KEY QUESTIONS: - "What's the difference between ray tracing and rasterization?" [02:57] - "What function are we integrating in Monte Carlo ray tracing?" [09:25] - "Which probability distribution has higher variance?" [19:03] - "Why is the image noisy despite our setup?" [47:12] - "How can we reduce that noise?" [48:04] - "When do we stop the path tracing procedure?" [1:07:49] NOTABLE QUOTES: - "Monte Carlo rendering is gonna let us render whatever we like" [01:43] - "The cool thing about Monte Carlo is it works in basically any scenario you can throw at it" [12:51] - "As you do more and more experiments, the variance is going to zero" [23:50] - "Important sampling is really really important" [54:32] - "Most of the time we're just getting a little bit brighter, it's not really helping our image" [1:08:59] - "Because the values we're getting out of tracing each path are unbiased, the average value over many many paths will converge to the correct value" [1:14:46] TIMESTAMPS: [00:00] - Introduction to Monte Carlo Rendering • Overview of photorealistic image generation techniques • Discussion of combining multiple graphics concepts • Comparison with previously covered rasterization approach • Introduction to quality vs speed tradeoffs in rendering [02:57] - Ray Tracing vs Rasterization Comparison • Analysis of ordering differences in processing samples • Explanation of primitive vs sample iteration approaches • Demonstration of illumination model differences • Comparison of shadow and reflection capabilities • Discussion of rendering quality tradeoffs [09:25] - Monte Carlo Integration Fundamentals • Introduction to integration concept for rendering • Explanation of path tracing and sample meanings • Overview of rendering equation components • Discussion of light path integration approaches • Connection to previous numerical integration concepts [14:52] - Understanding Expected Values • Definition and intuition behind expected values • Practical examples using coin flips • Explanation of linearity properties • Discussion of probability fundamentals • Application to rendering calculations [19:03] - Variance in Random Variables • Detailed explanation of variance concept • Analysis of probability distributions • Comparison of different variance examples • Discussion of practical implications • Connection to rendering quality [22:32] - Law of Large Numbers and Applications • Explanation of convergence properties • Practical example using Pi estimation • Discussion of sample count impact • Analysis of variance reduction • Application to Monte Carlo rendering [30:59] - Importance Sampling Concepts • Introduction to biased sampling approaches • Analysis of probability distribution choices • Discussion of BRDF sampling strategies • Explanation of image-based lighting sampling • Optimization techniques for Monte Carlo rendering [35:40] - Exploring Direct Lighting Examples • Introduction to visibility function concept • Discussion of irradiance calculation for ground points • Analysis of direct lighting computation approaches • Implementation of uniform sampling strategy • Explanation of hemisphere sampling techniques [39:48] - Understanding Hemisphere Sampling • Detailed exploration of rejection sampling • Application to hemisphere point selection • Discussion of uniform sampling methods • Practical implementation considerations • Connection to direct lighting calculations [46:51] - Analyzing Noise and Variance • Explanation of noise in rendered images • Discussion of random direction sampling impact • Analysis of variance in different approaches • Introduction to sampling efficiency concepts • Exploration of noise reduction techniques [54:55] - Evaluating Monte Carlo Techniques • Analysis of variance and noise relationship • Discussion of estimator efficiency measures • Comparison of different sampling strategies • Cost-benefit analysis of techniques • Introduction to importance sampling strategies [1:00:54] - Advancing Beyond Simple Light Transport • Extension to multiple bounce lighting • Introduction to recursive rendering equation • Explanation of path tracing concept • Analysis of indirect illumination • Implementation considerations for multiple bounces [1:05:05] - Visualizing Path Tracing Effects • Demonstration of direct vs indirect illumination • Analysis of multiple bounce impacts • Examination of glass and transparent materials • Comparison of different bounce depths • Discussion of diminishing returns [1:08:42] - Introducing Russian Roulette • Explanation of path termination strategies • Analysis of contribution thresholds • Discussion of unbiased estimation • Implementation of probability-based termination • Cost-efficiency considerations [1:14:03] - Summary and Looking Forward • Recap of rendering equation concepts • Review of Monte Carlo integration application • Discussion of path tracing fundamentals • Analysis of importance sampling impact • Preview of variance reduction techniques
Thanks for this awesome series! A question: Wouldn't the direct lighting example here not really be "importance sampling", as it seems we're just changing the domain of integration rather than sampling one portion of the domain with higher probability?
