How to convert a 24 Bit PNG to 3 Bit PNG using Floyd–Steinberg dithering? java.awt.image.BufferedImage should be used to get and set RGB values.
On wik
Use image.getRGB(x, y) and image.setRGB(x, y, color) and use the pseudocode from the wikipedia article. Note that code on the wiki does not say how to "subtract", "add" and "multiply" colors. (The T3 class below handles "color" manipulation.)
The code below will produce this screenshot:

class Test {
private static BufferedImage floydSteinbergDithering(BufferedImage img) {
C3[] palette = new C3[] {
new C3( 0, 0, 0),
new C3( 0, 0, 255),
new C3( 0, 255, 0),
new C3( 0, 255, 255),
new C3(255, 0, 0),
new C3(255, 0, 255),
new C3(255, 255, 0),
new C3(255, 255, 255)
};
int w = img.getWidth();
int h = img.getHeight();
C3[][] d = new C3[h][w];
for (int y = 0; y < h; y++)
for (int x = 0; x < w; x++)
d[y][x] = new C3(img.getRGB(x, y));
for (int y = 0; y < img.getHeight(); y++) {
for (int x = 0; x < img.getWidth(); x++) {
C3 oldColor = d[y][x];
C3 newColor = findClosestPaletteColor(oldColor, palette);
img.setRGB(x, y, newColor.toColor().getRGB());
C3 err = oldColor.sub(newColor);
if (x+1 < w) d[y ][x+1] = d[y ][x+1].add(err.mul(7./16));
if (x-1>=0 && y+1