Spherical Harmonic Gradients

Transcription

1 X Y X Y Z[ \ ]_^a`cbedafhgeij]_k Max-Planck-Institut Informatik Computer Graphics Group Saarbrücken, Germany rx}8~8 vxr $p8~ƒv}8 ; U ƒ{-ˆˆ, $Š =,vvtrœƒt$ }8Žz $p{! o ž / ª«$ K š± 5 - ²J² ¹ -¾Q ;ž è èæqé êëäo K! K ¹ï 8¾ç É -¹! á ( ç K /! K K - â/ - K ûõ üò!ûüõ ý5þ/ÿ!!ó5õ ±ý5ûöó5 õ ó /õ öý5 Jøoüþ ýoûrñ úùwüõ!ýkò-ñ /œÿ Kâ/âB ½ r 2 àj ¹/ (¹ K Q o Á ãæqé êëäo K! / $ - «ª«ƒ Massachusetts Institute of Technology Computer Graphics Group Cambridge, USA ÁÀKÂ8ÃOÄrÅ8ÃRÆÇÉÈ½Ê Ë Ä,Ä,Ì Ï( ƒ (FG=HIÎ4-6 I-9-7 3JHI I Í +,RFG=HIÎ4-6 I-9-7 3JH-L-4 Ð Ñ8ÒƒÓ8Ó8Ô8ÕÖ8 Ø8ÙƒÓÚ ÛÜÖƒ Ý5ÛRÞƒÒ Radiance Transfer µ;ß; _àj á - šãâw äo K Using Spherical Harmonic Gradients Master Thesis in Computer Science Thomas Annen á/ Supervision: Prof. Dr. Hans-Peter Seidel Max-Planck-Institut Informatik Computer Graphics Group Saarbrücken, Germany Prof. Dr. Frédo Durand Dr. Jan Kautz Massachusetts Institute of Technology Computer Graphics Group Cambridge, USA Begin: February 24, 24 End: September 24, 24

2 Contents Problem Statement and Overview 2 Introduction 4 3 Previous Work 6 4 Theory of Spherical Harmonic Gradients 7 4. Review: Shading with Spherical Harmonics Spherical Harmonic Gradients Gradient Discussion Rendering with SH Gradients 5. SH Coefficient Extrapolation using the Gradient SH Coefficient Interpolation using the Gradient Results 3 6. Discussion Conclusion and Future Work 8 8 Appendix 9 i

3 Figures. Buddha shaded with PRT. (Courtesy of Sloan et al.) PRT for localized area lights Change in angular domain Visibility changes Max-Planck model under local light changes Bird model rendered with only a single sample (no gradient), single sample with gradient, and ground truth Incident radiance reconstruction Incident radiance reconstruction using SH Gradients Single-sampling versus multi-sampling and gradients versus no gradients RMS error of extrapolated radiance for the tooth example ii

4 Chapter Problem Statement and Overview Computer graphics has been advancing continuously during the last decades. Rendering quality has improved due to innovative new algorithms on one side, and on the other side hardware manufacturer producing more flexible and powerful graphics chips for the broad market. Latest NVIDIA and ATI Graphics Processing Units (GPUs) allow to create astonishing real-time effects for both the highend visual effects community and for consumer level applications, such as games. However, despite of all this great algorithmic and engineering advancement, it is still not possible to perform real-time global illumination without using massive computational power of many CPUs. For instance, Wald et al. [WKB 2, Wal4] presented a highly optimized ray-tracing system running on a cluster of PCs. Depending on the resolution and number of cluster nodes they are able to compute global illumination in real-time even for dynamic scenes. The reason why global illumination requires so any resources is not that the algorithms are so complicated themselves, but that their evaluation has to be done at a dense set of samples over the scene which is therefore very costly. As a consequence, most interactive and real-time applications use simplified local lighting models, even neglecting visibility, and assume simple material properties to achieve high frame rates. Whereas in reality, most materials have complex reflection properties, which are usually not easy to model. Furthermore, lighting conditions are often to complex to be modeled by a simple point or directional light source, as often used by local lighting models. In order to account for global effects (like soft-shadows, object interreflections, refraction, etc.) it is necessary to calculate an expensive shading integral at every visible surface point. Because of these barriers, displaying realistic images quickly is so far limited to afore-mentioned high-performance rendering systems. However, in 22 Sloan et al. [SKS2] presented a novel transport operator called Precomputed Radiance Transfer (PRT) that alleviates these restrictions if certain criteria hold. PRT is applicable to rendering static scenes illuminated by

