Plotting a graph of the distance between the centroid of an irregular shaped particle and its contour (radius) versus angle (equal intervals of theta from zero to 360 degrees)

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Johnson 2024-3-29

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https://ww2.mathworks.cn/matlabcentral/answers/2100626-plotting-a-graph-of-the-distance-between-the-centroid-of-an-irregular-shaped-particle-and-its-contou

评论： Johnson 2024-5-18

I am trying to plot a graph between the radius of an irregular shaped particle (from the centroid to a point on the contour) at equal intervals of theta around the particle (from zero to 360 degrees).

My code (I actually got most of it from here) is not giving me what I want. I would like the graph to look like that of a typical signal but with angles (0 to 360 degrees) on the x-axis and radius on the y-axis.

I would really appreciate if someone could help me get this graph so that I can proceed with further operations using the fast fourier transform algorithm

Thanks!

%Image Threshholding
clc; clearvars; clear all;
img = imread('1_50.jpg'); %Read image
BW = im2bw(img,0.45); %binarize image with a threshold value of 0.45
img2= bwareaopen (BW, 1000); %Remove small objects
img2= imfill(BW, 'holes'); %fill holes
 
centriod_value= regionprops(img2, 'Centroid'); %Finds centroid of image 
centroid = centriod_value.Centroid; %Returns centroid in [row, column]
%Display the binary image with the centroid locations superimposed
subplot(1,3,1)
imshow(img2);
hold on
plot(centroid(:,1),centroid(:,2),'b.')
hold on
%tracing the boundary of image
p_boundary= bwboundaries(img2); % in row, column order, not x,y
number_of_boundaries = size(p_boundary,1); %one boundary
for k = 1 : length(p_boundary);
  thisBoundary = p_boundary{k};
  y = thisBoundary(:,2);
  x = thisBoundary(:,1);
  
  subplot(1,3,2)
  plot(x, y, 'g', 'LineWidth', 2);
  hold on
plot(centroid(:,1),centroid(:,2),'b.')
hold on 
end
hold on
% Calculate the angles in degrees
deltaY = thisBoundary(:,1) - centroid(1);
deltaX = thisBoundary(:,2) - centroid(2);
angles = atand(deltaY ./ deltaX);
% Calculate the distances.
distances = sqrt((thisBoundary(:,1) - centroid(1)).^2 + (thisBoundary(:,2) - centroid(2)).^2);
% Plot distance vs. angle.
angle=0:360/128:360;
for i=1:length(p_boundary);
    angles(i)+360/128;
   
subplot(1,3,3)
plot(angles,distances)
end 

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Voss 2024-3-29

Can you upload the image (using the paperclip button)?

Johnson 2024-3-29

编辑：Johnson 2024-3-29

Hi, I have uploaded the image. I snipped it from a scanning electron microscope (SEM) micrograph so it's quite small.

The second image illustrates what I'm trying to do.

Thanks!

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Answer 1

Voss 2024-3-29

编辑：Voss 2024-3-29

在 MATLAB Online 中打开

1_50.JPG

%Image Threshholding

clc; clearvars; clear all;

img = imread('1_50.JPG'); %Read image

BW = im2bw(img,0.45); %binarize image with a threshold value of 0.45

% this does nothing useful, since img2 is overwritten in the next line:

% img2= bwareaopen (BW, 1000); %Remove small objects

img2= imfill(BW, 'holes'); %fill holes

centriod_value= regionprops(img2, 'Centroid'); %Finds centroid of image

centroid = cat(1,centriod_value.Centroid);

%Display the binary image with the centroid locations superimposed

subplot(1,3,1)

imshow(img2);

hold on

plot(centroid(:,1),centroid(:,2),'b.')

% hold on % don't need to call hold on more than once on the same axes

% for plotting into multiple axes in one loop, it's convenient to set up

% all the axes first, before anything is plotted into them

% set up the 2nd axes:

ax2 = subplot(1,3,2);

hold on

box on

axis image

set(ax2,'YDir','reverse')

% set up the 3rd axes:

ax3 = subplot(1,3,3);

hold on

%tracing the boundary of image

p_boundary= bwboundaries(img2); % in row, column order, not x,y

number_of_boundaries = size(p_boundary,1); %one boundary

for k = 1 : number_of_boundaries % use number_of_boundaries, which was just calculated

thisBoundary = p_boundary{k};

y = thisBoundary(:,1); % rows

x = thisBoundary(:,2); % columns (these were swapped)

plot(ax2, x, y, 'g', 'LineWidth', 2);

plot(ax2, centroid(:,1),centroid(:,2),'b.')

