Chain Code for 2D Line

Chain code is a lossless compression technique used for representing an object in images. The co-ordinates of any continuous boundary of an object can be represented as a string of numbers where each number represents a particular direction in which the next point on the connected line is present. One point is taken as the reference/starting point and on plotting the points generated from the chain, the original figure can be re-drawn.

This article describes how to generate a 8-neighbourhood chain code of a 2-D straight line. In a rectangular grid, a point can have at most 8 surrounding points as shown below. The next point on the line has to be one of these 8 surrounding points. Each direction is assigned a code. Using this code we can find out which of the surrounding point should be plotted next.

The chain codes could be generated by using conditional statements for each direction but it becomes very tedious to describe for systems having large number of directions(3-D grids can have up to 26 directions). Instead we use a hash function. The difference in X() and Y() co-ordinates of two successive points are calculated and hashed to generate the key for the chain code between the two points.

Chain code list: 

Hash function: 

Hash table:- 

1	0	5	0
1	1	8	1
0	1	7	2
-1	1	6	3
-1	0	3	4
-1	-1	0	5
0	-1	1	6
1	-1	2	7

The function does not generate the value 4 so a dummy value is stored there.

Examples: 

Input : (2, -3), (-4, 2)
Output : Chain code for the straight line from (2, -3) to (-4, 2) is 333433

Input : (-7, -4), (9, 3)
Output : Chain code for the straight line from (-7, -4) to (9, 3) is 0101010100101010

C++

// C++ code for generating 8-neighbourhood chain
// code for a 2-D line
#include <iostream>
#include <math.h>
#include <vector>

std::vector<int> codeList = { 5, 6, 7, 4, -1, 0, 3, 2, 1 };

// This function generates the chaincode
// for transition between two neighbour points
int getChainCode(int x1, int y1, int x2, int y2)
{
    int dx = x2 - x1;
    int dy = y2 - y1;
    int hashKey = 3 * dy + dx + 4;
    return codeList[hashKey];
}

// This function generates the list of
// chaincodes for given list of points
std::vector<int> generateChainCode(
    std::vector<std::vector<int> > ListOfPoints)
{
    std::vector<int> chainCode;
    for (int i = 0; i < ListOfPoints.size() - 1; i++) {
        std::vector<int> a = ListOfPoints[i];
        std::vector<int> b = ListOfPoints[i + 1];
        chainCode.push_back(
            getChainCode(a[0], a[1], b[0], b[1]));
    }
    return chainCode;
}

// This function generates the list of points for
// a straight line using Bresenham's Algorithm
std::vector<std::vector<int> > Bresenham2D(int x1, int y1,
                                           int x2, int y2)
{
    std::vector<std::vector<int> > ListOfPoints;
    ListOfPoints.push_back({ x1, y1 });
    int xdif = x2 - x1;
    int ydif = y2 - y1;
    int dx = abs(xdif);
    int dy = abs(ydif);
    int xs;
    if (xdif > 0) {
        xs = 1;
    }
    else {
        xs = -1;
    }

    int ys;
    if (ydif > 0) {
        ys = 1;
    }
    else {
        ys = -1;
    }

    if (dx > dy) {
        // Driving axis is the X-axis
        int p = 2 * dy - dx;
        while (x1 != x2) {
            x1 += xs;
            if (p >= 0) {
                y1 += ys;
                p -= 2 * dx;
            }
            p += 2 * dy;
            ListOfPoints.push_back({ x1, y1 });
        }
    }
    else {
        // Driving axis is the Y-axis
        int p = 2 * dx - dy;
        while (y1 != y2) {
            y1 += ys;
            if (p >= 0) {
                x1 += xs;
                p -= 2 * dy;
            }
            p += 2 * dx;
            ListOfPoints.push_back({ x1, y1 });
        }
    }

    return ListOfPoints;
}

void DriverFunction()
{
    int x1 = -9;
    int y1 = -3;
    int x2 = 10;
    int y2 = 1;
    std::vector<std::vector<int> > ListOfPoints
        = Bresenham2D(x1, y1, x2, y2);
    std::vector<int> chainCode
        = generateChainCode(ListOfPoints);

    std::string chainCodeString = "";
    for (int i = 0; i < chainCode.size(); i++) {
        chainCodeString += std::to_string(chainCode[i]);
    }

    std::cout << "Chain code for the straight line from ("
              << x1 << "," << y1 << ") to (" << x2 << ","
              << y2 << ") is " << chainCodeString
              << std::endl;
}

int main()
{
    DriverFunction();
    return 0;
}

// This code is contributed by ishankhndelwals.