5 infinitely distant low-frequency environments. Depending on the scene complexity, it yields interactive or even real-time frame rates while displaying soft shadows, interreflections, color bleeding, etc. Figure. shows the result of shading a buddha statue with PRT. This also illustrates that PRT is not limited to solid models (see Figure.(a)), but can also handle translucent objects (see Figure.(b)). This is possible because PRT does all expensive lighting computations offline, and only the reconstruction of the shading integral is done during rendering, which turns out to be a cheap operation. Key to this is to split the shading integral into two parts: the incident radiance and the energy response over the object (this includes the material BRDF and the visibility function), and to project the result for both in lower order Spherical Harmonic (SH) basis. At the end of the precomputation, incident radiance as well as the light transfer over the surfaces are compactly described in one lighting vector and a transfer matrix (for diffuse objects the matrix reduces to a vector) for each vertex respectively. What makes PRT feasable is that the reconstruction during runtime is very cheap, because it is independent of the complexity of the precomputation. The reconstruction itself only involves a simple per-vertex matrix-vector product for glossy surfaces and even only a dot product for objects with a Lambertian BRDF. This can easily be done utilizing a vertex shader on todays commodity graphics hardware. The dimension of the SH lighting vector and the transfer matrix is determined by the SH band L used to reconstruct a given signal defined over the spherical domain. (a) Solid (b) Translucent Figure.: Buddha shaded with PRT. (Courtesy of Sloan et al.) The reason for the infinitely distant lighting environment assumption is made because it permits to compute the SH lighting coefficients only for one point (usually the centroid of an object) and to reuse this vector at all vertices. This further improves execution speed of PRT algorithms, but also exhibits shading artifacts when lighting is more localized. Currently, the only solution to achieve convincing image quality for such scenarios of nearby area lights is an expensive multisampling scheme, where irradiance is sampled at many points distibuted over the object [SKS2]. In contrast to the costly multisampling, we suggest in this master thesis to 2

6 compute the Spherical Harmonic Gradients of the lighting coefficients for a certain sampling position and to extrapolate a new SH lighting coefficient vector at different spacial locations. The main goal of this thesis is to derive an analytic formula to compute this gradient and to show that SH Gradients yield much better image quality than multisampling alone. We will discuss the small additional overhead for the gradient computation and show some results comparing multisampling and our new method. We assume that the reader has a basic understanding of SHs as we will only review the concept of using them for shading briefly. Please refer to [Kau3, Ram2, Edm6] for a detailed description and overview of SHs. The structure of this Master Thesis is as follows: First, we give a brief introduction before we discuss most relevant related work and compare some previous methods to our algorithm. The previous work chapter is followed by a review of the basic concept of shading with SHs in order to provide the reader a basic background to follow the main concept. This leads us to the theoretical framework presenting an analytic formula to derive the gradient. After the mathmatical contribution is elaborated, we describe how the gradient is built in the shading process that was described earlier. As our method is a first order approximation, the next chapter discusses the limitations, presents an error analysis, and proposes how visual artifacts can be further minimized in problematic cases. Finally, we show results for dynamic scenes, give the timings and compare our technique to the standard SH lighting and the multisampling scheme. At the end we conclude our work and present an outlook for future work and ideas we want to investigate. Acknowledgements The presented master thesis is based on research conducted during my three month visit in the Computer Graphics Group at MIT Cambridge, USA. I owe special thanks to my supervisors Dr. Jan Kautz, Prof. Dr. Frédo Durand from MIT and Prof. Dr. Hans-Peter Seidel from MPI Informatik for their enormous help and advice during this time. Futhermore, I want to thank all my colleagues of both computer graphics groups at MIT and MPI Informatik for their support and valuable discussions. I especially want to thank the following people (alphabetical order): Volker Blanz, Stefan Brabec (now at TomTec), Eric Chan, Michael Goesele, Paul Green, Eugene Hsu, Hendrik Lensch (now at Stanford University), Wojciech Matusik (now at MERL), Karol Myszkowski, and Matthias Zwicker. 3

