Finding center of symmetry between zones in a image

I have and image such as below (I am pasting only a sketch here) where I want to calculate the center of symmetry and the displacement between the 2 marked zones in the image(Marked in Red and blue). Could anyone suggest a simple algorithm for this (Basic code sniippet would also help). (Please note the signal is symmetric to 180 degree rotation). [The idea is to calculate the center of symmetry between the red and blue zones]

[Walter Roberson]

Will there ever be a time when the "center of symmetry" is not at the middle of the line segment joining the centroids of the regions ?

[Vivek]

No. The center of symmetry will be in the midddle of linesegment joiing the centroid in the sketch i have shown. But there could be images without a line segment as in the red zone. The region marked in red could be simliar to the blue zone , just thicker or thinner lets say.

[Walter Roberson]

You have not given us any idea how far off the ideal the situation could get.

At the moment, with what you have shown us, and assuming the red and blue lines are not actually present in the images, just regionprops() asking for 'Centroid', sort by x coordinate, then take the middle of each consecutive line segment joining the now-sorted coordinates.

Answer 1

[Image Analyst]

Try this. The centroid of each blob is found and a vertical line is drawn there.

% Demo by Image Analyst
clc;    % Clear the command window.
close all;  % Close all figures (except those of imtool.)
clear;  % Erase all existing variables. Or clearvars if you want.
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 16;
markerSize = 40;

%--------------------------------------------------------------------------------------------------------
%    READ IN IMAGE
folder = pwd;
baseFileName = 'mO7aP.png';
fullFileName = fullfile(folder, baseFileName);
% Check if file exists.
if ~exist(fullFileName, 'file')
    % The file doesn't exist -- didn't find it there in that folder.
    % Check the entire search path (other folders) for the file by stripping off the folder.
    fullFileNameOnSearchPath = baseFileName; % No path this time.
    if ~exist(fullFileNameOnSearchPath, 'file')
        % Still didn't find it.  Alert user.
        errorMessage = sprintf('Error: %s does not exist in the search path folders.', fullFileName);
        uiwait(warndlg(errorMessage));
        return;
    end
end
rgbImage = imread(fullFileName);
% Get the dimensions of the image.
% numberOfColorChannels should be = 1 for a gray scale image, and 3 for an RGB color image.
[rows, columns, numberOfColorChannels] = size(rgbImage)
if numberOfColorChannels > 1
    % It's not really gray scale like we expected - it's color.
    fprintf('It is not really gray scale like we expected - it is color\n');
    % Extract the blue channel.
    grayImage = rgbImage(:, :, 1);
else
    grayImage = rgbImage;
end

%--------------------------------------------------------------------------------------------------------
% Display the image.
subplot(2, 3, 1);
imshow(grayImage);
impixelinfo;
axis('on', 'image');
title('Original Gray Scale Image', 'FontSize', fontSize, 'Interpreter', 'None');
% Maximize window.
g = gcf;
g.WindowState = 'maximized';
drawnow;

%--------------------------------------------------------------------------------------------------------
% Display histogram
subplot(2, 3, 2);
histogram(grayImage, 256);
grid on;
title('Histogram of Image', 'FontSize', fontSize, 'Interpreter', 'None');

%--------------------------------------------------------------------------------------------------------
% Binarize the image to get a mask.
threshold = 242;
mask = grayImage < threshold;
% Display mask image.
subplot(2, 3, 3);
imshow(mask);
axis('on', 'image');
drawnow;
caption = sprintf('Mask using a threshold of %.1f', threshold);
title(caption, 'FontSize', fontSize, 'Interpreter', 'None');

%--------------------------------------------------------------------------------------------------------
% Fill the blobs
mask = imfill(mask, 'holes');
% Take 5 largest blobs.
mask = bwareafilt(mask, 5);
subplot(2, 3, 4);
imshow(mask);
axis('on', 'image');
drawnow;
title('Final Mask Image', 'FontSize', fontSize, 'Interpreter', 'None');

%--------------------------------------------------------------------------------------------------------
% Label each blob with 8-connectivity, so we can make measurements of it
[labeledImage, numberOfBlobs] = bwlabel(mask, 8);
% Apply a variety of pseudo-colors to the regions.
coloredLabelsImage = label2rgb (labeledImage, 'hsv', 'k', 'shuffle');
% Display the pseudo-colored image.
subplot(2, 3, 5);
imshow(coloredLabelsImage);
% Get all the blob properties.
blobMeasurements = regionprops(labeledImage, 'Area', 'Centroid');
numberOfBlobs = size(blobMeasurements, 1);
caption = sprintf('%d Individual Blobs', numberOfBlobs);
title(caption, 'FontSize', fontSize, 'Interpreter', 'None');
% Get the areas
allAreas = [blobMeasurements.Area];
% Get the centroids
xy = vertcat(blobMeasurements.Centroid);
xCentroids = xy(:, 1);
yCentroids = xy(:, 2);
% Draw lines over centroids.
hold on;
for k = 1 : numberOfBlobs
    xline(xCentroids(k), 'LineWidth', 2, 'Color', 'k');
    plot(xCentroids(k), yCentroids(k), 'y+', 'LineWidth', 2, 'MarkerSize', 30);
end
hold off;
% Show size of Areas.
subplot(2, 3, 6);
bar(allAreas);
grid on;
xlabel('Area in Pixels', 'FontSize', fontSize, 'Interpreter', 'None');
ylabel('Blob Index', 'FontSize', fontSize, 'Interpreter', 'None');
title('Blob Areas', 'FontSize', fontSize, 'Interpreter', 'None');

[Vivek]

Thanks you for the answer .I will work on it now.