% Calculate the angles in degrees

deltaY = thisBoundary(:,1) - centroid(k,2); % boundary(:,1) is y, but centroid(k,2) is y

deltaX = thisBoundary(:,2) - centroid(k,1); % boundary(:,2) is x, but centroid(k,1) is x

% angles = atand(deltaY ./ deltaX); % atand gives angles in range [-90,90]

angles = atan2d(deltaY,deltaX); % use atan2d to get angles in range [-180,180]

% Calculate the distances.

distances = sqrt(deltaY.^2 + deltaX.^2); % use deltaX and deltaY, which were just calculated

% add 360 degrees to negative angles so that range is [0,360] instead of [-180,180]

idx = angles < 0;

angles(idx) = angles(idx)+360;

% sort the angles, so the first one is smallest (closest to zero)

% and the last one is largest (closest to 360)

% this is useful for getting a smooth line that goes from 0 to 360 in the plot

[angles,idx] = sort(angles);

% reorder distances the same way angles was reordered

distances = distances(idx);

% Plot distance vs. angle.

plot(ax3,angles,distances)

end

In your original code, after the first for loop, thisBoundary has the value corresponding to the last boundary. In this image there's only one boundary, so it doesn't matter, but in general the calculations that depend on thisBoundary (i.e., distances and angles) need to be done in a loop so that you're calculating for each boundary, not just the last one. That's why I'm doing it all in one loop.

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Johnson 2024-4-24

scalebar.JPG

Hi Voss, I would like to change the units of the y-axis from pixels to microns. From the scalebar of the original SEM micrograph, I have been able to establish using the code below that 90 pixels represents 100 microns. How do I now integrate this information into the first code to convert pixels to microns? Thanks

% Read the image

image = imread('scalebar.jpg');

% Convert to grayscale if the image is colored

if size(image, 3) == 3

grayImage = rgb2gray(image);

else

grayImage = image;

end

% Apply edge detection

edges = edge(grayImage, 'Canny');

% Perform Hough transform to detect lines

[H,theta,rho] = hough(edges);

peaks = houghpeaks(H, 1); % You might need to adjust the number of peaks

% Find lines using Hough peaks

lines = houghlines(edges, theta, rho, peaks);

% Find the longest line

maxLineLength = 0;

for k = 1:length(lines)

xy = [lines(k).point1; lines(k).point2];

lineLength = norm(xy(2,:) - xy(1,:));

if lineLength > maxLineLength

maxLineLength = lineLength;

longestLine = lines(k);

end

% Display the result

imshow(image);

hold on;

plot([longestLine.point1(1), longestLine.point2(1)], [longestLine.point1(2), longestLine.point2(2)], 'LineWidth', 2, 'Color', 'r');

hold off;

disp(['Length of the object in pixels: ', num2str(maxLineLength)]);

Voss 2024-4-24

在 MATLAB Online 中打开

1_50.JPG

See the definition and use of the variable scale_factor below. Note that the first subplot didn't need to change since its x- and y-axis have no labels (so it doesn't matter what scale it's plotted with).

scale_factor = 100/90; % (units: microns/pixel)

% to convert pixels to microns,

% multiply # of pixels by scale_factor

%Image Threshholding

img = imread('1_50.JPG'); %Read image

BW = im2bw(img,0.45); %binarize image with a threshold value of 0.45

% this does nothing useful, since img2 is overwritten in the next line:

% img2= bwareaopen (BW, 1000); %Remove small objects

img2= imfill(BW, 'holes'); %fill holes

centriod_value= regionprops(img2, 'Centroid'); %Finds centroid of image

centroid = cat(1,centriod_value.Centroid);

%Display the binary image with the centroid locations superimposed

subplot(1,3,1)

imshow(img2);

hold on

plot(centroid(:,1),centroid(:,2),'b.')

% hold on % don't need to call hold on more than once on the same axes

% for plotting into multiple axes in one loop, it's convenient to set up

% all the axes first, before anything is plotted into them

% set up the 2nd axes:

ax2 = subplot(1,3,2);

hold on

box on

axis image

set(ax2,'YDir','reverse')

% set up the 3rd axes:

ax3 = subplot(1,3,3);

hold on

%tracing the boundary of image

p_boundary= bwboundaries(img2); % in row, column order, not x,y

number_of_boundaries = size(p_boundary,1); %one boundary

centroid = centroid*scale_factor;

for k = 1 : number_of_boundaries % use number_of_boundaries, which was just calculated

thisBoundary = p_boundary{k}*scale_factor;

y = thisBoundary(:,1); % rows

x = thisBoundary(:,2); % columns (these were swapped)

plot(ax2, x, y, 'g', 'LineWidth', 2);

plot(ax2, centroid(:,1),centroid(:,2),'b.')

