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Most Recent Posts:
2023-04-07 Random Sampling Experiments: Avoid The Sides!
2023-03-15 Euler’s Best Candidate – For Generating Blue Noise Sample Points, and More
2023-03-12 Euler’s Number & Probability
2023-03-06 Uniform 1D Red Noise & Blue Noise Part 2
2023-02-20 Uniform 1D Red Noise & Blue Noise

244 posts grouped by the following categories:

(Posts TODO – to help me remember things I’d like to write up at some point

Other Publications

2021-08-04 Ray Tracing Gems II Chapter 24 “Using Blue Noise for Ray Traced Soft Shadows”

Simple But Effective Verlet Game Physics – GDC India 2021
Youtube Video:

Spatiotemporal Blue Noise Masks – EGSR 2022
EGSR 2022 Presentation (Youtube):
EGSR 2022 Paper:
Code and pregenerated textures:
NVIDIA Blog post part 1:
NVIDIA Blog post part 2:
Arxiv Paper on scalar valued STBN:


2022-02-26 Image Sharpening Convolution Kernels
2022-02-01 Two Low Discrepancy Grids: Plus Shaped Sampling LDG, and R2 LDG
2022-01-01 Interleaved Gradient Noise: A Different Kind of Low Discrepancy Sequence
2021-12-29 Why Can’t You Design Noise in Frequency Space?
2021-12-27 Not All Blue Noise is Created Equal Part 2
2021-04-11 Perlin Noise Experiments
2020-11-04 Frequency Domain Image Compression and Filtering
2020-07-11 Interpolating Data Over Arbitrary Shapes With Laplace’s Equation and Walk on Spheres
2020-06-14 Casual Shadertoy Path Tracing 3: Fresnel, Rough Refraction & Absorption, Orbit Camera
2020-06-06 Casual Shadertoy Path Tracing 2: Image Improvement and Glossy Reflections
2020-05-25 Casual Shadertoy Path Tracing 1: Basic Camera, Diffuse, Emissive
2020-05-16 Using Blue Noise For Raytraced Soft Shadows
2020-05-10 Ray Marching Fog With Blue Noise
2019-12-07 Bezier Triangles
2019-06-25 Generating Blue Noise Textures With Void And Cluster
2018-11-16 How To: Data Lookups Via Raytracing
2018-08-12 Not All Blue Noise is Created Equal
2018-08-07 Tiled Blue Noise
2018-07-13 Blending an HDR color into a U8 Buffer
2018-07-04 Pathtraced Depth of Field & Bokeh
2018-04-16 Prefix Sums and Summed Area Tables
2018-03-24 An Idea: Raytracing Lookup Tables
2018-03-19 A Very Quick DirectX Raytracing API Primer
2018-03-10 Don’t Convert sRGB U8 to Linear U8!
2018-01-30 What the Heck is Blue Noise?
2017-11-26 Dissecting “Tiny Clouds”
2017-11-03 Animating Noise For Integration Over Time 2: Uniform Over Time
2017-10-31 Animating Noise For Integration Over Time
2017-10-25 Transmuting White Noise To Blue, Red, Green, Purple
2017-10-20 Generating Blue Noise Sample Points With Mitchell’s Best Candidate Algorithm
2017-10-01 Calculating the Distance Between Points in “Wrap Around” (Toroidal) Space
2017-07-10 WebGL PBR Implementation
2017-07-01 Why Are Some Shadows Soft And Other Shadows Hard?
2017-01-09 Raytracing Reflection, Refraction, Fresnel, Total Internal Reflection, and Beer’s Law
2016-09-21 Path Tracing – Getting Started With Diffuse and Emissive
2016-07-28 Fourier Transform (And Inverse) Of Images
2016-03-14 Using Wang Tiles to Simulate Turing Machines
2016-03-05 Matrix Form of Bezier Curves
2016-03-02 Actually Making Signed Distance Field Textures With JFA
2016-02-29 Fast Voronoi Diagrams and Distance Field Textures on the GPU With the Jump Flooding Algorithm
2016-02-21 G-Buffer Upsizing
2016-02-03 Failed Shadertoy “Dust” Game – Browsers Need To Compile Shaders Off the Main Thread!
2016-01-16 Making a Ray Traced Snake Game in Shadertoy
2015-08-19 Gaussian Blur
2015-08-18 Box Blur
2015-08-15 Resizing Images With Bicubic Interpolation
2015-08-09 Cubic Hermite Rectangles
2015-06-19 What is Pre-multiplied Alpha and Why Does it Matter?
2015-04-30 Bilinear Filtering & Bilinear Interpolation
2015-04-23 FlipQuad & FlipTri Antialiasing
2015-04-23 4-Rook Antialiasing (RGSS)
2015-04-22 Quincunx Antialiasing
2015-01-17 Bresenham’s Drawing Algorithms
2014-08-13 Wang Tiling
2014-08-11 Temporal supersampling, flipquads and real time raytracing
2014-06-30 Distance Field Textures
2014-06-22 Analytic Fog Density
2014-02-07 Lots of Skyboxes!
2014-02-03 Converting RGB to Grayscale
2014-01-27 A trip through the Graphics Pipeline
2014-01-22 Shadertoy – WebGLSL pixel shaders
2012-09-25 How to Render the Mandelbrot Set
2012-05-03 MoriRT: Pixel and Geometry Caching to Aid Real Time Raytracing

