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# Antialiasing

Antialiasing (AA) means preventing aliasing, i.e. distortion of signal (images, audio, video, ...) caused by discrete sampling. Popularly antialiasing is though of as "smooth edges in video game graphics", however that's an inaccurate normie simplification suggesting misunderstanding of the subject: yes, one of the most noticeable effects of 3D graphics antialiasing to a common folk is that of seeing smooth edges in video games, but smooth edges are not the primary goal, they are not the only effect and they are not even the most important effect of antialisng. Naturally, understanding antialiasing firstly requires understanding what aliasing is, which is not completely trivial (it's not the most difficult thing in the world either, but most people are just afraid of mathematics, so they prefer to stick with "antialiasing = smooth edges" simplification).

The basic summary is following: aliasing is an undesirable effect which may arise when we try to sample (capture) continuous signals potentially containing high frequencies (the kind of "infinitely complex" data we encounter in real world such as images or sounds) in discrete (non-continuous) ways by capturing the signal values at specific points in time (as opposed to capturing integrals of intervals), i.e. in ways native and natural to computers. Note that the aliasing effect is mathematical and is kind of a "punishment" for our "cheating" which we do by trying to simplify capturing of very complex signals, i.e. aliasing has nothing to do with noise or recording equipment imperfections, and it may occur not only when recording real world data but also when simulating real world, for example during 3D graphics rendering (which simulates capturing real world with a camera). A typical example of such aliasing effect is a video of car wheels rotating very fast (with high frequency) with a relatively low FPS camera, which then seem to be rotating very slowly and in opposite direction -- a high frequency signal (fast rotating wheels) caused a distortion (illusion of wheels rotating slowly in opposite direction) due to simplified discrete sampling (recording video as a series of photographs taken at specific points in time in relatively low FPS). Similar undesirable effects may appear e.g. on high resolution textures when they're scaled down on a computer screen (so called Moire effect), but also in sound or any other data. Antialiasing exploits the mathematical Nyquist–Shannon sampling theorem that says that aliasing cannot occur when the sampling frequency is high enough relatively to the highest frequency in the sampled data, i.e. antialiasing tries to prevent aliasing effects typically by either preventing high frequency from appearing in the sampled data (e.g. blurring textures, see MIP mapping) or by increasing the sampling frequency (e.g. multisampling). As a side effect of better sampling we also get things such as smoothly rendered edges etc.

Perhaps it's obvious, but the prefix anti in antialising signifies that some methods may not absolutely eliminate all aliasing, just suppress it. Completely preventing aliasing is usually possible but may be costly or otherwise inconvenient. For example the FXAA (fast approximate antialiasing) method is a postprocessing algorithm which works with an already rendered (potentially aliased) image and attempts to apply a kind of smoothing or "plastic surgery" to make it look as if it was properly rendered in ways correctly preventing aliasing, however by principle it can't always provide a completely ideal result as it simply doesn't know the original signal, it can't retrieve information that's been lost, all it can do is try to give us a good enough approximation.

Mkay, that's all nice, but how to actually DO antialiasing? There are several ways, depending on the kind of data (e.g. the number of dimensions of the signal or what frequencies you expect in them) or required quality (whether you want to prevent aliasing completely, how precisely you want to do it, if you're fine with just suppressing aliasing etc.). As already stated, everything revolves around the Nyquist–Shannon sampling theorem which says that aliasing cannot occur if the sampling frequency is at least twice as high as the highest frequency in the sampled signal. I.e. ensuring the sampling frequency is high enough relative to the highest frequency in the sampled signal completely prevents aliasing -- you can do this by either processing the input signal with a low pass filter (e.g. blurring an image) or by increasing your sampling frequency (e.g. rendering at higher resolution). Some specific antialiasing methods include:

- avoiding/hiding aliasing: A pretty straightforward way :) Aliasing can be avoided for instance simply by using low resolution textures. A way of hiding aliasing may be for example distance fog that decreases color contrast and makes aliasing less noticeable in spite of its presence.
- multisampling (MSAA), supersampling (SSAA) etc.: Increasing sampling frequency, typically in graphics rendering. The specific methods vary by where and how they increase the number of samples (some methods increase sampling uniformly everywhere, some try to detect aliasing areas and only put more samples there etc). A simple (but expensive) way of doing this is rendering the image at higher resolution and then scaling it back down.
- FXAA: Cheating, approximation of antialiasing with postprocessing (tl;dr: detecting lines and then smoothing them), usually implemented as a pixel/fragment shader, cheap but can be imperfect.
- MIP mapping: Way of preventing aliasing in rendering of scaled-down textures by keeping precomputed scaled-down antialiased versions of it.
- anisotrpic filtering: Improved version of MIP mapping.
- motion blur: Temporal antialiasing in video, basically increasing the number of samples in the time domain.
- ...

## Code Example

Here is a quite primitive example of supersampling (one of the simplest antialiasing methods) in C. We will draw a two dimensional "fish eye" distorted sine pattern (similar to checkerboard pattern but smooth, to show that aliasing happens even with smooth images!) that gets smaller towards the edges, i.e. the pattern is quite big in the center but near the edges the brightness oscillates with subpixel frequency which will lead to aliasing. First we'll draw the pattern as is, i.e. taking one sample per each pixel, letting aliasing happen; then we'll try to suppress aliasing by taking multiple samples per each pixel and averaging them -- this effectively increases our sampling frequency. It is basically equivalent to drawing the picture in increased resolution and then smoothly downsizing it (but in practice we don't do this as we'd waste a lot of RAM on storing the big resolution picture, which is completely unnecessary). Let's see the code.

    #include <stdio.h>
    #include <math.h>

    #define W 64  // image width
    #define H 32  // image height
    #define S 9.0 // pattern scale

    const char palette[] = "#OVaxsflc/!;,.- ";

    double sample(double x, double y) // function sampling our pattern
    {
      return sin(S / (x < 0 ? (x + 1) : (1 - x))) *
             sin(S / (y < 0 ? (y + 1) : (1 - y)));
    }

    char doubleToChar(double x) // maps <-1,1> brightness to palette character
    {
      int i = ((x + 1) / 2.0) * 15;
      return palette[i < 0 ? 0 : (i > 15 ? 15 : i)];
    }

    void draw(int antialiasSamples)
    {
      #define OFFSET 0.0001 // this tiny offset makes the pictures a bit nicer

      double
        x, y = -1 + OFFSET,
        stepX = 2.0 / W,
        stepY = 2.0 / H;

      double
        aaStepX = stepX / antialiasSamples,
        aaStepY = stepX / antialiasSamples;

      for (int j = 0; j < H; ++j) // draw rows
      {
        x = -1 + OFFSET;

        for (int i = 0; i < W; ++i) // draw columns
        {
          double r = 0;

          for (int l = 0; l < antialiasSamples; ++l)
            for (int k = 0; k < antialiasSamples; ++k)
              r += sample(x + k * aaStepX,y + l * aaStepY);

          putchar(doubleToChar(r / (antialiasSamples * antialiasSamples)));
          x += stepX;
        }

        y += stepY;
        putchar('\n');
      }
    }

    int main(void)
    {
      draw(1);
      putchar('\n');
      draw(8);
      return 0;
    }

Here are the results, first picture is without any antialiasing, second one with 8x8 supersampling (i.e. taking 64 samples per pixel):

