Measuring the Radius of a Role of Tape

Your objective is to measure the radius of a roll of tape, which is partially obscured by the tape dispenser. You will utilize bwtraceboundary in order to accomplish this task.

Step 1: Read image
Step 2: Threshold the image
Step 3: Extract initial boundary point location
Step 4: Trace the boundaries
Step 5: Fit a circle to the boundary

Step 1: Read image

Read in tape.png.

RGB = imread('tape.png');
imshow(RGB);

text(15,15,'Estimate radius of the roll of tape',...
     'FontWeight','bold','Color','y');

Step 2: Threshold the image

Convert the image to black and white for subsequent extraction of the edge coordinates using the bwtraceboundary routine.

I = rgb2gray(RGB);
threshold = graythresh(I);
BW = im2bw(I,threshold);
imshow(BW)

Step 3: Extract initial boundary point location

The bwtraceboundary routine requires that you specify a single point on a boundary. This point is used as the starting location for the boundary tracing process.

To find the edge of the tape, pick a column in the image and inspect it until a transition from a background pixel to the object pixel occurs.

dim = size(BW);

col = round(dim(2)/2)-90;

row = min(find(BW(:,col)));

Step 4: Trace the boundaries

The bwtraceboundary routine is used to find (X, Y) locations of the boundary points. In order to maximize the accuracy of the radius calculation, it is important to find as many points belonging to the tape boundary as possible. You should determine the number of points experimentally.

connectivity = 8;
num_points   = 180;
contour = bwtraceboundary(BW, [row, col], 'N', connectivity, num_points);

imshow(RGB);
hold on;

plot(contour(:,2),contour(:,1),'g','LineWidth',2);

Step 5: Fit a circle to the boundary

Rewrite basic equation for a circle:

(x-xc)^2 + (y-yc)^2 = radius^2,  where (xc,yc) is the center

in terms of parameters a, b, c as

x^2 + y^2 + a*x + b*y + c = 0,  where a = -2*xc, b = -2*yc, and
                                      c = xc^2 + yc^2 - radius^2

Solve for parameters a, b, c, and use them to calculate the radius.

x = contour(:,2);
y = contour(:,1);

% solve for parameters a, b, and c in the least-squares sense by
% using the backslash operator
abc=[x y ones(length(x),1)]\[-(x.^2+y.^2)];
a = abc(1); b = abc(2); c = abc(3);

% calculate the location of the center and the radius
xc = -a/2;
yc = -b/2;
radius  =  sqrt((xc^2+yc^2)-c)

% display the calculated center
plot(xc,yc,'yx','LineWidth',2);

% plot the entire circle
theta = 0:0.01:2*pi;

% use parametric representation of the circle to obtain coordinates
% of points on the circle
Xfit = radius*cos(theta) + xc;
Yfit = radius*sin(theta) + yc;

plot(Xfit, Yfit);

message = sprintf('The estimated radius is %2.3f pixels', radius);
text(15,15,message,'Color','y','FontWeight','bold');

radius =

   80.7256