E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Computer Graphics Proceedings, Annual Conference Series, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....or paths) from the sources which travel through the scene according to the Form Factors transition probabilities, either discrete patch to patch or continuous point to point Form Factors. On each bounce, the exit point can be the same impinging point, as in non discrete methods [2] 4] [12] and Particle Tracing [5] or can be take at random on the patch [7] The methods in [10] 3] can be considered as a breadth first approach to random walk, which would be a depth first approach. Also, shooting random walk estimators for radiosity can be classified according to the expected path ....

Eric Veach and Leonidas J. Guibas, "Optimally combining sampling techniques for monte carlo rendering", ACM Computer Graphics Proceedings, (ACM SIGGRAPH '95 Proceedings), pp. 419-- 428, 1995. 8

....p(x) q(x) cos(#) corresponds to moving the cosine power out of the integrand and into the sampling density. We will suppose that a choice of p has already been made. There is also the possibility of using a mixture of sampling densities p j as with the balance heuristic of Veach and Guibas [42, 43]. This case can be incorporated by increasing the dimension of x by one, and using that variable to select j from a discrete distribution. Monte Carlo sampling of x p over almost always uses points from a pseudo random number generator simulating the uniform distribution on the interval from ....

Eric Veach and Leonidas Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In SIGGRAPH '95 Conference Proceedings, pages 419--428. Addison-Wesley, August 1995.

....other directions important. The question that we shall address is how to construct an importance sampling method that accounts for all such hot spots by combining available sampling methods, but without introducing statistical bias. The following discussion closely parallels the work of Veach [13], who was the first to systematically explore this idea in the context of global illumination. To simplify the discussion, let us assume that we are attempting to approximate some quantity , which is given by the integral of an unknown and potentially illbehaved function f over the domain ....

....not depend on i, which indicates the pdfs they are distributed according to. This is precisely the formula we would obtain if we began with q as our pdf for importance sampling. Adopting Veach s terminology, we shall refer to this particular choice of weighting functions as the balance heuristic [13]. Other possibilities for the weighting functions, which are also based on convex combinations of the original pdfs, include w i (x) 1 if c i p i (x) max j c j p j (x) 0 otherwise , 2.12) and w i (x) c i p i (x) c j p j (x) 1 , 2.13) for some exponent m 1. Again, we ....

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Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, pages 419--428, August 1995. 34

....of the light source, which is computed identical to [PP98] The user only needs to specify the overall shadow ray number and does not need to take care of the sample rate for the single light sources. The technique also easily transfers to the bidirectional path tracing algorithm [LW93, VG94, VG95] where it is used to save shadow or connection rays. Then the photon map would serve as a point approximation of the radiance like in [Kel97] This would also improve the bidirectional mutations of the Metropolis light transport [VG97] algorithm. 4 Conclusion We presented new importance ....

E. Veach and L. Guibas. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In SIGGRAPH 95 Conference Proceedings, Annual Conference Series, pages 419--428, 1995. 3

....the backward estimates will have higher variance. Taking into account the two techniques with equal weights, we also inherit the disadvantages of backward estimates. Fortunately, more sophisticated combination of the two techniques is also possible as suggested by multi ple importance sampling [7, 1]. Balanced heuristic of multiple importance sampling [7] divides the integrand by the average probability density of the methods to be combined no matter which method has generated the given sample. 2 A bi directional global illumination algorithm Let us first sample 77 and o: with p(77, o: ....

....into account the two techniques with equal weights, we also inherit the disadvantages of backward estimates. Fortunately, more sophisticated combination of the two techniques is also possible as suggested by multi ple importance sampling [7, 1] Balanced heuristic of multiple importance sampling [7] divides the integrand by the average probability density of the methods to be combined no matter which method has generated the given sample. 2 A bi directional global illumination algorithm Let us first sample 77 and o: with p(77, o: that is proportional to L(77, o: cos 0g and then 3 ....

E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '9, pages 147-162, 1994.