Original Research / Inventions

2019-03-22 Linear Fit Search
2018-04-23 Taking a Stroll Between The Pixels
2017-05-07 Improved Storage Space Efficiency of GPU Texture Sampler Bezier Curve Evaluation
2016-12-16 Evaluating Points on Analytical Surfaces and in Analytical Volumes Using the GPU Texture Sampler
2016-12-10 Failed Experiment: The GPU Texture Sampler is Turing Complete But That Fact is Pretty Useless
2016-12-08 Evaluating Polynomials with the GPU Texture Sampler
2016-02-22 GPU Texture Sampler Bezier Curve Evaluation
2015-01-24 DAPSE Preview


2023-04-07 Random Sampling Experiments: Avoid The Sides!
2023-03-15 Euler’s Best Candidate – For Generating Blue Noise Sample Points, and More
2023-01-04 Generalizing Binary Search To Higher Dimensions
2022-10-30 Rapidly Solving Sudoku, N-Queens, Pentomino Placement, and More, With Knuth’s Algorithm X and Dancing Links.
2022-07-21 Rounding Modes For Integer Division
2020-10-25 When Life Gives You Lemons, Make Random Numbers
2020-06-23 Weighted Round Robin (Weighted Random Integers) Using The Golden Ratio Low Discrepancy Sequence
2019-08-10 Measuring Debug STL Container Perf Cost in MSVC
2017-11-21 Demystifying Floating Point Precision
2017-06-20 SIMD / GPU Friendly Branchless Binary Search
2016-12-06 The Secret to Writing Fast Code / How Fast Code Gets Slow
2016-11-13 Minimizing Code Complexity by Programming Declaratively
2016-10-14 A Data Point for MSVC vs Clang Code Generation
2016-10-03 Is Code Faster Than Data? Examining Hash Tables
2016-09-26 Is Code Faster Than Data? Switch Statements vs. Arrays
2016-09-23 Exploring Compile Time Hashing
2016-05-18 Who Cares About Dynamic Array Growth Strategies?
2016-04-09 Turning a Truth Table Into A digital Circuit (ANF)
2016-02-09 How and Why Cleaning Up Code or Processes Gives Multiplicative Benefits
2015-12-20 Simplifying Boolean Expressions With Karnaugh Maps
2015-12-18 What Happens When you Mix Hash Tables and Binary Searching?
2015-12-14 O(1) Data Lookups With Minimal Perfect Hashing
2015-03-09 HyperLogLog: Estimate Unique Value Counts Like The Pros
2015-02-25 Avoiding The Performance Hazzards of std::function
2015-02-22 Count Min Sketch: A Probabilistic Histogram
2015-02-13 Writing a Basic Search Engine AKA Calculating Similarity of Histograms with Dot Product
2015-02-08 Estimating Set Membership With a Bloom Filter
2015-02-05 Getting Strongly Typed Typedefs Using Phantom Types
2015-02-03 Estimating Counts of Distinct Values with KMV
2013-09-13 External C++ Header Guards
2013-09-09 Alloca and Realloc – Useful Tools, Not Ancient Relics
2013-07-08 Is pre-increment really faster than post increment? Part 2
2013-07-07 A Super Tiny Random Number Generator
2013-07-06 Fast & Lightweight Random “Shuffle” Functionality – FIXED!
2013-07-06 The Incredible Time Traveling Random Number Generator
2013-06-18 Why do you hate me rand()?!
2013-06-08 Fast & Lightweight Random “Shuffle” Functionality
2013-06-08 Efficiently Generate Random Numbers Without Repeats
2013-06-05 Is pre-increment really faster than post increment? Part 1
2013-05-21 The Black Art of sizeof() aka Compile Time Type Deduction
2013-05-11 Lists of Macro Lists
2013-05-07 Macro Lists For The Win – Side B
2013-05-03 Macro Lists For The Win
2013-05-01 Permutation Programming Without Maintenance Nightmares
2012-11-28 Mechanical Computer Quest Part I

