## A frequency analysis of light transport (2005)

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Citations: | 74 - 11 self |

### BibTeX

@MISC{Durand05afrequency,

author = {Frédo Durand and Nicolas Holzschuch and Cyril Soler and Eric Chan and François X. Sillion},

title = {A frequency analysis of light transport},

year = {2005}

}

### Years of Citing Articles

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### Abstract

We present a signal-processing framework for light transport. We study the frequency content of radiance and how it is altered by phenomena such as shading, occlusion, and transport. This extends previous work that considered either spatial or angular dimensions, and it offers a comprehensive treatment of both space and angle. We show that occlusion, a multiplication in the primal, amounts in the Fourier domain to a convolution by the spectrum of the blocker. Propagation corresponds to a shear in the space-angle frequency domain, while reflection on curved objects performs a different shear along the angular frequency axis. As shown by previous work, reflection is a convolution in the primal and therefore a multiplication in the Fourier domain. Our work shows how the spatial components of lighting are affected by this angular convolution. Our framework predicts the characteristics of interactions such as caustics and the disappearance of the shadows of small features. Predictions on the frequency content can then be used to control sampling rates for rendering. Other potential applications include precomputed radiance transfer and inverse rendering.

### Citations

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Citation Context ...ruction that exploits the higher-resolution depth and normal to reconstruct the shading, in the spirit of McCool’s filtering of Monte-Carlo ray tracing outputs [1999] but based on a bilateral filter [=-=Tomasi and Manduchi 1998-=-]. As demonstrated in Fig. 17, this results in a smooth reconstruction where needed and on sharp silhouettes. The spatial width of the bilateral filter is scaled according to the bandwidth prediction.... |

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Citation Context ...rvo 1994], and Hessians [Holzschuch and Sillion 1998], but only provide local information, which cannot easily be used for sampling control. Fourier analysis has also been extensively used in optics [=-=Goodman 1996-=-], but in the context of wave optics where phase and interferences are crucial. In contrast, we consider geometric optics and characterize frequency content in the visible spatial frequencies. The var... |

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Citation Context ...velets and frequency bases Wavelets and spherical harmonics have been used extensively as basis functions for lighting simulation [Gortler et al. 1993; Keller 2001] or pre-computed radiance transfer [=-=Sloan et al. 2002-=-; Ramamoorthi and Hanrahan 2002]. They are typically used in a data-driven manner and in the context of projection methods, where an oracle helps in the selection of the relevant components based on t... |

338 | Lambertian reflectance and linear subspaces
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Citation Context ...cal illumination has been characterized in terms of convolution and it was shown that the outgoing radiance is band-limited by the BRDF [Ramamoorthi and Hanrahan 2001b; Ramamoorthi and Hanrahan 2004; =-=Basri and Jacobs 2003-=-]. However the lighting is assumed to come from infinity and occlusion is ignored. Frolova et al. [2004] explored spatial lighting variations, but only for convex diffuse objects. We build on these ap... |

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Citation Context ...age using the same reconstruction algorithm as for the final image, which we describe shortly. The blocker criterion is shown Fig. 15. It is similar to Ward et al.’s criterion for irradiance caching [=-=Ward et al. 1988-=-], but expressing it in a unified frequency framework allows us to combine it with other bandwidth considerations such as BRDF roughness. Algorithm and image reconstruction Our proof-of-concept comput... |

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Citation Context ...ompute these 800 × 500 image. Note how our approach better captures the sharp detail in the shiny dinosaur’s head and feet. The criteria and sampling are shown in Fig. 16. Images rendered using PBRT [=-=Pharr and Humphreys 2004-=-] Compute visibility at full resolution Use finite-differences for curvature criterion Compute harmonic blocker distance for sparse samples Perform bilateral reconstruction Compute B’ based on blocker... |

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Citation Context ...yszkowski 1998]. Finally we note that the Fourier basis can separate different phenomena and thus facilitate inverse lighting [Ramamoorthi and Hanrahan 2001b; Basri and Jacobs 2003] depth from focus [=-=Pentland 1987-=-] and shape from texture [Malik and Rosenholtz 1997]. ℓR 2D light field (2D) around ray R x spatial dimension (distance to central ray) v directional dimension in 2-plane parameterization θ directiona... |

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Citation Context ...cular effects. Local illumination as a convolution Recently, local illumination has been characterized in terms of convolution and it was shown that the outgoing radiance is band-limited by the BRDF [=-=Ramamoorthi and Hanrahan 2001-=-b; Ramamoorthi and Hanrahan 2004; Basri and Jacobs 2003]. However the lighting is assumed to come from infinity and occlusion is ignored. Frolova et al. [2004] explored spatial lighting variations, bu... |

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Citation Context ...orresponds to the incoming radiance in the frame of the local normal: the frame varies for each point. In a sense, we have unrolled the local surface and warped the space of light ray in the process [=-=Wood et al. 2000-=-]. The direction of the shear depends on the sign of the curvature (concave vs. convex). Step 3: Cosine term and differential irradiance In the local frame of each point, we compute differential irrad... |

