# Bilinear Interpolation

Bilinear interpolation (also bilinear filtering) is a simple way of creating a smooth transition (interpolation) between discrete samples (values) in 2D, it is a generalization of linear interpolation to 2 dimensions. It is used in many places and popularly encountered e.g. in 3D computer graphics as a method of **texture filtering**; bilinear interpolation allows for upscaling textures to higher resolutions (i.e. insert new pixels in between existing pixels) whilst keeping their look smooth and "non-blocky" (even though blurry). On the scale of quality vs simplicity it is kind of a middle way between a simpler nearest neighbour interpolation (which creates the "blocky" look) and more complex bicubic interpolation (which uses yet smoother curves but also requires more samples). Bilinear interpolation can further be generalized to trilinear interpolation (in computer graphics trilinear interpolation is used to also additionally interpolate between different levels of a texture's mipamap) and perhaps even bilinear extrapolation. Many frameworks/libraries/engines come with bilinear filtering built-in as a standard feature (e.g. `GL_LINEAR` in OpenGL). Of course this method is not limited to upscaling textures and can be applied to any set of discrete samples such as terrain heightmaps or any kind of discrete mathematical function which we want to automatically generalize to all real numbers, it's not something encountered only in computer graphics, even though this article will still mostly view it from the graphics perspective.

Why is it named *bilinear*? Probably because it's doing linear interpolation twice: once in *X* direction, then in *Y* direction.

```
####OOOOVVVVaaaaxxxxssssffffllllcccc////!!!!;;;;,,,,....----
####OOOOVVVVaaaaxxxxxssssffffllllcccc////!!!!;;;;,,,,.....----
####OOOOVVVVaaaaaxxxxssssfffflllllcccc////!!!!!;;;;,,,,....-----
###OOOOOVVVVaaaaaxxxxsssssfffflllllcccc////!!!!!;;;;,,,,,....---
###OOOOVVVVVaaaaaxxxxsssssfffffllllccccc/////!!!!!;;;;,,,,,.....
##OOOOOVVVVVaaaaaxxxxxsssssffffflllllcccc/////!!!!!;;;;;,,,,,...
##OOOOOVVVVVaaaaaxxxxxsssssfffffflllllccccc/////!!!!!;;;;;,,,,,.
#OOOOOOVVVVVaaaaaxxxxxxsssssfffffflllllccccc//////!!!!!;;;;;;,,,
OOOOOOVVVVVVaaaaaaxxxxxssssssfffffflllllcccccc//////!!!!!;;;;;;,
OOOOOOVVVVVVaaaaaaxxxxxxssssssffffffllllllcccccc//////!!!!!!;;;;
OOOOOVVVVVVVaaaaaaxxxxxxsssssssfffffflllllllcccccc///////!!!!!!;
OOOOOVVVVVVaaaaaaaxxxxxxxsssssssffffffflllllllccccccc//////!!!!!
OOOOVVVVVVVaaaaaaaaxxxxxxxsssssssfffffffflllllllccccccc////////!
OOOVVVVVVVVaaaaaaaaxxxxxxxxssssssssffffffffllllllllcccccccc/////
OOVVVVVVVVVaaaaaaaaaxxxxxxxxsssssssssfffffffflllllllllcccccccc//
OVVVVVVVVVVaaaaaaaaaxxxxxxxxxssssssssssffffffffflllllllllccccccc
VVVVVVVVVVaaaaaaaaaaaxxxxxxxxxxssssssssssfffffffffffllllllllllcc
VVVVVVVVVVaaaaaaaaaaaxxxxxxxxxxxxsssssssssssffffffffffffllllllll
VVVVVVVVVVaaaaaaaaaaaaxxxxxxxxxxxxxsssssssssssssffffffffffffflll
VVVVVVVVVaaaaaaaaaaaaaaaxxxxxxxxxxxxxxsssssssssssssssfffffffffff
VVVVVVVVaaaaaaaaaaaaaaaaaxxxxxxxxxxxxxxxxxxsssssssssssssssssffff
VVVVVVVaaaaaaaaaaaaaaaaaaaaaxxxxxxxxxxxxxxxxxxxxssssssssssssssss
VVVVVVaaaaaaaaaaaaaaaaaaaaaaaaaxxxxxxxxxxxxxxxxxxxxxxxxxxsssssss
VVVaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaxxxxxxxxxxxxxxxxxxxxxxxxxxx
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaxxxxxxxxxxxxxxx
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaVVVVVVVVVVVVVVVVVVV
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
aaaaaaaaaaaaaaaaaaaaaaaaVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVOOOOOO
aaaaaaaaaaaaaaaaaaaaaVVVVVVVVVVVVVVVVVVVVVVVVVVOOOOOOOOOOOOOOOOO
aaaaaaaaaaaaaaaaaaaaVVVVVVVVVVVVVVVVVVVVOOOOOOOOOOOOOOOOOOOOO###
```

*The above image is constructed by applying bilinear interpolation to the four corner values.*

The principle is simple: first linearly interpolate in one direction (e.g. horizontal), then in the other (vertical). Mathematically the order in which we take the dimensions doesn't matter (but it may matter practically due to rounding errors etc.).