7 Chapter 2 Introduction In recent years, several methods have been proposed that permit the usage of global incident lighting in real-time rendering [SKS2, NRH3, SHHS3] (see Figure.). These approaches represent the incident radiance in spherical harmonics (SH). They, however, assume distant lighting. As a result, they are incapable of rendering scenes with mid-range lighting effects without visual error. Figure 2. depicts a hand model shaded using various means. Figure 2.(a) shows only one sample at the center. Figure 2.(b) illustrates the same model shaded using one sample and the analytical gradient, as we propose. The last image in Figure 2. shows a reference rendering where the incident radiance field is sampled per-vertex. To alleviate this problem, Sloan et al. [SKS2] sample the incident light field at multiple points over the object. While this was shown to be a possible solution, the high computational cost of multiple sampling leaves room for improvement. In this thesis, we propose to compute a first-order Taylor expansion of the spherical harmonic coefficients around a sampling point [AKDS4]. We show (a) Single sample (b) Analytic Gradient (c) Reference (per-vertex) Figure 2.: PRT for localized area lights. 4

8 how the gradient of the incident radiance (represented in SH) can be computed for little additional cost compared to the coefficients alone. A semi-analytical formula is introduced to calculate this gradient at run-time. The incident radiance can now be extrapolated to different positions around the original sample location. In case of multiple samples, the interpolation quality is greatly improved, thus requiring less samples. Extrapolation/interpolation can be easily performed in a vertex shader on the GPU. The extrapolated/interpolated incident radiance can then used together with any radiance transfer technique, e.g. [RH, SKS2]. 5

9 Chapter 3 Previous Work Our work uses the same framework as the recent precomputed radiance transfer (PRT) technique [SKS2]. This approach permits the illumination of objects with low-frequency incident lighting represented in spherical harmonics [Edm6]. The object can either be diffuse [SKS2] or glossy [KSS2, SHHS3, LK3]. Rendering can be performed in real-time, but requires precomputing the transfer for self-shadowing and other global illumination effects. PRT is limited to distant illumination, unless multiple incident radiance samples are taken and interpolated [SKS2]. We improve on this by computing the gradient of the spherical harmonics coefficients around a sample point. This enables extrapolation of the incident radiance to other points in space, which in turn can be used to improve interpolation of multiple samples. Our technique is similar in spirit to the irradiance gradients for ray-tracing by Ward and Heckbert [WH92]. They propose to compute gradients of the viewindependent irradiance at various sample points in order to improve interpolation. This involves a translational gradient (for the change of position) as well as rotational gradient (for the change of relative surface orientation). In contrast to their work, we choose to compute a gradient for the whole sphere of incident radiance (given in SH), independent of any incident surface orientation. Therefore we only need a translational gradient, but not the rotational gradient. However, our gradient is higher-dimensional, as we encode the directional radiance information through the vector of lighting coefficients. Essentially, we trade dependencies on the receiver orientation for a more comprehensive directional treatment of incident radiance. Furthermore, we focus on real-time rendering, whereas their application area was offline rendering. Irradiance gradients were generalized by Arvo [Arv94], who additionally accounted for occlusions. As Ward and Heckbert, we have to decided to neglect occlusion changes. This is motivated by implementation robustness and simplicity goals, and is justified by the use of low-frequency incident illumination, which is not very susceptible to visibility changes. 6

10 Chapter 4 Theory of Spherical Harmonic Gradients 4. Review: Shading with Spherical Harmonics Computing exit radiance at a diffuse surface is usually computed by the following integral: L p = Ω I p (ω) V p (ω) dω, (4.) V p (ω) = V p (ω)(n p ω), (4.2) where L p is the exitant radiance at point p, I p is the incident radiance at p, V p is the visibility function at p, V p is the cosine-weighted visbility, and n p is the normal at p. Integration is performed over all directions ω. This is an expensive integral, and it needs to be computed at every point p on an object. Recently, techniques [SKS2, SHHS3] were introduced to speed up the computation of this integral under the following assumptions: Lighting is assumed to be low-frequency, the object is static, and the illumination is distant (i.e., I p (ω) = I(ω) is the same for all p). If we now project I(ω) and V p (ω) into the spherical harmonics (SH) basis y i, we get two coefficient vectors I = (c, c,...) T and V p, and exitant radiance can then be easily computed with a dot-product [SKS2]: L p = I V p. (4.3) Multiple Samples. For close- and mid-range illumination, the incident radiance does change at every p. Hence, the above distant-illumination assumption is not valid. 7

