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# Anti-aliasing

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Anti-aliasing in digital signal processing is the technique of minimizing aliasing when representing a high-resolution signal at a lower resolution.

The term aliasing is used here in its radio-engineering sense, originally used to describe to the phenomenon causing the same transmission to be heard at multiple dial positions on a superheterodyne radio if pre-filtering is not used.

In most cases, anti-aliasing means removing data at too high a frequency to represent. When such data is left in a signal, it causes unpredictable artifacts such as the black-and-white noise in figure 1.

 (a) (b) Figure 1 Compare the diiamond on the left with the antialiased one on the right Enlarged view Figure 2

Figure 1-a illustrates visual distortion which occurs when anti-aliasing is not used. Notice that near the top of the image, where the checkerboard is very distant, the image is impossible to recognize, and is displeasing to the eye. By contrast, figure 1-b is anti-aliased. The checkerboard near the top blends into gray, which is usually the desired effect when the resolution is insufficient to show the detail. Even near the bottom of the image, the edges appear much smoother in the anti-aliased image. Fig 2 shows how anti-aliasing affects small black and white images. Text would be affected in this way. The enlarged image shows how anti aliasing adds gray pixels around the boarder between black and white. This visually smooths the outline.

The idealized image has infinite detail, and we represent it using a continuous function f(x,y) where x and y are real numbers defining co-ordinates[?].

There are infinitely many such functions. However, the computer screen is capable of displaying only finitely many different images. Indeed, an ordinary computer screen has no more than a few million pixels, and each pixel only has a finite number of colors it can display.

Hence, to convert an image f(x,y) into something that the screen can display, we must simplify it. By the pigeonhole principle, sometimes two ideal images f(x,y) and g(x,y) will be converted to the same picture on the screen. This cannot be avoided. The question is how to choose the reduced image so that it looks better.

An example of a poor choice is illustrated in Figure 1-a. The most direct way to simplify an image for display is to use a sample of the image at f(i,j) for each pixel (i,j). That is how figure 1-a was generated. At the top of the checkerboard, multiple black and white tiles may be represented by a single pixel. But since only black and white points occur in the ideal image, an area containing both colors in similar proportion will be represented with a strange pattern of black and white. This type of aliasing is called a Moiré effect.

A better approach is to use the average intensity of a rectangular area in the scene corresponding to the parts of the scene nearer to this pixel than any other. This gives a better, but not yet ideal, "anti-aliased" appearance; figure 1-b was generated this way. To see why this works better, it helps to look at the problem from a signal-processing perspective.

In this approach, the ideal image is regarded as a signal, the image displayed on the screen is viewed as a certain filtering of the signal. Ideally, we would understand how the human brain would process the original signal, and provide the image on the screen which most resemble the original signal according to the human brain.

The most widely accepted method is to use the Fourier transform. The Fourier transform decomposes our signal into basic waves, or harmonics, and gives us the amplitude of each wave in our signal. The waves are of the form:

$\cos (2j \pi x) \cos (2k \pi y)$

where j and k are arbitrary non-negative integers. (In fact, there are also waves involving the sine, but for the purpose of this discussion, the cosine will suffice; see Fourier transform for technical details.)

The number j and k together are the frequency of the wave: j is the frequency in the x direction, and k is the frequency in the y direction.

It has been observed that to measure a signal of frequency n, you need at least n sample points, and they need to be well-placed. If your sample points occur near the zeros of the signal, you will be led to the believe that the signal is in fact zero.

And so, in signal processing, we chose to eliminate all high frequencies from the signal, keeping only the frequencies that are low enough to be sampled correctly by our sample rate.

Strictly speaking, there is no justification why this form of aliasing is less troublesome than some other form of averaging, such as the uniform averaging algorithm. However, experimentation has suggested that the Fourier approach is superior and somehow matches what the brain would expect to see.

The basic waves need not be cosine waves. See, for instance, wavelets. If one uses basic waves which are not cosine waves, one obtains a slightly different image. Some basic waves yield anti-aliasing algorithms which are not so good (for instance, the Haar wavelet gives the uniform averaging algorithm.) However, some wavelets are good, and it is possible that some wavelets are better at approximating the functioning of the human brain than the cosine basis.