I have a question. Can i project the region of interest (Areas marked in red and blue) to 1D and flip the regions and perform a correlation. That should also give the center of symmetry right ?

Lets say we the projection of zones to 1D as - P0

Projection of flipped zones to 1D as P1

Center of symmetry = xcorr(P0,P1)

[Image Analyst]

I don't understand your question.

In general you can convert any 2-D matrix to a 1-D vector like this

rowVector = reshape(matrix, 1, []);
columnVector = reshape(matrix, [], 1);

Not sure what you mean by center of symmetry. I found the center of each of the oblong gray blobs in your image.

[Vivek]

Please note , my goal is the calculate the center of symmtery and then calculate the overlay error between the 2 features marked in red and blue. The image from the microsope can have other patterns as well. Thats the reason I have to use zones (regionprops) .

[Image Analyst]

I'm sorry I wasted my time then. I computed the center of symmetry for each blob. But I don't know what you mean by "overlap". The black outlined blobs with blue dashed lines do not overlap the gray blobs with the red dashed lines. They are in separate parts of the image.

[Vivek]

What I was trying to say is to accurately caluclate the center of gravity for each of the zone , should't flipping each zone and doing a correlation should give us the center of symmetry. The reasoning is the edge profile of each zone is different . Take a look at the example below. The two region marked with dotted lines.

[Walter Roberson]

The center of gravity of an object is exactly at its centroid, unless the object is different densities (for example there are cases where the intensity at a point should be understood to be its weight.)

The center of symmetry of an object is always at its centroid.

Flipping and correlation are unnecessary steps.

This does, however, presume that the entire object is visible in the image. If the task has to do with estimating the center of symmetry of an object that is truncated by image boundaries, then approaches such as correlation might be worth doing.

[Vivek]

Thank you Walter and Image Analyst.I will try your suggestions . How can i project the zone as 1D plot in matlab ?

I have few more questions (unrelated to finding the COG).

Whats the best way in matlab to select mutiple rectangular regions of interest to such as the above once I plot the image ?

Also in Matlab there is an app called imageBatchProcessor which accepts a function to perform batch operations on images. Is it possible to modify and compile the app as an exe ?

[Walter Roberson]

None of the App provided by Mathworks can be compiled, if I recall correctly.

https://www.mathworks.com/products/compiler/compiler_support.html

Image Processing Toolbox

Not Supported:

• Prebuilt Apps and UIs included in the toolbox
• cpselect

[Image Analyst]

@Walter Roberson is right. None of the Applets on the Apps tab of the tool ribbon can be compiled into your own standalone executable.

You should use the FAQ:

https://matlab.fandom.com/wiki/FAQ#How_can_I_process_a_sequence_of_files?

or

https://www.mathworks.com/matlabcentral/fileexchange/24224

The "center of gravity" is what regionprops calls "WeightedCentroid" and takes the gray levels in the blob into account. The "Centroid" just gives the centroid based on the blob's shape alone -- it ignores the gray levels in the blob.

Not sure still what you mean when you talk about line of symmetry. Do you want the centerline of the blob so that you essentially have a centroid location line-by-line?

And the flipping stuff you talked about would/should not happen. We're not doing convolution or correlation here, where that is done internally and automatically. We're simply doing blob analysis based on shape and possibly gray level so don't worry about correlation or convolution.

All that said, I still don't know what you want. My demo code did find the centroid of the blobs and drew a line at that location, though it is not accurate on a line-by-line basis (if that's what you want). And I still don't know what you mean by overlap. None of the five or six dark regions overlap. Do you mean anything to do with dice?

https://en.wikipedia.org/wiki/S%C3%B8rensen%E2%80%93Dice_coefficient

[Vivek]

.Thanks you for your explnations.Just to explain what I am trying to caluclate. Lets take the example image below.The image in the left has 2 regions selected and the image in the right has 2 regions selected.

The error I want to calculate is the planar distance from the center of gravity of the 2 boxes in the left image to the center of the gravity of the 2 boxes in the right image. In this case I am trying to the centerline of each structure along both the X . Pleas omit the 6 in the right image.

Thats why I was asking if we take 1D prfile of the regions (original and fliiped) (Something like below) specified and a correlation we can determine the placement error of the 2 structures .(Subtraction should give us the pixel difference)

[Image Analyst]

You can crop the image so that the new image starts at the top of the box and the last line of the image is the last line of the box. Then find the centroids as I showed you. To find the center of the two boxes (#'s 2 and 5) in the left image you can do

% Find "center of gravity of the 2 boxes in the left image"
xctr = (xCentroid(2) + xCentroid(5)) / 2;

To do the same thing for the right image for blobs 3 and 4:

xctr = (xCentroid(3) + xCentroid(4)) / 2;