Implementing the CCL (Connected Component Labeling) algorithm without the union-find data structure?

Question

Hello fellow programmers!

A week ago I have been asigned the task of implementing the Connected Components Algorithm, mainly to extract the number of objects from an image.

You can read more about the algorithm here (https://en.wikipedia.org/wiki/Connected-component_labeling), the variant I am trying to implement is the two pass one.

This is my current attempt:

% ------------------------------------------------------------------------------
% -> Connected Component Labeling (CCL) Algorithm
% -> 4-Connectivity Version
% ------------------------------------------------------------------------------

% ------------------------------------------------------------------------------
% - [ Pre-Scan Code To Get Everything Ready ] -
% ------------------------------------------------------------------------------
% Starting With A Clean (Workspace) And (Command Window).
clear, clc;

% Instead Of Loading An Actual Image, We Are Creating A Matrix Of Zeros And Ones, Representing A Binary Image.
originalImage = [ ...
                0 1 0
                1 0 1
                0 1 0 ];

% Creating A Bigger Matrix That We Will Use To Store The Original Image In Its Middle, This Will Help Us Eliminate Border Checking In The Raster Scan.
binaryImage = zeros(size(originalImage) + 2);

% Copying The Pixels From The Original Image Into The Middle Of The Larger Matrix We Created.
binaryImage(2:size(originalImage, 1) + 1, 2:size(originalImage, 2) + 1) = originalImage;

% Getting The Number Of Rows (Height) And Number Of Columns (Width) Of The Binary Image.
[imageRows, imageColumns] = size(binaryImage);

% Creating A Matrix The Same Dimensions As The Binary Image In Which The Labeling Will Happen.
labeledImage = zeros(imageRows, imageColumns);

% Creating A Label Counter That We Will Use To Assign When We Create New Labels.
labelCounter = 1;

% ------------------------------------------------------------------------------
% - [First Scan: Assigning Labels To Indices] -
% ------------------------------------------------------------------------------
% Going Over Each Row In The Image One By One.
for r = 1:imageRows
   % Going Over Each Column In The Image One By One.
   for c = 1:imageColumns
       % If The Pixel Currently Being Scanned Is A Foreground Pixel (1).
       if (binaryImage(r, c) == 1)
           % Since We Are Working With 4-Connectivity We Only Need To Read 2 Labels, Mainly The (Left) And (Top) Labels.
           % Storing Them In Variables So Referencing Them Is Easier.
           left = labeledImage(r, c - 1);
           top = labeledImage(r - 1, c);
           % If Left == 0 And Top == 0 -> Create A New Label, And Increment The Label Counter, Also Add The Label To The Equivalency List.
           if (left == 0 && top == 0)
               labeledImage(r, c) = labelCounter;
               labelCounter = labelCounter + 1;
           % If Left == 0 And Top >= 1 -> Copy The Top Label.
           elseif (left == 0 && top >= 1)
               labeledImage(r, c) = top;
           % If Left >= 1 And Top == 0 -> Copy The Left Label.
           elseif (left >= 1 && top == 0)
               labeledImage(r, c) = left;
           % If Left >= 1 And Top >= 1 -> Find The Minimum Of The Two And Copy It, Also Add The Equivalent Labels To The Equivalency List, So We Can Fix Them In The Second Scan.
           elseif (left >= 1 && top >= 1)
               labeledImage(r, c) = min(left, top);
           end
       end
   end
end

% ------------------------------------------------------------------------------
% - [Second Scan: Fixing The Connected Pixels But Mismatched Labels] -
% ------------------------------------------------------------------------------
for r = 1:imageRows
   for c = 1:imageColumns

   end
end

This first pass is going through without any issues, I have tried multiple tests on it, however I have no idea how to implement the second pass, in which I have to fix the equivalent labels in the labeled matrix.

I did do my research online, and the preferred way to do it is to use the union-find (disjoint set) data structure to store the equivalences between the labels.

However, since I am using MATLAB and the union-find data structure is not implemented, I have to implement it myself, which is cumbersome and requires massive time and hard work due to MATLAB being an interpreted language.

So, I am open to ideas on implementing the second pass without having to use the union-find data structure.

Thanks in advance!

Answer 1

If you don't want to use union-find, then you should really do the flood-fill-each-component algorithm (the other one in Wikipedia).

Union-find is neither difficult nor slow, however, and it's what I would use to solve this problem. To make a simple and fast implementation:

• Use a matrix of integers the same shape as your image to represent the sets -- each integer in the matrix represents the set of the corresponding pixel.
• An integer x represents a set as follows: If x < 0, then it is a root set of size -x. If x>=0 then it's a child set with parent set x (i.e., row x/width, column x%width)). Initialize all sets to -1 since they all start off as roots with size 1.
• Use union-by-size and path compression