% Calculate the angles in degrees

deltaY = thisBoundary(:,1) - centroid(k,2); % boundary(:,1) is y, but centroid(k,2) is y

deltaX = thisBoundary(:,2) - centroid(k,1); % boundary(:,2) is x, but centroid(k,1) is x

% angles = atand(deltaY ./ deltaX); % atand gives angles in range [-90,90]

angles = atan2d(deltaY,deltaX); % use atan2d to get angles in range [-180,180]

% Calculate the distances.

distances = sqrt(deltaY.^2 + deltaX.^2); % use deltaX and deltaY, which were just calculated

% add 360 degrees to negative angles so that range is [0,360] instead of [-180,180]

idx = angles < 0;

angles(idx) = angles(idx)+360;

% sort the angles, so the first one is smallest (closest to zero)

% and the last one is largest (closest to 360)

% this is useful for getting a smooth line that goes from 0 to 360 in the plot

[angles,idx] = sort(angles);

% reorder distances the same way angles was reordered

distances = distances(idx);

% Plot distance vs. angle.

plot(ax3,angles,distances)

end

Voss 2024-5-15

编辑：Voss 2024-5-15

在 MATLAB Online 中打开

1_50.JPG

Yeah, no problem. You can use linspace to generate a vector of equally-spaced angles and then use interp1 to interpolate distances based on those angles. See the code below for an example.

Note that N points between between 0 and 360, including the end points, implies N-1 "spaces" between the points, giving a spacing of 360/(N-1). So if you want 128 points, the distance between two adjacent points would be 360/127. And an increment of 360/128 would generate 129 points. (See: fencepost error)

The code below uses 129 points including 0 and 360, and I guess that you should use only the first 128 of those in your FFT since 0 and 360 are really the same angle.

scale_factor = 100/90; % (units: microns/pixel)

% to convert pixels to microns,

% multiply # of pixels by scale_factor

%Image Threshholding

img = imread('1_50.JPG'); %Read image

BW = im2bw(img,0.45); %binarize image with a threshold value of 0.45

% this does nothing useful, since img2 is overwritten in the next line:

% img2= bwareaopen (BW, 1000); %Remove small objects

img2= imfill(BW, 'holes'); %fill holes

centriod_value= regionprops(img2, 'Centroid'); %Finds centroid of image

centroid = cat(1,centriod_value.Centroid);

%Display the binary image with the centroid locations superimposed

subplot(1,3,1)

imshow(img2);

hold on

plot(centroid(:,1),centroid(:,2),'b.')

% hold on % don't need to call hold on more than once on the same axes

% for plotting into multiple axes in one loop, it's convenient to set up

% all the axes first, before anything is plotted into them

% set up the 2nd axes:

ax2 = subplot(1,3,2);

hold on

box on

axis image

set(ax2,'YDir','reverse')

% set up the 3rd axes:

ax3 = subplot(1,3,3);

hold on

%tracing the boundary of image

p_boundary= bwboundaries(img2); % in row, column order, not x,y

number_of_boundaries = size(p_boundary,1); %one boundary

centroid = centroid*scale_factor;

% use 129 points to have an angle spacing of 360/128

N_angles = 129;

interp_angles = linspace(0,360,N_angles).';

for k = 1 : number_of_boundaries % use number_of_boundaries, which was just calculated

thisBoundary = p_boundary{k}*scale_factor;

y = thisBoundary(:,1); % rows

x = thisBoundary(:,2); % columns (these were swapped)

plot(ax2, x, y, 'g', 'LineWidth', 2);

plot(ax2, centroid(:,1),centroid(:,2),'b.')

