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)

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 

[Voss]

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

[Johnson]

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!

Answer 1

[Voss]

%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.

[Johnson]

Thank you so much for your insights and help. This is exactly what I need

[Voss]

You're welcome!

[Johnson]

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

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]

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

[Johnson]

Thank you so much for this

[Voss]

You're welcome!

[Johnson]

Hi Voss, I'd like to ask for another favor.

When i run the code and look at the specific angles used to generate the curve, they are not equally spaced (e.g. the first angle is 3.1451, second 7.1563, third 11.6560...)

Is it possible to have the angles equally spaced out? Ideally, I would like the total number of points in the plot to be 128, where the first angle is 0 degrees, the second (0 + (360/128)) degrees, the third (360/128) + (360/128), ......, and the last angle 360 degrees (that is, an increment of 360/128 from 0 to 360 degrees).

I have realized i need to specify the angles and the increment in my next operation (using the fft algorithm).

Much Thanks!

[Voss]

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]

Thank you so much for this!

[Voss]

You're welcome!

[Johnson]

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
     -0.75802-0.54169i        66        0.93168        0.0003675       -2.5211
      -1.4739-1.8772i         67         2.3867       0.00094141       -2.2364
      0.83253-0.69215i        68         1.0827       0.00042706      -0.69359
      0.40332-0.63473i        69        0.75202       0.00029663       -1.0047
       2.4396-2.7008i         70         3.6395        0.0014356      -0.83617
     -0.31294+1.3413i         71         1.3773       0.00054328           1.8
       1.3453+0.60543i        72         1.4753       0.00058191       0.42288
      0.35185-1.0271i         73         1.0857       0.00042823       -1.2407
     -0.71874-0.13201i        74        0.73076       0.00028824       -2.9599
       1.1117+0.96505i        75         1.4721       0.00058068        0.7149
       2.4134-3.1935i         76         4.0029        0.0015789      -0.92364
     -0.67805+0.83992i        77         1.0795       0.00042579          2.25
       3.0466+0.99287i        78         3.2043        0.0012639       0.31504
      0.83255-2.1538i         79         2.3091       0.00091083       -1.2019
      -1.6067+1.1573i         80         1.9801       0.00078106        2.5174
       1.1535+1.8886i         81          2.213       0.00087291        1.0225
       1.8684-0.58354i        82         1.9574        0.0007721      -0.30272
       0.6727+3.35i           83         3.4168        0.0013478        1.3726
     -0.48522+0.36734i        84        0.60858       0.00024005        2.4936
      0.63203-0.26403i        85        0.68496       0.00027018      -0.39571
     -0.39951+1.9583i         86         1.9986       0.00078836         1.772
      -2.9289-3.2315i         87         4.3614        0.0017203       -2.3071
     -0.10762+3.0166i         88         3.0185        0.0011906        1.6065
       3.9696-1.8349i         89         4.3732         0.001725      -0.43298
     -0.71473-1.6764i         90         1.8224       0.00071885       -1.9738
      0.75775+5.1181i         91         5.1739        0.0020408        1.4238
       5.0802-1.0899i         92         5.1957        0.0020494      -0.21133
      -2.6008-5.8739i         93         6.4239        0.0025339       -1.9876
    0.0080565+3.8057i         94         3.8057        0.0015011        1.5687
       1.0544+4.2003i         95         4.3306        0.0017082        1.3249
      -3.0106-2.7811i         96         4.0986        0.0016167       -2.3958
       2.5774+4.6633i         97         5.3282        0.0021017        1.0659
       1.4931+0.59173i        98         1.6061       0.00063351       0.37733
       1.5435-1.5387i         99         2.1795       0.00085968      -0.78383
      -2.7055+1.7054i        100         3.1981        0.0012615        2.5792
       1.4489+3.1397i        101         3.4579         0.001364        1.1384
      -3.2769-1.8264i        102         3.7515        0.0014798       -2.6331
     -0.55218+7.9741i        103         7.9932        0.0031529        1.6399
      -4.3927-1.9717i        104         4.8149        0.0018992       -2.7197
       4.1541-5.3612i        105         6.7823        0.0026752      -0.91158
       2.6487+4.4635i        106         5.1902        0.0020473        1.0352
       6.1491-3.5726i        107         7.1116        0.0028052      -0.52632
      -3.4387-2.0494i        108         4.0031         0.001579       -2.6041
      -2.5284+6.0568i        109         6.5633        0.0025889        1.9663
      -5.5082-0.23644i       110         5.5133        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]

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.

[Voss]

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]

That's okay. Thanks!