....whereas the right border describes the paths started from the detectors. The inner terms of the pyramid scheme are functionals linking segments independently started on the light sources and the detectors. The original stochastic algorithm for bidirectional path tracing has been introduced in [VG95, Laf96] For the deterministic simulation, the Halton vector x i = u i,1 , u i,4j 2 ) is used in an interleaved way to exploit the good discrepancy in the lower dimensions: Choosing the pixel functional (1.6) y # 0 is the first point hit when shooting a ray from the eye point through a ....

....the geometric term G(y # j # , y n ) cos # y # j # n cos # yn y # j # n y n which results from a domain change from# to S using d# x = cos #y x y 2 dy. A sample image of the algorithm with the simple choice of w jn = j 1 (although better choices for w jn exist, see [VG95] is shown in figure 3b. Note, that this scheme is a single pass algorithm, working without discretization error. Due to the scene complexity, a very high sampling rate (N 1000) Printer: Opaque this is required for each pixel of the image, which can be reduced by techniques proposed in [VG95, ....

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E. Veach and L. Guibas. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In SIGGRAPH 95 Conference Proceedings, Annual Conference Series, pages 419--428, 1995. 4.2

....W . Lin. Lr Pi j sj The method discussed so far applied a static weighting of different techniques. However, it seems worth using a weighting scheme that de pends on the generated direction as well [2] The formal basis of such combination is given by the theory of multiple importance sampling [9]. Multiple importance sampling combines different sampling techniques in a way that their advantages are preserved, i.e. the variance of the com bined estimator is smaller than the individual es timators and not far from the optimum. One of such weighting schemes, called the balance heuristic, ....

E. Veach and L. Guibas. Optimally combin- ing sampling techniques for Monte Carlo rendering. In Rendering Techniques '93, pages 147-162, 1994.

....question. 1 Introduction Computing global illumination solutions for general scenes is a difficult job. Scenes can be very complex, and the materials used can have arbitrary reflection and refraction properties. Pure Monte Carlo methods, like path tracing[11] or bidirectional path tracing[12, 20], are capable of computing light transport for such general scenes. They do not store any information in the scene and are therefore capable of rendering very complex geometry. However, not storing the illumination means that it has to be recomputed every time when needed. This can be very ....

Eric Veach and Leonidas J. Guibas. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In Computer Graphics Proceedings, Annual Conference Series, 1995 (ACM SIGGRAPH '95 Proceedings), pages 419--428, 1995.

....strategy tailored to each subset to achieve good performance over the entire sampling domain. See Figure 4 for an illustration. This approach is related to the stratification methods for Monte Carlo integration [KW86] and the multiple importance sampling for ray tracing photo realistic images [VG95] The significance of a hybrid distribution is not about putting together two distributions, but rather about identifying distributions that complement each other s strengths and combining them so that their individual strengths are preserved. Our approach differs from that of the previous work ....

E. Veach and L.J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In SIGGRAPH 95 Conference Proceedings, pages 419-- 428, 1995.

....random from each cluster and used for BRDF importance calculations as well as for Russian roulette path termination. Antialiasing was implemented by integration over the support of a filter function centered at each pixel. The balance heuristic was used to combine samples from multiple estimators [20]. Depth of field and area sources were not used in our test images. Adaptive sampling was not used, nor were the images shown here postprocessed to reduce noise. We implemented the following four variations for comparison purposes: CMC: Crude Monte Carlo. A pseudorandom number generator was used ....

E. Veach and L. Guibas. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In Proc. SIGGRAPH, pages 419--428, August 1995.

.... and two pass methods which combine radiosity and ray tracing [26,40,38] To reduce the variance of the Monte Carlo integration, most of these methods incorporate some form of importance sampling [31] The importance can be based on the local BRDFs [8,14] on the direct illumination [28] on both [35,15], or can even be explored adaptively [37] Expansion techniques generate random walks independently. It can be an advantage, since these algorithms can be parallelized easily. However, it also means that these methods forget the previous history of walks and cannot reuse the visibility ....

