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 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 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 Optimization of Illumination through Windows for MCRT

2019 ◽  
Author(s):  
Ильдар Валиев ◽  
Ildar Valiev ◽  
Дмитрий Жданов ◽  
Dmitriy Zhdanov ◽  
Сергей Ершов ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Ray Tracing ◽  
Light Source ◽  
Light Sources ◽  
Camera Image ◽  
Accelerate Convergence ◽  
Run Time ◽  
The Given

The paper presents an improvement of Monte-Carlo ray tracing which changes ray emission from a light source to accelerate convergence i.e. reduce the noise remained after the given run time. It is mainly intended for interior scenes illuminated from outside (e.g. skylight) through windows or other holes. The rays from light sources are generated so that they are directed to these windows. In other words, the number of rays is increased for directions that contribute to the camera image. It is shown that the proposed method allows calculating image with desirable quality several times faster.

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.

Importance Sampling in Monte Carlo Ray Tracing Solutions Applied to Radiation in Hypersonic Entry Flows

2011 ◽  
Author(s):  
Andrew Feldick ◽  
Michael F. Modest
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Probability Distribution ◽  
Ray Tracing ◽  
Importance Sampling ◽  
Wavelength Selection ◽  
Weighting Functions ◽  
Statistical Precision

A method for the application of importance sampling to wavelength selection in ray tracing Monte Carlo solvers is introduced. In this method individual lines are treated independently, and probability distribution and weighting functions are applied to individual lines. The method reduces the number of photons required to reach the same statistical precision as the standard Monte Carlo method.

scholarly journals Stochastic Order and Generalized Weighted Mean Invariance

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 662
Author(s):  
Mateu Sbert ◽  
Jordi Poch ◽  
Shuning Chen ◽  
Víctor Elvira
Keyword(s):  
Monte Carlo ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Arithmetic Mean ◽  
Stochastic Orders ◽  
Arithmetic Means ◽  
Increasing Convex Order ◽  
Monte Carlo Techniques ◽  
Multiple Importance Sampling ◽  
Invariance Properties

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

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