Floyd-Steinberg dithering is a technique for reducing the colour palette of an image (for example, to reduce its file size) whilst keeping as much of the perceived detail as possible. For each pixel in the original image, the nearest colour to that pixel is chosen from a restricted palette and any "error" (difference in pixel colour value, original - new) is distributed across the neighbouring pixels as follows:

$$ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & * & \frac{7}{16} \\ \frac{3}{16} & \frac{5}{16} & \frac{1}{16} \end{array} $$

The pixel under consideration is labelled $*$ in the above. Since the pixels above and to the left of this pixel are not affected, the algorithm can be applied by scanning the image once, top-to-bottom, left-to-right.

The Python code below demonstrates the Floyd-Steinberg dithering technique, in comparison with simply choosing the nearest colour value from the restricted palette for each pixel. The image used is Vermeer's *Girl with a Pearl Earring* (either a tronie masterpiece or a depiction of gratuitous cultural appropriation depending on your point of view):

This image is available from Wikimedia as 1665_Girl_with_a_Pearl_Earring.jpg.

The original uses one byte for each of the red, green and blue channels giving a palette of $256^3 = 16,777,216$ colours. Setting `nc=2`

in the `fs_dither`

method below will restrict the palette to one bit (two colours) per channel and $2^3 = 8$ colours. The dithering algorithm (right hand side, below) clearly gives a better representation of the orignal image than simply replacing each pixel's colour with its nearest representation in the restricted palette (left hand side):

`nc = 2`

(two colours per RGB channel):

```
import numpy as np
from PIL import Image
GREYSCALE = False
img_name = '1665_Girl_with_a_Pearl_Earring.jpg'
# Read in the image, convert to greyscale.
img = Image.open(img_name)
if GREYSCALE:
img = img.convert('L')
width, height = img.size
new_width = 400
new_height = int(height * new_width / width)
img = img.resize((new_width, new_height), Image.ANTIALIAS)
def get_new_val(old_val, nc):
"""
Get the "closest" colour to old_val in the range [0,1] per channel divided
into nc values.
"""
return np.round(old_val * (nc - 1)) / (nc - 1)
# For RGB images, the following might give better colour-matching.
#p = np.linspace(0, 1, nc)
#p = np.array(list(product(p,p,p)))
#def get_new_val(old_val):
# idx = np.argmin(np.sum((old_val[None,:] - p)**2, axis=1))
# return p[idx]
def fs_dither(img, nc):
"""
Floyd-Steinberg dither the image img into a palette with nc colours per
channel.
"""
arr = np.array(img, dtype=float) / 255
for ir in range(new_height):
for ic in range(new_width):
# NB need to copy here for RGB arrays otherwise err will be (0,0,0)!
old_val = arr[ir, ic].copy()
new_val = get_new_val(old_val, nc)
arr[ir, ic] = new_val
err = old_val - new_val
# In this simple example, we will just ignore the border pixels.
if ic < new_width - 1:
arr[ir, ic+1] += err * 7/16
if ir < new_height - 1:
if ic > 0:
arr[ir+1, ic-1] += err * 3/16
arr[ir+1, ic] += err * 5/16
if ic < new_width - 1:
arr[ir+1, ic+1] += err / 16
carr = np.array(arr/np.max(arr, axis=(0,1)) * 255, dtype=np.uint8)
return Image.fromarray(carr)
def palette_reduce(img, nc):
"""Simple palette reduction without dithering."""
arr = np.array(img, dtype=float) / 255
arr = get_new_val(arr, nc)
carr = np.array(arr/np.max(arr) * 255, dtype=np.uint8)
return Image.fromarray(carr)
for nc in (2, 3, 4, 8, 16):
print('nc =', nc)
dim = fs_dither(img, nc)
dim.save('dimg-{}.jpg'.format(nc))
rim = palette_reduce(img, nc)
rim.save('rimg-{}.jpg'.format(nc))
```

More images with different values of `nc`

:

`nc = 3`

(three colours per RGB channel):

`nc = 4`

(four colours per RGB channel):

`nc = 8`

(eight colours per RGB channel):

`nc = 16`

(16 colours per RGB channel):

## Comments

Comments are pre-moderated. Please be patient and your comment will appear soon.

## Uri 1 year, 12 months ago

256 * 256 * 256 = 16777216 colours

Link | Reply## christian 1 year, 12 months ago

You're right: fixed!

Link | Reply## Matteo 1 year ago

Awesome. The commented out alternative `get_new_val` is great for a custom palette, which is essential for generating an image for a 7-colour Waveshare e-paper or Pimoroni e-ink etc. Namely, the calculated np.array `p` is the palette, which can be replaced with a custom one (i.e. the RGB tuples as floats of black, white, green, blue, red, yellow, orange) disregarding `nc`.

Link | Reply## christian 1 year ago

Great – good to know it's being put to good use!

Link | Reply## Laksiri 6 months, 3 weeks ago

Hi this is very interesting, I want to know a further explanation on how the number of colours get restricted per channel as the given explanation is not understood by me. You mention that each colour channel in the original is represented using 1 byte and the setting (nc-2) reduces it to one bit, can you please explain this in more details please. get_new_val function multiplies the old pixel value by (nc-1) and divides by the same (nc-1), wont this operation always have the same result because you multiply and divide by the same value. How does it limit the colour channels as you say to two.

Link | Reply## christian 6 months, 3 weeks ago

Dear Laksiri,

Link | ReplyYes – it should be "nc=2" (i.e. if number of colours is 2 then instead of 3 bytes = 24 bits per pixel (1 byte for each RGB component) you only need 1 bit for each pixel. (I've fixed the typo).

The point about the get_new_val function is that it maps a real number between 0 and 1 to the nearest point on a grid of nc real numbers: e.g. if you have a grid of 11 real numbers 0, 0.1, 0.2, 0.3, ... 1.0 and you want to know which of those the number 0.52 is closest to, one way is the approach taken in this function: multiply by 10 (to get 5.2), round to the nearest integer (5), and then normalize back to a real number by dividing by 10 (to get 0.5). Note that because of the rounding, the multiplication and division do not cancel.

I hope that helps, Christian

## mischl 1 month, 4 weeks ago

thankfully used and mentioned the code: https://github.com/mischlrebl/zpl-grayscale-prep

Link | Reply## New Comment