E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

....state, and tracked to a predetermined fixed length. The covering path is terminated after it has visited all the states plus the fixed length (thus a single physical path conveys many different logical paths) As a mixture of non discrete shooting and gathering we find bidirectional ray tracing [VG95, LW93]. Shooting random walk methods have also been investigated in the limiting, non discrete case [DLW94, DW95] Variances and complexities of shooting and gathering random walk are discussed in [Sbe97a, Sbe97d, Sbe97b] It must be remarked that for the dual methods just discussed, the traced paths ....

Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for monte carlo rendering. In Computer Graphics Proceedings, Annual Conference Series,

....can lead to high variance because area measure (on the planar triangle) and projected solid angle measure are not related and may differ too much, specially for triangles covering a big solid angle when projected onto p. This problem can be solved by using a combination of sampling strategies [10]. We consider here the solid angle covered on the hemisphere by any planar region when projected over p. Projected solid angle measure is equal to solid angle measure times the cosine factor. A better technique is to use a mapping from U to the triangle, such that regions with equal areas are ....

E. Veach and L.J. Guibas. Optimally combining sampling techniques for montecarlo rendering. In Computer Graphics Proceedings, Annual Conference Series, pages 419--428. ACM Siggraph, 1995.

....on the respective paths are connected by means of shadow rays, which determine the contribution to the estimated flux of the pixel. 4 Bidirectional path tracing Bidirectional path tracing has been introduced by us [14, 15] and has been presented independently and improved by Veach and Guibas [16, 17]. It combines the ideas of path tracing and light tracing by creating random walks, starting not only from the eye point, but also from the light sources. Figure 4 shows the basic idea. Eye paths start from the eye point, through the pixel that is being computed, as in classical path tracing. ....

....has proven to be particularly effective for rendering scenes with indirect illumination, compared to ordinary path tracing [18] The different illumination transport paths present different estimators for the flux, each one with its own potential strength. A judicious combination of the estimators [17] yields a robust estimator with a generally lower variance. For this purpose the alternative estimators for illumination transport with the same number of reflections have to be considered in the same parameter space, for instance as the sequence of points x 0 ; x k ; y l ; y ....

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E. Veach and L. Guibas, "Optimally combining sampling techniques for Monte Carlo rendering," Computer Graphics, vol. 29, pp. 419--428, Aug. 1995.

....might be quite inaccurate. In a single step the importance is usually selected according to the BRDF [12, 25] or according to the direction of the direct lightsources [46] Combined methods that find the important directions using both the BRDF and the incident illumination have been proposed in [66, 17, 26, 56]. Just recently, Veach and Guibas[68] proposed the Metropolis method to be used in the solution of the rendering equation. Unlike other approaches, Metropolis sampling[29] can assign importance to a complete walk not just to the steps of this walk, and it explores important regions of the domain ....

E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

....methods that include the full directionally depended reflection [21, 3, 32, 6, 24] is still cubic in the number of scene objects, which rules this method out even for simple environments with more than a few non diffuse surfaces. On the other hand, Monte Carlo path or photon tracing techniques [10, 23, 45, 46, 19, 28] simulate illumination by recursive stochastic sampling of illumination in the environment, starting with rays from the virtual camera or the light sources, respectively. Although this technique solves the general form of the radiance equation and can therefore account for any lighting effect, it ....

VEACH, E., AND GUIBAS, L. J. Optimally combining sampling techniques for Monte Carlo rendering. Computer Graphics (SIGGRAPH '95 Proceedings) (August 1995), 419--428.

