Radon Transform Line Detection - matlab

Question

I'm trying to detect lines in a grayscale image. For that purpose, I'm using Radon transform in MATLAB. An example of my m-file is like below. I can detect multiple lines using this code. I also draw lines using shift and rotation properties for lines. However, I didn't understand how to get the start and end points of the detecting lines after getting rho and theta values.
It is easy for Hough transform since there is a function called houghlines() that returns the list of the lines for the given peaks. Is there any function that i can use for Radon transform similar to this function?
% Radon transform line detection algorithm
clear all; close all;
% Determine the path of the input image
str_inputimg = '3_lines.png' ;
% Read input image
I = imread(str_inputimg) ;
% If the input image is RGB or indexed color, convert it to grayscale
img_colortype = getfield(imfinfo(str_inputimg), 'ColorType') ;
switch img_colortype
case 'truecolor'
I = rgb2gray(I) ;
case 'indexedcolor'
I = ind2gray(I) ;
end
figure;
subplot(2,2,1) ;
imshow(I) ;
title('Original Image') ;
% Convert image to black white
%BW = edge(I,'Sobel');
BW=im2bw(I,0.25) ;
subplot(2,2,2) ;
imshow(BW);
title('BW Image') ;
% Radon transform
% Angle projections
theta = [0:179]' ;
[R, rho] = radon(BW, theta) ;
subplot(2,2,3) ;
imshow(R, [], 'XData', theta, 'YData', rho, 'InitialMagnification', 'fit');
xlabel('\theta'), ylabel('\rho');
axis on, axis normal, hold on;
% Detect the peaks of transform output
% Threshold value for peak detection
threshold_val = ceil(0.3*max(R(:))) ;
% Maximum nof peaks to identify in the image
max_nofpeaks = 5 ;
max_indexes = find(R(:)>threshold_val) ;
max_values = R(max_indexes) ;
[sorted_max, temp_indexes] = sort(max_values, 'descend') ;
sorted_indexes = max_indexes(temp_indexes) ;
% Get the first highest peaks for the sorted array
if (length(sorted_max) <= max_nofpeaks)
peak_values = sorted_max(1:end) ;
peak_indexes = sorted_indexes(1:end) ;
else
peak_values = sorted_max(1:max_nofpeaks) ;
peak_indexes = sorted_indexes(1:max_nofpeaks) ;
end
[y, x] = ind2sub(size(R), peak_indexes ) ;
peaks = [rho(y) theta(x)] ;
plot(peaks(:,2), peaks(:,1), 's', 'color','white');
title('Radon Transform & Peaks') ;
% Detected lines on the image
subplot(2,2,4), imshow(I), title('Detected lines'), hold on
x_center = floor(size(I, 2)/2) ;
y_center = floor(size(I, 1)/2) ;
for p=1:length(peaks)
x_1 = [-x_center, x_center] ;
y_1 = [0, 0] ;
% Shift at first
x_1_shifted = x_1 ;
y_1_shifted = [y_1(1)-peaks(p,1), y_1(2)-peaks(p,1)] ;
% Rotate
peaks(p,2) = 90 - peaks(p,2) ;
t=peaks(p,2)*pi/180;
rotation_mat = [ cos(t) -sin(t) ; sin(t) cos(t) ] ;
x_y_rotated = rotation_mat*[x_1_shifted; y_1_shifted] ;
x_rotated = x_y_rotated(1,:) ;
y_rotated = x_y_rotated(2,:) ;
plot( x_rotated+x_center, y_rotated+y_center, 'b', 'linewidth', 2 );
end
hold off;

Answer 1

There's a suggestion at math.SE which might help. Then there's a rather complicated-looking research paper "Sharp endpoint estimates for the X-ray transform and the Radon
transform in finite fields", which appears just to show certain bounds on estimation accuracy.
From skimming other papers, it appears that it's a nontrivial problem. I suspect it may be simpler (if less accurate) to use some adaptation of a Sobel-operation to identify high gradient points along the discovered line, and claim those as endpoints.