7.1 Sampling Theory
A digital image is represented as a set of pixel values, typically aligned on a rectangular grid. When a digital image is displayed on a physical device, these values are used to determine the spectral power emitted by pixels on the display. When thinking about digital images, it is important to differentiate between image pixels, which represent the value of a function at a particular sample location, and display pixels, which are physical objects that emit light with some distribution. (For example, in an LCD display, the color and brightness may change substantially when the display is viewed at oblique angles.) Displays use the image pixel values to construct a new image function over the display surface. This function is defined at all points on the display, not just the infinitesimal points of the digital image’s pixels. This process of taking a collection of sample values and converting them back to a continuous function is called reconstruction.
In order to compute the discrete pixel values in the digital image, it is necessary to sample the original continuously defined image function. In pbrt, like most other ray-tracing renderers, the only way to get information about the image function is to sample it by tracing rays. For example, there is no general method that can compute bounds on the variation of the image function between two points on the film plane. While an image could be generated by just sampling the function precisely at the pixel positions, a better result can be obtained by taking more samples at different positions and incorporating this additional information about the image function into the final pixel values. Indeed, for the best quality result, the pixel values should be computed such that the reconstructed image on the display device is as close as possible to the original image of the scene on the virtual camera’s film plane. Note that this is a subtly different goal from expecting the display’s pixels to take on the image function’s actual value at their positions. Handling this difference is the main goal of the algorithms implemented in this chapter.
Because the sampling and reconstruction process involves approximation, it introduces error known as aliasing, which can manifest itself in many ways, including jagged edges or flickering in animations. These errors occur because the sampling process is not able to capture all of the information from the continuously defined image function.
As an example of these ideas, consider a 1D function (which we will interchangeably refer to as a signal), given by , where we can evaluate at any desired location in the function’s domain. Each such is called a sample position, and the value of is the sample value. Figure 7.1 shows a set of samples of a smooth 1D function, along with a reconstructed signal that approximates the original function . In this example, is a piecewise linear function that approximates by linearly interpolating neighboring sample values (readers already familiar with sampling theory will recognize this as reconstruction with a hat function). Because the only information available about comes from the sample values at the positions , is unlikely to match perfectly since there is no information about ’s behavior between the samples.
Fourier analysis can be used to evaluate the quality of the match between the reconstructed function and the original. This section will introduce the main ideas of Fourier analysis with enough detail to work through some parts of the sampling and reconstruction processes but will omit proofs of many properties and skip details that aren’t directly relevant to the sampling algorithms used in pbrt. The “Further Reading” section of this chapter has pointers to more detailed information about these topics.
7.1.1 The Frequency Domain and the Fourier Transform
One of the foundations of Fourier analysis is the Fourier transform, which represents a function in the frequency domain. (We will say that functions are normally expressed in the spatial domain.) Consider the two functions graphed in Figure 7.2. The function in Figure 7.2(a) varies relatively slowly as a function of , while the function in Figure 7.2(b) varies much more rapidly. The more slowly varying function is said to have lower frequency content.
Figure 7.3 shows the frequency space representations of these two functions; the lower frequency function’s representation goes to 0 more quickly than does the higher frequency function.
Most functions can be decomposed into a weighted sum of shifted sinusoids. This remarkable fact was first described by Joseph Fourier, and the Fourier transform converts a function into this representation. This frequency space representation of a function gives insight into some of its characteristics—the distribution of frequencies in the sine functions corresponds to the distribution of frequencies in the original function. Using this form, it is possible to use Fourier analysis to gain insight into the error that is introduced by the sampling and reconstruction process and how to reduce the perceptual impact of this error.
The Fourier transform of a 1D function is
(Recall that , where .) For simplicity, here we will consider only even functions where , in which case the Fourier transform of has no imaginary terms. The new function is a function of frequency, . We will denote the Fourier transform operator by , such that . is clearly a linear operator—that is, for any scalar , and .