....We are most interested in cases where d is moderate to large, and f is spiky. Our motivating examples come from particle physics and Bayesian statistics. Spiky integrands also arise in computational finance (Glasserman, 1 Heidelberger Shahabuddin 1999, Owen Zhou 1999) computer graphics (Veach Guibas 1995) and reliability (Hesterberg 1995) For d = 1 and smooth f , classical methods, such as those in Davis Rabinowitz (1984) provide excellent accuracy with only a handful of function evaluations. For small d, tensor products of 1 dimensional rules work very well on smooth functions. For large d, ....

Veach, E. & Guibas, L. (1995), Optimally combining sampling techniques for Monte Carlo rendering, in `SIGGRAPH '95 Conference Proceedings', Addison-Wesley, pp. 419--428.

....) where samples are generated using bidirectional path tracing [Lafor93, Veach94] Note that we need to reconstruct an arbitrary illumination function and not a pure pdf. Also note that that the samples on screen originate from a number of different pdf s using multiple importance sampling 1 [Veach95]. Moreover do the samples obtained in the image plane originate from a much higher dimensional sampling procedure (tracing paths) so that even samples with the same image plane position can have a vastly different value. The image plane function radiance ( that we want to reconstruct is ....

....( and do this by taking , samples or paths ) from a set of pdf s 0 , Due to the multiple importance sampling, each sample ) has to be assigned a weight 1 ,2 ) 3 to get an unbiased solution. The constraint 4 1 , ensures this unbiasedness (see [Veach95] for more information) Now the modified variable kernel density estimator becomes: 5 , 6, 1 , 7 ) 0 ,2 ) 8 9 ; # ) 2) Using 4 , and , A ABDCFE=BHG I J K=CFE=BHG ....

Eric Veach and Leonidas J. Guibas. Optimally Combining Sampling Techniques for Monte Carlo Rendering. In Computer Graphics Proceedings, Annual Conference Series, 1995 (ACM SIGGRAPH '95 Proceedings), pages 419--428, 1995.

....but only slowly; this is known as the law of diminishing returns of Monte Carlo sampling. State of the art path tracing methods use several different sampling strategies, each targeted towards a particular aspect of the global illumination problem, and combine them in ways that are provably good [Veac94, Veac95]. These methods do improve results for the middle ground of semi diffuse scenes measurably, but pure diffuse scenes still suffer from noise. Again, see Figure 7. Finite Element Methods As discussed previously, finite element methods work directly on the mesh of the input scene. The input scene ....

Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for monte carlo rendering. Proceedings of SIGGRAPH 95, pages 419--428, August 1995.

....inaccurate. In a single step the importance is usually selected according to the BRDF [Dutre93, Lafor93] or according to the direction of the direct lightsources [Shirl96] Combined methods that find the important directions using both the BRDF and the incident illumination have been proposed in [Veach94, Lafor96]. Just recently, Veach and Guibas [Veach97] proposed the Metropolis method [Metro53] to be used in the solution of the rendering equation. Unlike other approaches, Metropolis sampling can assign importance to a complete walk not just to the steps of this walk, and it explores important regions of ....

E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

....they suggest to make a linear combination of the PDFs corresponding to the individual luminaires. Again a single sample is taken according to the resulting PDF. Unfortunately many choices in these techniques are intuitive and one has to rely on rules of thumb to select acceptable parameters. In [125, 126] Veach and Guibas present a novel approach in the contexts of direct illumination computations and of bidirectional path tracing. Although the results seem quite fundamental we have not found them in the general Monte Carlo literature. As it has proven to be very useful for the algorithms ....

.... Z 1 0 p i (x) P n j=1 p j (x) f(x)dx # 2 : 3.20) This choice is optimal in the sense that it minimises the sum of the second order moments of the individual terms of the estimator, i.e. the first term of the expression for the variance (3. 15) For a more general case it is proven in [126] that the variance obtained using other weight functions can only improve the balance heuristic by a limited amount: oe 2 combine oe 2 balance Gamma 1 Gamma 1 n I 2 : 44 CHAPTER 3. MONTE CARLO METHODS ffl w i (x) p fi i (x) P n j=1 p fi j (x) is a generalisation ....