% Calculate the angles in degrees

deltaY = thisBoundary(:,1) - centroid(k,2); % boundary(:,1) is y, but centroid(k,2) is y

deltaX = thisBoundary(:,2) - centroid(k,1); % boundary(:,2) is x, but centroid(k,1) is x

% angles = atand(deltaY ./ deltaX); % atand gives angles in range [-90,90]

angles = atan2d(deltaY,deltaX); % use atan2d to get angles in range [-180,180]

% Calculate the distances.

distances = sqrt(deltaY.^2 + deltaX.^2); % use deltaX and deltaY, which were just calculated

% add 360 degrees to negative angles so that range is [0,360] instead of [-180,180]

idx = angles < 0;

angles(idx) = angles(idx)+360;

% sort the angles, so the first one is smallest (closest to zero)

% and the last one is largest (closest to 360)

% this is useful for getting a smooth line that goes from 0 to 360 in the plot

[angles,idx] = sort(angles);

% reorder distances the same way angles was reordered

distances = distances(idx);

% in case any angles are repeated, use only the unique set and

% corresponding distances in interp1 (in this data, one angle is

% repeated twice, with the same distance each time)

[~,idx] = unique(angles);

% perform linear interpolation/extrapolation:

interp_distances = interp1(angles(idx),distances(idx),interp_angles,'linear','extrap');

% Plot distance vs. angle.

plot(ax3,angles,distances)

hold(ax3,'on')

plot(ax3,interp_angles,interp_distances)

end

Johnson 2024-5-17

在 MATLAB Online 中打开

Hi Voss,

I'm very grateful for your help thus far.

But I am kind of stuck.

I have used the fft algorithm to obtain the descriptors/coefficients and corrresponding phase angles and plotted them. The last step involves using the first eight descriptors/coefficients to reconstruct the original shape of the particle. I don't know which function I need to use to achieve that. I would appreciate your help.

I have attached an image to demostrate exactly what I need. Thanks!

scale_factor = 100/90; % (units: microns/pixel)

% to convert pixels to microns,

% multiply # of pixels by scale_factor

%1. Open and show image of particle

img = imread('1_50.JPG'); %Read image

subplot (3,3,1)

imshow(img) %show image

%2. Binarize and show image

BW = im2bw(img,0.45); %binarize image with a threshold value of 0.45

img1= bwareaopen (BW, 1000); %Remove small objects

img2= imfill(img1, 'holes'); %fill holes

subplot(3,3,2)

imshow(img2) %show binarized image

%3. Setting up the axes

% for plotting into multiple axes in one loop, it's convenient to set up

% all the axes first, before anything is plotted into them

% 3.1. set up the 2nd axes:

ax3 = subplot(3,3,3);

hold on

box on %draws a box that encompasses the entire axes area

axis image %sets the aspect ratio of the axes (ax2) to be equal and

%adjusts the limits of the axes so that the data units are

%of equal size along both the x-axis and y-axis.

set(ax3,'YDir','reverse') %reverses the y-axis to begin from the top

%since the origin of the image is at the top left-corner

% 3.2. set up the 3rd axes:

ax4 = subplot(3,3,4);

hold on

%4. Compute centroid of binarized image

centriod_value= regionprops(img2, 'Centroid'); %Finds centroid of image

centroid = cat(1,centriod_value.Centroid); %Stores centroid coordinates in a two-column matrix

%5. Tracing the boundary of image

p_boundary= bwboundaries(img2); % trace the exterior boundaries of particle

%(in row, column order, not x,y)

number_of_boundaries = size(p_boundary,1); %specify only one boundary

centroid = centroid*scale_factor;

% 6. Setting up equally spaced angles before initing the loop

%use 129 points to have an angle spacing of 360/128

N_angles = 129;

interp_angles = linspace(0,360,N_angles).';

%7. Initiate for loop to plot boundary, centroid, & a graph of radius vs angle

for k = 1 : number_of_boundaries % initiate loop to go round the selected boundary

thisBoundary = p_boundary{k}*scale_factor;

y = thisBoundary(:,1); % rows

x = thisBoundary(:,2); % columns

plot(ax3, x, y, 'g', 'LineWidth', 2);

plot(ax3, centroid(:,1),centroid(:,2),'b.')

% 7.1. Calculate the angles in degrees

deltaY = thisBoundary(:,1) - centroid(k,2); % boundary(:,1) is y, but centroid(k,2) is y

deltaX = thisBoundary(:,2) - centroid(k,1); % boundary(:,2) is x, but centroid(k,1) is x

% angles = atand(deltaY ./ deltaX); % atand gives angles in range [-90,90]

angles = atan2d(deltaY,deltaX); % use atan2d to get angles in range [-180,180]

% 7.2. Calculate the radii

radius = sqrt(deltaY.^2 + deltaX.^2); % use deltaX and deltaY, which were just calculated

