Floyd–Steinberg dithering

Left: original picture; middle: no dithering; right: Floyd–Steinberg dithering;

An image of the Statue of David, dithered with Floyd–Steinberg algorithm

Floyd–Steinberg dithering is an image dithering algorithm first published in 1976 by Robert W. Floyd and Louis Steinberg. It is commonly used by image manipulation software, for example when an image is converted into GIF format that is restricted to a maximum of 256 colors.

The algorithm achieves dithering using error diffusion, meaning it pushes (adds) the residual quantization error of a pixel onto its neighboring pixels, to be dealt with later. It spreads the debt out according to the distribution (shown as a map of the neighboring pixels):

{\begin{bmatrix}&&*&{\frac {\displaystyle 7}{\displaystyle 16}}&\ldots \\\ldots &{\frac {\displaystyle 3}{\displaystyle 16}}&{\frac {\displaystyle 5}{\displaystyle 16}}&{\frac {\displaystyle 1}{\displaystyle 16}}&\ldots \\\end{bmatrix}}

The pixel indicated with a star (*) indicates the pixel currently being scanned, and the blank pixels are the previously-scanned pixels. The algorithm scans the image from left to right, top to bottom, quantizing pixel values one by one. Each time the quantization error is transferred to the neighboring pixels, while not affecting the pixels that already have been quantized. Hence, if a number of pixels have been rounded downwards, it becomes more likely that the next pixel is rounded upwards, such that on average, the quantization error is close to zero.

The diffusion coefficients have the property that if the original pixel values are exactly halfway in between the nearest available colors, the dithered result is a checkerboard pattern. For example 50% grey data could be dithered as a black-and-white checkerboard pattern. For optimal dithering, the counting of quantization errors should be in sufficient accuracy to prevent rounding errors from affecting the result.

In some implementations, the horizontal direction of scan alternates between lines; this is called "serpentine scanning" or boustrophedon transform dithering.

In the following pseudocode we can see the algorithm described above. The values of the input image's pixels are normalized in floating point format to [0,1] with 0 (black) and 1 (white).

for each y from top to bottom
   for each x from left to right
      oldpixel  := pixel[x][y]
      newpixel  := find_closest_palette_color(oldpixel)
      pixel[x][y]  := newpixel
      quant_error  := oldpixel - newpixel
      pixel[x + 1][y    ] := pixel[x + 1][y    ] + quant_error * 7 / 16
      pixel[x - 1][y + 1] := pixel[x - 1][y + 1] + quant_error * 3 / 16
      pixel[x    ][y + 1] := pixel[x    ][y + 1] + quant_error * 5 / 16
      pixel[x + 1][y + 1] := pixel[x + 1][y + 1] + quant_error * 1 / 16

When converting 16 bit greyscale to 8 bit, find_closest_palette_color() may perform just a simple rounding, for example:

find_closest_palette_color(oldpixel) = round(oldpixel / 256)

References

Floyd–Steinberg Dithering (Graphics course project, Visgraf lab, Brazil)
R.W. Floyd, L. Steinberg, An adaptive algorithm for spatial grey scale. Proceedings of the Society of Information Display 17, 75–77 (1976).

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