Equation (7.1) is called the Fourier analysis equation, or sometimes just the Fourier transform. We can also transform from the frequency domain back to the spatial domain using the Fourier synthesis equation, or the inverse Fourier transform:
Table 7.1 shows a number of important functions and their frequency space representations. A number of these functions are based on the Dirac delta distribution, a special function that is defined such that , and for all , . An important consequence of these properties is that
The delta distribution cannot be expressed as a standard mathematical function, but instead is generally thought of as the limit of a unit area box function centered at the origin with width approaching 0.
|Spatial Domain||Frequency Space Representation|
|Box: if , otherwise||Sinc:|
7.1.2 Ideal Sampling and Reconstruction
Using frequency space analysis, we can now formally investigate the properties of sampling. Recall that the sampling process requires us to choose a set of equally spaced sample positions and compute the function’s value at those positions. Formally, this corresponds to multiplying the function by a “shah,” or “impulse train,” function, an infinite sum of equally spaced delta functions. The shah is defined as
where defines the period, or sampling rate. This formal definition of sampling is illustrated in Figure 7.4. The multiplication yields an infinite sequence of values of the function at equally spaced points:
These sample values can be used to define a reconstructed function by choosing a reconstruction filter function and computing the convolution
where the convolution operation is defined as
For reconstruction, convolution gives a weighted sum of scaled instances of the reconstruction filter centered at the sample points:
We have gone through a process that may seem gratuitously complex in order to end up at an intuitive result: the reconstructed function can be obtained by interpolating among the samples in some manner. By setting up this background carefully, however, Fourier analysis can now be applied to the process more easily.
We can gain a deeper understanding of the sampling process by analyzing the sampled function in the frequency domain. In particular, we will be able to determine the conditions under which the original function can be exactly recovered from its values at the sample locations—a very powerful result. For the discussion here, we will assume for now that the function is band limited—there exists some frequency such that contains no frequencies greater than . By definition, band-limited functions have frequency space representations with compact support, such that for all . Both of the spectra in Figure 7.3 are band limited.
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:
These properties are derived in the standard references on Fourier analysis. Using these ideas, the original sampling step in the spatial domain, where the product of the shah function and the original function is found, can be equivalently described by the convolution of with another shah function in frequency space.
We also know the spectrum of the shah function from Table 7.1; 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.
Thus, the frequency domain representation of the sampled signal is given by the convolution of and this new shah function. Convolving a function with a delta function just yields a copy of the function, so convolving with a shah function yields an infinite sequence of copies of the original function, with spacing equal to the period of the shah (Figure 7.6). This is the frequency space representation of the series of samples.
Now that we have this infinite set of copies of the function’s spectrum, how do we reconstruct the original function? Looking at Figure 7.6, the answer is obvious: just discard all of the spectrum copies except the one centered at the origin, giving the original .
In order to throw away all but the center copy of the spectrum, we multiply by a box function of the appropriate width (Figure 7.7). The box function of width is defined as
This multiplication step corresponds to convolution with the reconstruction filter in the spatial domain. This is the ideal sampling and reconstruction process. To summarize:
This is a remarkable result: we have been able to determine the exact frequency space representation of , purely by sampling it at a set of regularly spaced points. Other than knowing that the function was band limited, no additional information about the composition of the function was used.
Applying the equivalent process in the spatial domain will likewise recover exactly. Because the inverse Fourier transform of the box function is the sinc function, ideal reconstruction in the spatial domain is
where , and thus
Unfortunately, because the sinc function has infinite extent, it is necessary to use all of the sample values to compute any particular value of in the spatial domain. Filters with finite spatial extent are preferable for practical implementations even though they don’t reconstruct the original function perfectly.
A commonly used alternative in graphics is to use the box function for reconstruction, effectively averaging all of the sample values within some region around . This is a very poor choice, as can be seen by considering the box filter’s behavior in the frequency domain: This technique attempts to isolate the central copy of the function’s spectrum by multiplying by a sinc, which not only does a bad job of selecting the central copy of the function’s spectrum but includes high-frequency contributions from the infinite series of other copies of it as well.
Beyond the issue of the sinc function’s infinite extent, one of the most serious practical problems with the ideal sampling and reconstruction approach is the assumption that the signal is band limited. For signals that are not band limited, or signals that aren’t sampled at a sufficiently high sampling rate for their frequency content, the process described earlier will reconstruct a function that is different from the original signal.
The key to successful reconstruction is the ability to exactly recover the original spectrum by multiplying the sampled spectrum with a box function of the appropriate width. Notice that in Figure 7.6, the copies of the signal’s spectrum are separated by empty space, so perfect reconstruction is possible. Consider what happens, however, if the original function was sampled with a lower sampling rate. Recall that the Fourier transform of a shah function with period is a new shah function with period . This means that if the spacing between samples increases in the spatial domain, the sample spacing decreases in the frequency domain, pushing the copies of the spectrum closer together. If the copies get too close together, they start to overlap.