    c//xfs/c!fcs/lxf//cfssflc/!//cllfllc//!/clfssfc//fxl/scf!c/sfscl
    /,!Vsa;c,x!a,cVs;,cxVVxl!;,,;/cfsfc/;,,;!lxVVxc,;sVc,a!x,/;afVcc
    fss/c/sfscf/sl/cssfc//clfssssfllcllfssssflc//cfssc/ls/fcsfs/l/fl
    /,;OsV;/.x!V,cOs;,/xVVxl!,.,;!cfsfc!;,.,!lxVVx/,;sOc,V!x./;VfV/c
    !-,#sO./-a;O-c#x.-/a##al;.--.;cfxfc;.--.;la##a/-.x#c-O;a-/.#f#/c
    c!!afx!c;s/x!caf!!csaxsl/!;;!/clslc/!;;!/lsxasc!!fac!x/s;c!xfacl
    /.,#sO,/-a!O.c#s,./a#Oxl;.-.,!cfxfc!,.-.;lxO#a/.,s#c.O!a-/,Of#/c
    x#V-/.Os#;a.#f-!O#s;--;laO##Oafc!cfaO##Oal;--;s#O!-f#.a;#sO-c-sf
    /,;OsV;/.x!V,cOs;,/xVVxl!,.,;!cfsfc!;,.,!lxVVx/,;sOc,V!x./;VfV/c
    c/csfs/c/fcs/lsf//cfssflc////cllfllc////clfssfc//fsl/scf/c/slscl
    s#V-/.Vs#;a.#f-/V#s;--;laO##Vafc!cfaO##Oal;--;s#V!-f#.a;#sO.c-sf
    fxx;c!xfa/s!xf;cxaf/;!/lsxaaxsfc/lfsxaaxsl/!;/faxc;fx!s/afx!c;fl
    c;!afx!c;s/x;caf!;csaasl/!;;!/cfsfc/!;;!/lsaasc;!fac;a/s;c!afacl
    !-,#sO./-a;O-c#x.-/a##al;.--.;cfxfc;.--.;la##a/-.x#c-#;a-/.#f#/c
    /,;OsV;/.x!V,cOs;,/xVVxl!,.,;!cfsfc!;,.,!lxVVx/,;sOc,V!x./;VfV/c
    lccflfclcfcfclflcclfffflccccccllfllcccccclfffflcclflcfcfclcflfll
    fxs!c!sfx/s!xl!csxf/!!/lsxxxsfflclffsxxxsl/!!/fxsc!fx!s/xfs!c!fl
    lccflfclcfcfclflcclfffflccccccllfllcccccclfffflcclflcfcfclcflfll
    /,;OsV;/.x!V,cOs;,/xOVxl!,.,;!cfsfc!;,.,!lxVVx/,;sOc,V!x./;VfV/c
    !-,#sO./-a;O-c#x.-/a##al;.--.;cfxfc;.--.;la##a/-.x#c-#;a-/.#f#/c
    c;!afx!c;s/x;caf!;csaasl/!;;!/cfsfc/!;;!/lsaasc;!fac;x/s;c!afacl
    fax;c!xfa/s!xf;cxaf/;!/lsxaaxsfc/cfsxaaxsl/!;/faxc;fx!s/afx!c;fl
    s#V-/.Vs#;a.#f-/V#s;--;laO##Vafc!cfaV##Oal;--;s#V!-f#.a;#sO.c-sf
    c/csfs/c/fcs/lsf//cfssflc////cllfllc////clfssfc//fsl/scf/c/slscl
    /,;OsV;/.x!V,cOs;,/xVVxl!,.,;!cfsfc!;,.,!lxVVx/,;sOc,V!x./;VfV/c
    x#V-/.Os#;a.#f-!O#s;--;laO##Oafc!cfaO##Oal;--;s#O!-f#.a;#sO-c-sf
    /.,#sO,/-a!O.c#s,./a#Oxl;.-.,!cfxfc!,.-.;lxO#a/.,s#c.O!a-/,Of#/c
    c;!afx!c;s/x!caf!;csaxsl/!;;!/cfsfc/!;;!/lsxasc;!fac!x/s;c!xfacl
    !-,#sO./-a;O-c#x.-/a##al;.--.;cfxfc;.--.;la##a/-.x#c-O;a-/.#f#/c
    /,;OsV;/.x!V,cOs;./xOVxl!,..;!cfsfc!;..,!lxVOx/.;sOc,V!x./,VfO/c
    fffclcflfcfcflccfflcccclffffffllcllfffffflcccclffcclfcfcflfclcll
    c/csfs/c/fcs/lsf//cfssflc////cllfllc////clfssfc//fsl/scf/c/slscl

