Searching Across a Line in a Matrix in Octave

Question

The attached image has a line with a break in it.

My code finds the line using a hough transform resulting in r=32 and theta=2.3213 . The hough transform isn't perfect, the angle (especially with a more complex image) is always off by a little bit, and in this case, because of the edge detection, the line is offset. I want to read values across the line to find the breaks in it. In order to do this, I will need to be able to sample values on either side of the line to find where the maximum density of the line is.

Further explanation (if you want it): If you look closely at the image you can see areas where the line crosses a pixel pretty much dead on resulting in a value of nearly 1/white. Other areas have two pixels side by side with values of about .5/gray. I need to find a solution that takes into account the anti-aliasing of the line, and allows me to extract the breaks in it.

%Program Preparation
clear ; close all; clc  %clearing command window
pkg load image %loading image analyzation suite
pkg load optim  

%Import Image
I_original = imread("C:/Users/3015799/Desktop/I.jpg");

%Process Image to make analysis quicker and more effective
I = mat2gray(I_original);   %convert to black and white
I = edge(I, 'sobel');

%Perform Hough Transform
angles = pi*[-10:189]/180;
hough = houghtf(I,"line",angles);

%Detect hot spots in hough transform
detect = hough>.5*max(hough(:));

%Shrink hotspots to geometric center, and index
detect = bwmorph(detect,'shrink',inf);
[ii, jj] = find(detect);
r = ii - (size(hough,1)-1)/2;
theta = angles(jj);

%Cull duplicates. i.e outside of 0-180 degrees
dup = theta<-1e-6 | theta>=pi-1e-6;
r(dup) = [];
theta(dup) = [];

%Compute line parameters (using Octave's implicit singleton expansion)
r = r(:)'
theta = theta(:)'
x = repmat([1;1133],1,length(r)); % 2xN matrix, N==length(r)
y = (r - x.*cos(theta))./sin(theta); % solve line equation for y

%The above goes wrong when theta==0, fix that:
horizontal = theta < 1e-6;
x(:,horizontal) = r(horizontal);
y(:,horizontal) = [1;:];

%Plot
figure
imshow(I)
hold on
plot(y,x,'r-','linewidth',2)

Answer 1

If you are only interested in the length of the gap, this would be very easy:

clear all
pkg load image

img_fn = "input.jpg";

if (! exist (img_fn, "file"))
  urlwrite ("https://i.stack.imgur.com/5UnpO.jpg", img_fn);
endif

Io = imread(img_fn);
I = im2bw (Io);

r = max(I);
c = max(I');

ri = find (diff(r));
ci = find (diff(c));

## both should have 4 elements (one break)
assert (numel (ri) == 4);
assert (numel (ci) == 4);

## the gap is in the middle
dx = diff(ri(2:3))
dy = diff(ci(2:3))

# the length is now easy
l = hypot (dy, dx)

gives

dx =  5
dy =  5
l =  7.0711

without any hogh transform. Of course you have to also check the corener cases for horizontal and vertical lines but this should give you an idea