NeuQuant: Fast High-Quality Image Quantization

The NeuQuant Neural-Net image quantization algorithm (© Anthony Dekker 1994) is a replacement for the common Median Cut algorithm. It is described in the article Kohonen neural networks for optimal colour quantization in Volume 5, pp 351-367 of the journal Network: Computation in Neural Systems, Institute of Physics Publishing, 1994 (PDF version available).

The Java version of the code, together with a test driver, are at the bottom of the page.

What does the algorithm do?

The algorithm performs quantization of 24-bit colour images to e.g. 8-bit colour. By adjusting a sampling factor, the network can either produce extremely high-quality images slowly, or produce good images in reasonable times. With a sampling factor of 1, the entire image is used in the learning phase, while with a factor of e.g. 10, a pseudo-random subset of 1/10 of the pixels are used in the learning phase. A sampling factor of 10 gives a substantial speed-up, with a small quality penalty. Careful coding and a novel indexing scheme are used to make the algorithm efficient. This confounds the frequent observation that Kohonen Neural Networks are necessarily slow.

The algorithm operates using a one-dimensional self-organizing Kohonen Neural Network, typically with 256 neurons, which self-organises through learning to match the distribution of colours in an input image. Taking the position in RGB-space of each neuron gives a high-quality colour map in which adjacent colours are similar. This map is then used to quantise the image.

The image above is an example neural network for the test image shown lower down on this page. The colours along the network show the RGB values inside the cubical RGB-space. The corresponding colour map (with the colours shown in sequence along the neural network) is:

As an alternative viewpoint, this diagram overlays the neural network on the distribution of colours in the test image at the bottom of this page (this was the most challenging image from our test suite, and performance was not quite as good as for photographic images, such as that shown in the paper:

It can be seen in this diagram that some colours in the image have no close match in the neural network, but these are very rare colours (pale yellow, red, and blue) only occurring in tiny highlights (see the test images below). Tiny distortions in these highlights introduced by quantization are not noticeable, although examination of the enlargements in the test images shows that NeuQuant does nevertheless minimise these distortions (compared with the other methods).

How good is the algorithm?

Very good! We use two quality measures: the average error per pixel, and the average error for 2×2 blocks centred on each pixel (which takes into account dithering). We measure speed using the number of seconds to quantise one million pixels on a (very old) SUN SPARC workstation. We compare with two versions of the common Median Cut algorithm: one is fast and dithered, the other is slower, better, and non-dithered. Results are as follows:

  1. NeuQuant with a sampling factor of 1: time=875, average error=5.34, block error=2.96.
  2. NeuQuant with a sampling factor of 10: time=119, average error=5.97, block error=3.71.
  3. Fast dithered Median Cut: time=36, average error=10.39, block error=4.20.
  4. Slower, improved Median Cut: time=122, average error=9.41, block error=5.94.

Visually the dithered Median Cut algorithm suffers from grain, while the slower Median Cut algorithm suffers from banding artifacts. NeuQuant with a sampling factor of 10 is slightly faster than the slower Median Cut, but with much higher quality. In addition, the space usage for NeuQuant is 8kB, plus the space needed for the image (as compared to 432kB plus the image for Median Cut). This means NeuQuant will benefit significantly from the use of a processor with a copy-back cache.

Compare the following test images. Each image contains an enlargement in the top left which was added after quantization:

NeuQuant with a sampling factor of 1

Time=875, average error=5.34, block error=2.96 (Highest quality).

NeuQuant with a sampling factor of 10

Time=119, average error=5.97, block error=3.71 (Good compromise).

Fast Dithered Median Cut

Time=36, average error=10.39, block error=4.20 (Grainy image).

Slower, Improved Median Cut

Time=122, average error=9.41, block error=5.94 (Banding artifacts).

Original Image

A PNG version of the original test image is also available.

Frequently Asked Questions

Who wrote NeuQuant?

Anthony (Tony) Dekker

Is there an electronic version of the paper?

Yes! See the top of this page.

The algorithm doesn’t work for 8 (or 4, or 16) colours

Well, it wasn’t meant to. It was intended to produce 6 to 8-bit images (64 to 256 colours) from 24-bit images. But one technique that works is:

  1. Run the algorithm with N=100 (or perhaps 30) colours and a sampling factor of 10 (or perhaps 30).
  2. Create an NxN matrix of the distances between the colours.
  3. Repeatedly find the two most similar colours, and eliminate one, until only the desired number of colours are left.
  4. For each image pixel, find the best-matching colour in this set (dithering will probably be required).