Game Dev

2020-03-01 Using Low Discrepancy Sequences & Blue Noise in Loot Drop Tables for Games
2017-09-30 Half Tile Offset Streaming World Grids
2016-04-01 Game Development Needs Data Pipeline Middleware
2014-01-11 Getting Ideal Disk Reads on the Cheap
2012-09-24 Bias And Gain Are Your Friend
2012-09-21 Anatomy of a Skeletal Animation System Part 3
2012-09-21 Anatomy of a Skeletal Animation System Part 2
2012-09-21 Anatomy of a Skeletal Animation System Part 1
2012-09-09 Recording lagless demo videos of a laggy game


2023-03-12 Euler’s Number & Probability
2023-03-06 Uniform 1D Red Noise & Blue Noise Part 2
2023-02-20 Uniform 1D Red Noise & Blue Noise
2023-02-17 Fibonacci Word Sampling: A Sorted Golden Ratio Low Discrepancy Sequence
2022-08-09 Calculating Discrete Sums With Umbral Calculus
2022-08-02 Calculating the Similarity of Histograms or PDFs & Interpolating Them Using the p-Wasserstein Distance
2022-07-12 Calculating SVD and PCA in C++
2022-07-10 Programming PCA From Scratch In C++
2022-06-29 Piecewise Least Squares Curve Fitting
2022-06-06 Fitting Data Points With Weighted Least Squares
2022-03-02 Sampling Importance Resampling
2022-03-01 Picking Fairly From A List of Unknown Size With Reservoir Sampling
2022-02-26 The Discrete Cosine Transform, and Derivatives
2022-02-20 Adjusting Point Sets in Frequency Space Using a Differentiable Fourier Transform
2022-01-29 Distance Between Points on a “Toroidal Disk”
2021-04-01 Mean Squared Error is Variance
2020-11-25 Multiple Importance Sampling in 1D
2020-07-26 Irrational Numbers
2020-07-17 Using Low Discrepancy Sequences With Rejection Sampling
2020-06-04 A Link Between Russian Roulette and Rejection Sampling / Importance Sampling
2020-05-03 Guess a Number Between 1 and 10
2020-03-17 Basic Methods For Finding Zeroes and Mins / Maxes of Functions
2020-03-12 Using White Noise to Choose Between Red Noise and Blue Noise
2020-03-10 How Do I Calculate Variance in 1 Pass?
2020-02-09 A Fun 2d Rotation Matrix Derivation
2019-12-01 Calculating Information Entropy
2019-10-25 Bayes’ Theorem Intuition
2019-07-30 Dice, Distributions & Noise Colors
2019-06-1 Taking the Max of Uniform Random Numbers
2019-05-25 Generating Random Numbers From a Specific Distribution With The Metropolis Algorithm (MCMC)
2018-06-12 Monte Carlo Integration Explanation in 1D
2017-08-08 Generating Random Numbers From a Specific Distribution With Rejection Sampling
2017-08-05 Generating Random Numbers From a Specific Distribution By Inverting the CDF
2017-07-25 Counting Bits & The Normal Distribution
2017-05-29 When Random Numbers Are Too Random: Low Discrepancy Sequences
2017-04-10 Solving N equations and N unknowns: The Fine Print (Gauss Jordan Elimination)
2017-03-31 Orthogonal Projection Matrix Plainly Explained
2017-03-13 Neural Network Gradients: Backpropagation, Dual Numbers, Finite Differences
2017-02-20 Multivariable Dual Numbers & Automatic Differentiation
2017-02-07 A Geometric Interpretation of Neural Networks
2017-01-02 Incremental Least Squares Surface and Hyper-Volume Fitting
2016-12-22 Incremental Least Squares Curve Fitting
2016-08-23 Incremental Averaging
2016-08-11 Understanding The Discrete Fourier Transform
2016-03-11 A Sixth Way To Calculate Sine Without Trig
2016-02-19 Normalized Vector Interpolation TL;DR
2015-11-13 Hiding a Lookup Table in a Modulus Operation
2015-09-15 Solving Nested Modulus Equations
2015-09-12 Solving Simultaneous Congruences (Chinese Remainder Theorem)
2015-09-10 Modular Multiplicative Inverse
2015-08-08 Cubic Hermite Interpolation
2015-08-05 Lagrange Rectangles
2015-08-02 Finite Differences
2015-07-28 Rectangular Bezier Patches
2015-07-08 Lagrange Interpolation
2015-07-05 The De Casteljau Algorithm for Evaluating Bezier Curves
2015-06-14 A Fifth Way to Calculate Sine Without Trig
2015-05-25 Easy Binomial Expansion & Bezier Curve Formulas
2015-01-24 Programmatically Calculating GCD and LCM
2014-12-30 Dual Numbers & Automatic Differentiation
2014-12-27 Using Imaginary Numbers To Rotate 2D Vectors
2014-11-04 Four Ways to Calculate Sine Without Trig
2014-08-28 One Dimensional Bezier Curves
2014-08-14 Counting in Binary is a Fractal
2014-08-10 What if My Equation DOESN’T Equal Zero??
2014-03-22 Bezier Curves Part 2 (and Bezier Surfaces)
2014-03-09 Implicit vs Parametric vs Explicit Surfaces
2014-03-04 Bezier Curves
2014-02-20 Soft Maximum vs Hard Maximum
2013-11-13 Transformation Matrix Basics
2013-10-12 Converting To and From Polar / Spherical Coordinates Made Easy
2013-09-17 How to Test Randomness of Numbers