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E. Veach and L. Guibas, "Optimally combining sampling techniques for Monte Carlo rendering," Computer Graphics, vol. 29, pp. 419--428, Aug. 1995.

....rays) Shooting random walk shoots paths carrying energy from the sources, to update the visited patches [11] 1] The techniques in [20, 4] can be seen as a breadthfirst approach to a shooting random walk estimator, which in turn would be the depth first approach. Bidirectional ray tracing [23, 9] is a mixture of non discrete shooting and gathering. The random walk proceeds according to the Form Factor probability transitions [20, 11, 4] or to biased ones [12, 10] The survival (or not absorption) probability on a patch has usually been considered equal to its reflectivity. An exception ....

Eric Veach and Leonidas J. Guibas, "Optimally combining sampling techniques for Monte Carlo rendering ", ACM Computer Graphics Proceedings, (ACM SIGGRAPH '95 Proceedings), pp. 419--428, 1995.

....on each level, it is a lot easier to compare the representations even if they do not use the same mesh as a support. We will further investigate some variants of the new hierarchical approach as indicated above, and explore the possibilities of combining it with bidirectional methods [VG95] and importance driven methods [SAS92] as these algorithms yield additional information on the necessary mesh resolution in different parts of the scene. We will also investigate the combination of the presented algorithm with discontinuity meshing algorithms similar to the method introduced by ....

Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for monte carlo rendering. Computer Graphics (SIGGRAPH '95 Proceedings), pages 419--428, August 1995.

....illumination contribution is computed for each transport path. These illumination contributions are combined to obtain an unbiased estimator for the radiance reaching the eye, taking into account the probability densities for generating the transport paths used. Concretely, the balance heuristic [22] is used to obtain the weights of the illumination contributions. Random walks (both light and eye rays) are traced computing interaction points within the media as in [14] Simple Absorption case) For the scattering direction computation the Schlick phase function is used. 5 Discussion 5.1 ....

E. Veach and L. Guibas. Optimally Combining Sampling Techniques for Monte Carlo Rendering. Computer Graphics Proc., Annual Conf. Series: SIGGRAPH '95, pp. 419--428, 1995.

....so any acceleration technique is important. The view dependent distribution ray tracing methods are the most used non diffuse techniques [1] 4] 5] 13] Optimal connection of rays starting from the view point and from the light sources is supported by increasingly efficient sampling techniques [15]. Other techniques using non diffuse radiosity methods need very large memory and are view independent [8] 9] 12] Non diffuse radiosity methods are efficient for diffuse and specular decomposition. In practice, they can be used as part of hybrid methods combined with appropriate final gathering ....

Veach E., Guibas, L. J., Optimally Combining Sampling Techniques for Monte Carlo Rendering, Computer Graphics (SIGGRAPH'95), 29, 419 - 428 (1995).

....the phenomena of incoherent emission and scattering at surfaces dominate the transport. These phenomena can be described in terms of geometric optics. In the following discussion several terms will be used to represent light and other quantities. The discussion follows previous treatments [7, 105, 113]. The quantities of interest are as follows. L(x; 0 ) radiance (brightness) measured in w= m 2 sr) Phi(x) irradiance (power) measured in w=m 2 . x; x 0 points in a geometric domain. n(x) surface normal vector at x. 0 (x) angle of incident light at x. 0 (x) ....

....0 is close to F . When p(x) is a poor match for f(x) the estimate will be poor. In image generation this results in defects that appear as pixels that are overly bright or excessively dark. Monte Carlo methods have been applied to the global illumination problem and have exhibited these defects [53, 103, 113, 124]. Despite these defects Monte Carlo methods are appealing because they avoid the most severe problems plaguing radiosity methods, namely explicit construction of the transport operator M and the problems associated with discretization. The transport operator is never explicitly formed, nor is the ....

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Veach, E. & Guibas, L. "Optimally combining sampling techniques for Monte Carlo rendering." Proc. ACM SIGGRAPH (1995) pp. 419-428.