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Citation Context ...lex blockers at an arbitrary distance of the blocker and receiver. Wavelets and frequency bases Wavelets and spherical harmonics have been used extensively as basis functions for lighting simulation [=-=Gortler et al. 1993-=-; Keller 2001] or pre-computed radiance transfer [Sloan et al. 2002; Ramamoorthi and Hanrahan 2002]. They are typically used in a data-driven manner and in the context of projection methods, where an ... |

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Citation Context ...xture pre-filtering based on a first-order Taylor expansion of the perspective transform. The effect of perspective is also studied in the contexts of holography and light field sampling [Halle 1994; =-=Isaksen et al. 2000-=-; Chai et al. 2000; Stewart et al. 2003], mostly ignoring visibility and specular effects. Local illumination as a convolution Recently, local illumination has been characterized in terms of convoluti... |

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Citation Context ...re filtering [Igehy 1999]. Chen and Arvo use perturbation theory to exploit ray coherence [2000]. Authors have also exploited on-the-fly the frequency content of the image to make better use of rays [=-=Bolin and Meyer 1998-=-; Myszkowski 1998; Keller 2001]. Our work is complementary and provides a framework for frequency-content prediction. Illumination differentials have been used to derive error bounds on radiance varia... |

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Citation Context ...er 2001]. Our work is complementary and provides a framework for frequency-content prediction. Illumination differentials have been used to derive error bounds on radiance variations (e.g. gradients [=-=Ward and Heckbert 1992-=-; Annen et al. 2004], Jacobians [Arvo 1994], and Hessians [Holzschuch and Sillion 1998], but only provide local information, which cannot easily be used for sampling control. Fourier analysis has also... |

109 | Sillion. Fast calculation of soft shadow textures using convolution
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Citation Context ...ss the cosine term for BRDFs such as Phong. On convolution It might come as a surprise that two phenomena that have been expressed by previous work as convolutions in the primary space, soft shadows [=-=Soler and Sillion 1998-=-] and the cosine term [Ramamoorthi and Hanrahan 2004] correspond in our framework to convolutions in the frequency domain. We show here that our formulation in fact extends these previous work and tha... |

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Citation Context ...ize frequency content in the visible spatial frequencies. The varying contrast sensitivity of humans to these spatial frequencies can be exploited for efficient rendering, e.g. [Bolin and Meyer 1995; =-=Ferwerda et al. 1997-=-; Bolin and Meyer 1998; Myszkowski 1998]. Finally we note that the Fourier basis can separate different phenomena and thus facilitate inverse lighting [Ramamoorthi and Hanrahan 2001b; Basri and Jacobs... |

90 | Frequency space environment map rendering - Ramamoorthi, Hanrahan - 2002 |

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Citation Context ...he Fourier basis can separate different phenomena and thus facilitate inverse lighting [Ramamoorthi and Hanrahan 2001b; Basri and Jacobs 2003] depth from focus [Pentland 1987] and shape from texture [=-=Malik and Rosenholtz 1997-=-]. ℓR 2D light field (2D) around ray R x spatial dimension (distance to central ray) v directional dimension in 2-plane parameterization θ directional dimension in plane-sphere paramerization ̂f Fouri... |

78 | Image-based reconstruction of spatially varying materials
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Citation Context ...f the spectrum of ρ is separable (texture mapping), then the spatially-varying BRDF case is a multiplication followed by a convolution. The special case of a a spatially-varying combination of BRDFs [=-=Lensch et al. 2001-=-] can be handled more simply as the superposition of multiple BRDFs with weights encoded as textures. 5 Extension to 3D We now show how our framework extends to 3D scenes. 5.1 Light-field parameteriza... |

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68 |
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Citation Context ... of additional local shading effects such as refraction, bumpmapping, and local shadowing [Ramamoorthi et al. 2004]. We hope to study the frequency cutoff for micro, meso, and macro-geometry effects [=-=Becker and Max 1993-=-]. The study of participating media is promising given the ability of Fourier analysis to model differential equations. The spectral analysis of light interaction in a full scene is another challengin... |

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Citation Context ...of concept and that further development is necessary to make it fully general, as we discuss below. We are motivated by the rendering of glossy materials, which despite effective recent developments [=-=Lawrence et al. 2004-=-] remains computationally expensive. We observe, however, that glossy objects appear blurry, so it should be possible to reduce the image sampling rate. Our framework permits a quantitative expression... |

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Citation Context ...e problems. Ray footprint A number of techniques use notions related to bandwidth in a ray’s neighborhood and propagate a footprint for adaptive refinement [Shinya et al. 1987] and texture filtering [=-=Igehy 1999-=-]. Chen and Arvo use perturbation theory to exploit ray coherence [2000]. Authors have also exploited on-the-fly the frequency content of the image to make better use of rays [Bolin and Meyer 1998; My... |

53 | The irradiance Jacobian for partially occluded polyhedral sources
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Citation Context ...ework for frequency-content prediction. Illumination differentials have been used to derive error bounds on radiance variations (e.g. gradients [Ward and Heckbert 1992; Annen et al. 2004], Jacobians [=-=Arvo 1994-=-], and Hessians [Holzschuch and Sillion 1998], but only provide local information, which cannot easily be used for sampling control. Fourier analysis has also been extensively used in optics [Goodman ... |