Java

import java.util.ArrayList;
import java.util.List;

public class Main {
  static List<Integer> codeList
    = new ArrayList<Integer>();
  static
  {
    codeList.add(5);
    codeList.add(6);
    codeList.add(7);
    codeList.add(4);
    codeList.add(-1);
    codeList.add(0);
    codeList.add(3);
    codeList.add(2);
    codeList.add(1);
  }

  // This function generates the chaincode
  // for transition between two neighbour points
  static int getChainCode(int x1, int y1, int x2, int y2)
  {
    int dx = x2 - x1;
    int dy = y2 - y1;
    int hashKey = 3 * dy + dx + 4;
    return codeList.get(hashKey);
  }

  // This function generates the list of
  // chaincodes for given list of points
  static List<Integer>
    generateChainCode(List<List<Integer> > ListOfPoints)
  {
    List<Integer> chainCode = new ArrayList<Integer>();
    for (int i = 0; i < ListOfPoints.size() - 1; i++) {
      List<Integer> a = ListOfPoints.get(i);
      List<Integer> b = ListOfPoints.get(i + 1);
      chainCode.add(getChainCode(a.get(0), a.get(1),
                                 b.get(0), b.get(1)));
    }
    return chainCode;
  }

  // This function generates the list of points for
  // a straight line using Bresenham's Algorithm
  static List<List<Integer> > Bresenham2D(int x1, int y1,
                                          int x2, int y2)
  {
    List<List<Integer> > ListOfPoints
      = new ArrayList<List<Integer> >();
    ListOfPoints.add(List.of(x1, y1));
    int xdif = x2 - x1;
    int ydif = y2 - y1;
    int dx = Math.abs(xdif);
    int dy = Math.abs(ydif);
    int xs;
    if (xdif > 0) {
      xs = 1;
    }
    else {
      xs = -1;
    }

    int ys;
    if (ydif > 0) {
      ys = 1;
    }
    else {
      ys = -1;
    }

    if (dx > dy) {
      // Driving axis is the X-axis
      int p = 2 * dy - dx;
      while (x1 != x2) {
        x1 += xs;
        if (p >= 0) {
          y1 += ys;
          p -= 2 * dx;
        }
        p += 2 * dy;
        ListOfPoints.add(List.of(x1, y1));
      }
    }
    else {
      // Driving axis is the Y-axis
      int p = 2 * dx - dy;
      while (y1 != y2) {
        y1 += ys;
        if (p >= 0) {
          x1 += xs;
          p -= 2 * dy;
        }
        p += 2 * dx;
        ListOfPoints.add(List.of(x1, y1));
      }
    }

    return ListOfPoints;
  }

  static void DriverFunction()
  {
    int x1 = -9;
    int y1 = -3;
    int x2 = 10;
    int y2 = 1;
    List<List<Integer> > ListOfPoints
      = Bresenham2D(x1, y1, x2, y2);
    List<Integer> chainCode
      = generateChainCode(ListOfPoints);

    StringBuilder chainCodeString = new StringBuilder();
    for (int i = 0; i < chainCode.size(); i++) {
      chainCodeString.append(
        Integer.toString(chainCode.get(i)));
    }

    System.out.println(
      "Chain code for the straight line from (" + x1
      + "," + y1 + ") to (" + x2 + "," + y2 + ") is "
      + chainCodeString);
  }

  public static void main(String[] args)
  {
    DriverFunction();
  }
}

// This code is contributed by ishankhandelwals.