....a given scene. An image is then a snapshot of this light field at the point of the virtual camera. All solution techniques in use today for solving the problem, which is described mathematically by the radiance or rendering equation [9] can be roughly classified into two groups: the Monte Carlo [9, 25, 18], and the finite element approach [6, 8, 16] Of course, a number of hybrid techniques that combine aspects of both approaches are also available [26, 3] 2.2 Stages The process of rendering can be divided into three distinct stages, which also roughly correspond to three different phases during ....

.... includes finite element algorithms like Progressive Refinement Radiosity [2] Galerkin Radiosity [29] Hierarchical and Wavelet Radiosity [8, 7] as well as three Wavelet Radiance algorithms [16, 1] From the class of Monte Carlo algorithms simple path tracing [9] and Bidirectional Estimators [25] have been implemented. Several hybrid techniques like Irradiance Gradients [28, 27] and Backward BeamTracing [3] are also available. This collection of algorithms is the basis for much of our current work in this area. a) Monte Carlo (b) Wavelet Radiosity Fig. 3. Two example images computed ....

E. Veach and L. J. Guibas. Optimally combining sampling techniques for monte carlo rendering. Computer Graphics (SIGGRAPH '95 Proceedings), pages 419--428, August 1995.

.... Refinement Radiosity [CCWG88] Galerkin Radiosity [Zat93] Hierarchical and Wavelet Radiosity [HSA91, GSCH93] as well as three Wavelet Radiance algorithms [SH94, CSSD94] From the class of Monte Carlo algorithms we have implemented simple path tracing [Kaj86] and Bidirectional Estimators [VG95]. Several hybrid techniques like Irradiance Gradients [WR88, WH92] and Backward Beam tracing [Col94] are also available. This collection of algorithms is the basis for much of our current work in this area, which focuses on better radiance and clustering algorithms [Sil94, SSS96] as well as on ....

Veach, E. and Guibas, L. J. Optimally combining sampling techniques for Monte Carlo rendering. Computer Graphics (SIGGRAPH '95 Proceedings) , pages 419--428, August 1995.

....from single best mixture component. Theorem 2 proves this assuming optimal control variate coefficients, and Section 4.1 gives mild conditions under which we can expect our sample coefficients to behave like the optimal ones. Section 5 presents the method of multiple importance sampling due to Veach Guibas (1995). Like defensive importance sampling, multiple importance sampling is motivated by the desire to pool importance sampling methods and get nearly the best performance. Section 6 is a simulation of Examples 1 and 2. The proposed hybrid methods are nearly best on both examples. Section 7 introduces a ....

....earlier in a dissertation (Hesterberg 1988) Our contribution is to extend the theoretical result to more general mixture samples, to describe conditions under which estimated coefficients approach the true ones, and to combine it with positivisation. Multiple importance sampling was proposed by Veach Guibas (1995) for problems in computer graphics. Our positivisation technique is based upon it. 2 Importance sampling This section reviews importance sampling and introduces our notation. Integrals without an explicit domain are assumed to be over D. For brevity, we sometimes omit the arguments of functions, ....

[Article contains additional citation context not shown here]

Veach, E. & Guibas, L. (1995), Optimally combining sampling techniques for Monte Carlo rendering, in `SIGGRAPH '95 Conference Proceedings', Addison-Wesley, pp. 419--428.

....each frame, and the polygons are drawn as is. In contrast, scenes containing glossy surfaces cannot yet be treated in an interactive context. To generate images with glossy surfaces, ray tracing based approaches are typically used, such as the RADIANCE system [27] or pathtracing algorithms (e.g. [11, 23]) Some finite element approaches have been presented, but can only treat trivial scenes (e.g. 13, 1] or require a second, ray casting pass to generate an image [3] Two approaches have been proposed which are capable of interactive viewing [16, 25] but they are limited in their capacity to ....

