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## A weighted error metric and optimization method for antialiasing patterns (2006)

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Venue: | COMPUTER GRAPHICS FORUM |

Citations: | 11 - 1 self |

### Citations

467 | Distributed ray tracing
- Cook, Porter, et al.
- 1984
(Show Context)
Citation Context ... 2 uniform supersampling, 2 × 2 jittered, rotated grid, Quincunx, Flipquad and Fliptri. Sampling patterns The quality of uniform sampling patterns (Figure 1a) can be improved by jittering (Figure 1b) =-=[CPC84]-=-. N-rooks [Shi91] patterns divide a pixel into N×N grid and place one sample to each row and column. A special case with four samples per pixel is the rotated grid supersampling (Figure 1c). Kollig an... |

334 |
L.: Stochastic sampling in computer graphics
- COOK
- 1986
(Show Context)
Citation Context ... samples per pixels [AM03], and to high-quality offlinesLaine and Aila / A Weighted Error Metric and Optimization Method for Antialiasing Patterns patterns that consist of dozens of samples per pixel =-=[Coo86]-=-. As the number of samples per pixel increases, the importance of carefully choosing the positions and weights of the sampling points diminishes. Therefore, the presented metric is primarily useful fo... |

253 |
Principles of Digital Image Synthesis.
- GLASSNER
- 1994
(Show Context)
Citation Context ...use deterministic low-discrepancy sequences (e.g. [Hal70]) for generating the sampling patterns. Dobkin et al. [DEM96] use simulated annealing for creating low-discrepancy sampling patterns. Glassner =-=[Gla95]-=- points out that commonly used simple discrepancy metrics do not directly correlate with the quality of images generated with the patterns. Also, discrepancy does not take reconstruction filters into ... |

186 | Generating antialiased images at low sampling densities,” - Mitchell - 1987 |

130 | Anti aliasing Through Stochastic Sampling,"
- Dippe, Wold
- 1985
(Show Context)
Citation Context ...t lowdiscrepancy patterns that combine the strengths of jittered and N-rooks sampling. Poisson disk patterns have good spectral characteristics, and thus also desirable properties for image synthesis =-=[DW85]-=-. Molnar [Mol91] places unweighted sampling points so that the minimum distance of a Poisson disk process is locally modified according to the impulse response of a reconstruction filter. Hierarchical... |

102 | P.-M.: Fast hierarchical importance sampling with blue noise properties.
- OSTROMOUKHOV, DONOHUE, et al.
- 2004
(Show Context)
Citation Context ...the impulse response of a reconstruction filter. Hierarchical construction of blue noise sampling patterns have been studied by McCool and Fiume [MF92], Hiller et al. [HDK01], and Ostromoukhov et al. =-=[ODJ04]-=-. Poisson disk patterns are often advocated as theoretically optimal in the sense of signal processing. Nevertheless, Goss and Wu [GW00] observe that the spectra of Poisson disk patterns exhibit clear... |

77 | Geometric Discrepancy; An Illustrated Guide
- Matousek
- 1999
(Show Context)
Citation Context ...[Shi91] computes discrepancy by sampling the area with rectangles of different sizes and aspect ratios, and Dobkin et al. [DM93] measure discrepancy based on arbitrary edges through a pixel. Matousek =-=[Mat99]-=- offers a more complete discussion on different types of discrepancy. Quasi Monte Carlo methods [Kel01] typically use deterministic low-discrepancy sequences (e.g. [Hal70]) for generating the sampling... |

77 | Discrepancy as a quality measure for sample distributions. In Computer Graphics Forum:
- SHIRLEY
- 1991
(Show Context)
Citation Context ...uities are difficult to take into account. Analysis using discrepancy Discrepancy is a quantity that measures the ability of a sampling pattern to estimate areas of subregions inside a pixel. Shirley =-=[Shi91]-=- computes discrepancy by sampling the area with rectangles of different sizes and aspect ratios, and Dobkin et al. [DM93] measure discrepancy based on arbitrary edges through a pixel. Matousek [Mat99]... |

60 |
Hierarchical Poisson disk sampling distributions.
- MCCOOL, FIUME
- 1992
(Show Context)
Citation Context ... Poisson disk process is locally modified according to the impulse response of a reconstruction filter. Hierarchical construction of blue noise sampling patterns have been studied by McCool and Fiume =-=[MF92]-=-, Hiller et al. [HDK01], and Ostromoukhov et al. [ODJ04]. Poisson disk patterns are often advocated as theoretically optimal in the sense of signal processing. Nevertheless, Goss and Wu [GW00] observe... |

56 |
The aliasing problem in computer-generated shaded images
- CROW
- 1977
(Show Context)
Citation Context ... 2. Related Work The four major sources that contribute to aliasing in rendered images are discontinuity edges, shader undersampling, intersecting surfaces and objects falling between sampling points =-=[Cro77]-=-. Supersampling is a general technique that reduces all aliasing artifacts by increasing the sampling rate. Multisampling differs from supersampling in that the shader is not evaluated separately in a... |

