Hough Transform - Introduction

 So what is hough lines or the transform ,

 Hough Line Transform is a transform used to detect straight lines.

consider a image where u need to find the lines straight lines ,could be horizontal , vertical , diagonal lines in the image u can do that via the hough transform

 

before that there is something called

 Cartesian coordinate system: Parameters: (m,b).

Polar coordinate system: Parameters: (r,θ) (r,θ)

 

if the line equation is y = m * x +b  in the cartesian 

x, y 

in the polar it is 

x = r cos(𝛳)

y = r sin(𝛳) .

we will see about this in detail 

from the diagram u can see the inclined line and 𝛳 are related as

x = r cos(𝛳) , the adjacent side of 𝛳

y = r sin(𝛳), the opposite side of 𝛳 

' r ' the distance from the origin the reference point to the line which

hits the line at 90 degrees 

ok so by Pythagoras 

r = sqrt ( x ^2 + y^2)

substitute for x and y 

r = sqrt  (r^2 *x^2 cos^2𝛳 + r^2 * y^2sin^2𝛳)

cos^2𝛳 + ^sin^2𝛳 = 1

r = r  

ok what we have proved is the value of x and y is correct  moving forward

u will not get this satisfying until u play with a use case ,

the use case would be a image .we will see and example with an image for the 

implementation .

consider a point x, y (4, 3) this is the cartesian system

we will transform this point into something in terms of (r(𝛳 ) ,𝛳)

stay with me for now

r𝛳 = x cos 𝛳 +y sin𝛳 it means for the value of x,y 

we need to find r ,so what is ' 𝛳 ' it could be any value 𝛳 >0 and 𝛳 <2𝞹 

for x ,y [4, 3] 

plot values of 𝛳 and  r 

from the diagram u can see the family of lines that pass through x ,y which is 4,3

for  now i have take only 2 lines the 𝛳 =50 and  r 50   

 𝛳 =10 and r10  

if you plot this as a graph of 𝛳 and r  

 

u might wonder what is the use of this curve  ,we will see with two point

examples and the intersection of these curves will give us results for line computation.

reference -

https://docs.opencv.org/3.4/d9/db0/tutorial_hough_lines.html