% 7.3. Add 360 degrees to negative angles so that range is [0,360] instead of [-180,180]

idx = angles < 0; %idx reps angles less than 0 degreees

angles(idx) = angles(idx)+360; %360 degrees is added to idx angles

%7.4. Sort angles in ascending order

% sort the angles, so the first one is smallest (closest to zero)

% and the last one is largest (closest to 360)

% this is useful for getting a smooth line that goes from 0 to 360 in the plot

% I returns the corresponding indices of the angles

[angles,I] = sort(angles);

%7.5. Reorder distances the same way angles was reordered using I

radius = radius(I);

%7.6. Use only unique angles in case an angle is repeated

% in case any angles are repeated, use only the unique set and

% corresponding distances in interp1 (in this data, one angle is

% repeated twice, with the same distance each time)

%The ~ (tilde) here is a placeholder indicating that we're not interested

%in capturing the actual unique elements, only their indices

[~,I] = unique(angles);

% 7.7. perform linear interpolation/extrapolation:

%angles(I)= sample points, radius(I)= correspoinding points

%and interp_angles = coordinates of the querry points (QP)

%In general, the functions returns interpolated values at specific QP

interp_distances = interp1(angles(I),radius(I),interp_angles,'linear','extrap');

% 7.8 Plot distance vs. angle.

plot(ax4,interp_angles,interp_distances)

end

%8. Apply fft algorithm to values of radius

L = length(interp_distances); %Define the number of the radiuses (points)

f = 0:L-1; %Define the sampling frequencies

A = fft(interp_distances); %Transforming the radius function into frequency domain using FFT

B = (abs(A)/2535.2); % Normalize the amplitude of fourier descriptors using the first descriptors

%9. Plot fft descriptor vs. frequency

subplot(3,3,5)

bar(f,B,3) %plot the frequencies vs. the nornalised descriptors

xlim([0 128])

%10. Plot phase angle vs. frequency

subplot(3,3,6)

bar(f, angle(A),3) % plot the frequencies vs phase angles

xlim([0 128])

ylim([-4 4])

%11. Create a table of values

table1 = table(A, f', abs(A), B, angle(A));

table1.Properties.VariableNames = {'FFT_coeffs', 'Frequency', 'Amplitude', 'Norm_Amplitude' 'Phase'};

disp(table1);