The FreeImage port doesn’t work

The code as written assumes that pixels are not word aligned, while FreeImage assumes that they are. See the fixes for 4-byte boundary alignment below.

Do you know of any similar work?

Yes, see:

Are people using NeuQuant?

Yes indeed, for example:

Do I need to pay a license fee to use NeuQuant?

No, but you must include the copyright notice at the top of the program in the source code and documentation of the program using NeuQuant.

What about routines to read and write image formats?

I’m afraid you’ll have to supply your own, based on your preferred formats. See the test driver code and Jürgen Weigert’s netpbm wrapper for example code.

How do I fix the code to handle 4-byte boundary alignment?

It’s easy. Either do it in image load/store, or another way is:

  1. Fix the the calculation of samplepixels in learn().
  2. Fix the pixel access code in learn().
  3. Fix code to update p in learn().
  4. Fix the loop that calls inxsearch(b,g,r).

Can the code be adapted for the CIE Lab color space?

Sure! Most of the algorithm just operates on triples of sub-pixels, and doesn’t care whether they are RGB or something else. The only required change is in the inxbuild() and inxsearch() functions, which index the colour map on the green sub-pixels first (relevant lines of code are marked), because the human eye is more sensitive to green. In Lab space, indexing on L first might be more appropriate.

Have you done other work with Kohonen neural networks?

Yes, they are very powerful. See, for example:

Java source code

// Constructors:
//    public NeuQuant (Image im, ImageObserver obs) throws IOException -- default sample = 1
//    public NeuQuant (int sample, Image im, ImageObserver obs) throws IOException
//    public NeuQuant (Image im, int w, int h) throws IOException -- default sample = 1
//    public NeuQuant (int sample, Image im, int w, int h) throws IOException

// Initialisation method: call this first
//    public void init ()

// Methods to look up pixels (use in a loop)
//    public int convert (int pixel)
//    public int lookup (int pixel)
//    public int lookup (Color c)
//    public int lookup (boolean rgb, int x, int g, int y)

// Method to write out colour map (used for GIFs, with "true" parameter)
//    public int writeColourMap (boolean rgb, OutputStream out) throws IOException

// Other methods to interrogate colour map
//    public int getColorCount ()
//    public Color getColor (int i)

import java.util.*;
import java.io.*;
import java.awt.Image;
import java.awt.Color;
import java.awt.image.*;

/* NeuQuant Neural-Net Quantization Algorithm
 * ------------------------------------------
 *
 * Copyright (c) 1994 Anthony Dekker
 *
 * NEUQUANT Neural-Net quantization algorithm by Anthony Dekker, 1994.
 * See "Kohonen neural networks for optimal colour quantization"
 * in "Network: Computation in Neural Systems" Vol. 5 (1994) pp 351-367.
 * for a discussion of the algorithm.
 * See also  http://www.acm.org/~dekker/NEUQUANT.HTML
 *
 * Any party obtaining a copy of these files from the author, directly or
 * indirectly, is granted, free of charge, a full and unrestricted irrevocable,
 * world-wide, paid up, royalty-free, nonexclusive right and license to deal
 * in this software and documentation files (the "Software"), including without
 * limitation the rights to use, copy, modify, merge, publish, distribute, sublicense,
 * and/or sell copies of the Software, and to permit persons who receive
 * copies from any such party to do so, with the only requirement being
 * that this copyright notice remain intact.
 */

public class NeuQuant {
    
    public static final int ncycles	=	100;			// no. of learning cycles

    public static final int netsize  = 256;		// number of colours used
    public static final int specials  = 3;		// number of reserved colours used
    public static final int bgColour  = specials-1;	// reserved background colour
    public static final int cutnetsize  = netsize - specials;
    public static final int maxnetpos  = netsize-1;

    public static final int initrad	 = netsize/8;   // for 256 cols, radius starts at 32
    public static final int radiusbiasshift = 6;
    public static final int radiusbias = 1 << radiusbiasshift;
    public static final int initBiasRadius = initrad*radiusbias;
    public static final int radiusdec = 30; // factor of 1/30 each cycle

