tracing, main ray generation and the rendering algorithm. 4.3 Organisation All project related files lies under the A5 directory. Source, executable, documentation and data are organised according to the following: All source files lies under A5/src. The command make will compile and link the entire project, creating the executable grsh. The executable grsh lies under the A5 directory. The scene files demonstrating the different features and images produced by these scene files lies in A5/data. The README file contains a simple user guide, and instructions on how to run the different test cases. 5. Implementation issues This section covers interesting issues in an implementation point of view. (The) first issue covered is the extensions of the scene language needed for the path tracing as well as the advanced material functions, followed by a pseudo code description of the

10 path tracing algorithm. A simple description of the hierarchical spatial uniform grid optimisation is included, as well as a limited description of the median filter used on the final image. 5.1 Scene language extensions The scene language extensions used for controlling the new features are listed below Material extensions The material command is extended to support the different material types listed in the technical issues section. The basic material command is: gr_material name type options Where name is the name of the material, as in the original command. Type being one of the material types described above, diffuse, dielectric or shirley. The options field consists of different material type dependant options, all possible options are listed below: diffuse, requires a RGB value, i.e. {r g b}, and sets the diffuse reflectance of the material. Only valid for diffuse and shirley material types. specular, requires a RGB value, i.e. {r g b}, and sets the specular reflectance of the material. Only valid for shirley and dielectric material types. exp, requires a two value tuple, i.e. {nu nv}, defining the two exponents of the shirley type material model, nu and nv. n, requires a double value. Defines the index of refraction for the dielectric material type. sampling, requires a four value tuple, i.e. {a b c d}. Defines the probability of the path using each of the above specified path sampling methods when scattering at a surface intersection point. They are defined in the order, diffuse sampling, reflective sampling, refractive sampling and shirley sampling. Valid for all material types. beers, requires a three value tuple, i.e. {ar ag ab}. Defining the exponent used when applying Beer s law in the dielectric material type Area light sources Instead of using the simple point light sources implemented in the ray tracing assignment, the project will use area light sources. These are included in the scene graph as any other geometry would be, and they are defined by the following scene graph description language command. gr_als_planar pname name {r g b} {x y z} size This command will create a planar light source at position {x y z} spanning in the x-zplane the amount of size, with emitted radiance {r g b}. The light source will be inserted into the hierarchy at pname:name.

11 5.1.3 Rendering options The gr_render command has been extended with options controlling the extra features of the path tracer. The command has the same basic functionality as in the original ray tracer but with the following extensions: accelerator, <on/off>, activates or deactivates the hierarchical spatial subdivision acceleration scheme. Default is off. globalilllumination, <on/off>, activates or deactivates the recursive path tracing algorithm, if deactivated only direct illumination and primary rays are used. Default is off. antialiasing, <on/off>, activates or deactivates the jitter grid antialiasing process. Default is off. pathlength, defines the path length used by the recursive path tracing algorithm. A path is terminated if it hits the light source or if the length reaches the value defined by the option pathlength. Default is 2. depthoffield, <on/off>, activates or deactivates thin-lens camera simulation, i.e. depth of field effects. focaldistance, sets the focal distance of the camera model i.e. the distance along the view direction where all objects are in perfect focus. Default is 1. aperature, sets the radius of the thin-lens used by the camera mode. Default is startline, sets the first scanline to render. Can be used to controll which segment of an image is to be rendered. If not specified the image will start rendering from the first scanline. stopline, set the last scanline to render. Can be used to controll which segment of an image is to be rendered. If not specified the image will stop rendering after the last scanline. samplesperpixel, sets the number of samples per pixel used when approximating the rendering equation. Default is 16. samplesperlightsource, sets the number of samples per light source in the scene used when computing direct illumination using Monte-Carlo integration. Default is The path tracing algorithm The basic path tracing algorithm is presented below. The details of the algorithm, such as direct illumination, path sampling and thin-lens simulation, are covered above in the technical issues section.

12 Render the image using path tracing For each pixel Clear color For each sample Pick ray on lens through random position in pixel Color = color + pathtrace (ray) Pixel = color / #samples Pathtrace(ray) Find nearest intersection with scene Compute intersection point and normal Color = shade(point,normal) Return color Shade(point,normal) Clear Color For each light source Test visibility of random position of light source If visible Color = color + direct illumination Color = color + pathtrace(randomly selected ray) Return color 5.3 Heirarchical spatial uniform grid optimisation When rendering images using path tracing, a great number of rays are traced. To optimise the time consuming procedure, a hierarchical spatial uniform grid optimisation is implemented. The basic algorithm is described in [7]. The basic algorithm is extended by a two level hierarchy, one level covering the entire scene and one level for each polyhedron mesh where required. Tests with and without the optimisation scheme activeated where performed resulting in Figure 5.1, below # seconds # spheres Figure 5.1. Results from acceleration testing, dotted line being with acceleration activated and filled line without acceleration.