Python3

# Python3 code for generating 8-neighbourhood chain
# code for a 2-D line

codeList = [5, 6, 7, 4, -1, 0, 3, 2, 1]

# This function generates the chaincode
# for transition between two neighbour points
def getChainCode(x1, y1, x2, y2):
    dx = x2 - x1
    dy = y2 - y1
    hashKey = 3 * dy + dx + 4
    return codeList[hashKey]

'''This function generates the list of
chaincodes for given list of points'''
def generateChainCode(ListOfPoints):
    chainCode = []
    for i in range(len(ListOfPoints) - 1):
        a = ListOfPoints[i]
        b = ListOfPoints[i + 1]
        chainCode.append(getChainCode(a[0], a[1], b[0], b[1]))
    return chainCode


'''This function generates the list of points for
a straight line using Bresenham's Algorithm'''
def Bresenham2D(x1, y1, x2, y2):
    ListOfPoints = []
    ListOfPoints.append([x1, y1])
    xdif = x2 - x1
    ydif = y2 - y1
    dx = abs(xdif)
    dy = abs(ydif)
    if(xdif > 0):
        xs = 1
    else:
        xs = -1
    if (ydif > 0):
        ys = 1
    else:
        ys = -1
    if (dx > dy):

        # Driving axis is the X-axis
        p = 2 * dy - dx
        while (x1 != x2):
            x1 += xs
            if (p >= 0):
                y1 += ys
                p -= 2 * dx
            p += 2 * dy
            ListOfPoints.append([x1, y1])
    else:

        # Driving axis is the Y-axis
        p = 2 * dx-dy
        while(y1 != y2):
            y1 += ys
            if (p >= 0):
                x1 += xs
                p -= 2 * dy
            p += 2 * dx
            ListOfPoints.append([x1, y1])
    return ListOfPoints

def DriverFunction():
    (x1, y1) = (-9, -3)
    (x2, y2) = (10, 1)
    ListOfPoints = Bresenham2D(x1, y1, x2, y2)
    chainCode = generateChainCode(ListOfPoints)
    chainCodeString = "".join(str(e) for e in chainCode)
    print ('Chain code for the straight line from', (x1, y1),
            'to', (x2, y2), 'is', chainCodeString)

DriverFunction()

C#

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace CPP_to_C_Sharp
{
  class Program
  {
    static List<int> codeList = new List<int> { 5, 6, 7, 4, -1, 0, 3, 2, 1 };

    // This function generates the chaincode
    // for transition between two neighbour points
    static int getChainCode(int x1, int y1, int x2, int y2)
    {
      int dx = x2 - x1;
      int dy = y2 - y1;
      int hashKey = 3 * dy + dx + 4;
      return codeList[hashKey];
    }

    // This function generates the list of
    // chaincodes for given list of points
    static List<int> generateChainCode(
      List<List<int>> ListOfPoints)
    {
      List<int> chainCode = new List<int>();
      for (int i = 0; i < ListOfPoints.Count - 1; i++)
      {
        List<int> a = ListOfPoints[i];
        List<int> b = ListOfPoints[i + 1];
        chainCode.Add(getChainCode(a[0], a[1], b[0], b[1]));
      }
      return chainCode;
    }

    // This function generates the list of points for
    // a straight line using Bresenham's Algorithm
    static List<List<int>> Bresenham2D(int x1, int y1,
                                       int x2, int y2)
    {
      List<List<int>> ListOfPoints = new List<List<int>>();
      ListOfPoints.Add(new List<int> { x1, y1 });
      int xdif = x2 - x1;
      int ydif = y2 - y1;
      int dx = Math.Abs(xdif);
      int dy = Math.Abs(ydif);
      int xs;
      if (xdif > 0)
      {
        xs = 1;
      }
      else
      {
        xs = -1;
      }

      int ys;
      if (ydif > 0)
      {
        ys = 1;
      }
      else
      {
        ys = -1;
      }