    public static final int alphabiasshift = 10;			// alpha starts at 1
    public static final int initalpha      = 1<<alphabiasshift; // biased by 10 bits

    public static final double gamma = 1024.0;
    public static final double beta = 1.0/1024.0;
    public static final double betagamma = beta * gamma;
    
    private double [] [] network = new double [netsize] [3]; // the network itself
    protected int [] [] colormap = new int [netsize] [4]; // the network itself
    
    private int [] netindex = new int [256]; // for network lookup - really 256
    
    private double [] bias = new double [netsize];  // bias and freq arrays for learning
    private double [] freq = new double [netsize];

    // four primes near 500 - assume no image has a length so large
    // that it is divisible by all four primes
   
    public static final int prime1	=	499;
    public static final int prime2	=	491;
    public static final int prime3	=	487;
    public static final int prime4	=	503;
    public static final int maxprime=	prime4;
    
    protected int [] pixels = null;
    private int samplefac = 0;


    public NeuQuant (Image im, int w, int h) throws IOException {
        this (1);
        setPixels (im, w, h);
		setUpArrays ();
    }
    
    public NeuQuant (int sample, Image im, int w, int h) throws IOException {
        this (sample);
        setPixels (im, w, h);
		setUpArrays ();
    }

    public NeuQuant (Image im, ImageObserver obs) throws IOException {
        this (1);
        setPixels (im, obs);
		setUpArrays ();
    }
    
    protected NeuQuant (int sample) throws IOException {
        if (sample < 1) throw new IOException ("Sample must be 1..30");
        if (sample > 30) throw new IOException ("Sample must be 1..30");
        samplefac = sample;
        // rest later
    }
    
    public NeuQuant (int sample, Image im, ImageObserver obs) throws IOException {
        this (sample);
        setPixels (im, obs);
		setUpArrays ();
    }

    public int getColorCount () {
    	return netsize;
    }

    public Color getColor (int i) {
    	if (i < 0 || i >= netsize) return null;
	    int bb = colormap[i][0];
    	int gg = colormap[i][1];
    	int rr = colormap[i][2];
    	return new Color (rr, gg, bb);
    }

    public int writeColourMap (boolean rgb, OutputStream out) throws IOException {
    	for (int i=0; i<netsize; i++) {
		    int bb = colormap[i][0];
	    	int gg = colormap[i][1];
	    	int rr = colormap[i][2];
	    	out.write (rgb ? rr : bb);
	    	out.write (gg);
	    	out.write (rgb ? bb : rr);
    	}
    	return netsize;
    }

    protected void setUpArrays () {
    	network [0] [0] = 0.0;	// black
    	network [0] [1] = 0.0;
    	network [0] [2] = 0.0;
    	
    	network [1] [0] = 255.0;	// white
    	network [1] [1] = 255.0;
    	network [1] [2] = 255.0;
		
    	// RESERVED bgColour	// background
    	
        for (int i=0; i<specials; i++) {
		    freq[i] = 1.0 / netsize;
		    bias[i] = 0.0;
        }
        
        for (int i=specials; i<netsize; i++) {
		    double [] p = network [i];
		    p[0] = (255.0 * (i-specials)) / cutnetsize;
		    p[1] = (255.0 * (i-specials)) / cutnetsize;
		    p[2] = (255.0 * (i-specials)) / cutnetsize;

		    freq[i] = 1.0 / netsize;
		    bias[i] = 0.0;
        }
    }    	

    private void setPixels (Image im, ImageObserver obs) throws IOException {
        if (im == null) throw new IOException ("Image is null");
        int w = im.getWidth(obs);
        int h = im.getHeight(obs);
        setPixels (im, w, h);
    }
    
    private void setPixels (Image im, int w, int h) throws IOException {
        if (w*h < maxprime) throw new IOException ("Image is too small");
        pixels = new int [w * h];
        java.awt.image.PixelGrabber pg
            = new java.awt.image.PixelGrabber(im, 0, 0, w, h, pixels, 0, w);
       	try {
	        pg.grabPixels();
	   	} catch (InterruptedException e) { }
	    if ((pg.getStatus() & java.awt.image.ImageObserver.ABORT) != 0) {
	        throw new IOException ("Image pixel grab aborted or errored");
	  	}
    }
    

