scholarly journals Optimal weights for bidirectional ray tracing with photon maps while mixing 3 strategies

2021 ◽  
pp. 1-46
Author(s):  
Sergey Valentinovich Ershov ◽  
Mikhail Sergeevich Kopylov ◽  
Alexey Gennadievich Voloboy
Keyword(s):  
Closed Form ◽  
Ray Tracing ◽  
Integral Equations ◽  
Importance Sampling ◽  
Powerful Method ◽  
System Of Integral Equations ◽  
Multiple Importance Sampling ◽  
Optimal Weights ◽  
Algebraic Formula

Bidirectional stochastic ray tracing with photon maps is a powerful method but suffers from noise. It can be reduced by the Multiple Importance Sampling which combines results of different “strategies”. The “optimal weights” minimize the noise functional thus providing the best quality of the results. In the paper we derive and solve the system of integral equations that determine the optimal weights. It has several qualitative differences from the previously investigated case of mixing two strategies, but further increase of their number beyond 3 retains the qualitative features of the system. It can be solved in a closed form i.e. as an algebraic formula that include several integrals of the known functions that can be calculated in ray tracing.

scholarly journals Calculation of MIS Weights for Bidirectional Path Tracing with Photon Maps

2020 ◽  
Author(s):  
Sergey Ershov ◽  
Alexey Voloboy ◽  
Dmitriy Zhdanov ◽  
Andrey Zhdanov
Keyword(s):  
Integral Equation ◽  
Monte Carlo ◽  
Integral Operator ◽  
Closed Form ◽  
Ray Tracing ◽  
Importance Sampling ◽  
Multiple Importance Sampling ◽  
Path Tracing ◽  
Light Rays ◽  
Sphere Radius

A widely used method for noise reduction in Monte-Carlo ray tracing is combing different means of sampling, known as multiple importance sampling (MIS). For bi-directional Monte-Carlo ray tracing with photon maps (BDPM), the join paths are obtained by merging camera and light sub-paths, and since several light paths are checked again the same camera path, and vice versa, the join paths obtained are not statistically independent. Thus the noise in this method obeys laws different from those in simple classic Monte-Carlo with independent samples so the weights that minimize that noise must also be calculated differently. This paper drives that weights for the simplest case when we mix contribution from only two vertices of camera ray. It shows that the weights obey an integral equation which is qualitatively different from the well-known MIS formulae for uncorrelated samples. Besides that, even if forget the integral operator, the weights depend on the integration sphere radius and the number of light rays used. The integral equation is solved analytically in a closed form and it is demonstrated how to perform the necessary calculations in BDPM.

scholarly journals Direct calculation of the optimal weight for MIS

2020 ◽  
Author(s):  
Sergey Ershov ◽  
Alexey Voloboy ◽  
Dmitriy Zhdanov ◽  
Andrey Zhdanov ◽  
Vladimir Frolov
Keyword(s):  
Monte Carlo ◽  
Simple Model ◽  
Ray Tracing ◽  
Light Source ◽  
Importance Sampling ◽  
Direct Calculation ◽  
Multiple Importance Sampling ◽  
Lighting Simulation ◽  
Numerical Minimization ◽  
Realistic Images

A Monte-Carlo ray tracing is nowadays standard approach for lighting simulation and generation of realistic images. A widely used method for noise reduction in Monte-Carlo ray tracing is combing different means of sampling, known as Multiple Importance Sampling (MIS). For bi-directional Monte-Carlo ray tracing with photon maps (BDPM) the join paths are obtained by merging camera and light sub-paths. Since several light paths are checked against the same camera path and vice versa, the join paths obtained are not statistically independent. Thus the noise in this method does not obey the laws which are correct in simple classic Monte-Carlo with independent samples. And, correspondingly, the MIS weights that minimize that noise must also be calculated differently. In this paper we calculate these weights for a simple model scene directly minimizing the noise of calculation. This is a pure direct numerical minimization that does not involve any doubtful hypothesis or approximations. We show that the weights obtained are qualitatively different from those calculated from classic “balance heuristic” for Monte-Carlo with independent samples. They depend on the scene distance, but not only on scattering properties of the surfaces and the distribution of light source emission.

scholarly journals Drone Ground Impact Footprints with Importance Sampling: Estimation and Sensitivity Analysis

