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7. Sampling and Reconstruction

Radiance在胶平面上是连续函数，但是渲染的输出是离散的像素。

But what happens when we need the value of s signal at a location that we didn’t sample at? In such cases, we can use a process known as reconstruction to derive an approximation of the original continuous function. With this approximation we can then discretely sample its values at a new set of sample points, which is referred to as resampling. signal processing primer

7.1 Sampling Theory

  • 7.1.1 The Frequency Domain and the Fourier Transform

Definition of the Fourier (and Inverse) Transform (synthesis and analysis) 书里给的是第二种定义方式Symmetric Form: Hertz Frequency

  • 7.1.2 Ideal Sampling and Reconstruction 使用shah fucntion、impulse train function也叫Dirac comb，记作来乘以函数来进行采样。

采样过程如下：

有个疑问，对于Dirac delta函数，有，积分形式才得到，采样过程为啥不是积分形式？

来自the dirac comb and its fourier transform

An important idea used in Fourier analysis is the fact that the Fourier transform of the product of two functions can be shown to be the convolution of their individual Fourier transforms and : .

It is similarly the case that convolution in the spatial domain is equivalent to multiplication in the frequency domain:

空域的周期和频域的周期互为相反数，这是理解混叠的关键，空域中采样间隔越大，在频域中间隔就越小，就导致频域的互相挤压，一起混叠（aliasing）。

the Fourier transform of a shah function with period is another shah function with period . This reciprocal relationship between periods is important to keep in mind: it means that if the samples are farther apart in the spatial domain, they are closer together in the frequency domain.

  • 7.1.4 Antialiasing Techniques

NonUniform Sampling 以前是在整数倍周期位置采样，现在在整数倍附近周期的位置随机选择一个采样点。采样率不够的情况下，无论均匀采样还是非均匀采样，都会产生重建走样，但是非均匀采样倾向于把规则的走样转变为噪声，相对来说噪声对视觉的干扰更小。 Adaptive Sampling adaptive supersampling，难实现 Prefiltering 滤波（模糊）移除高频，模糊也没有走样使人讨厌 - 7.1.6 Sources of Aliasing in Rendering 1. 几何（Geometry aliasing）：模型边界（step function）、完美重建滤波器（sinc filter）应用到混叠采样（aliased samples，or 走样采样？，会产生振铃走样，ring artifacts，也叫Gibbs phenomenon）、特别小的物体（在两个采样之间，就会在动画的不同帧间出现消失的反复闪烁） 2. 着色（Shading aliasing）：贴图、材质、锐利的阴影（step function）

  • 7.1.7 Understanding Pixels
  1. Pixel是图像函数上的采样点，并不关联面积的概念。
  2. 图像定义再离散坐标上，采样函数定义在连续浮点型坐标位置上，（这里的离散值应该是离散的序号，连续值是物理意义的位置，可以映射，但是不能等同）。可以使用不同的映射，1）取整到最近的整数，即离散点正好落在整数位置上，2）向下取整，即离散点落在整数点距离0.5的位置。

7.2 Sampling Interface

  • Evaluating Sample Patterns: Discrepancy
  1. 使用discrepancy评估采样样式的质量。采样范围，discrepancy的定义如下： Note “sup”是数学中常用的符号，代表“上确界”(supremum)。它表示一个集合或函数的最小上界。
  2. 采样点的聚集性，使用点之间的最小距离来度量。Sobol采样器就存在采样点集中聚集的问题，太近的采样点能提供的额外信息更少。 > Intuitively, samples that are too close together aren’t a good use of sampling resources: the closer one sample is to another, the less likely it is to give useful new information about the function being sampled. Therefore, computing the minimum distance between any two samples in a set of points has also proved to be a useful metric of sample pattern quality; the higher the minimum distance, the better.

Poisson disk sampling采样模式在距离度量上表现优异。Studies have shown that the rods and cones in the eye are distributed in a similar way, which further validates the idea that this distribution is a good one for imaging. 实际使用中，Poisson disk sampling更适合2D图像的采样，在更高维采样时就没那么有效了。 - low-discrepancy

7.4 The Halton Sampler

HaltonSampler直接生成low-discrepancy的点集，保证点不聚集，且在每一维上都良好分布。