less_retarded_wiki/bilinear.md

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, popularly e.g. in 3D computer graphics for texture filtering; bilinear interpolation allows to upscale textures to higher resolutions (i.e. compute new pixels between existing pixels) while 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 have bilinear filtering built-in (e.g. GL_LINEAR in OpenGL). Of course this method may be used to smooth not just textures but anything, for example terrain heightmaps or just any discrete mathematical function that we simply want to have defined everywhere, it's not just graphics thing, but here we will focus on its application in graphics.

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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