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48 |
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Citation Context ...c optics and characterize frequency content in the visible spatial frequencies. The varying contrast sensitivity of humans to these spatial frequencies can be exploited for efficient rendering, e.g. [=-=Bolin and Meyer 1995-=-; Ferwerda et al. 1997; Bolin and Meyer 1998; Myszkowski 1998]. Finally we note that the Fourier basis can separate different phenomena and thus facilitate inverse lighting [Ramamoorthi and Hanrahan 2... |

46 |
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Citation Context ...ight paths can exhibit an infinite number of bounces (i.e. in terms of signal processing, the system has dense feedback). To address the above challenges, we focus on the neighborhood of light paths [=-=Shinya et al. 1987-=-]. This restriction to local properties is both a price to pay and a fundamental difficulty with the problem we study: characteristics such as reflectance or presence and size of blockers are non-stat... |

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Citation Context ...he intersection v with a parallel plane at unit distance, expressed in the local frame of x (Fig. 3-a). In the plane-sphere parameterization, we use the angle θ with the central direction (Fig. 3-b) [=-=Camahort et al. 1998-=-]. These two parameterizations are linked by v = tan θ and are equivalent around the origin thanks to a linearization of the tangent. We study the Fourier spectrum of the radiance field ℓR, which we d... |

37 | Holographic stereograms as discrete imaging systems
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Citation Context ...width for texture pre-filtering based on a first-order Taylor expansion of the perspective transform. The effect of perspective is also studied in the contexts of holography and light field sampling [=-=Halle 1994-=-; Isaksen et al. 2000; Chai et al. 2000; Stewart et al. 2003], mostly ignoring visibility and specular effects. Local illumination as a convolution Recently, local illumination has been characterized ... |

36 | Feature-based control of visibility error: A multi-resolution clustering algorithm for global illumination
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Citation Context ...e selection of the relevant components based on the local frequency characteristics of radiance. Refinement criteria for multiresolution calculations often implicitly rely on frequency decomposition [=-=Sillion and Drettakis 1995-=-]. In our framework we study the frequency effect of the equations of light transport in the spirit of linear systems, and obtain a more explicit characterization of frequency effects. Our results on ... |

34 | A signal-processing framework for reflection
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Citation Context ...n as a convolution Recently, local illumination has been characterized in terms of convolution and it was shown that the outgoing radiance is band-limited by the BRDF [Ramamoorthi and Hanrahan 2001b; =-=Ramamoorthi and Hanrahan 2004-=-; Basri and Jacobs 2003]. However the lighting is assumed to come from infinity and occlusion is ignored. Frolova et al. [2004] explored spatial lighting variations, but only for convex diffuse object... |

28 | Theory and application of specular path perturbation
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(Show Context)
Citation Context ...er to extend these analyzes to arbitrary input signals, and in particular to non-Lambertian objects and occlusion. Ray footprint Approaches based on ray differentials [Shinya et al. 1987; Igehy 1999; =-=Chen and Arvo 2000-=-] capture the shear transforms due to transport and curvature, and our first-order Taylor expansion for curvature corresponds to the same differentials. The approach by Igehy [1999] only uses 2D deriv... |

26 |
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Citation Context ...mplementary and provides a framework for frequency-content prediction. Illumination differentials have been used to derive error bounds on radiance variations (e.g. gradients [Ward and Heckbert 1992; =-=Annen et al. 2004-=-], Jacobians [Arvo 1994], and Hessians [Holzschuch and Sillion 1998], but only provide local information, which cannot easily be used for sampling control. Fourier analysis has also been extensively u... |

24 | Fourier theory for cast shadows
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Citation Context ...ray tracer according to frequency predictions. Future work On the theory side, we are working on the analysis of additional local shading effects such as refraction, bumpmapping, and local shadowing [=-=Ramamoorthi et al. 2004-=-]. We hope to study the frequency cutoff for micro, meso, and macro-geometry effects [Becker and Max 1993]. The study of participating media is promising given the ability of Fourier analysis to model... |

24 | A new reconstruction filter for undersampled light fields
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(Show Context)
Citation Context ...der Taylor expansion of the perspective transform. The effect of perspective is also studied in the contexts of holography and light field sampling [Halle 1994; Isaksen et al. 2000; Chai et al. 2000; =-=Stewart et al. 2003-=-], mostly ignoring visibility and specular effects. Local illumination as a convolution Recently, local illumination has been characterized in terms of convolution and it was shown that the outgoing r... |

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6 | An exhaustive errorbounding algorithm for hierarchical radiosity
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Citation Context ...t prediction. Illumination differentials have been used to derive error bounds on radiance variations (e.g. gradients [Ward and Heckbert 1992; Annen et al. 2004], Jacobians [Arvo 1994], and Hessians [=-=Holzschuch and Sillion 1998-=-], but only provide local information, which cannot easily be used for sampling control. Fourier analysis has also been extensively used in optics [Goodman 1996], but in the context of wave optics whe... |