    public void init () {
        learn ();
        fix ();
        inxbuild ();
    }

    private void altersingle(double alpha, int i, double b, double g, double r) {
        // Move neuron i towards biased (b,g,r) by factor alpha
        double [] n = network[i];				// alter hit neuron
    	n[0] -= (alpha*(n[0] - b));
    	n[1] -= (alpha*(n[1] - g));
    	n[2] -= (alpha*(n[2] - r));
    }

    private void alterneigh(double alpha, int rad, int i, double b, double g, double r) {
        
    	int lo = i-rad;   if (lo<specials-1) lo=specials-1;
    	int hi = i+rad;   if (hi>netsize) hi=netsize;

    	int j = i+1;
    	int k = i-1;
    	int q = 0;
    	while ((j<hi) || (k>lo)) {
    	    double a = (alpha * (rad*rad - q*q)) / (rad*rad);
    	    q ++;
    		if (j<hi) {
    			double [] p = network[j];
    			p[0] -= (a*(p[0] - b));
    			p[1] -= (a*(p[1] - g));
    			p[2] -= (a*(p[2] - r));
    			j++;
    		}
    		if (k>lo) {
    			double [] p = network[k];
    			p[0] -= (a*(p[0] - b));
    			p[1] -= (a*(p[1] - g));
    			p[2] -= (a*(p[2] - r));
    			k--;
    		}
    	}
    }
    
    private int contest (double b, double g, double r) {    // Search for biased BGR values
    	// finds closest neuron (min dist) and updates freq 
    	// finds best neuron (min dist-bias) and returns position 
    	// for frequently chosen neurons, freq[i] is high and bias[i] is negative 
    	// bias[i] = gamma*((1/netsize)-freq[i]) 

    	double bestd = Float.MAX_VALUE;
    	double bestbiasd = bestd;
    	int bestpos = -1;
    	int bestbiaspos = bestpos;
        
    	for (int i=specials; i<netsize; i++) {
    		double [] n = network[i];
    		double dist = n[0] - b;   if (dist<0) dist = -dist;
    		double a = n[1] - g;   if (a<0) a = -a;
    		dist += a;
    		a = n[2] - r;   if (a<0) a = -a;
    		dist += a;
    		if (dist<bestd) {bestd=dist; bestpos=i;}
    		double biasdist = dist - bias [i];
    		if (biasdist<bestbiasd) {bestbiasd=biasdist; bestbiaspos=i;}
    		freq [i] -= beta * freq [i];
    		bias [i] += betagamma * freq [i];
    	}
    	freq[bestpos] += beta;
    	bias[bestpos] -= betagamma;
    	return bestbiaspos;
    }
    
    private int specialFind (double b, double g, double r) {
    	for (int i=0; i<specials; i++) {
    		double [] n = network[i];
    		if (n[0] == b && n[1] == g && n[2] == r) return i;
    	}
    	return -1;
    }
    
    private void learn() {
        int biasRadius = initBiasRadius;
    	int alphadec = 30 + ((samplefac-1)/3);
    	int lengthcount = pixels.length;
    	int samplepixels = lengthcount / samplefac;
    	int delta = samplepixels / ncycles;
    	int alpha = initalpha;

	    int i = 0;
    	int rad = biasRadius >> radiusbiasshift;
    	if (rad <= 1) rad = 0;
	
    	System.err.println("beginning 1D learning: samplepixels=" + samplepixels + "  rad=" + rad);

        int step = 0;
        int pos = 0;
        
    	if ((lengthcount%prime1) != 0) step = prime1;
    	else {
    		if ((lengthcount%prime2) !=0) step = prime2;
    		else {
    			if ((lengthcount%prime3) !=0) step = prime3;
    			else step = prime4;
    		}
    	}
	
    	i = 0;
    	while (i < samplepixels) {
    	    int p = pixels [pos];
	        int red   = (p >> 16) & 0xff;
	        int green = (p >>  8) & 0xff;
	        int blue  = (p      ) & 0xff;

    		double b = blue;
    		double g = green;
    		double r = red;

    		if (i == 0) {   // remember background colour
    			network [bgColour] [0] = b;
    			network [bgColour] [1] = g;
    			network [bgColour] [2] = r;
    		}

