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Jaroslav Křivánek

James Ferwerda

Kavita Bala

Effects of Global Illumination Approximations on Material

Appearance

Cornell University &

Charles University, Prague Rochester Institute of

Technology Cornell University

(2)

• Required for accurate appearance, but slow

2

Global illumination rendering

scene: Autodesk | rendering: Edgar Velázquez-Armendáriz

(3)

• Fast GI algorithms are inaccurate

3

Global illumination rendering

scene: Autodesk | rendering: Edgar Velázquez-Armendáriz

(4)

• VPL rendering (Instant Radiosity [Keller 1997] )

– Fast & popular

– Image artifacts & energy losses

• Impact of VPL rendering on visual fidelity – Systematic perceptual study

4

Overview

(5)

• Perceptually-based rendering

– Visible Difference Predictor

[Mitchell 87, Bolin and Meyer 95/98, Myszkowski 02, …]

– Illumination components

[Stokes et al. 04; Debattista et al 05]

– Higher-level processing

[Yee et al. 01, O’Sullivan et al. 04]

– Material appearance

[Pellacini et al. 00; Westlund and Meyer 01; Fleming et al. 03;

Khan et al. 06; Vangorp et al. 07/08]

– Visual Equivalence [Ramanarayanan et al. 2007]

5

Related work

(6)

• Visually equivalent =

– Same scene appearance – Visibly different

• Foundation of our work

– Apply visual equivalence to VPL rendering

6

Related work – Visual equivalence

(7)

• Based on Instant Radiosity [Keller 1997]

• Approximate indirect illumination by 1. Generate VPLs

7

Related work – VPL rendering

2. Render with VPLs

(8)

1. Interactive GI (≈ 1,000 VPLs)

[Wald et al. 02; Segovia et al. 07; Laine et al. 07;

Ritschel et al. 08; Dong et al. 09; Yu et al. 09 ; …]

2. Preview-quality (≈ 100,000 VPLs)

[Hašan et al. 07/09]

3. High-quality ( > 1,000,000 VPLs)

[Walter et al. 05/06]

8

Related work – VPL rendering

(9)

VPL rendering is fast, but…

9

(10)

Q: When do VPL methods produce high-fidelity renderings?

A: Systematic perceptual study

– trade-offs : VPL parameters vs. visual fidelity

10

VPLs for high-fidelity rendering

(11)

VPL count

clamping level

VPL Rendering Parameters

VPL count

cla m pin g le ve l

VPL rendering

parameters

(12)

VPL count

clamping level

VPL Rendering Parameters

VPL count

cla m pin g le ve l

VPL rendering

parameters

(13)

Space of rendering parameters

cla m pin g le ve l

VPL count

(14)

• How are the thresholds affected by

– shape complexity?

– material?

– illumination?

• Outline

– Experiment design – Results & validation – Applications

14

Psychophysical experiments

cla m pi ng le ve l

VPL count

(15)

15

Test objects – Shape complexity

• Same as [ Ramanarayanan et al. 07]

• G0 ... sphere

• G1-G3 ... bumpy spherical objects

(16)

16

Test objects – Materials

Ward-Dür BRDF:

(17)

17

Test objects – Materials

17

Ward-Dür BRDF:

(18)

18

Test objects – Materials

Ward-Dür BRDF:

(19)

Test objects – Materials

19

Ward-Dür BRDF:

(20)

• Art gallery café

• Studied object on a pedestal

20

Scene

(21)

• Different VPL rendering parameters

– 3 VPL counts

– 11 clamping levels

cla m pi ng le ve l

VPL count

21

Stimulus images

(22)

22

Stimulus images –VPL count

1,000 (1k) Interactive

rendering

100,000 (100k) Preview-quality

rendering

5,000,000 (5M) High-quality

rendering

(23)

• 11 levels from none to severe (C0 – C10)

23

Stimulus images – Clamping level

(24)

• Standard two-alternative forced choice method

• 480 trials, 12 participants 24

Experiment 1: Artifact visibility

cla m pi ng le ve l

VPL count

(25)

• Standard two-alternative forced choice method

• 520 trials, 14 participants 25

Experiment 2: Material change

VPL count

cla m pi ng le ve l

(26)

VPL count

cl am pi ng lev el

• Extract thresholds from subjects’ answers

– Standard 75% 2 AFC threshold criterion

Data analysis

26

1k 100k 5M

C0 C2 C4 C6 C8 C10

C1

C3

C5

C7

C9

(27)

27

Experiment results

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

(28)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

28

Trends – VPL count

(29)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

29

Trends – VPL count

(30)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

30

Trends – VPL count

(31)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

31

Trends – VPL count

(32)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

32

Trends – VPL count

(33)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

33

Trends – Material contrast gloss

(34)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

34

Trends – Shape complexity

(35)

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M 1k 100k 5M

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9 C0

C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

C0 C2 C4 C6 C8 C10

C1 C3 C5 C7 C9

35

Trends – Shape complexity

(36)

36

Trends – Illumination

indirect-only direct-and-indirect

• Does accurate direct illumination help preserve material appearance?

– No significant improvement measured

– Further investigation needed

(37)

1. Real-world geometry

– Trends confirmed

2. New material (diffuse)

– Most forgiving material

– Need more than 1k VPL to achieve equivalence

37

Validation

(38)

• Per-object clamping

• Luminance normalization

38

Applications

(39)

• Energy compensation [Kollig & Keller 04]

39

Application – Per-object clamping

+ =

clamping (VPL) compensation (PT)

+ = ≡

(40)

• Visual equivalence in space of VPL parameters

• Trends in equivalence

– VPL count

– Shape complexity

– Contrast gloss decrease

40

Summary of results

cla m pi ng le ve l

VPL count VPL count

(41)

• Adequacy of VPL methods for high fidelity rendering

• Take-home messages

– VPL methods produce equivalent renderings for a wide range of scene settings

– 1k VPLs used in interactive apps → no equivalence – Smooth metal & simple shape → no equivalence

• Solid perceptual foundation for VPL methods

41

Conclusions

(42)

• Model for visual equivalence in VPL rendering

• Effects of accurate direct illumination

– Ambiguity between artifacts and highlights

• Scalable & equivalent VPL methods

42

Future work

(43)

• Marie Curie Fellowship PIOF-GA-2008-221716

• NSF CAREER 0644175

• NSF CPA 0811680

• Intel

• Microsoft

• Experiment participants

43

Acknowledgements

(44)

Thank you

(45)

Additional Slides

(46)

46

Ambiguity: highlights vs. artifacts

(47)

• Re-introduce energy removed by clamping

47

match avg.

luminance

Apps: Luminance normalization

• Validation

– Can increase

equivalence

(48)

48

Apps: Luminance normalization

(49)

49

Trends – Material roughness

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In the very beginning of this paper, the reader was informed about the key questions this paper sought to answer: could the widely accepted theory concerning the oil price

Trefethen, [GMRES/CR and Arnoldi/Lanczos as matrix approximation problems, Iterative methods in numerical linear algebra (Copper Mountain Resort, CO, 1992), SIAM J.. Tichý, [On

Faber, [On worst-case GMRES, ideal GMRES, and the polynomial numerical hull of a Jordan block, Electronic Transactions on Numerical Analysis (ETNA), Volume 26, pp.. Tichý, [On

 We presented a scheme for automatic remapping of uniform BRDFs without access to the BRDF implementations.  The remapping procedure has been tested with both dielectric and

The fatigue limit (zero mean stress) of each material was obtained under alternating bending, alternating tor- sion, and five different combinations of alternating in-phase bending