The russion roulette section feels very incoherent: 1) 1:10:00 Is the "tentative contribution" being tested, COMMULATIVE across the entire path up-to-end-including the current event/vertex, or is it just the contribution of the CURRENT event by itself that is being tested? 2) In either of the above cases, should it also include the direct illumination component (NEE) or exclude it? 3) 1:11:00 Anything multiplied by 0 is 0, and the expected value of a constant is that constant, so the right part of the equation would be "E[(something)*0]" which is "E[0]" whichi is just 0 so the whole thing would always be 0. So - what's the point? What is that notation meant to convey?
the calulation at 1:11:00 is meant to show that the *new* random variable obtained by introducing the russian roulette procedure, has the same expected value as the original random variable. So it's valid for our Monte Carlo estimation
When you say "terminate 50% of all contributions with luminance less than 0.25" in the Roussian roulette example, how do you do that? In the Roussian roulette explanation you only showed how to terminate paths with a certain probability at each bounce, but how can you make it luminance-dependent?
or is the idea that there is some relation between the russian roulette probability p_rr and the percentage of paths below a certain luminance that you discard? Also why talk about luminance rather than radiance here?
compute radiance. generate a random number. if random number is smaller than some predecided value and computed radiance is smaller than 0.25 kill that ray
100 out of 100 for this course. Thank you so much @KeenanCrane.
KEY QUESTIONS:
- "What's the difference between ray tracing and rasterization?" [02:57]
- "What function are we integrating in Monte Carlo ray tracing?" [09:25]
- "Which probability distribution has higher variance?" [19:03]
- "Why is the image noisy despite our setup?" [47:12]
- "How can we reduce that noise?" [48:04]
- "When do we stop the path tracing procedure?" [1:07:49]
NOTABLE QUOTES:
- "Monte Carlo rendering is gonna let us render whatever we like" [01:43]
- "The cool thing about Monte Carlo is it works in basically any scenario you can throw at it" [12:51]
- "As you do more and more experiments, the variance is going to zero" [23:50]
- "Important sampling is really really important" [54:32]
- "Most of the time we're just getting a little bit brighter, it's not really helping our image" [1:08:59]
- "Because the values we're getting out of tracing each path are unbiased, the average value over many many paths will converge to the correct value" [1:14:46]
TIMESTAMPS:
[00:00] - Introduction to Monte Carlo Rendering
• Overview of photorealistic image generation techniques
• Discussion of combining multiple graphics concepts
• Comparison with previously covered rasterization approach
• Introduction to quality vs speed tradeoffs in rendering
[02:57] - Ray Tracing vs Rasterization Comparison
• Analysis of ordering differences in processing samples
• Explanation of primitive vs sample iteration approaches
• Demonstration of illumination model differences
• Comparison of shadow and reflection capabilities
• Discussion of rendering quality tradeoffs
[09:25] - Monte Carlo Integration Fundamentals
• Introduction to integration concept for rendering
• Explanation of path tracing and sample meanings
• Overview of rendering equation components
• Discussion of light path integration approaches
• Connection to previous numerical integration concepts
[14:52] - Understanding Expected Values
• Definition and intuition behind expected values
• Practical examples using coin flips
• Explanation of linearity properties
• Discussion of probability fundamentals
• Application to rendering calculations
[19:03] - Variance in Random Variables
• Detailed explanation of variance concept
• Analysis of probability distributions
• Comparison of different variance examples
• Discussion of practical implications
• Connection to rendering quality
[22:32] - Law of Large Numbers and Applications
• Explanation of convergence properties
• Practical example using Pi estimation
• Discussion of sample count impact
• Analysis of variance reduction
• Application to Monte Carlo rendering
[30:59] - Importance Sampling Concepts
• Introduction to biased sampling approaches
• Analysis of probability distribution choices
• Discussion of BRDF sampling strategies
• Explanation of image-based lighting sampling
• Optimization techniques for Monte Carlo rendering