11 da(x) x n(x) s da(x) x s n(x) p p d p Figure 4.: At a point p, we see point x in direction s := x p. The point x has a differential solid area da(x). When we move from p to p along d, the direction s changes, as well as the angle between s and n. To alleviate this problem, Sloan et al. [SKS2] propose to take multiple incident radiance samples at several p j and interpolate between the I pj. We also propose to use multiple samples in these cases, but increase interpolation quality using our gradient-based technique. 4.2 Spherical Harmonic Gradients Consider the configuration visualized in Figure 4.. A single sample of incident radiance is taken at point p. We want to estimate the incident radiance I p (ω) at a new point p = p + d, where d is simply the translation vector. More precisely, we want the coefficient vector I p. We approximate the coefficients I p = (c, c,...) T at point p with a firstorder Taylor expansion: c i = c i + ( c i d). (4.4) As we will show in the next section, the gradients c i can be computed efficiently and allows for fast rendering. Assumptions. Before we derive the gradient formulation, we introduce the assumptions we make. We neglect specular surfaces in the environment and assume that exitant radiance at a 3D point is independent of direction. This assumption is made only for the environment contribution, and the illuminated object can be specular. Furthermore, we assume that visibility does not change when we move away from p. That is, the set of visible points remains unchanged. Although this assumption almost never holds for real scenes, artifacts are likely to be negligible, 8

12 since we consider only low-frequency incident illumination. If visibility changes are dramatic and artifacts might occur, multiple samples and gradient-based interpolation can be used to circumvent potential problems (see Section 5.2) Gradient We need to compute the gradient of the coefficients c i. Projecting the incident radiance into spherical harmonics is done by integrating it against the SH basis functions: c i = y i (ω) I(ω) dω. (4.5) Ω Similar to the rendering equation, this formula can be written as an integral over the sphere of directions Ω or over visible scene surfaces S. For gradient computation, this choice will influence which terms of the integrand need to be derived and which ones are constant. Both approaches yield similar algebraic complexity, but different numerical integration. In the surface-based case, differential surface elements are assumed fixed and we need to derive the various angular terms (see Figure 4.). In contrast, in the angle integration, we assume that angles are fixed but the derivatives of radiance over the scene surfaces needs to be computed. The former approach is simpler because it makes a direct use of the environment-map-sampled representation of the surface elements and because the angular terms are simple to derive, as shown below. We thus rewrite this integral so that integration is performed over visible surfaces S. We first define the non-unit vector s = (x p), where x is a point on a visible surface, substituting it into Equation 4.5: c i = S y i ( s ) I(x) ds. (4.6) s The main difference is that the incident radiance now depends on the points x of the visible surfaces S. This is useful for taking the gradient, since the radiance leaving x remains the same, even when the sample location p moves to p. The measure ds can be rewritten as the differential surface area da at x weighted by the squared distance to that surface point and by the angle between the surface normal n(x) and the direction s towards p [CW93]: ds = da n(x) (s/ s ) s 2. (4.7) We explain how to compute da in practice in the next section. We then write the gradient of c i : ( c i = y i ( s ) (s/ s ) )n(x) I(x) da (4.8) S s s 2 9

13 Note, that we take the gradient in s instead of p, since it is easier to write. The only difference between the two gradients is the sign (when p moves along d, then s moves along d). Note also, that because we use the integration on surfaces, the incident radiance I(x) does not vary under translation of the incident point. To simplify notations we define the geometric term g(s) = n(x) (s/ s ) s 2 (4.9) and Equation 4.8 becomes ( c i = y i ( s ) S s ) g(s) I(x) da, (4.) which we expand using the derivative of a product to ( c i = y i ( s S s ) g(s) + y i( s ) s ) g(s) I(x) da. (4.) The gradients of the geometric term is simply: g(s) = n(x) (n(x) s) 3s. (4.2) s 3 s 5 The gradient of the spherical harmonics y i can be easily derived analytically. It is convenient to use the Cartesian formulation [VMK88], as we compute the translational gradient in Cartesian coordinates. The recursion on Legendre polynomials can also be exploited. We provide C code for these terms online. Note, that the integrand of Equation 4. is analytical. The actual integration is performed numerically, as some quantities (I(x), the normal n(x), and da) come from sampled representations (cube maps in our case) Discussion The gradient formulation in Equation 4. has the additional advantage that one can tabulate the translational gradient of the spherical harmonics for each texel location. By applying the chain-rule and factoring / s 2 outside this term, the integrand can be precomputed for unit directions ω and does not need to be recomputed. In this thesis we only show examples of objects with diffuse surfaces and precomputed radiance transfer. Glossy surfaces [SKS2, SHHS3] as well as non-shadowed objects [RH] can be incorporated the same way. Just the final dot-product (Equation 4.3) is replaced with a different operation.

