Transcript Presentazione di PowerPoint

Edge Detection
27th November 2012
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Computer Vision Group
Edge Detection in Images
 Goal: Automatically find the contour of objects in a scene.
 What For: Edges are significant descriptors, useful for classification,
compression…
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Edge Detection in Images
 What is an object?
It is one of the goals of computer vision to identify objects in
scenes.
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Edges May Have Different Sources




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Object Boundaries
Occlusions
Textures
Shading
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What is an Edge
 Lets define an edge to be a discontinuity in image intensity
function.
 Several Models
• Step Edge
• Ramp Edge
• Roof Edge
• Spike Edge
 They can be
thus detected as
discontinuities
of image
Derivatives
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Differentiation and convolution

Recall
f
 f x     f x  
 lim



0
x



 Now this is linear and shift
invariant. Therefore, in
discrete domain, it will be
represented as a convolution
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Differentiation and convolution


Recall
f
 f x     f x  
 lim



0
x



We could approximate
this as
f  xn 1   f  x 
f

x
x
 Now this is linear and shift (which is obviously a
convolution with Kernel
invariant. Therefore, in
1  1 ; it’s not a very
discrete domain, it will be
represented as a convolution
good way to do things, as
we shall see)
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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



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
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Horizontal
1
 1
Vertical
1
1
 
Convolution Kernels
A 1D Example
 Take a line on a grayscale image
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A 1D Example (II)
 Filter the discrete image values, convolution against
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[1 -1]
Differentiating Filters
1D Discrete derivatives
fx  f *
1
0
f x ( x)  f ( x 1)  f ( x  1)
-1
2D Discrete derivatives (separable)
fy  f *
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1
0
-1
t
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Classical Operators : Prewitt
Horizontal
1 1
1 1


1 1
Smooth
1
 1
1 0  1
1 0  1


1 0  1
Differentiate
Vertical
1 1 1
1 1 1


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1
1
 
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1 1 1
0 0 0


 1  1  1
Classical Operators: Sobel
Horizontal
1 1 
2 2


1 1 
1
 1
1 0  1 
2 0  2


1 0  1 
Differentiate
Smooth
Vertical
1 2 1
1 2 1


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1
1
 
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2
1
1
0

0
0


 1  2  1
Another famous test image - cameraman
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Horizontal Derivatives using Sobel
y1  y * h 
t
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Vertical Derivatives using Sobel
y2  y * h 
t
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Gradient Magnitude and edge detectors

| y | y * h
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t
   y * h
2
2

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Finite differences
I x  I *1 1
I
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1
Iy  I * 
 1
The Gradient Orientation
 Like for continuous function, the gradient at each pixel points at the
steepest intensity growth direction.
 The gradient norm indicates the inclination of the intensity growth.
 Matlab…..
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Finite differences responding to noise
Increasing noise ->
(this is zero mean additive Gaussian noise)
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Finite differences responding to noise
Increasing noise ->
(this is zero mean additive Gaussian noise)
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Finite differences responding to noise
Increasing noise ->
(this is zero mean additive Gaussian noise)
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Smoothing reduces noise

Generally expect pixels to “be like” their neighbors
•
surfaces turn slowly
•
relatively few reflectance changes

Generally expect noise processes to be independent from pixel
to pixel and with zero mean

Implies that smoothing suppresses noise, (i.i.d. noise!)

Gaussian Filtering
•
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 somehow suppressed
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Smoothing reduces noise

Generally expect pixels to “be like” their neighbors
•
surfaces turn slowly
•
relatively few reflectance changes

Generally expect noise processes to be independent from pixel
to pixel and with zero mean

Implies that smoothing suppresses noise, (i.i.d. noise!)

Gaussian Filtering
•
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 somehow suppressed
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Low - Pass
The effects of smoothing
Each row shows smoothing
with gaussians of different
width; each column shows
different realisations of
an image of gaussian noise.
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Gradient Magnitude and edge detectors

| y | y * h
t
   y * h
2
2

Gradient Magnitute
is not a binary
image -> “shows
edges” but “does
not allow to
identify them” yet
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Detecting Edges in Image
 Sobel Edge Detector
Discrete Derivatives
*
1 0  1 
2 0  2


1 0  1 
d
I
dx
d  d 
 I    I 
 dx   dy 
2
Image I
*
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2
1
1
0

0
0


 1  2  1
Threshold
Gradient Norms
d
I
dy
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2
Edges
Threshold
any alternative ?
Canny Edge Detector Criteria

Good Detection: The optimal detector must minimize the probability
of false positives as well as false negatives.

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. similar to good detection but requires an
ad-hoc formulation to get rid of multiple responses to a single edge
True Edge
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Poor robustness to noise Poor localization Too many responses
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Canny Edge Detector
 Basically 3 steps
• Convolution with derivative of Gaussian
• Non-maximum Suppression
• Hysteresis Thresholding
J. Canny “A Computational Approach to Edge Detection” IEEE PAMI vol
8, no. 6, Nov. 1986
http://perso.limsi.fr/Individu/vezien/PAPIERS_ACS/canny1986.pdf
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Canny Edge Detector

Smooth by Gaussian

1
G 
e
2 
S  G * I

Compute x and y derivatives

S   S
 x

 
S   Sx
y 

Sy
Compute gradient magnitude and orientation
S  S  S
2
x
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T
2
y
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  tan
1
Sy
Sx

T
x2  y2
2 2
Canny Edge Operator
S  G * I   G * I
 G
G   
 x
G 
y 
 G
S  
*I
 x
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T

