 | [TOC] Chapter 6: Sampling and Reconstruction |  |
Effective sampling and reconstruction methods are essential for generating high-quality images with minimal noise. This section covers various sampling theories, techniques, and reconstruction processes that help achieve realistic rendering results. We'll discuss sampling theory, sampling interfaces, stratified sampling, and various sampling methods, along with image reconstruction concepts and the imaging pipeline.
| Sampling Theory | |
Sampling theory deals with the principles of converting a continuous signal (like light) into a discrete one (sampled data). The key concept is the Nyquist-Shannon sampling theorem, which states that to avoid aliasing, a signal must be sampled at least twice its highest frequency.
In computer graphics, sampling techniques are used to discretize the process of gathering light information from a scene. This helps approximate the color and intensity values that contribute to the final image.
The mathematical representation of a sampled function can be expressed as:
\[
f(x) = \sum_{n=-\infty}^{\infty} f(nT) \cdot \delta(x - nT)
\]
where:
\( f(x) \) is the continuous function,
\( T \) is the sampling interval,
\( \delta \) is the Dirac delta function, which samples \( f(x) \) at discrete points.
| Sampling Interface | |
A sampling interface serves as a common structure to handle various sampling techniques. It defines the methods for generating random samples and managing sample distributions.
Here's an example of a JavaScript sampling interface:
class Sampler {
constructor(sampleCount) {
this.sampleCount = sampleCount; // Number of samples to generate
}
getSamples() {
throw new Error("getSamples() must be implemented by subclasses.");
}
}
| Stratified Sampling | |
Stratified sampling improves the sampling process by dividing the sample space into distinct strata or segments, ensuring that each segment is adequately represented. This approach minimizes variance and improves convergence in rendering.
In stratified sampling, the sample space is divided into \( N \) regions, and a fixed number of samples are taken from each region.
Here's a simple JavaScript implementation of stratified sampling:
class StratifiedSampler extends Sampler {
constructor(sampleCount, strataCount) {
super(sampleCount);
this.strataCount = strataCount;
}
getSamples() {
const samples = [];
const samplesPerStratum = Math.floor(this.sampleCount / this.strataCount);
for (let i = 0; i < this.strataCount; i++) {
const offset = i / this.strataCount;
for (let j = 0; j < samplesPerStratum; j++) {
const x = offset + Math.random() / this.strataCount;
samples.push(x);
}
}
return samples;
}
}
| The Halton Sampler | |
The Halton sampler is a quasi-random sampling method that produces low-discrepancy sequences. It uses different prime bases to generate samples in a multi-dimensional space, making it useful for reducing visual artifacts in rendered images.
The Halton sequence generates numbers using a base \( b \):
\[
H(n) = \sum_{k=1}^{\infty} \frac{d_k(n)}{b^k}
\]
where \( d_k(n) \) is the \( k \)-th digit of \( n \) in base \( b \).
Here's a simple implementation of a Halton sampler in JavaScript:
function haltonSequence(n, base) {
let result = 0;
let f = 1 / base;
while (n > 0) {
result += (n % base) * f;
n = Math.floor(n / base);
f /= base;
}
return result;
}
class HaltonSampler extends Sampler {
constructor(sampleCount) {
super(sampleCount);
}
getSamples() {
const samples = [];
for (let i = 0; i < this.sampleCount; i++) {
const x = haltonSequence(i, 2); // Base 2 for x-coordinate
const y = haltonSequence(i, 3); // Base 3 for y-coordinate
samples.push({ x, y });
}
return samples;
}
}
| Maximized Minimal Distance Sampler | |
The Maximized Minimal Distance (MMD) sampler ensures that the distance between any two samples is maximized, helping to minimize clustering and ensure better coverage of the sample space.
The MMD technique can be useful for generating sample points in a way that they are uniformly distributed across a defined area.
Here's a basic implementation of MMD sampling in JavaScript:
class MMD_Sampler extends Sampler {
constructor(sampleCount) {
super(sampleCount);
this.samples = [];
}
distance(p1, p2) {
return Math.sqrt((p1.x - p2.x) ** 2 + (p1.y - p2.y) ** 2);
}
generateSample() {
return { x: Math.random(), y: Math.random() }; // Random sample in unit square
}
getSamples() {
while (this.samples.length < this.sampleCount) {
const newSample = this.generateSample();
if (this.samples.every(existingSample => this.distance(existingSample, newSample) > 0.1)) {
this.samples.push(newSample);
}
}
return this.samples;
}
}
| Sobol' Sampler | |
The Sobol' sampler is another low-discrepancy sequence generator that creates quasi-random samples, making it well-suited for numerical integration and rendering applications.
Sobol' sequences can be generated using a linear transformation matrix specific to the dimension of the sample space. The method is based on generating binary representations for sample indices.
Here's a simple JavaScript implementation of a Sobol' sampler:
function sobolSequence(n) {
// Placeholder for the actual Sobol' generation algorithm
return { x: n * 0.5, y: n * 0.5 }; // Replace with actual Sobol' logic
}
class SobolSampler extends Sampler {
constructor(sampleCount) {
super(sampleCount);
}
getSamples() {
const samples = [];
for (let i = 0; i < this.sampleCount; i++) {
samples.push(sobolSequence(i));
}
return samples;
}
}
| Image Reconstruction | |
Image reconstruction is the process of converting sampled data back into a continuous image. This is typically done using reconstruction filters to smooth out the pixel values and reduce aliasing artifacts.
One common filter used in image reconstruction is the box filter, which averages the values of the pixels in a specific region.
Mathematically, the reconstruction of an image \( I \) can be represented as:
\[
I(x, y) = \int \int f(x', y') \cdot R(x - x', y - y') \, dx' \, dy'
\]
where \( R \) is the reconstruction filter.
| Imaging Pipeline Mediums (Film Quality) | |
In ray tracing, the film medium is the output target image for the rendering process. The imaging pipeline includes various stages that process the data from the scene to create a final image.
1. Ray Generation: Rays are cast from the camera through pixels on the film.
2. Scene Intersection: Rays intersect with scene geometry to gather color and intensity information.
3. Shading and Lighting: Based on the material properties and light sources, color values are calculated.
4. Image Reconstruction: Samples are combined using reconstruction filters to produce the final image.
5. Output: The final image is displayed or saved to a file.
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