14 Chapter 5 Rendering with SH Gradients 5. SH Coefficient Extrapolation using the Gradient Rendering an object using the SH gradient, involves the following steps. First, we pick a single sample location (usually the centroid of an object) and render a cube map containing exit radiances I(x) of the surrounding objects/emitters. Additionally, we read back the depth-buffer, as we need to know the distance r from p to x for each texel. In a separate pass we render the emitters again but with color-coded normals, which we also read back yielding n(x). We go over each texel t of the cube maps and analytically evaluate the integrand. The numerical integration over all texels provides the gradients c i. The term da can be computed from the read-back data as follows: da = dω t r 2 n(x) ω, (5.) where dω t is the solid angle of the texel t, ω is the unit direction from p towards x, and r is the distance from p to x. At each vertex of the object, we now compute the estimated coefficients: c i = c i + ( c i d). (5.2) These coefficients are then used to compute the final exitant radiance using the dot-product between the coefficients c i and the cosine-weighted visibility coefficient vector V p (see Equation 4.3). We currently compute the gradients on the CPU. The new coefficients c i as well as the final dot-product are computed in a vertex shader on the GPU.

15 5.2 SH Coefficient Interpolation using the Gradient If light is emitted from nearby or overlapping sources, a single extrapolated sample might not be sufficient due to parallax changes. Such a scenario is sketched in Figure 5.. If SH coefficients are computed only at object O i s center, then vertex v would not be shaded properly, because object O i+2 is becomes visible from this v. Shading for objects with larger extend requires interpolation between multiple sample points in order to achieve faithful results [WH92]. In such a case, we Object Oi+2 is visible from vertex v Object Oi exhibits shading artifacts due to occlusion Oi+2 Oi+ O i+2 v Oi Object Oi+ occludes Oi+2 Figure 5.: Due to visibility changes, a previously hidden object becomes visible. choose N sample points p j over the object (using Sloan et al. s method [SKS2]). The incident radiance coefficients c j i as well as the gradient c j i is computed for each sample point p j. At each vertex of the object, we compute the estimated coefficients as a weighted sum of all extrapolated samples: c i = j w j ( c j i + c j i d j ), (5.3) where w j is a weight based on the distance between the vertex position v and the sample point position p j. We use a simplified version of Ward and Heckbert s formula [WH92], which was also employed by Sloan et al. [SKS2]: w j = (/ v p j ) b i(/ v p i ) b. (5.4) If only a few samples are used, then rendering can be done in a shader on the GPU. Note, that the gradient-based interpolation scheme could also be used to improve interpolation of the irradiance volumes technique of Greger et al. [GSHG98]. 2

16 Chapter 6 Results For all the results, presented in this section we have used a 3 GHz PC, equipped with 2 GB RAM and an NVIDIA Quadro FX 3 (NV35-GL) graphics board. The viewport size for all renderings was The size of a single cube map face, into which we read back the necessary data from the frame buffer, was The strength of the gradient method can be illustrated by moving an object below a small and local area light, see Figure 6.. In this example, we compute the original incident lighting sample and the gradient only once at the center of the head in the middle image. We then move the head from left to right underneath the light. Shading is computed with our gradient method based on the single sample. Shading responds correctly to the change of location. Figure 6.: Max-Planck model moved underneath a local emitter from left to right. The incident lighting is only sampled once at the center of the head (middle image). Figure 6.2 shows a bird model lit by three different colored area light sources. It compares a single sample with the gradient-based technique and ground truth. The same comparison is made in Figure 2.. These comparisons highlights the quality improvement due to our method. One can see that our gradient method produces results very similar to the actual ground truth. 3