Because the copies are added together, the resulting spectrum no longer looks like many copies of the original (Figure 7.8). When this new spectrum is multiplied by a box function, the result is a spectrum that is similar but not equal to the original : high-frequency details in the original signal leak into lower frequency regions of the spectrum of the reconstructed signal. These new low-frequency artifacts are called aliases (because high frequencies are “masquerading” as low frequencies), and the resulting signal is said to be aliased.
Figure 7.9 shows the effects of aliasing from undersampling and then reconstructing the 1D function .
A possible solution to the problem of overlapping spectra is to simply increase the sampling rate until the copies of the spectrum are sufficiently far apart to not overlap, thereby eliminating aliasing completely. In fact, the sampling theorem tells us exactly what rate is required. This theorem says that as long as the frequency of uniform sample points is greater than twice the maximum frequency present in the signal , it is possible to reconstruct the original signal perfectly from the samples. This minimum sampling frequency is called the Nyquist frequency.
For signals that are not band limited (), it is impossible to sample at a high enough rate to perform perfect reconstruction. Non-band-limited signals have spectra with infinite support, so no matter how far apart the copies of their spectra are (i.e., how high a sampling rate we use), there will always be overlap. Unfortunately, few of the interesting functions in computer graphics are band limited. In particular, any function containing a discontinuity cannot be band limited, and therefore we cannot perfectly sample and reconstruct it. This makes sense because the function’s discontinuity will always fall between two samples and the samples provide no information about the location of the discontinuity. Thus, it is necessary to apply different methods besides just increasing the sampling rate in order to counteract the error that aliasing can introduce to the renderer’s results.
7.1.4 Antialiasing Techniques
If one is not careful about sampling and reconstruction, myriad artifacts can appear in the final image. It is sometimes useful to distinguish between artifacts due to sampling and those due to reconstruction; when we wish to be precise we will call sampling artifacts prealiasing and reconstruction artifacts postaliasing. Any attempt to fix these errors is broadly classified as antialiasing. This section reviews a number of antialiasing techniques beyond just increasing the sampling rate everywhere.
Although the image functions that we will be sampling are known to have infinite-frequency components and thus can’t be perfectly reconstructed from point samples, it is possible to reduce the visual impact of aliasing by varying the spacing between samples in a nonuniform way. If denotes a random number between 0 and 1, a nonuniform set of samples based on the impulse train is
For a fixed sampling rate that isn’t sufficient to capture the function, both uniform and nonuniform sampling produce incorrect reconstructed signals. However, nonuniform sampling tends to turn the regular aliasing artifacts into noise, which is less distracting to the human visual system. In frequency space, the copies of the sampled signal end up being randomly shifted as well, so that when reconstruction is performed the result is random error rather than coherent aliasing.
Another approach that has been suggested to combat aliasing is adaptive supersampling: if we can identify the regions of the signal with frequencies higher than the Nyquist limit, we can take additional samples in those regions without needing to incur the computational expense of increasing the sampling frequency everywhere. It can be difficult to get this approach to work well in practice, because finding all of the places where supersampling is needed is difficult. Most techniques for doing so are based on examining adjacent sample values and finding places where there is a significant change in value between the two; the assumption is that the signal has high frequencies in that region.
In general, adjacent sample values cannot tell us with certainty what is really happening between them: even if the values are the same, the functions may have huge variation between them. Alternatively, adjacent samples may have substantially different values without any aliasing actually being present. For example, the texture-filtering algorithms in Chapter 10 work hard to eliminate aliasing due to image maps and procedural textures on surfaces in the scene; we would not want an adaptive sampling routine to needlessly take extra samples in an area where texture values are changing quickly but no excessively high frequencies are actually present.
Another approach to eliminating aliasing that sampling theory offers is to filter (i.e., blur) the original function so that no high frequencies remain that can’t be captured accurately at the sampling rate being used. This approach is applied in the texture functions of Chapter 10. While this technique changes the character of the function being sampled by removing information from it, blurring is generally less objectionable than aliasing.
Recall that we would like to multiply the original function’s spectrum with a box filter with width chosen so that frequencies above the Nyquist limit are removed. In the spatial domain, this corresponds to convolving the original function with a sinc filter,
In practice, we can use a filter with finite extent that works well. The frequency space representation of this filter can help clarify how well it approximates the behavior of the ideal sinc filter.
Figure 7.10 shows the function convolved with a variant of the sinc with finite extent that will be introduced in Section 7.8. Note that the high-frequency details have been eliminated; this function can be sampled and reconstructed at the sampling rate used in Figure 7.9 without aliasing.