G
*I
y

T
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Canny Edge Detector
Sx
I
Sy
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An Overview on Threshold
 According to the way the Threshold T is used/determined they are
divided into
• Global Threshold
• Local (or) Adaptive Threshold
 According to the output they can be classified in
• Binary Threshold
• Hard Threshold
• Soft Threshold
 Matlab…
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Canny Edge Detector
S  S x2  S y2
I
S  Threshold  25
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Non-Maximum Suppression: The Idea
 We wish to determine the points along the curve where the gradient
magnitude is largest.
 Non-maximum suppression: we look for a maximum along a slice
orthogonal to the curve. These points form a 1D curve.
 There are two issues:
• which point is the maximum,
• and where is the next one?
Original Image
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Gradient Magnitude
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Segment orthogonal
Non-Maximum Suppression: Quantize Gradient
Directions
 For each pixel compute gradient direction and quantize it in 4 main
direction, each covering 45° (orientation is not considered)
 For each pixel buld up a segment following the quantized directions
0,1,2,3
2
tanθ 
3
Sy
Sx
0 : -0.4142 tanθ  0.4142
1 : 0.4142 tanθ  2.4142
2 : tanθ  2.4142
3 : - 2.4142 tanθ  0.4142
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1
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0
Non-Maximum Suppression – Idea (II)
A segment orthogonal
to the gradient direction
in a pixel
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The intensity profile along the
segment
Non-Maximum Suppression - Threshold
 Suppress the pixels in ‘Gradient Magnitude Image’ which are not
local maximum

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 
x, y
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 x, y and  x, y  are the
neighbors of  x, y  in S
 These have to be taken on a line along the
direction orthogonal to the gradient in (x,y)
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Non-Maximum Suppression
S  S x2  S y2
M  Threshold  25
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M
Hysteresis Thresholding

Use of two different threshold High and Low for
• For new edge starting point
• For continuing edges

In such a way the edges continuity is preserved
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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 ‘nonedge-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’.
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Hysteresis Thresholding
M  Threshold  25
M
High  35
Low  15
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Examples
an image
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its detected edges
In matlab..
 Canny edge detector is implemented with the
edge.m command
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A brief overview on Morphological
Operators in Image Processing
Giacomo Boracchi
24/11/2010
[email protected]
home.dei.polimi.it/boracchi/teaching/IAS.htm
Computer Vision Group
An overview on morphological operations
 Erosion, Dilation
 Open, Closure
 We assume the image being processed is binary, as these operators
are typically meant for refining “mask” images.
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AND operator in Binary images
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OR in Binary Images
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Erosion
 General definition:
Nonlinear Filtering procedure that replace to each pixel value
the minimum on a given neighbor
 As a consequence on binary images
E(x)=1 iff the image in the neighbor is constantly 1
 This operation reduces thus the boundaries of binary images
 It can be interpreted as an AND operation of the image and the
neighbor overlapped at each pixel
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Erosion Example
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Erosion Example
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Dilation
 General definition:
Nonlinear Filtering procedure that replace to each pixel value
the maximum on a given neighbor
 As a consequence on binary images
E(x)=1 iff at least a pixel in the neighbor is 1
 This operation grows fat the boundaries of binary images
 It can be interpreted as an OR operation of the image and the
neighbor overlapped at each pixel
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Dilation Example
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Dilation Example
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In matlab
 They are performed using the
bwmorph.m script passing the name of the operation as a parameter
Examples…
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Open and Closure
 Open Erosion followed by a Dilation
 Closure Dilation followed by an Erosion
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Open
 Open Erosion followed by a Dilation
• Smoothes the contours of an object
• Typically eliminate thin protrusions
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Open
 Figure to Open,
 Structuring element a Disk
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Open
 First Erode,
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Open

Then Dilate,

Open results, the bridge has been eliminate from the first erosion and cannot
be replaced by the dilation.

Edges are smoothed

Corners are rounded
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Closure
 Closure Dilation followed by an Erosion
• Smoothes the contours of an object, typically creates
• Generally fuses narrow breaks
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Open
 Figure to Open,
 Structuring element a Disk
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Close
 First Dilate,
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Close

Then Erode,

Close results, the bridge has been preserved

Edges are smoothed

Corners are rounded
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There are several other Non Linear Filters
 Ordered Statistic based
• Median Filter
• Weight Ordered Statistic Filter
• Trimmed Mean
• Hybrid Median
 Thresholding
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Fitting – A brief Introduction
 We assume that the images are generated from a “straight lines”
word.
 By performing edge detection we are only able to select those pixels
that belongs to an edge
 We are not able to determine if such an edge is a straight line, and in
case it is, which are the parameters of such line
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Fitting – A brief Introduction
 Let now assume we are interested in detecting lines only, and we
want to determine the parameters of such a lines
 Line Fitting can be performed via
• Hough Transform
• Error minimization
• Maximum Likelihood
 But we have to determine first how the process generating data is:
i.e.
• Observation are noisy, how is noise affecting straight lines?
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Fitting – A brief Introduction
 There are different noise models
1. Noise displaces line pixels orthogonally to the straight line and it
is gaussian distributed
2. Noise displaces line pixels only along the y coordinate and it is
gaussian distributed
 Then, in both case there exist a closed form solution
 Or it can be performed via numerical minimization … matlab
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Fitting and Convolution
 One can prove that the least square fit of polynomial of 0-th order (i.e
constant) is given by
where
and thus
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Frequency Response of Differential Kernel
f x 

f x 
x
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Fourier
Transform
Fourier
Transform
F  f u 
uF f u 
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Canny Edge Detector
 Difficult to find closed-form solutions.
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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
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
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
• obvious reason: image noise results in pixels
that look very different from their neighbors

The more relevant is the noise, the stronger the
response
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Finite differences and noise

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What is to be done?
• intuitively, most pixels in images look quite a lot
like their neighbors.
• 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 neighbors
(=noise pixels?) to look more like neighbors.
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Gaussian Filter

 x2  y2
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
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