    		int j = specialFind (b, g, r);
    		j = j < 0 ? contest (b, g, r) : j;

			if (j >= specials) {   // don't learn for specials
            	double a = (1.0 * alpha) / initalpha;
    			altersingle (a, j, b, g, r);
    			if (rad > 0) alterneigh (a, rad, j, b, g, r);   // alter neighbours
    		}

    		pos += step;
    		while (pos >= lengthcount) pos -= lengthcount;
	        
    		i++;
    		if (i%delta == 0) {	
    			alpha -= alpha / alphadec;
    			biasRadius -= biasRadius / radiusdec;
    			rad = biasRadius >> radiusbiasshift;
    			if (rad <= 1) rad = 0;
    		}
    	}
    	System.err.println("finished 1D learning: final alpha=" + (1.0 * alpha)/initalpha + "!");
    }

    private void fix() {
        for (int i=0; i<netsize; i++) {
            for (int j=0; j<3; j++) {
                int x = (int) (0.5 + network[i][j]);
                if (x < 0) x = 0;
                if (x > 255) x = 255;
                colormap[i][j] = x;
            }
            colormap[i][3] = i;
        }
    }

    private void inxbuild() {
        // Insertion sort of network and building of netindex[0..255]

    	int previouscol = 0;
    	int startpos = 0;

    	for (int i=0; i<netsize; i++) {
    		int[] p = colormap[i];
    		int[] q = null;
    		int smallpos = i;
    		int smallval = p[1];			// index on g
    		// find smallest in i..netsize-1
    		for (int j=i+1; j<netsize; j++) {
    			q = colormap[j];
    			if (q[1] < smallval) {		// index on g
    				smallpos = j;
    				smallval = q[1];	// index on g
    			}
    		}
    		q = colormap[smallpos];
    		// swap p (i) and q (smallpos) entries
    		if (i != smallpos) {
    			int j = q[0];   q[0] = p[0];   p[0] = j;
    			j = q[1];   q[1] = p[1];   p[1] = j;
    			j = q[2];   q[2] = p[2];   p[2] = j;
    			j = q[3];   q[3] = p[3];   p[3] = j;
    		}
    		// smallval entry is now in position i
    		if (smallval != previouscol) {
    			netindex[previouscol] = (startpos+i)>>1;
    			for (int j=previouscol+1; j<smallval; j++) netindex[j] = i;
    			previouscol = smallval;
    			startpos = i;
    		}
    	}
    	netindex[previouscol] = (startpos+maxnetpos)>>1;
    	for (int j=previouscol+1; j<256; j++) netindex[j] = maxnetpos; // really 256
    }

    public int convert (int pixel) {
        int alfa = (pixel >> 24) & 0xff;
	    int r   = (pixel >> 16) & 0xff;
	    int g = (pixel >>  8) & 0xff;
	    int b  = (pixel      ) & 0xff;
	    int i = inxsearch(b, g, r);
	    int bb = colormap[i][0];
	    int gg = colormap[i][1];
	    int rr = colormap[i][2];
	    return (alfa << 24) | (rr << 16) | (gg << 8) | (bb);
    }

    public int lookup (int pixel) {
	    int r   = (pixel >> 16) & 0xff;
	    int g = (pixel >>  8) & 0xff;
	    int b  = (pixel      ) & 0xff;
	    int i = inxsearch(b, g, r);
	    return i;
    }

    public int lookup (Color c) {
	    int r   = c.getRed ();
	    int g = c.getGreen ();
	    int b  = c.getBlue ();
	    int i = inxsearch(b, g, r);
	    return i;
    }

    public int lookup (boolean rgb, int x, int g, int y) {
	    int i = rgb ? inxsearch (y, g, x) : inxsearch (x, g, y);
	    return i;
    }

    private int not_used_slow_inxsearch(int b, int g, int r) {
        // Search for BGR values 0..255 and return colour index
    	int bestd = 1000;		// biggest possible dist is 256*3
    	int best = -1;
    	for (int i = 0; i<netsize; i++) {
 			int [] p = colormap[i];
			int dist = p[1] - g;
			if (dist<0) dist = -dist;
			int a = p[0] - b;   if (a<0) a = -a;
			dist += a;
			a = p[2] - r;   if (a<0) a = -a;
			dist += a;
			if (dist<bestd) {bestd=dist; best=i;}
		}
		return best;
    }
    