There are only a handful of primitives used at the lowest level in a real-time rendering engine (either software or hardware accelerated.) These include "points", "lines" and "triangles". If one is to draw such a primitive in white against a black background, it is possible to design such a primitive to have fuzzy edges, achieving some sort of anti-aliasing. However, this approach has difficulty dealing with adjacent primitives (such as triangles that share an edge.)

To approximate the uniform averaging algorithm, it was suggested to have an extra buffer for sub-pixel data. The initial, and least memory-hungry approach, used 16 extra bits per pixel, in a 4x4 grid. If one renders the primitives in a careful order, for instance front-to-back, it is possible to create a reasonable image.

Since this requires that the primitives be in some order, and hence interacts poorly with an application programming interface such as OpenGL, the latest attempts simply have two or more full sub-pixels per pixel, including full color information for each sub-pixel. Some information may be shared between the sub-pixels (such as the Z-buffer.)

Aside from multiplying the number of sub-pixels per pixel, there is an older approach specialized for texture mapping called mip-mapping which does not require any sub-pixel samples, and in fact can improve the speed of the rendering by making the memory accesses more local, hence improving the performance of any cache system.

Mip-mapping works by creating lower resolution, prefiltered versions of the texture map. When rendering the image, the appropriate resolution mip-map is chosen and hence the texture pixels (texels) are already filtered when they arrive on the screen.

Typically, if a texture is of resolution 256x256 (for instance) then mip-maps will be created with the following resolutions: 128x128, 64x64, 32x32, 16x16, 8x8, 4x4, 2x2 and 1x1. The simplest way to generate these textures is by successive averaging, however more sophisticated algorithms (perhaps based on signal processing and Fourier transforms) can also be used.

The advantage here is that having such a hierarchy of texture maps requires only 1/3 more memory than having simply the base 256x256 texture map. However, in many instances, the filtering should not be uniform in each direction (it should be anisotropic, as opposed to isotropic), and a compromise resolution is used. If a higher resolution is used, the cache coherence goes down, and the aliasing is increased in one direction, but the image tends to be clearer. If a lower resolution is used, the cache coherence is improved, but the image is overly blurry, to the point where it becomes difficult to identify.

To help with this problem, nonuniform mip-maps (also known as rip-maps) are sometimes used. With a 16x16 base texture map, the rip map resolutions would be 16x8, 16x4, 16x2, 16x1, 8x16, 8x8, 8x4, 8x2, 8x1, 4x16, 4x8, 4x4, 4x2, 4x1, 2x16, 2x8, 2x4, 2x2, 2x1, 1x16, 1x8, 1x4, 1x2 and 1x1.

The unfortunate problem with this approach is that rip-maps require four times as much memory as the base texture map, and so rip-maps have been very unpopular.

To reduce the memory requirement, and simultaneously give more resolutions to work with, summed-area tables were conceived. Given a texture $(t_{jk})$, we can build a summed area table $(s_{jk})$ as follows. The summed area table has the same number of entries as there are texels in the texture map. Then, define

$s_{mn}:=\sum _{1 \leq j \leq m,\ 1 \leq k \leq n} t_{jk}$

Then, the average of the texels in the rectangle (a1,b1] × (a2,b2] is given by

$s_{a_2b_2}-s_{a_1b_2}-s_{a_2b_1}+s_{a_1b_1} \over {(a_2-a_1)(b_2-b_1)}$

However, this approach tends to exhibit poor cache behavior. Also, a summed area table needs to have wider types to store the partial sums $s_{jk}$ than the word size used to store $t_{jk}$. For these reasons, there isn't any hardware that implements summed-area tables today.

A compromise has been reached today, called anisotropic mip-mapping. In the case where an anisotropic filter is needed, a higher resolution mip-map is used, and several texels are averaged in one direction to get more filtering in that direction. This has a somewhat detrimental effect on the cache, but greatly improves image quality.

• Dithered ray tracing and anti-aliasing
• Statistical sampling and anti-aliasing
• Temporal anti-aliasing or motion blur
• Measure theory and anti-aliasing
• Font rasterization[?] and anti-aliasing
• In most real-world systems, gamma correction is required to linearise the response curve of the sensor and display systems. If this is not taken into account, the resultant non-linear distortion will defeat the purpose of anti-aliasing calculations based on the assumption of a linear system response.
• Color theory for certain physical details pertinent to images which are not grayscale.

Important early works in the history of anti-aliasing include:

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