7.1.5 Application to Image Synthesis
The application of these ideas to the 2D case of sampling and reconstructing images of rendered scenes is straightforward: we have an image, which we can think of as a function of 2D image locations to radiance values :
The good news is that, with our ray tracer, we can evaluate this function at any point that we choose. The bad news is that it’s not generally possible to prefilter to remove the high frequencies from it before sampling. Therefore, the samplers in this chapter will use both strategies of increasing the sampling rate beyond the basic pixel spacing in the final image as well as nonuniformly distributing the samples to turn aliasing into noise.
It is useful to generalize the definition of the scene function to a higher dimensional function that also depends on the time and lens position at which it is sampled. Because the rays from the camera are based on these five quantities, varying any of them gives a different ray and thus a potentially different value of . For a particular image position, the radiance at that point will generally vary across both time (if there are moving objects in the scene) and position on the lens (if the camera has a finite-aperture lens).
Even more generally, because many of the integrators defined in Chapters 14 through 16 use statistical techniques to estimate the radiance along a given ray, they may return a different radiance value when repeatedly given the same ray. If we further extend the scene radiance function to include sample values used by the integrator (e.g., values used to choose points on area light sources for illumination computations), we have an even higher dimensional image function
Sampling all of these dimensions well is an important part of generating high-quality imagery efficiently. For example, if we ensure that nearby positions on the image tend to have dissimilar positions on the lens, the resulting rendered images will have less error because each sample is more likely to account for information about the scene that its neighboring samples do not. The Sampler classes in the next few sections will address the issue of sampling all of these dimensions effectively.
7.1.6 Sources of Aliasing in Rendering
Geometry is one of the most common causes of aliasing in rendered images. When projected onto the image plane, an object’s boundary introduces a step function—the image function’s value instantaneously jumps from one value to another. Not only do step functions have infinite frequency content as mentioned earlier, but, even worse, the perfect reconstruction filter causes artifacts when applied to aliased samples: ringing artifacts appear in the reconstructed function, an effect known as the Gibbs phenomenon. Figure 7.11 shows an example of this effect for a 1D function. Choosing an effective reconstruction filter in the face of aliasing requires a mix of science, artistry, and personal taste, as we will see later in this chapter.
Very small objects in the scene can also cause geometric aliasing. If the geometry is small enough that it falls between samples on the image plane, it can unpredictably disappear and reappear over multiple frames of an animation.
Another source of aliasing can come from the texture and materials on an object. Shading aliasing can be caused by texture maps that haven’t been filtered correctly (addressing this problem is the topic of much of Chapter 10) or from small highlights on shiny surfaces. If the sampling rate is not high enough to sample these features adequately, aliasing will result. Furthermore, a sharp shadow cast by an object introduces another step function in the final image. While it is possible to identify the position of step functions from geometric edges on the image plane, detecting step functions from shadow boundaries is more difficult.
The key insight about aliasing in rendered images is that we can never remove all of its sources, so we must develop techniques to mitigate its impact on the quality of the final image.
7.1.7 Understanding Pixels
There are two ideas about pixels that are important to keep in mind throughout the remainder of this chapter. First, it is crucial to remember that the pixels that constitute an image are point samples of the image function at discrete points on the image plane; there is no “area” associated with a pixel. As Alvy Ray Smith (1995) has emphatically pointed out, thinking of pixels as small squares with finite area is an incorrect mental model that leads to a series of errors. By introducing the topics of this chapter with a signal processing approach, we have tried to lay the groundwork for a more accurate mental model.
The second issue is that the pixels in the final image are naturally defined at discrete integer coordinates on a pixel grid, but the Samplers in this chapter generate image samples at continuous floating-point positions. The natural way to map between these two domains is to round continuous coordinates to the nearest discrete coordinate; this is appealing since it maps continuous coordinates that happen to have the same value as discrete coordinates to that discrete coordinate. However, the result is that given a set of discrete coordinates spanning a range , the set of continuous coordinates that covers that range is . Thus, any code that generates continuous sample positions for a given discrete pixel range is littered with offsets. It is easy to forget some of these, leading to subtle errors.
If we instead truncate continuous coordinates to discrete coordinates by
and convert from discrete to continuous by
then the range of continuous coordinates for the discrete range is naturally and the resulting code is much simpler (Heckbert 1990a). This convention, which we will adopt in pbrt, is shown graphically in Figure 7.12.