    protected int inxsearch(int b, int g, int r) {
        // Search for BGR values 0..255 and return colour index
    	int bestd = 1000;		// biggest possible dist is 256*3
    	int best = -1;
    	int i = netindex[g];	// index on g
    	int j = i-1;		// start at netindex[g] and work outwards

    	while ((i<netsize) || (j>=0)) {
    		if (i<netsize) {
    			int [] p = colormap[i];
    			int dist = p[1] - g;		// inx key
    			if (dist >= bestd) i = netsize;	// stop iter
    			else {
    				if (dist<0) dist = -dist;
    				int a = p[0] - b;   if (a<0) a = -a;
    				dist += a;
    				if (dist<bestd) {
    					a = p[2] - r;   if (a<0) a = -a;
    					dist += a;
    					if (dist<bestd) {bestd=dist; best=i;}
    				}
    				i++;
    			}
    		}
    		if (j>=0) {
    			int [] p = colormap[j];
    			int dist = g - p[1]; // inx key - reverse dif
    			if (dist >= bestd) j = -1; // stop iter
    			else {
    				if (dist<0) dist = -dist;
    				int a = p[0] - b;   if (a<0) a = -a;
    				dist += a;
    				if (dist<bestd) {
    					a = p[2] - r;   if (a<0) a = -a;
    					dist += a;
    					if (dist<bestd) {bestd=dist; best=j;}
    				}
    				j--;
    			}
    		}
    	}

    	return best;
    }

}

Test driver

import java.util.*;
import java.io.*;
import java.awt.*;
import java.awt.image.*;

public class NQCanvas extends Canvas {

    private NeuQuant nq = null;
	private int pixels[] = null;
    private int w = 0;;
    private int h = 0;
    private Image image = null;
    private String url = null;
    
    public NQCanvas (String url) {
        this.url = url;
    }
    
    public void set () {
        try {
            System.err.println ("Fetching " + url + " ...");
            Image img = null;
            try {
                java.net.URL u = new java.net.URL (url);
                img = Toolkit.getDefaultToolkit ().getImage (u);
            } catch (Exception ee) {
                // filename
                img = Toolkit.getDefaultToolkit ().getImage (url);
            }
            java.awt.MediaTracker tracker = new java.awt.MediaTracker(this);
            tracker.addImage(img, 0);
            try {
		        tracker.waitForID(0);
	        } catch (InterruptedException e) { }
            System.err.println ("w = " + img.getWidth (this));
            System.err.println ("h = " + img.getHeight (this));
            set (img);
        } catch (Exception ex) {
            System.err.println(ex);
        }
    }
    
    public void set (Image img) throws IOException {
        nq = new NeuQuant (img, this);
        nq.init();
        w = img.getWidth (this);
    	h = img.getHeight (this);
        pixels = new int [w * h];
        java.awt.image.PixelGrabber pg
            = new java.awt.image.PixelGrabber(img, 0, 0, w, h, pixels, 0, w);
       	try {
	        pg.grabPixels();
	   	} catch (InterruptedException e) { }
	    if ((pg.getStatus() & java.awt.image.ImageObserver.ABORT) != 0) {
	        throw new IOException ("Image pixel grab aborted or errored");
	  	}
	  	for (int i = 0; i < w*h; i++) pixels[i] = nq.convert(pixels[i]);
	  	this.image = this.createImage(new MemoryImageSource(w, h, pixels, 0, w));
    }
    
    public void paint (Graphics graphics) {
        if (image == null) set ();
        if (image != null) graphics.drawImage (image, 0, 0, this);
    } 

    public static void main (String [] args) throws java.io.IOException {
        NQCanvas canvas = new NQCanvas (args[0]);
        java.awt.Frame frame = new java.awt.Frame("NeuQuant Test");
	    frame.setSize (500, 500);
	    frame.add (canvas);
	    frame.show ();
	    frame.addWindowListener (new java.awt.event.WindowAdapter() {
	        public void windowClosing(java.awt.event.WindowEvent e) { 
		        System.err.println("Closing program");		
		        System.exit(0); }
	    } );

    }
}

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