2021 ◽  
Vol 11 (9) ◽  
pp. 3871
Author(s):  
Jérôme Morio ◽  
Baptiste Levasseur ◽  
Sylvain Bertrand
Keyword(s):  
Sensitivity Analysis ◽  
Monte Carlo ◽  
Importance Sampling ◽  
Loss Of Control ◽  
Minimum Volume ◽  
Multiple Importance Sampling ◽  
Impact Position ◽  
Engine Failure ◽  
Main Engine ◽  
Ground Impact

This paper addresses the estimation of accurate extreme ground impact footprints and probabilistic maps due to a total loss of control of fixed-wing unmanned aerial vehicles after a main engine failure. In this paper, we focus on the ground impact footprints that contains 95%, 99% and 99.9% of the drone impacts. These regions are defined here with density minimum volume sets and may be estimated by Monte Carlo methods. As Monte Carlo approaches lead to an underestimation of extreme ground impact footprints, we consider in this article multiple importance sampling to evaluate them. Then, we perform a reliability oriented sensitivity analysis, to estimate the most influential uncertain parameters on the ground impact position. We show the results of these estimations on a realistic drone flight scenario.

scholarly journals Solvability of an infinite system of integral equations on the real half-axis

2020 ◽  
Vol 10 (1) ◽  
pp. 202-216
Author(s):  
Józef Banaś ◽  
Weronika Woś
Keyword(s):  
Integral Equations ◽  
Measure Of Noncompactness ◽  
Infinite System ◽  
Main Tool ◽  
Supremum Norm ◽  
Nonlinear Integral Equations ◽  
Measures Of Noncompactness ◽  
System Of Integral Equations ◽  
The Real ◽  
Half Axis

Abstract The aim of the paper is to investigate the solvability of an infinite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions defined, continuous and bounded on the real half-axis with values in the space l∞ consisting of real bounded sequences endowed with the standard supremum norm. The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of a measure of noncompactness in the mentioned sequence space l∞. An example illustrating our result will be included.

Constant-Sign Lp Solutions for a System of Integral Equations

2004 ◽  
Vol 46 (3-4) ◽  
pp. 195-219 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O’Regan ◽  
Patricia J. Y. Wong
Keyword(s):  
Integral Equations ◽  
Constant Sign ◽  
System Of Integral Equations ◽  
Lp Solutions

MGF-Based Effective Capacity for Generalized Fading Channels

2014 ◽  
Vol 519-520 ◽  
pp. 929-933 ◽  
Author(s):  
Zhe Ji ◽  
You Zheng Wang ◽  
Jian Hua Lu
Keyword(s):  
Closed Form ◽  
Fading Channels ◽  
Mathematical Expression ◽  
Moment Generating Function ◽  
Effective Capacity ◽  
Multiple Antenna ◽  
Method Of Moment ◽  
Single Antenna ◽  
Generalized Fading Channels

In this paper, we study the effective capacity (EC) which was proposed to measure the quality of service (QoS) for fading channels. A unified expression for the effective capacity based on the method of moment generating function (MGF) is proposed. The unified expression applies to various fading channels and is derived for both single antenna and multiple antenna diversity system. The mathematical expression is illustrated with Nakagami-m fading channels and closed form expressions are derived in this case. The simulation results verify the consistence of the closed-form expressions with numerical evaluations.

scholarly journals Simple and Exact Closed-Form Expression for Determination of a Penetrated Ray Path in a Ray Tracing

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hyun Wook Moon ◽  
Woojoong Kim ◽  
Sewoong Kwon ◽  
Jaeheung Kim ◽  
Young Joong Yoon
Keyword(s):  
Closed Form ◽  
Approximate Method ◽  
Ray Tracing ◽  
Conventional Method ◽  
Closed Form Expression ◽  
Indoor Environments ◽  
Straight Line ◽  
Ray Path ◽  
Ray Tracing Simulation

A simple and exact closed-form equation to determine a penetrated ray path in a ray tracing is proposed for an accurate channel prediction in indoor environments. Whereas the penetrated ray path in a conventional ray tracing is treated as a straight line without refraction, the proposed method is able to consider refraction through the wall in the penetrated ray path. Hence, it improves the accuracy in ray tracing simulation. To verify the validation of the proposed method, the simulated results of conventional method, approximate method, and proposed method are compared with the measured results. The comparison shows that the proposed method is in better agreement with the measured results than the conventional method and approximate method, especially in high frequency bands.

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