HOUGH TRANSFORM INEL 6088 - Fall 2018 - M. Toledo Jain et.al. - PowerPoint PPT Presentation

HOUGH TRANSFORM INEL 6088 - Fall 2018 - M. Toledo Jain et.al. section 6.8.4 Davies, section 9.2 and chapter 10

y = mx + b = − x cos θ ρ ρ = x cos θ + y sin θ sin θ + sin θ tan θ = − cos θ 1 slope = m = − sin θ ρ y-intercept = b = sin θ θ θ line is a dot in parameter plane ρ ρ 90- θ c θ =cos θ ; s θ =sin θ θ

For a point (x 1 , y 1 ) you get a curve in the ρ - θ plane. Each point in the curve correspond to a line passing thru (x 1 ,y 1 ) y ρ = x 1 cos θ + y 1 sin θ θ ρ x

For co-linear points the curves intersect. y θ ρ x

ρ = x 1 cos θ + y 1 sin θ Puntos (2,5) y (4,1) y ⇒ ρ , x ⇒ θ

Example: θ versus ρ ; points (2,5) and (4,1) 90" 80" 70" 60" 50" θ" Series2" 40" 30" 20" θ =27; ρ =4 10" 0" 0" 1" 2" 3" 4" 5" 6" y= m x + b = - x/tan(27) + 4/sin(27) = -2x + 9

Hough Transform for line detection The input is E, and M x N binary image in which each pixel E(i,j) is 1 if an edge pixel, 0 otherwise. Let ρ d , θ d be arrays containing discretized intervals of the ρ , θ parameter spaces ( ρ is within the image, θ is between 0 and π ), and R and T, respectively, their number of elements. 1. Discretize the parameter space of ρ and θ using sampling steps δρ , δθ , which must yield acceptable resolution and be of manageable size for ρ d and θ d. 2. Let A(R,T) be an array of integer counters. Initialize all elements to 0. 3.For each pixel E(i,j) = 1 and for h=1...T i. let ρ = i cos θ d (h) + j sin θ d (h) ii. find the index k of the element ρ d closest to ρ iii. increment A(k,h) by one 4. find all local maxima (k p , h p ) for which A(k p , h p ) > threshold. The output is a set of pairs ( ρ d (k p , h p ), θ d (k p , h p )) describing the detected lines in polar form.

im = zeros(256,256); im = mat2gray(im); [nim xyp] = makeline(im, 3, 75, 255, 4, 1); figure; imshow(nim); [ints, slopes]=mc_hough(xyp); figure; plot(slopes, ints); [rhos angles]=hough(xyp, 1); figure; plot(angles, rhos); vm = votes(rhos, angles); % [newim, coords] = makeline(im, slope, yint, value, radious, period) % Creates a line in an 8-bit image. figure; mesh(vm); % newim: name of new image % im: initial image (without the line) % slope: line slope % yint: y intercept % value: line pixel graylevel % radious: radious of point (num of pixels) % 1 for single pixel % period: number of x points to omit between % calculations; use 0 for max points in line %

200 im = zeros(256,256); im = mat2gray(im); 100 [nim xyp] = makeline(im, 3, 75, 255, 4, 1); 0 figure; imshow(nim); -100 [ints, slopes]=mc_hough(xyp); -200 figure; plot(slopes, ints); -300 [rhos angles]=hough(xyp, 1); figure; plot(angles, rhos); -400 1 2 3 4 5 6 7 8 9 10 vm = votes(rhos, angles); figure; mesh(vm); function [ints, slopes] = mc_hough(pts) % function [intercepts slopes] = mc_hough(pts) % Draw an image for parameter space of hough transform % pts = matrix of point coordinates, size 2x(number of points) % % for each point draw a line;

im = zeros(256,256); im = mat2gray(im); 300 250 [nim xyp] = makeline(im, 3, 75, 255, 4, 1); 200 figure; imshow(nim); 150 [ints, slopes]=mc_hough(xyp); 100 figure; plot(slopes, ints); 50 0 [rhos angles]=hough(xyp, 1); -50 figure; plot(angles, rhos); -100 0 20 40 60 80 100 120 140 160 180 vm = votes(rhos, angles); figure; mesh(vm); function [rhos, angles] = hough(pts, ar) % function [rhos angles] = hough(pts, ar, rr) % Draw an image for parameter space of hough transform % pts = matrix of point coordinates, size 2x(number of % points) % ar = theta (angle) resolution (example 10 degrees) %

im = zeros(256,256); im = mat2gray(im); [nim xyp] = makeline(im, 3, 75, 255, 4, 1); figure; imshow(nim); [ints, slopes]=mc_hough(xyp); figure; plot(slopes, ints); [rhos angles]=hough(xyp, 1); figure; plot(angles, rhos); vm = votes(rhos, angles); figure; mesh(vm); function vmat = votes(rhos, angles) % function vmat = votes(rhos, angles) % Find the votes for two parameter space hough transform % rhos = matrix of point distances % % angles = vector of theta (angle) %

