1
Lecture-12
Hough Transform
Examples
Hough Space
Theta is from -90 to +90
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Hough Transform and Image Processing: Lecture Notes, Study notes of Computer Science
Lecture notes on hough transform and its applications in fitting lines and circles in images. It also covers the concept of pyramids, gaussian pyramids, and convolution. The notes include explanations, examples, and formulas.
Typology: Study notes
Pre 2010
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Lecture-
Hough Transform
Examples
Hough Space
Theta is from -90 to +
Fitting Lines In an Image
Fitting Lines In an Image
Fitting Circles
Detecting Lines in Gray Level
Images
Detect yellow line in the middle
Use gray levels instead of edges
Increment the parameter space by gray level at a pixel instead of by 1.
Pyramids
- Very useful for representing images.
- Pyramid is built by using multiple copies of
image.
- Each level in the pyramid is 1/4 of the size
of previous level.
- The lowest level is of the highest resolution.
- The highest level is of the lowest resolution.
Pyramid
Gaussian Pyramids
[ ] , , - 1
= l n ln
g EXPAND g
2
2
2
2
, , 1
i p j q
g i j w p q g
p q
ln ln
= (^) Â Â
=- =-
Reduce (1D)
() ˆ( ) ( 2 )
2
2
1
g i w m g i m
m
l l
= + Â
=-
1 1 1
1 1
- -l l l
l l l
w g w g w g
g w g w g w
1 1 1
1 1
- -l l l
l l l
w g w g w g
g w g w g w
Reduce
Expand (1D)
2
2
, , 1
i p
g i w p g
p
ln ln
 =-
, 1 , 1 , 1
, , 1 , 1
- -ln ln l n
ln ln ln
w g w g w g
g w g w g
gl , n ( 4 )= w ˆ(- 2 ) gl , n - 1 ( 1 )+ w ˆ( 0 ) gl , n - 1 ( 2 )+ w ˆ( 2 ) gl , n - 1 ( 3 )
Convolution Mask
[ w ( - 2 ), w (- 1 ), w ( 0 ), w ( 1 ), w ( 2 )]
Convolution Mask
- Separable
w ( m , n ) = w ˆ( m ) w ˆ( n )
w ˆ^ ( i ) = w ˆ(- i )
[ c , b , a , b , c ]
•Symmetric
Convolution Mask
- The sum of mask should be 1.
a + 2 b + 2 c = 1
a + 2 c = 2 b
•All nodes at a given level must contribute the
same total weight to the nodes at the next
higher level.
c c
b
b
a
Approximate Gaussian
0
0.
0.
0.
0.
0.
0.
0.
0.
c b a b c
Gaussian
Gaussian
2
2
( )
o
x
g x e
=
Gaussian
2
2
( )
o
x
g x e
=
x
g(x)
Gaussian Pyramid
Laplacian Pyramids
- Similar to edge detected images.
- Most pixels are zero.
- Can be used for image compression.
[ ] 2 2 3 L = g - EXPAND g
[ ] 3 3 4 L = g - EXPAND g
[ ] 1 1 2 L = g - EXPAND g
Coding using Laplacian Pyramid
•Compute Gaussian pyramid
1 2 3 4
g , g , g , g
•Compute Laplacian pyramid
4 4
3 3 4
2 2 3
1 1 2
[ ]
[ ]
[ ]
L g
L g EXPANDg
L g EXPANDg
L g EXPANDg
=
= -
= -
= -
•Code Laplacian pyramid
Image Compression (Entropy)
.
Huffman Coding (Example-1)
A 1
A 2
A 3
A 4
P=.
P=.
P=.
P=.
0
1
0
1
0
(^1) A 1
A 2 10
A 3 110
A
4
Huffman Coding
()log ( )
255
0
H pi 2 pi
i
Â
. 125 log. 125 1. 75
. 5 log. 5. 25 log. 25. 125 log. 125
H =- - - -
Entropy
Image Compression
1
.