Machine Learning

2019-05-11 Markov Chain Text Generation
2017-12-23 C++ Differentiable Programming: Searching For An Optimal Dither Pattern
2017-03-15 Neural Network Recipe: Recognize Handwritten Digits With 95% Accuracy
2017-03-09 How to Train Neural Networks With Backpropagation
2012-10-23 B.A.M. Neural Networks


2016-10-19 Low Tech Homomorphic Encryption
2016-04-30 Shamir’s Quest: Collect Any 3 Keys To Unlock The Secret!
2015-09-05 Improving the Security of the Super Simple Symmetric Leveled Homomorphic Encryption Implementation
2015-09-05 Super Simple Symmetric Leveled Homomorphic Encryption Implementation
2014-05-18 Feistel Networks – Do They Have to use XOR?
2012-09-17 Encryption 101: Realistic Security
2012-09-16 Cryptography 101: Encryption – Asymmetric Keys
2012-09-15 Cryptography 101: Encryption – Symmetric Keys
2012-09-15 Cryptography 101: Encryption – One Time Pad
2012-09-12 Cryptography 101: Hashing


2020-01-24 IIR Audio & Data Filters – Featuring Biquads
2020-01-15 FIR Audio & Data Filters

Audio Synth / Audio Programming

2018-03-05 Granular Audio Synthesis
2016-07-01 Intro To Audio Synthesis For Music Presentation
2016-06-23 The Beating Effect
2016-06-16 Synthesizing a Plucked String Sound With the Karplus-Strong Algorithm
2015-04-19 Frequency Domain Audio Synthesis – With IFFT and Oscillators
2015-04-14 Decibels (dB) and Amplitude
2015-03-23 DIY Synth: Convolution Reverb & 1D Discrete Convolution of Audio Samples
2015-03-17 DIY Synth: Multitap Reverb
2015-03-17 DIY Synth: Delay Effect (Echo)
2015-03-16 DIY Synth: Flange Effect
2015-03-14 DIY Synth: Basic Drum
2012-06-18 DIY Synth 3: Sampling, Mixing, and Band Limited Wave Forms
2012-05-19 DIY Synth 2: Common Wave Forms
2012-05-14 DIY Synth 1: Sound Output

Quantum Programming

2015-11-06 Quantum Computing For Programmers Part 2: Multiple Qubits
2015-10-17 Quantum Computing For Programmers Part Ib: Bloch Sphere
2015-10-17 Quantum Computing For Programmers Part I: One Qubit
2015-10-12 Quantum Computing References

Network Programming

2018-6-04 A Neat Trick For Compressing Networked State Data

Soft Skills / Not Directly About Tech

2017-11-16 My Old Master: On Optimism
2017-07-03 A Tool To Debug Teams (Knoster)
2017-03-17 Plastic Bag Ban – Semi Reusable Bag Kiosks a Better Solution?
2017-01-24 My Old Master: How to Correct as a Mentor or a Teacher
2015-06-18 Adventures in Learning How to Publish a Research Paper Part 1
2015-05-24 No Bad Code, Creeping Normality and Social Structure Code Organization
2014-07-21 Situational Leadership


2021-07-11 Inverting a Button Press (Featuring Current Dividers)
2021-06-27 Resistance and Voltage Dividers
2021-06-15 Voltage, Amps, Resistance and LEDs (Ohm’s Law)

Random Sampling Experiments: Avoid The Sides!

I’ve heard that when sampling in 1D, such as , you should keep your samples away from the edges of the sampling domain (0 and 1). I’ve also heard that when sampling in 2D, such as the square, you should keep the samples away from the origin and presumably the edges as well. That came … Continue reading

Generalizing Binary Search To Higher Dimensions

Hey all, I’m currently on vacation in Aruba, having a blast snorkeling, hanging out with the family, and working on some stuff on a kindle scribe I got for Xmas. The scribe is an excellent paper replacement, and I’m digging it. I used it a lot while working through things in this post, but I … Continue reading