Post on 23-Mar-2016
description
transcript
Hough TransformProcedure to find a shape in an imageShape can be described in parametric formShapes in image correspond to a family of parametric solutionsA voting scheme is used to determine the correct parameters
Hough Transform for Linesy ax b b y ax
y
x
3 3( , )x y
2 2( , )x y
1 1( , )x y
a
b
3 3b y ax
0 0( , )a b
0 0y a x b
2 2b y ax 1 1b y ax
Hough Transform for LinesQuantize spaceEach bin represents a line defined by Accumulate values in bin according to edge points Each bin contains the amount of points in image that would form a line defined by
b
3 3b y ax
( , )i ia b
2 2b y ax
1 1b y ax
( , )i ia b
( , )j jx y
( , )a b
( , )i ia b
Hough Transform for LinesProblem with vertical lines Normal form
a
cos( ) sin( )r x y y
x
( , )r
0 0
0
cos( )sin( )
r xy
0r
11r
1 1( , )x y
0
2 2( , )x y
0 0( , )x y1 1cos( ) sin( )r x y
Hough Transform for Lines
y
x
0r1 1( , )x y
0
2 2( , )x y
0 0( , )x y
1 1cos( ) sin( )r x y
-3 -2 -1 1 2 3
-2
-1
1
2
r
1 1( , )x y
0 0( , )x y
2 2( , )x y
0 0( , )r
0 0( , )r
Hough Transform for Lines
y
x1
1r1 1( , )x y
3 3( , )x y4 4( , )x y
-3 -2 -1 1 2 3
-4
-2
2
4
1 1( , )x y
r1 1cos( ) sin( )r x y
3 3( , )x y
4 4( , )x y 1 1( , )r
1 1( , )r
Pick in the parametric space that corresponds
to the line
Hough Transform for LinesDomain of the parametric space:
M and N image resolution
Not just lines, any parametric curve!
However increase of dimensions of the parametric space
2 2 2 2, , ,2 2
r M N M N
Hough Transform of CirclesCircle is define by its center and radius -3D Hough parametric space
2 2 2( ) ( )x a y b r
( , )a br
0r
1r0 0( , )a b
1 1( , )a b
0 0( , )x y2 2 2
0 0 0( ) ( )x a y b r
2 2 21 1 1( ) ( )x a y b r
( , , )a b r
Hough Transform of Circles
0r
1r0 0( , )a b
1 1( , )a b
y
x
a
br
0r r
0 0( , )x y0 0( , )x y
1 1( , )x y
1 1( , )x y
0 0 0( , , )a b r
cosa x r sinb y r
a
br
Hough Transform of Circles
0r
1r0 0( , )a b
1 1( , )a b
y
x
1r r
0 0( , )x y
1 1( , )x y0 0( , )x y
1 1( , )x y
1 1 1( , , )a b rcosa x r sinb y r
Hough Transform
Find parametric shapes in an imageSlow but robust to noise and occlusionQuantization of the parameters controls accuracy and speed