E. Veach and L. J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. Computer Graphics (SIGGRAPH '95), pages 419--428, aug 1995.

....of primitives full modelling will require. The speed of modern computers means that many more particles can be traced than can fit in main memory, however, so some kind of information condensation must be used. We also think some kind of importance or bidirectional approach is needed. Veach [17] has made it clear that sometimes working from the light source is good, and sometimes working from the view point is good. Fournier s [4] insight into how geometry and radiance can be interchanged might provide a way to utilize Rushmeier s geometric simplification ideas. We have also found that ....

Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for monte carlo rendering. In Robert Cook, editor, SIGGRAPH 95 Conference Proceedings, Annual Conference Series, pages 419--428. ACM SIGGRAPH, Addison Wesley, August 1995. held in Los Angeles, California, 06-11 August 1995.

....Similar problems occur when there are glossy surfaces, caustics, strong indirect lighting, etc. Several techniques have been proposed to sample these di#cult paths more e#ciently. One is bidirectional path tracing, developed independently by Lafortune and Willems [12, 13] and Veach and Guibas [24, 25]. These methods generate one subpath starting at a light source and another starting at the lens, then they consider all the paths obtained by joining every prefix of one subpath to every su#x of the other. This leads to a family of di#erent importance sampling techniques for paths, which are then ....

....starting at a light source and another starting at the lens, then they consider all the paths obtained by joining every prefix of one subpath to every su#x of the other. This leads to a family of di#erent importance sampling techniques for paths, which are then combined to minimize variance [25]. This can be an e#ective solution for certain kinds of indirect lighting problems. Another idea is to build an approximate representation oftheradianceinascene,whichisthenusedtomodify the directional sampling of the basic path tracing algorithm. This can be done with a particle tracing prepass ....

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Veach, E., and Guibas, L. J. Optimally combining sampling techniques for Monte Carlo rendering. In SIGGRAPH 95 Proceedings (Aug. 1995), Addison-Wesley, pp. 419--428.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Computer Graphics Proceedings, Annual Conference Series, 1995.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

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E. Veach and L. J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In SIGGRAPH 95 Conference Proceedings, pages 419--428, August 1995.

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E. Veach and L. J. Guibas. Optimally combining sampling techniques for monte carlo rendering. SIGGRAPH '95 Conference Proceedings, pp. 419--428, August 1995.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Computer Graphics Proceedings, Annual Conference Series, 1995.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

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Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, pages 419--428, August 1995.

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Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for monte carlo rendering. In Computer Graphics Proceedings, Annual Conference Series, 1995 (ACM SIGGRAPH '95 Proceedings), pages 419--428, 1995.

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E. Veach and L. J. Guibas, "Optimally Combining Sampling Techniques for Monte Carlo Rendering", in Computer Graphics Proceedings, Annual Conference Series, 1995 (ACM SIGGRAPH '95 Proceedings), pp. 419--428, (1995).

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994. 6

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E. Veach and L. J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In SIGGRAPH 95 Conference Proceedings, pages 419--428, August 1995.

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Veach E., L. Guibas L.; Optimally Combining Sampling Techniques for Monte Carlo Rendering; SIGGRAPH 95

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Eric Veach and Leonidas J. Guibas, "Optimally Combining Sampling Techniques for Monte Carlo Rendering ", Computer Graphics (ACM SIGGRAPH '95 Proceedings) , pp. 419--428 (1995).

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Eric Veach and Leonidas J. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Computer Graphics (SIGGRAPH '95 Proceedings), pages 419--428, 1995.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

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E. Veach and L. Guibas. Optimally combining sampling techniques for Monte Carlo rendering. In Rendering Techniques '94, pages 147--162, 1994.

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Eric Veach and Leonidas J. Guibas. "Optimally combining sampling techniques for monte carlo rendering." In Robert Cook, editor, SIGGRAPH 95 Conference Proceedings, Annual Conference Series, pages 419--428. ACM SIGGRAPH, 1995.

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