13 As expected the time required to render a scene without acceleration increases linearly with the number of objects in the scene. If the described acceleration scheme is activated the time required is more or less constant, of course it will increase but not as a factor of the number of objects but as a factor of their distribution in space. 5.4 Median filtering To produce images with low noise variance a great number of samples per pixel and light source is required. Images with high noise can be post-processed with a median filter to reduce noise variance. A very basic median filter was implemented in MATLAB and applied to the final image. The simple MATLAB function for grayscale images is presented below. function out=medfilt2(img,n) for y=1:size(img,1), for x=1:size(img,2), cell_red = img(max(yn/2,1):min(y+n/2,size(img,1)),max(xn/2,1):min(x+n/2,size(img,2))); out(y,x) = median(median(cell)); end; end; 6. Goals The goal of this project was to extend the ray tracer with features essential for photorealistic rendering. The ray tracer shall be able to produce convincing light simulation, both indirect and direct illumination. Henrik Van Jensen has produced several close to photo-realistic images using different global illumination techniques. Below is an example of global illumination simulation of diffuse surfaces such as processed clay or wood, courtesy of Henrik Van Jensen. Figure 1. Left, jaguar rendered using path tracing, courtesy of Henrik Van Jensen. Right, polygonal model that will be rendered using the path tracer implemented as the project. One of my goals was to be able to produce close to the same realism with my ray tracer using the model presented above to the right. The model will be placed in a suitable environment and rendered with global illumination and camera lens simulation, i.e. depth of field.

14 To clearly show that all objectives has been reached, a Cornell box test image will be created and rendered in several versions according to the objectives section. It will contain shiny surfaces, refractive surfaces and depth of field effects. 7. Conclusion I believe all objectives were reached in an satisfactory fashion. Although I had hoped to be able to conduct more genuine testing and buf fixing. Most algorithms work for reasonable scenes. I have found some problem in the uniform grid acceleration scheme leaving glitches in polygon meshes most certainly due to numerical errors in the bounding box computations. Although I believe the purpose of this project was to demonstrate the different features and successfully implement all stated algorithms. For the final image I produced, according to the goals stated above, a scene composed of the modelled character staged in a suitable environment. The scene contains most of the features implemented such as global illumination, reflection, refraction and depth of field. Figure 7.1. The final image.

15 8. Bibliography [1] Peter Shirley. Realistic Ray Tracing. A K Peters, Ltd. pp [Contains a detailed description of the Monte Carlo integration technique for direct illumination.] [2] Peter Shirley. Realistic Ray Tracing. A K Peters, Ltd. pp [Contains information on the path-tracing technique.] [3] Michael Ashikhmin and Peter Shirley. An anisotropic Phong BRDF model. Journal of Graphics Tools, 5(2):25-32, 2000 [Describes the lighting model to be used.] [4] James T. Kajiya. The rendering equation. Proceedings of the 13th annual conference on Computer graphics,1986.[describes the theory behind the rendering equation.] [5] Peter Shirley. Realistic Ray Tracing. A K Peters, Ltd. pp [Description of camera models and depth of field] [6] Peter Shirley. Realistic Ray Tracing. A K Peters, Ltd. pp [Description of basic materials, Beer s law, Fresnel effect.] [7] Peter Shirley. Realistic Ray Tracing. A K Peters, Ltd. pp [Uniform grid acceleration scheme.]

16 Objectives: Project Name: Mikael Persson UsedID: mpersson StudentID: Objective 1: A hierarchical spatial subdivision scheme is introduced as described in the specification. Performance measures are performed with and without the optimisation. Objective 2: A scene including the character presented in the specification is modelled, exported to the scene description language format and rendered using the ray tracer. Objective 3: Antialiasing via supersampling is performed by the jittered grid approach. Sample images with and without the supersampling are presented. Objective 4: Area light sources produce soft shadows using Monte Carlo integration. Soft shadows are clearly visible in the Cornell box test image. Objective 5: Indirect illumination via reflection, so called colour bleeding, is performed via path tracing and is clearly visible in the Cornell box test image. Objective 6: Depth of field is implemented [5] and clearly visible in a version of the Cornell box test image. Objective 7: Light transmission is implemented via refraction. The feature is clearly visible in the Cornell box test image. Objective 8: Beer s law is implemented for attenuation through transmissive objects as described in [6]. The results are clearly visible in the Cornell box test image. Objective 9: The BRDF described in [3] is used for lighting computations. Examples of the materials presented in [3] is clearly visibile in the Cornell box test image. Objective 10: The BRDF described in [3] is used for importance sampling. The Cornell box image with and without importance sampling is available. For the ray tracing assignment a spatial subdivision optimisation scheme for polyhedron meshes was implemented. Declaration: I have read the statements regarding cheating in the CS488/688 course handouts. I affirm with my signature that I have worked out my own solution to this project, and the code I am handing in is my own. Signature:

17 sum is: /usr/bin/sum Dec 2 11: Checksum for A5 for mpersson on ethane.math Page 1 total drwxrwx--x 4 mpersson cs Dec 2 00:36./ drwxrwx--x 8 mpersson cs Sep 14 14:30../ rw-r--r-- 1 mpersson cs Dec 2 00:28 README drwxrwx--x 2 mpersson cs Dec 2 11:16 data/ rwxr-x--x 1 mpersson cs Dec 2 00:33 grsh* drwxrwx--x 2 mpersson cs Dec 2 00:33 src/ A5/data: total drwxrwx--x 2 mpersson cs Dec 2 11:16./ drwxrwx--x 4 mpersson cs Dec 2 00:36../ rw-r--r-- 1 mpersson cs Dec 1 22:25 cornell1-aa.gr rw-r--r-- 1 mpersson cs Dec 1 22:15 cornell1-aa.png rw-r--r-- 1 mpersson cs Dec 1 22:25 cornell1.gr rw-r--r-- 1 mpersson cs Dec 1 18:57 cornell1.png rw-r--r-- 1 mpersson cs Dec 1 22:26 cornell2.gr rw-r--r-- 1 mpersson cs Dec 1 18:57 cornell2.png rw-r--r-- 1 mpersson cs Dec 1 22:26 cornell3-noimp.gr rw-r--r-- 1 mpersson cs Dec 1 22:16 cornell3-noimp.png rw-r--r-- 1 mpersson cs Dec 1 21:37 cornell3.gr rw-r--r-- 1 mpersson cs Dec 1 21:32 cornell3.png rw-r--r-- 1 mpersson cs Dec 1 22:26 cornell4.gr rw-r--r-- 1 mpersson cs Dec 1 22:04 cornell4.png rw-r--r-- 1 mpersson cs Dec 1 22:26 dof.gr rw-r--r-- 1 mpersson cs Dec 1 18:57 dof.png rw-r--r-- 1 mpersson cs Dec 1 22:26 final.gr rw-r--r-- 1 mpersson cs Dec 1 18:56 final.png rw-r--r-- 1 mpersson cs Dec 1 18:56 final_median.png rw-r--r-- 1 mpersson cs Dec 2 00:35 nodof.gr rw-r--r-- 1 mpersson cs Dec 2 11:16 nodof.png rw-r--r-- 1 mpersson cs Dec 1 21:35 performance2x2.gr rw-r mpersson cs Dec 1 22:14 performance2x2.png rw-r--r-- 1 mpersson cs Dec 1 21:35 performance3x3.gr rw-r mpersson cs Dec 1 22:18 performance3x3.png rw-r--r-- 1 mpersson cs Dec 1 21:35 performance4x4.gr rw-r mpersson cs Dec 1 22:19 performance4x4.png rw-r--r-- 1 mpersson cs Dec 1 21:35 performance5x5.gr rw-r mpersson cs Dec 1 22:20 performance5x5.png A5/src: total drwxrwx--x 2 mpersson cs Dec 2 00:33./ drwxrwx--x 4 mpersson cs Dec 2 00:36../ rw-r-x--- 1 mpersson cs Nov 20 19:27 Makefile* rw-r--r-- 1 mpersson cs Dec 1 21:24 gr.c rw-r--r-- 1 mpersson cs Dec 1 21:14 gr.h rw-r--r-- 1 mpersson cs Dec 1 21:21 grsh.c rw-r--r-- 1 mpersson cs Nov 20 19:36 pic.c rw-r--r-- 1 mpersson cs Nov 20 19:36 pic.h rw-r--r-- 1 mpersson cs Dec 1 21:23 render.c rw-r--r-- 1 mpersson cs Dec 1 21:14 render.h rw-r--r-- 1 mpersson cs Dec 1 21:23 util.c rw-r--r-- 1 mpersson cs Dec 1 21:14 util.h A5

18 Dec 2 11: Checksum for A5 for mpersson on ethane.math Page 2 A5/src A5/src/gr.c A5/src/gr.c A5/src/grsh.c A5/src/grsh.c A5/src/pic.c A5/src/pic.c A5/src/render.c A5/src/render.c A5/src/util.c A5/src/util.c A5/src/gr.h A5/src/gr.h A5/src/pic.h A5/src/pic.h A5/src/render.h A5/src/render.h A5/src/util.h A5/src/util.h A5/src/Makefile A5/src/Makefile A5/data A5/data/final.png A5/data/final.png A5/data/final_median.png A5/data/final_median.png A5/data/cornell3.gr A5/data/cornell3.gr A5/data/dof.png A5/data/dof.png A5/data/cornell2.png A5/data/cornell2.png A5/data/cornell4.png A5/data/cornell4.png A5/data/performance2x2.gr A5/data/performance2x2.gr A5/data/performance3x3.gr A5/data/performance3x3.gr A5/data/performance4x4.gr A5/data/performance4x4.gr A5/data/performance5x5.gr A5/data/performance5x5.gr A5/data/cornell3.png A5/data/cornell3.png A5/data/cornell3-noimp.png A5/data/cornell3-noimp.png A5/data/cornell1-AA.png A5/data/cornell1-AA.png A5/data/cornell1.png A5/data/cornell1.png A5/data/performance2x2.png A5/data/performance2x2.png A5/data/performance3x3.png A5/data/performance3x3.png A5/data/cornell3-noimp.gr A5/data/cornell3-noimp.gr

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