      if (dx > dy)
      {
        // Driving axis is the X-axis
        int p = 2 * dy - dx;
        while (x1 != x2)
        {
          x1 += xs;
          if (p >= 0)
          {
            y1 += ys;
            p -= 2 * dx;
          }
          p += 2 * dy;
          ListOfPoints.Add(new List<int> { x1, y1 });
        }
      }
      else
      {
        // Driving axis is the Y-axis
        int p = 2 * dx - dy;
        while (y1 != y2)
        {
          y1 += ys;
          if (p >= 0)
          {
            x1 += xs;
            p -= 2 * dy;
          }
          p += 2 * dx;
          ListOfPoints.Add(new List<int> { x1, y1 });
        }
      }

      return ListOfPoints;
    }

    static void DriverFunction()
    {
      int x1 = -9;
      int y1 = -3;
      int x2 = 10;
      int y2 = 1;
      List<List<int>> ListOfPoints
        = Bresenham2D(x1, y1, x2, y2);
      List<int> chainCode
        = generateChainCode(ListOfPoints);

      string chainCodeString = "";
      for (int i = 0; i < chainCode.Count; i++)
      {
        chainCodeString += chainCode[i].ToString();
      }

      Console.WriteLine("Chain code for the straight line from ("
                        + x1 + "," + y1 + ") to (" + x2 + ","
                        + y2 + ") is " + chainCodeString);
    }

    static void Main(string[] args)
    {
      DriverFunction();
    }
  }
}

// This code is contributed by ishankhandelwals.

Javascript

// JavaScript code for generating 8-neighbourhood chain
// code for a 2-D line

let codeList = [5, 6, 7, 4, -1, 0, 3, 2, 1];

// This function generates the chaincode
// for transition between two neighbour points
function getChainCode(x1, y1, x2, y2){
    let dx = x2 - x1;
    let dy = y2 - y1;
    let hashKey = 3 * dy + dx + 4;
    return codeList[hashKey];
}


// This function generates the list of
// chaincodes for given list of points
function generateChainCode(ListOfPoints){
    let chainCode = [];
    for(let i = 0; i < ListOfPoints.length-1; i++){
        let a = ListOfPoints[i];
        let b = ListOfPoints[i + 1];
        chainCode.push(getChainCode(a[0], a[1], b[0], b[1]));

    }
    return chainCode;
}

// This function generates the list of points for
// a straight line using Bresenham's Algorithm
function Bresenham2D(x1, y1, x2, y2){
    let ListOfPoints = [];
    ListOfPoints.push([x1, y1]);
    let xdif = x2 - x1;
    let ydif = y2 - y1;
    let dx = Math.abs(xdif);
    let dy = Math.abs(ydif);
    if(xdif > 0){
        xs = 1;
    }
    else{
        xs = -1;
    }

    if (ydif > 0){
        ys = 1;
    }
    else{
        ys = -1;
    }

    if (dx > dy){
        // Driving axis is the X-axis
        let p = 2 * dy - dx;
        while (x1 != x2){
            x1 += xs
            if (p >= 0){
                y1 += ys;
                p -= 2 * dx;
            }
            p += 2 * dy;
            ListOfPoints.push([x1, y1]);
        }

    }
    else{
        // Driving axis is the Y-axis
        let p = 2 * dx-dy;
        while(y1 != y2){
             y1 += ys;
            if (p >= 0){
                x1 += xs;
                p -= 2 * dy;
            }
            p += 2 * dx;
            ListOfPoints.push([x1, y1]);
        }
    }

    return ListOfPoints
}


function DriverFunction(){
    let x1 = -9;
    let y1 = -3;
    let x2 = 10;
    let y2 = 1;
    let ListOfPoints = Bresenham2D(x1, y1, x2, y2);
    let chainCode = generateChainCode(ListOfPoints);

    let chainCodeString = "";
    for(let i = 0; i < chainCode.length; i++){
            chainCodeString += chainCode[i].toString();
    }

    console.log("Chain code for the straight line from (", x1,",", y1,")","to","(", x2,",", y2,")", "is", chainCodeString);
}

DriverFunction();
// The code is contributed by Nidhi goel

Output:

Chain code for the straight line from (-9, -3) to (10, 1) is 0010000100010000100