Edge detection
Edge Detection in Images
• Finding the contour of objects in a scene
Edge Detection in Images
• What is an object?
It is one of the goals of computer vision to
identify objects in scenes.
Edge Detection in Images
• Edges have different sources.
What is an Edge
• Lets define an edge to be a discontinuity in
image intensity function.
• Edge Models
–
–
–
–
Step Edge
Ramp Edge
Roof Edge
Spike Edge
Detecting Discontinuities
• Discontinuities in signal can be detected by
computing the derivative of the signal.
Differentiation and convolution
• Recall
f
 f x     f x  
 lim 



0
x



• Now this is linear and
shift invariant, so must
be the result of a
convolution.
•
We could approximate
this as
f xn 1   f x 
f

x
x
(which is obviously a
convolution with Kernel
1 ; 1it’s
 not a very
good way to do things, as
we shall see)
Finite Difference in 2D
f x, y 
 f  x   , y   f  x, y  
 lim 



0
x



f x, y 
 f  x, y     f  x, y  
 lim 



0
y



Definition
f x, y  f xn 1 , ym   f xn , ym 

x
x
f x, y  f xn , ym1   f xn , ym 

y
x
Discrete Approximation
1
1
1
 1
 
Convolution Kernels
Finite differences
I x  I * 1  1
I
1
Iy  I * 
 1
Frequency Response of
Differential Kernel
f x 

f x 
x
Fourier
Transform
Fourier
Transform
F  f u 
uF  f u 
Noise
• Simplest noise model
– independent stationary
additive Gaussian
noise
– the noise value at each
pixel is given by an
independent draw from
the same normal
probability distribution
•
Issues
–
–
–
–
this model allows noise
values that could be
greater than maximum
camera output or less
than zero
for small standard
deviations, this isn’t too
much of a problem - it’s
a fairly good model
independence may not be
justified (e.g. damage to
lens)
may not be stationary
(e.g. thermal gradients in
the ccd)
Finite differences and noise
•
Finite difference filters
respond strongly to noise
–
•
•
What is to be done?
–
obvious reason: image
noise results in pixels
that look very different
from their neighbours
–
Generally, the larger the
noise the stronger the
response
–
intuitively, most pixels in
images look quite a lot
like their neighbours
this is true even at an
edge; along the edge
they’re similar, across
the edge they’re not
suggests that smoothing
the image should help,
by forcing pixels
different to their
neighbours (=noise
pixels?) to look more
like neighbours
Finite differences responding to
noise
Increasing noise ->
(this is zero mean additive gaussian noise)
Smoothing reduces noise
• Generally expect
pixels to “be like”
their neighbours
– surfaces turn slowly
– relatively few
reflectance changes
• Generally expect
noise processes to be
independent from
pixel to pixel
•
•
Implies that smoothing
suppresses noise, for
appropriate noise models
Scale
–
–
–
–
the parameter in the
symmetric Gaussian
as this parameter goes
up, more pixels are
involved in the average
and the image gets more
blurred
and noise is more
effectively suppressed
The effects of smoothing
Each row shows smoothing
with gaussians of different
width; each column shows
different realisations of
an image of gaussian noise.
Classical Operators
Prewitt’s Operator
1 1
1 1


1 1
Smooth
1 1 1
1 1 1


1
1
1 0  1
1 0  1


1 0  1
Differentiate
1
 1
 
1 1 1
0 0 0


 1  1  1
Classical Operators
Sobel’s Operator
1 1 
2 2


1 1 
Smooth
1
1
1 0  1 
2 0  2


1 0  1 
Differentiate
1 2 1
1 2 1


1
 1
 
2
1
1
0

0
0


 1  2  1
Gaussian Filter

 x2  y 2
G x, y  
exp  
2
2
2

2 

1




 i  k  12   j  k  12

H i, j  
exp
2

2

2  2

1
where H i, j  is 2k  1 2k  1 array



Detecting Edges in Image
• Sobel Edge Detector
*
1 0  1 
2 0  2


1 0  1 
d
I
dx
Image I
*
2
1
1
0

0
0


 1  2  1
d
I
dy
d  d 
 I    I 
 dx   dy 
2
2
Edges
Threshold
any alternative ?
Quality of an Edge Detector
• Robustness to Noise
• Localization
• Too Many/Too less Responses
Poor robustness to noise
Poor localization
True Edge
Too many responses
Canny Edge Detector
• Criterion 1: Good Detection: The optimal
detector must minimize the probability of false
positives as well as false negatives.
• Criterion 2: Good Localization: The edges
detected must be as close as possible to the true
edges.
• Single Response Constraint: The detector
must return one point only for each edge point.
Canny Edge Detector
• Difficult to find closed-form solutions.
Canny Edge Detector
• Convolution with derivative of Gaussian
• Non-maximum Suppression
• Hysteresis Thresholding
Canny Edge Detector
• Smooth by Gaussian

1
G 
e
2 
S  G * I
• Compute x and y derivatives

S   S
 x
T
 
S   Sx
y 

Sy

T
• Compute gradient magnitude and orientation
S  S x2  S y2
  tan
1
Sy
Sx
x2  y 2
2 2
Canny Edge Operator
S  G * I   G * I
 G
G  
 x
 G
S  
*I
 x
G 
y 
T
G 
*I
y

T
Canny Edge Detector
Sx
I
Sy
Canny Edge Detector
S  S x2  S y2
I
S  Threshold  25
Non-Maximum Suppression
We wish to mark points along the curve where the magnitude is biggest.
We can do this by looking for a maximum along a slice normal to the curve
(non-maximum suppression). These points should form a curve. There are
then two algorithmic issues: at which point is the maximum, and where is the
next one?
Non-Maximum Suppression
• Suppress the pixels in ‘Gradient Magnitude
Image’ which are not local maximum
x, y

if S x, y   S x, y
 S x, y 
M x, y   
& S x, y   S x, y
 0
otherwise

x, y 
x,y and x,y are the neighbors of x,y in
x, y along the direction normal to an edge
S
Non-Maximum Suppression
tan θ 
Sy
Sx
2
3
1
0 : -0.4142  tan θ  0.4142
1 : 0.4142  tan θ  2.4142
2 : tan θ  2.4142
3 : - 2.4142  tan θ  0.4142
0
Non-Maximum Suppression
S  S x2  S y2
M  Threshold  25
M
Hysteresis Thresholding
Hysteresis Thresholding
• If the gradient at a pixel is above ‘High’, declare
it an ‘edge pixel’
• If the gradient at a pixel is below ‘Low’, declare
it a ‘non-edge-pixel’
• If the gradient at a pixel is between ‘Low’ and
‘High’ then declare it an ‘edge pixel’ if and only
if it is connected to an ‘edge pixel’ directly or via
pixels between ‘Low’ and ‘ High’
Hysteresis Thresholding
M  Threshold  25
M
High  35
Low  15
an image
its detected edges