FFT_coeffs Frequency Amplitude Norm_Amplitude Phase __________________ _________ _________ ______________ ________ 2535.2+0i 0 2535.2 1 0 -9.0317+15.358i 1 17.817 0.0070278 2.1024 26.845+253.25i 2 254.67 0.10045 1.4652 13.688+20.778i 3 24.882 0.0098145 0.98829 -121.97+49.187i 4 131.51 0.051874 2.7583 -19.682+68.058i 5 70.847 0.027945 1.8523 24.087-5.3486i 6 24.674 0.0097324 -0.21851 -18.809+11.409i 7 21.999 0.0086773 2.5964 10.666-4.2899i 8 11.497 0.0045348 -0.38239 24.768-24.963i 9 35.165 0.013871 -0.78932 -11.726+1.5935i 10 11.833 0.0046677 3.0065 -3.3974-0.96347i 11 3.5313 0.0013929 -2.8653 2.1882-7.7423i 12 8.0456 0.0031735 -1.2953 -5.7579+7.612i 13 9.5444 0.0037648 2.2184 2.2416-4.7407i 14 5.244 0.0020685 -1.1291 6.7123-4.3787i 15 8.0143 0.0031612 -0.57802 -5.5518+5.4896i 16 7.8075 0.0030797 2.3618 5.1405-1.1483i 17 5.2672 0.0020776 -0.21977 -0.67655+0.29538i 18 0.73822 0.00029119 2.7299 -5.5082+0.23644i 19 5.5133 0.0021747 3.0987 -2.5284-6.0568i 20 6.5633 0.0025889 -1.9663 -3.4387+2.0494i 21 4.0031 0.001579 2.6041 6.1491+3.5726i 22 7.1116 0.0028052 0.52632 2.6487-4.4635i 23 5.1902 0.0020473 -1.0352 4.1541+5.3612i 24 6.7823 0.0026752 0.91158 -4.3927+1.9717i 25 4.8149 0.0018992 2.7197 -0.55218-7.9741i 26 7.9932 0.0031529 -1.6399 -3.2769+1.8264i 27 3.7515 0.0014798 2.6331 1.4489-3.1397i 28 3.4579 0.001364 -1.1384 -2.7055-1.7054i 29 3.1981 0.0012615 -2.5792 1.5435+1.5387i 30 2.1795 0.00085968 0.78383 1.4931-0.59173i 31 1.6061 0.00063351 -0.37733 2.5774-4.6633i 32 5.3282 0.0021017 -1.0659 -3.0106+2.7811i 33 4.0986 0.0016167 2.3958 1.0544-4.2003i 34 4.3306 0.0017082 -1.3249 0.0080565-3.8057i 35 3.8057 0.0015011 -1.5687 -2.6008+5.8739i 36 6.4239 0.0025339 1.9876 5.0802+1.0899i 37 5.1957 0.0020494 0.21133 0.75775-5.1181i 38 5.1739 0.0020408 -1.4238 -0.71473+1.6764i 39 1.8224 0.00071885 1.9738 3.9696+1.8349i 40 4.3732 0.001725 0.43298 -0.10762-3.0166i 41 3.0185 0.0011906 -1.6065 -2.9289+3.2315i 42 4.3614 0.0017203 2.3071 -0.39951-1.9583i 43 1.9986 0.00078836 -1.772 0.63203+0.26403i 44 0.68496 0.00027018 0.39571 -0.48522-0.36734i 45 0.60858 0.00024005 -2.4936 0.6727-3.35i 46 3.4168 0.0013478 -1.3726 1.8684+0.58354i 47 1.9574 0.0007721 0.30272 1.1535-1.8886i 48 2.213 0.00087291 -1.0225 -1.6067-1.1573i 49 1.9801 0.00078106 -2.5174 0.83255+2.1538i 50 2.3091 0.00091083 1.2019 3.0466-0.99287i 51 3.2043 0.0012639 -0.31504 -0.67805-0.83992i 52 1.0795 0.00042579 -2.25 2.4134+3.1935i 53 4.0029 0.0015789 0.92364 1.1117-0.96505i 54 1.4721 0.00058068 -0.7149 -0.71874+0.13201i 55 0.73076 0.00028824 2.9599 0.35185+1.0271i 56 1.0857 0.00042823 1.2407 1.3453-0.60543i 57 1.4753 0.00058191 -0.42288 -0.31294-1.3413i 58 1.3773 0.00054328 -1.8 2.4396+2.7008i 59 3.6395 0.0014356 0.83617 0.40332+0.63473i 60 0.75202 0.00029663 1.0047 0.83253+0.69215i 61 1.0827 0.00042706 0.69359 -1.4739+1.8772i 62 2.3867 0.00094141 2.2364 -0.75802+0.54169i 63 0.93168 0.0003675 2.5211 -2.0057+0.36636i 64 2.0389 0.00080424 2.9609 -2.0057-0.36636i 65 2.0389 0.00080424 -2.9609 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0.0021747 -3.0987 -0.67655-0.29538i 111 0.73822 0.00029119 -2.7299 5.1405+1.1483i 112 5.2672 0.0020776 0.21977 -5.5518-5.4896i 113 7.8075 0.0030797 -2.3618 6.7123+4.3787i 114 8.0143 0.0031612 0.57802 2.2416+4.7407i 115 5.244 0.0020685 1.1291 -5.7579-7.612i 116 9.5444 0.0037648 -2.2184 2.1882+7.7423i 117 8.0456 0.0031735 1.2953 -3.3974+0.96347i 118 3.5313 0.0013929 2.8653 -11.726-1.5935i 119 11.833 0.0046677 -3.0065 24.768+24.963i 120 35.165 0.013871 0.78932 10.666+4.2899i 121 11.497 0.0045348 0.38239 -18.809-11.409i 122 21.999 0.0086773 -2.5964 24.087+5.3486i 123 24.674 0.0097324 0.21851 -19.682-68.058i 124 70.847 0.027945 -1.8523 -121.97-49.187i 125 131.51 0.051874 -2.7583 13.688-20.778i 126 24.882 0.0098145 -0.98829 26.845-253.25i 127 254.67 0.10045 -1.4652 -9.0317-15.358i 128 17.817 0.0070278 -2.1024

%12. Reconstruct particle boundary using FFT descriptors ?????

Voss 2024-5-17

I'm not sure. I would ask your professor/advisor what exactly is involved in that.

There are others on here who know more than I do about signal processing and could probably help. But it seems to me that you probably should be using both the distances and the angles, not just the distances, in the FFT calculation. I recommend talking to your advisor and/or posting a new question about the FFT reconstruction part.

Johnson 2024-5-18