52 | W.: Interleaved sampling
- KELLER, HEIDRICH
(Show Context)
Citation Context ... good set of samples is typically determined for only one pixel, and the rest of the sets are then obtained by mirroring along the axis of translation. This implicitly results in interleaved sampling =-=[KH01]-=-. This is a special case because in general s distinct sets of samples must be considered. The simplest pattern family with a larger reconstructionsLaine and Aila / A Weighted Error Metric and Optimiz... |

51 |
A retrospective and prospective survey of the Monte Carlo method.
- Halton
- 1970
(Show Context)
Citation Context ...es through a pixel. Matousek [Mat99] offers a more complete discussion on different types of discrepancy. Quasi Monte Carlo methods [Kel01] typically use deterministic low-discrepancy sequences (e.g. =-=[Hal70]-=-) for generating the sampling patterns. Dobkin et al. [DEM96] use simulated annealing for creating low-discrepancy sampling patterns. Glassner [Gla95] points out that commonly used simple discrepancy ... |

41 | Graphics for the Masses: A Hardware Rasterization Architecture for Mobile Phones.
- Akenine-Moller, Strom
- 2003
(Show Context)
Citation Context ...techniques use the same samples in the reconstruction of multiple output pixels. Quincunx [BTB91, NVI01] generates two samples per pixel and uses five samples for reconstruction (Figure 1d). Flipquad =-=[AMS03]-=- results in higher quality images, while also creating two samples per pixel and using only four samples for reconstruction (Figure 1e). Due to mirroring along the x- and y-axes there are four distinc... |

40 | A.: Tiled blue noise samples
- Hiller, Deussen, et al.
(Show Context)
Citation Context ...is locally modified according to the impulse response of a reconstruction filter. Hierarchical construction of blue noise sampling patterns have been studied by McCool and Fiume [MF92], Hiller et al. =-=[HDK01]-=-, and Ostromoukhov et al. [ODJ04]. Poisson disk patterns are often advocated as theoretically optimal in the sense of signal processing. Nevertheless, Goss and Wu [GW00] observe that the spectra of Po... |

40 | Efficient Multidimensional Sampling
- Kollig, Keller
(Show Context)
Citation Context ... [Shi91] patterns divide a pixel into N×N grid and place one sample to each row and column. A special case with four samples per pixel is the rotated grid supersampling (Figure 1c). Kollig and Keller =-=[KK02]-=- construct lowdiscrepancy patterns that combine the strengths of jittered and N-rooks sampling. Poisson disk patterns have good spectral characteristics, and thus also desirable properties for image s... |

32 | Computing the discrepancy with applications to supersampling patterns
- DP, Eppstein, et al.
(Show Context)
Citation Context ...iscussion on different types of discrepancy. Quasi Monte Carlo methods [Kel01] typically use deterministic low-discrepancy sequences (e.g. [Hal70]) for generating the sampling patterns. Dobkin et al. =-=[DEM96]-=- use simulated annealing for creating low-discrepancy sampling patterns. Glassner [Gla95] points out that commonly used simple discrepancy metrics do not directly correlate with the quality of images ... |

32 |
Reconstruction filters in computer-graphics.
- MITCHELL, NETRAVALI
- 1988
(Show Context)
Citation Context ...sampling strategies for higher-dimensional integrals, global light transport, soft shadows, and other such phenomena are outside the scope of this article. Psychovisual results Mitchell and Netravali =-=[MN88]-=- observe that the theoretically optimal sinc is not a good reconstruction filter in practice, and define a family of bi-cubic filters that are parameterized by two parameters. Optimal values for these... |

29 |
Spectral consequences of photoreceptor sampling in the rhesus retina,”
- Yellott
- 1983
(Show Context)
Citation Context ...[DW85, Coo86, Mit87, HDK01] have studied the spectral characteristics of sampling patterns. Human eye rarely aliases even though the density of cones in extrafoveal regions is relatively low. Yellott =-=[Yel83]-=- observes that the cones (sampling points) are organized into a Poisson disk distribution, i.e., the points are spread randomly so that no two points are closer to each other than a specified distance... |

20 |
Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples
- Chen, Allebach
- 1987
(Show Context)
Citation Context ...rs arising from reconstruction can be more significant than the ones resulting from sampling. Our method takes reconstruction filters explicitly into account. Other analysis methods Chen and Allebach =-=[CA87]-=- analyze sampling patterns for estimation of band-limited signals, without employing reconstruction filters. Their method is based on the mean square error between the reference signal and a sampled v... |

16 | Random-edge discrepancy of supersampling patterns. Graphic Interface
- Dobkin, Mitchell
- 1993
(Show Context)
Citation Context ...of a sampling pattern to estimate areas of subregions inside a pixel. Shirley [Shi91] computes discrepancy by sampling the area with rectangles of different sizes and aspect ratios, and Dobkin et al. =-=[DM93]-=- measure discrepancy based on arbitrary edges through a pixel. Matousek [Mat99] offers a more complete discussion on different types of discrepancy. Quasi Monte Carlo methods [Kel01] typically use det... |