Equation of a circle r 2 = ( x − a ) 2 + ( y − b ) 2 x = a + r cos θ y = b + r sin θ a = x − r cos θ b = y − r sin θ Hough transform for circles 1. Select a value of r 2. Create a vector of quantized angles 3. For each θ , a) edge point (x,y), calculate a and b b) increment the votes for (a,b,r) 4. Threshold the (a,b,r) table to find the circles

function value = count_coins(fname) % value = count_coins(fname) % input: filename containing image % output: value of coins % % will also display an image of the file % indicating the coins found % warning off;close all; I=imread(fname); I=rgb2gray(I); %ime=edge(I,'sobel',0.12); ime=edge(I,'canny',[0.15 0.25],2); figure;imshow(ime) r=[141 125 110 103]; %radios to considering vr=[0.25 0.05 0.01 0.1]; tic; %Hough Transform for each radio cont=0;

>> count_coins('fig002.jpg') Elapsed time is 8.986488 seconds. ans = 0.9100 %******Calculate the value of money in coins******************* CC1=length(find(C1(1:nm(1),1))); CC2=length(find(C1(nm(1)+1:nm(1)+nm(2),1))); CC3=length(find(C1(nm(1)+nm(2)+1:nm(1)+nm(2)+nm(3),1))); CC4=length(find(C1(nm(1)+nm(2)+nm(3)+1:nm(1)+nm(2)+nm(3)+nm(4),1))); value=CC1*0.25+CC2*0.05+CC3*0.01+CC4*0.1; y=find(C1(:,1)); C1=C1(y,:); %*************************************************** %Plot of centers ans circles figure;imshow(I);hold on; theta1 = 0:0.01:2*pi; plot(C1(:,2), C1(:,1), 'b+'); %Center of circles for i = 1:size(C1,1) plot(C1(i,2)+C1(i,3)*sin(theta1),C1(i,1)+C1(i,3)*cos(theta1),'r'); %plot of the Circle(s) end toc; end

%Hough Transform for each radio cont=0; for k=1:size(r,2), [C]=hough_circ(ime,r(k)); temp=cont+size(C,1); C1(cont+1:temp,:)=C; cont=temp; for i=nm(1)+1:nm(1)+nm(2), nm(k)=size(C,1); for j=nm(1)+nm(2)+1:size(C1,1) end if C1(i,1)==0 %***End Hough Transform calculation continue; end % Remove repeated coins temp1=abs(C1(i,1)-C1(j,1)); for i=1:nm(1), temp2=abs(C1(i,2)-C1(j,2)); if (temp1<50)&(temp2<50) for j=nm(1)+1:size(C1,1) C1(j,:)=0; temp1=abs(C1(i,1)-C1(j,1)); end temp2=abs(C1(i,2)-C1(j,2)); end if (temp1<50)&(temp2<50) end C1(j,:)=0; end if length(find(C1(nm(1)+nm(2)+nm(3)+1:nm(1)+nm(2)+nm(3)+nm(4),1)))>1 for i=nm(1)+nm(2)+1:nm(1)+nm(2)+nm(3), end for j=nm(1)+nm(2)+nm(3)+1:size(C1,1) end if C1(i,1)==0 continue; end temp1=abs(C1(i,1)-C1(j,1)); temp2=abs(C1(i,2)-C1(j,2)); if (temp1<20)&(temp2<20) C1(j,:)=0; end end end elseif length(find(C1(nm(1)+nm(2)+1:nm(1)+nm(2)+nm(3),1)))>=1 C1(nm(1)+nm(2)+1:nm(1)+nm(2)+nm(3),1)=0; end %***************************** repeated coins removed

Fitting a line to a set of points To fit a line to several points: − x 1 cos θ = y 1 sin θ − ρ − x 1 = y 1 tan θ − ρ / cos θ � ✓ ρ / cos θ ◆ � 1 x 1 = − y 1 tan θ For 3 points 0 1 0 1 x 1 1 − y 1 ✓ ρ / cos θ ◆ x 2 = 1 − y 2 @ A @ A tan θ x 3 1 − y 3 X = MA ✓ ρ / cos θ ◆ � − 1 M t X � M t M A = = tan θ = arctan A (2) θ = A (1) cos θ ρ The points are on the same line if the rms error is below some threshold.

Fitting a Circle ( x − x c ) 2 + ( y − y c ) 2 r 2 = x 2 − 2 x c x + x 2 c + y 2 − 2 y c y + y 2 c − r 2 0 = x 2 + ax + y 2 + by + c 0 = This form is the implicit parametric equation of the circle. To fit a circle to n data points with coordinates ( x i , y i ), − x 2 ax i + y 2 = i + by i + c i 2 3 x i y 2 − x 2 6 7 ⇥ ⇤ i 1 = a b c 6 7 i y i 4 5 1 = − X BM − XM T BMM T = MM T � − 1 = − XP − XM T � = B Notice that a = B (1) = − 2 x c , b = B (3) = − 2 y c and c = B (4) = x 2 c + y 2 c − r 2 .

measured fitted and true circles measured 1 fitted true 0.8 0.6 0.4 0.2 0 center (0.0224336 , -0.00316806 ); R=1.10003 y -0.2 -0.4 -0.6 -0.8 -1 -1 -0.5 0 0.5 1 x