17 (a) Center sample (b) SH gradient (c) Ground truth Figure 6.2: Bird model rendered with only a single sample (no gradient), single sample with gradient, and ground truth. In Figure 6.3(a) we visualize a cubemap rendered at the center of the bird model, which represents incoming lighting (red, green, and yellow area lights). In Figure 6.3 (b), (c), and (d) we plot the reconstruction of the incident lighting using SH band L= 3, 4, and 5 respectively. In Figure 6.4, we show again the cubemap rendered at the center (see Figure 6.4(a)) and a sampling result at a different location (see Figure 6.4(b)), to (a) Reference cubemap (b) Reconstruction L=3 (c) Reconstruction L=4 (d) Reconstruction L=5 Figure 6.3: Incident radiance reconstruction. 4

18 (a) Object center (b) Translated by x = 5 (c) SH Gradients reconstruction (d) Reference reconstruction Figure 6.4: Incident radiance reconstruction using SH Gradients. which we want to extrapolate a new coefficient vector. Figure 6.4(c) visualizes the irradiance reconstruction at this location using a SH vector extrapolated by our gradient. As one can see it compares very well to the reference reconstruction shown in Figure 6.4(d), where the SH coefficients were computed at the new spatial location. In Table 6., we list the rendering times for the previous technique (no gradient) and our proposed technique (with gradient). Performance is very similar for both methods, but quality is increased by using our method. single with Model #vertices sample gradient Planck 25K Bird 32K Tooth 2.5K Table 6.: Rendering times in frames per second for different models and different techniques. In Figure 6.5, a tooth model is illuminated by partially overlapping emitters. Different renderings compare extrapolation of a single sample against interpolation of eight samples. In Figure 6.5(a) we see the standard method of using just a center sample with no additional gradient information. Whereas Figure 6.5(b) uses a single sample with a gradient which does similarly well as eight samples 5

19 (a) single sample (b) single gradient (c) ground truth (d) eight samples (e) eight samples (f) eight gradients Figure 6.5: Single-sampling versus multi-sampling and gradients versus no gradients. without gradients (see Figure 6.5(e)). Eventhough the single sample plus the gradient is close to the eight samples result, it still does not get close to the ground truth, shown in Figure 6.5(c). The only noticeable difference is the top of the tooth. Here visibility changes should make it mainly bluish, but the extrapolated single sample is purple. That is caused by the blue light source, which occludes the red one. In such cases where visibility changes we can circumvent visual artifacts, by adding gradient information to the multiple samples distributed over the tooth as depicted in Figure 6.5(d). As Figure 6.5(f) proofes, interpolation of eight samples with gradients produces virtually the same result as the reference image which is almost indistinguishable from the ground trouth. 6

20 6. Discussion In our experience, a single sample using the gradient is often sufficient. Only for very nearby emitters where nearby means closer than the distance from the sample position to the object s bounding box (empirically determined) we have found that one needs multiple samples. In these cases the /r 2 falloff is very important and cannot be modeled by our first-order approximation. In the case of overlapping emitters and occlusion changes, multiple samples are needed as well. We have only experimented with a fixed number of samples for now. For all the cases we have tried, up to 8 samples seem to be sufficient. Figure 6.6 depicts the RMS error between ground truth and extrapolated incident radiance from a single sample (tooth example, see Figure 6.5). The red curve includes occlusion changes, the green curve does not (blue blocker removed from scene) The error is shown for the red color channel only. The red error curve includes the blue emitter as a blocker, whereas the green error curve is computed with the blue emitter removed from the scene (no occlusion changes). As expected, the error is higher when occlusion changes occur, but not significantly RMS R occlusion RMS R no occlusion 4 35 RMS Error to Sample Figure 6.6: RMS error of extrapolated radiance for the tooth example. 7

21 Chapter 7 Conclusion and Future Work We have presented a method for efficient shading of objects that are lit with nearby area sources. We extrapolate/interpolate incident radiance based on a first-order approximation. To this end, we have derived an analytic formula for the gradient, which can be efficiently computed at run-time. Using this additional information, we achieve results close to a per-vertex reference while the additional cost is small. Simple extrapolation has limitations in terms of accuracy, namely when extrapolation is done at locations far from the initial sample position. Interpolation of multiple samples maintains good image quality in these cases. We would like to implement a robust and automatic scheme for selecting sample positions in dynamic scenes. New samples should be generated or removed based on the spatial relationship between objects. Furthermore, we would like to experiment with higher-order Taylor expansions. This should allow even better quality without taking multiple samples of the incident radiance. It would be interesting to find out at which order the computation of the Taylor expansion becomes more expensive than taking multiple samples. 8