13 |
Efficient Supersampling Antialiasing for High-performance Architectures
- Molnar
- 1991
(Show Context)
Citation Context ...y patterns that combine the strengths of jittered and N-rooks sampling. Poisson disk patterns have good spectral characteristics, and thus also desirable properties for image synthesis [DW85]. Molnar =-=[Mol91]-=- places unweighted sampling points so that the minimum distance of a Poisson disk process is locally modified according to the impulse response of a reconstruction filter. Hierarchical construction of... |

9 | The SAGE graphics architecture
- Deering, Naegle
- 2002
(Show Context)
Citation Context ...ity, whereas the ( 3 2 ,− 1 4 ) notch filter strongly suppresses postaliasing patterns but results in blurry images. The resulting filters have been recently implemented in high-end graphics hardware =-=[DN02]-=-. We use the ( 1 3 , 1 3 ) filter as the reference reconstruction filter in our test measurements in Section 6. Naiman [Nai98] observes that jaggedness is more disturbing in some orientations of edges... |

9 | Strictly deterministic sampling methods in computer graphics
- Keller
(Show Context)
Citation Context ... and Dobkin et al. [DM93] measure discrepancy based on arbitrary edges through a pixel. Matousek [Mat99] offers a more complete discussion on different types of discrepancy. Quasi Monte Carlo methods =-=[Kel01]-=- typically use deterministic low-discrepancy sequences (e.g. [Hal70]) for generating the sampling patterns. Dobkin et al. [DEM96] use simulated annealing for creating low-discrepancy sampling patterns... |

7 |
C.: Jagged edges: when is filtering needed
- Naiman
- 1998
(Show Context)
Citation Context ...aring a given sampling pattern to an arbitrary reference reconstruction filter. We take into account the fact that the observed amount of aliasing depends on the screen-space slope of a sharp feature =-=[Nai98]-=-. We also present an efficient optimization method for generating patterns with good characteristics. The analysis is equally applicable to modern hardware-friendly sample sharing patterns that take a... |

4 |
Low sampling densities using a psychovisual approach. Eurographics ’91
- Bouatouch, Bouville, et al.
(Show Context)
Citation Context ...tions of edges or features. He provides a table of psychovisual acuities for each direction. For example, almost vertical or horizontal features require the most antialiasing efforts. Bouville et al. =-=[BTB91]-=- utilize psychovisual results that resemble Naiman’s acuity factors. They argue that sampling density can be halved by replacing a regular sampling grid with a Quincunx pattern (Figure 1d). Spectral a... |

3 |
An extremely inexpensive multisampling scheme
- Akenine-Möller
- 2003
(Show Context)
Citation Context ...timization method for generating patterns with good characteristics. The analysis is equally applicable to modern hardware-friendly sample sharing patterns that take as few as 1.25 samples per pixels =-=[AM03]-=-, and to high-quality offlinesLaine and Aila / A Weighted Error Metric and Optimization Method for Antialiasing Patterns patterns that consist of dozens of samples per pixel [Coo86]. As the number of ... |

3 |
Study of Supersampling Methods for Computer Graphics Hardware Antialiasing
- Goss, Wu
- 1999
(Show Context)
Citation Context ...d Fiume [MF92], Hiller et al. [HDK01], and Ostromoukhov et al. [ODJ04]. Poisson disk patterns are often advocated as theoretically optimal in the sense of signal processing. Nevertheless, Goss and Wu =-=[GW00]-=- observe that the spectra of Poisson disk patterns exhibit clearly visible banding. Sample sharing techniques use the same samples in the reconstruction of multiple output pixels. Quincunx [BTB91, NVI... |

1 | S.: A Family of Inexpensive Sampling Schemes
- HASSELGREN, AKENINE-MÖLLER, et al.
- 2005
(Show Context)
Citation Context ...he current choices are the best ones. The L∞ norm gives perfectly plausible results in most cases, but some extremely low cost patterns such as Fliptri currently rank higher than they probably should =-=[HAML05]-=-. This is because only the maximum angular error is taken into account, and the high average error is disregarded. This could perhaps be solved by employing Lp norm with 2 < p < ∞ in E2. Futhermore, i... |

1 |
Comments on "Stochastic Sampling in Computer Graphics
- PAVLIDIS
- 1990
(Show Context)
Citation Context ...cs do not directly correlate with the quality of images generated with the patterns. Also, discrepancy does not take reconstruction filters into account. In a comment on Cook’s work [Coo86], Pavlidis =-=[Pav90]-=- emphasizes that errors arising from reconstruction can be more significant than the ones resulting from sampling. Our method takes reconstruction filters explicitly into account. Other analysis metho... |