Example: let's say we want to compute the value *x* between the four following given corner values:

```
1 . . . . . . 5
. . . . . . . .
. . . . . . . . 
. . . . . . . .
. . . . . . . .
. . . . x . . .
. . . . . . . .
8 . . . . . . 3
```

Let's say we first interpolate horizontally: we'll compute one value, *a*, on the top (between 1 and 5) and one value, *b*, at the bottom (between 8 and 3). When computing *a* we interpolate between 1 and 5 by the horizontal position of *x* (4/7), so we get *a = 1 + 4/7 * (5 - 1) = 23/7*. Similartly *b = 8 + 4/7 * (3 - 8) = 36/7*. Now we interpolate between *a* and *b* vertically (by the vertical position of *x*, 5/7) to get the final value *x = 23/7 + 5/7 * (36/7 - 23/7) = 226/49 ~= 4.6*. If we first interpolate vertically and then horizontally, we'd get the same result (the value between 1 and 8 would be 6, the value between 5 and 3 would be 25/7 and the final value 226/49 again).

Here is a C code to compute all the inbetween values in the above, using fixed point (no float):

```
#include <stdio.h>

#define GRID_RESOLUTION 8

int interpolateLinear(int a, int b, int t)
{
  return a + (t * (b - a)) / (GRID_RESOLUTION - 1);
}

int interpolateBilinear(int topLeft, int topRight, int bottomLeft, int bottomRight,
  int x, int y)
{
#define FPP 16 // we'll use fixed point to prevent rounding errors

#if 1 // switch between the two versions, should give same results:
  // horizontal first, then vertical
  int a = interpolateLinear(topLeft * FPP,topRight * FPP,x);
  int b = interpolateLinear(bottomLeft * FPP,bottomRight * FPP,x);
  return interpolateLinear(a,b,y) / FPP;
#else
  // vertical first, then horizontal
  int a = interpolateLinear(topLeft * FPP,bottomLeft * FPP,y);
  int b = interpolateLinear(topRight * FPP,bottomRight * FPP,y);
  return interpolateLinear(a,b,x) / FPP;
#endif
}

int main(void)
{
  for (int y = 0; y < GRID_RESOLUTION; ++y)
  {
    for (int x = 0; x < GRID_RESOLUTION; ++x)
      printf("%d ",interpolateBilinear(1,5,8,3,x,y));

    putchar('\n');
  }

  return 0;
}
```

The program outputs:

```
1 1 2 2 3 3 4 5 
2 2 2 3 3 4 4 5 
3 3 3 3 4 4 4 5 
4 4 4 4 4 4 4 5 
5 5 5 5 5 5 5 4 
6 6 6 6 5 5 5 4 
7 7 7 6 6 5 5 4 
8 8 7 6 6 5 4 3
```

**Cool hack to improve bilinear interpolation** (from https://iquilezles.org/articles/texture): bilinear interpolation doesn't looks as good as bicubic but bicubic is a lot more complex on hardware and bandwidth as it requires fetching more texels -- there is one trick which shader programmers use to improve the look of bilinear filtering while not requiring fetching more texels. They use the `smoothstep` function on the interpolation parameter which eliminates instant "jumps" at edges between texels, it replaces straight lines with a smoother curve and so makes the derivative of the result continuous -- basically it looks a lot better. Still not as good as bicubic but close enough.

TODO: code for the above

For suckless programs that do their own software rendering an issue of bilinear interpolation, as compared with nearest neighbor, might be that it **creates new colors** by averaging colors in the filtered image, i.e. image filtered this way may have new colors introduced and this may become a problem e.g. if we are using palettes (indexed mode) with limited number of colors and possible operations with them. This may also complicated e.g. using precomputed scaling tables (used in old games like wolf 3D) that simply store mapping of the original image to pixels in an upscaled image. A possible attempt at a "fix" -- or rather more of a poor man's bilinear interpolation -- may be in dithering the colors rather than averaging; perhaps once we sample in between pixels we assign probabilities to the 4 nearest pixels, based on their distance to the sample position, and then take one of the four pixels at random with those probabilities using some pseudorandom generator.

TODO: test the above