22 Chapter 8 Appendix In this chapter we want to present 25 plots which illustrate how each single SH coefficient behaves, when it is computed along a line in space, which was starting from sampling position (p d) and expanding to (p+d), where d = (5,, ). As expected, higher order SH coefficients oscillate more and are therefore not well approximated using a st oder Taylor approximation. R of SH-C G of SH-C B of SH-C R of SH-C G of SH-C B of SH-C.5.5 SH Coefficient SH Coefficient R of SH-C 2 G of SH-C 2 B of SH-C 2 R of SH-C 3 G of SH-C 3 B of SH-C SH Coefficient 2 SH Coefficient

23 - R of SH-C 4 G of SH-C 4 B of SH-C 4 R of SH-C 5 G of SH-C 5 B of SH-C SH Coefficient 4 SH Coefficient R of SH-C 6 G of SH-C 6 B of SH-C 6 R of SH-C 7 G of SH-C 7 B of SH-C SH Coefficient 6 SH Coefficient R of SH-C 8 G of SH-C 8 B of SH-C 8 R of SH-C 9 G of SH-C 9 B of SH-C SH Coefficient 8 SH Coefficient R of SH-C G of SH-C B of SH-C R of SH-C G of SH-C B of SH-C.5.5 SH Coefficient SH Coefficient

24 - R of SH-C 2 G of SH-C 2 B of SH-C 2 R of SH-C 3 G of SH-C 3 B of SH-C SH Coefficient 2 SH Coefficient R of SH-C 4 G of SH-C 4 B of SH-C 4 R of SH-C 5 G of SH-C 5 B of SH-C SH Coefficient 4 SH Coefficient R of SH-C 6 G of SH-C 6 B of SH-C 6 R of SH-C 7 G of SH-C 7 B of SH-C SH Coefficient 6 SH Coefficient R of SH-C 8 G of SH-C 8 B of SH-C 8 R of SH-C 9 G of SH-C 9 B of SH-C SH Coefficient 8 SH Coefficient

25 - R of SH-C 2 G of SH-C 2 B of SH-C 2 R of SH-C 2 G of SH-C 2 B of SH-C SH Coefficient 2 SH Coefficient R of SH-C 22 G of SH-C 22 B of SH-C 22 R of SH-C 23 G of SH-C 23 B of SH-C SH Coefficient 22 SH Coefficient R of SH-C 24 G of SH-C 24 B of SH-C 24.5 SH Coefficient

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27 [Ram2] RAMAMOORTHI R.: A Signal-Processing Framework for Forward and Inverse Rendering. PhD thesis, Stanford University, August 22. [RH] RAMAMOORTHI R., HANRAHAN P.: An Efficient Representation for Irradiance Environment Maps. In Proceedings of ACM SIGGRAPH 2 (August 2), ACM Press, pp [SHHS3] SLOAN P.-P., HALL J., HART J., SNYDER J.: Clustered Principal Components for Precomputed Radiance Transfer. ACM Transactions on Graphics 22, 3 (23), [SKS2] [VMK88] [Wal4] [WH92] SLOAN P.-P., KAUTZ J., SNYDER J.: Precomputed Radiance Transfer for Real-Time Rendering in Dynamic, Low-Frequency Lighting Environments. ACM Transactions on Graphics 2, 3 (22), VARSHALOVICH D., MOKSALEV A., KHERSONSKII V.: Quantum Theory of Angular Momentum. World Scientific Publishing Co., Singapore, 988. WALD I.: Realtime Ray Tracing and Interactive Global Illumination. PhD thesis, Computer Graphics Group, Saarland University, 24. Available at wald/phd/. WARD G., HECKBERT P.: Irradiance Gradients. In Third Eurographics Workshop on Rendering (992), pp [WKB 2] WALD I., KOLLIG T., BENTHIN C., KELLER A., SLUSALLEK P.: Interactive global illumination. In 3th Eurographics Workshop on Rendering (June 22), pp

28 Eidesstattliche Erklärung Hiermit erkläre ich an Eides Statt, dass ich die vorliegende Masterarbeit selbständig und ohne fremde Hilfe verfasst habe. Ich habe dazu keine weiteren als die angeführten Hilfsmittel benutzt und die aus anderen Quellen entnommenen Stellen als solche gekennzeichnet. Saarbrücken, September 24, 24 Thomas Annen