[35:40] - Exploring Direct Lighting Examples
• Introduction to visibility function concept
• Discussion of irradiance calculation for ground points
• Analysis of direct lighting computation approaches
• Implementation of uniform sampling strategy
• Explanation of hemisphere sampling techniques
[39:48] - Understanding Hemisphere Sampling
• Detailed exploration of rejection sampling
• Application to hemisphere point selection
• Discussion of uniform sampling methods
• Practical implementation considerations
• Connection to direct lighting calculations
[46:51] - Analyzing Noise and Variance
• Explanation of noise in rendered images
• Discussion of random direction sampling impact
• Analysis of variance in different approaches
• Introduction to sampling efficiency concepts
• Exploration of noise reduction techniques
[54:55] - Evaluating Monte Carlo Techniques
• Analysis of variance and noise relationship
• Discussion of estimator efficiency measures
• Comparison of different sampling strategies
• Cost-benefit analysis of techniques
• Introduction to importance sampling strategies
[1:00:54] - Advancing Beyond Simple Light Transport
• Extension to multiple bounce lighting
• Introduction to recursive rendering equation
• Explanation of path tracing concept
• Analysis of indirect illumination
• Implementation considerations for multiple bounces
[1:05:05] - Visualizing Path Tracing Effects
• Demonstration of direct vs indirect illumination
• Analysis of multiple bounce impacts
• Examination of glass and transparent materials
• Comparison of different bounce depths
• Discussion of diminishing returns
[1:08:42] - Introducing Russian Roulette
• Explanation of path termination strategies
• Analysis of contribution thresholds
• Discussion of unbiased estimation
• Implementation of probability-based termination
• Cost-efficiency considerations
[1:14:03] - Summary and Looking Forward
• Recap of rendering equation concepts
• Review of Monte Carlo integration application
• Discussion of path tracing fundamentals
• Analysis of importance sampling impact
• Preview of variance reduction techniques
Great Lecture!
A question: at 52:00, shouldn't dw = dA' * cos(theta')/... instead of dA' * cos(theta)/... ?
you are right. it should be dA'cosθ' / |p - p'|^2
The variance of sum of random variables is equal to the sum of variances (22:10) if they are independent no?
in 51:30, the expression for d omega should say cos theta'
@@GuillermoValleCosmos yes to all
yes, if and only iff cov(x, y) = 0. And cov(x, y) = 0 if x and y are independent.
Thanks for this awesome series! A question: Wouldn't the direct lighting example here not really be "importance sampling", as it seems we're just changing the domain of integration rather than sampling one portion of the domain with higher probability?
The russion roulette section feels very incoherent:
1) 1:10:00 Is the "tentative contribution" being tested, COMMULATIVE across the entire path up-to-end-including the current event/vertex, or is it just the contribution of the CURRENT event by itself that is being tested?
2) In either of the above cases, should it also include the direct illumination component (NEE) or exclude it?
3) 1:11:00 Anything multiplied by 0 is 0, and the expected value of a constant is that constant, so the right part of the equation would be "E[(something)*0]" which is "E[0]" whichi is just 0 so the whole thing would always be 0.
So - what's the point? What is that notation meant to convey?
the calulation at 1:11:00 is meant to show that the *new* random variable obtained by introducing the russian roulette procedure, has the same expected value as the original random variable. So it's valid for our Monte Carlo estimation
When you say "terminate 50% of all contributions with luminance less than 0.25" in the Roussian roulette example, how do you do that?
In the Roussian roulette explanation you only showed how to terminate paths with a certain probability at each bounce, but how can you make it luminance-dependent?
or is the idea that there is some relation between the russian roulette probability p_rr and the percentage of paths below a certain luminance that you discard? Also why talk about luminance rather than radiance here?
compute radiance. generate a random number. if random number is smaller than some predecided value and computed radiance is smaller than 0.25 kill that ray
@@GuillermoValleCosmos I think most of the time luminance and radiance refer to same thing