    llllllllllllflclfflcccllfffffllllllfffffllccclfflccflcflllllllll
    llllllllllllflclfflcccllfffffllllllfffffllccclfflccflcflllllllll
    llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
    llllllllllllclflcclfffllccccclllllllccccllffflcclflclfllllllllll
    lllllllllfclfcclfflcccllfffffllllllfffffllccclfflccflcfcllllllll
    lllllllccs/lx/!fxsl!!!cfsxxxsflcclfsxxxxfc/!!csxf!/sf/scllfllcll
    ffflllcff!xl,xVc.;fVOas/;..,;/lssl/;,..;/faOVf;./aa;ca;sllcflflf
    ccclllfccx!lV!,fOac,.,/saOOVasl//lsaVOOVsc;.,caOs,;af;a/llfcfclc
    ffflllcff!xl.xOc.,fV#Vs/,--.;/lssl/;.--,/fV#Of;-/Va,ca;xflcfcflf
    llllllllllllflclflllcllllfffllllllllfffllllccllfllllllllllllllll
    ccclllfcca!lO!.f#Vc.-./sV##OVxl//lsaO##Vsc,-.cV#s,;Vf,a!clscfclc
    lllllllllfclsc/lsflc/ccffsssfflcclffsssfflc//lfsfccflcfcllllllll
    ffflllcff!sl;sac,;laVafc;,,,!/lfsl/!;,,;/faVaf!,/ax;cx!sllcflflf
    ffflllcff;xl.aOc-,fO#Os/,--.,!lssl/,.--,/fV#Of,-/OV,cV,xfl/fcflf
    ffflllcff/sl;sac;!laVafc!,,;!/lffl/!;,,;/fxVaf!,cax!cx!sllcflflf
    llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
    lllllllllfclsc/lssl///cfssssfflcclffssssfl///lfsf/cslcfcllllllll
    lllllllllcfl/fsl//lsssfc/////clfflcc////clssslc/csf/lfcfllclllll
    ffflllcff!xl,xVc.;fVOVs/;...;/lssl/;,..,/faOVf;./Va,ca;sllcfcflf
    ffflllcff!xl.xOc-,fV#Vs/,--.,!lssl/;.--,/fV#Of;-/Va,ca;xfl/fcflf
    lllllllllcfl/fsl//lsssfc/////clfflc/////clsssl//css/ls/fllclllll
    ccclllfccx/la/;fVal;,;cfaVVVxslc/lsxaVVasc;,;cxVs;!af!x/llfclclc
    ccclllfccx!lV!,fOac,.,/saOOVasl//lsaVOOasc;.,caOs,;af;a/llfclclc
    ffflllcff/sl;sac;!laVxfc!;,;!/lfflc!;,;!/fxVaf!;cxx!cx!sllcllflf
    lllllllllcfl/fsl//lsssfc////cclfflcc////clssslc/csf/lfcfllllllll
    cccllllccs/la/;faxl!;!cfxaVaxslcclfxaaaxsc!;;cxaf!!xf!x/llfllclc
    lllllllllcflcfflcclfffllcccccllllllcccccllffflcclffclfclllllllll
    fffllllff/sl;sac;!lxaxfc!;,;!/lfflc!;;;!/fxaaf!;cxx!cx!sllcllflf
    llllllllllllllllflllcllllffflllllllllffllllclllfllllllllllllllll
    lllllllllcflcfflcclfsfllc//cccllllccc//cclfsflc/lffclfcfllllllll
    llllllllllllllflclllfllllcccllllllllcccllllflllcllllllllllllllll
    llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

It's a bit harder to see with ASCII art but still it is noticeable even here (the more it will be seen in "real" graphics) -- the first version of the image is much noisier, despite the underlying pattern itself being smooth, just sampled differently. Notice the images really significantly differ near the edges where aliasing appears, the center is basically the same -- here we can spot an obvious weakness of supersampling: that we have wasted computational power on supersampling the part which didn't need it. You can think of ways how this could be improved. Also think for example about placing our samples within one pixel differently than in a uniform grid -- what effect would it have? Here are things for you to explore.

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