Lecture 06 - image processingcourse1.pptx

Digital Image Processing
Lecture 6
Image Segmentation

Segmentation
Image segmentation is the process of partitioning the
digital image into multiple regions
Segmentation accuracy determines the eventual success
or failure of the following image processing tasks:
Pattern recognition
Computer Aided Diagnosis systems
For this reason, considerable care should be maintained
to segmentation procedure

Region/Segment
 What is a region or a segment?
 It is an aggregation of pixels
 Properties like gray level, color, texture, or shape are used to build groups
of regions having a particular meaning.
 In mathematical sense the segmentation of the image I, which is a set of
pixels, is the partition of I into n disjoint sets R1,R2, . . . , Rn, called
segments or regions such that their union of all regions equals I.
I= R1 U R2 U….. U Rn
 Image segmentation is based on either one of two gray level intensity
properties:
 Gray level discontinuity (e.g. edge detection)
 Gray level similarity (e.g. thresholding)

Detection of Discontinuity
The most common method to detect discontinuity in an
image is by running a spatial mask of specific width
and number of coefficients
The window mask is multiplied by the original image
pixels and the sum of product is calculated
Discontinuity could be applied on:
An isolated point
A line
An edge

Discontinuity: Point Detection
An isolated point: it is a point (several aggregate of pixels)
whose intensity differs significantly from the its homogeneous
background.
The idea for isolated point detection consists of assuming a mask
with equal coefficients everywhere except at the center, where the
coefficient has a higher value with opposite sign
The center of the mask is moved from one pixel to another within
an image.
At each location, the sum of the product of the mask coefficients
and the pixels gray levels is calculated.
An isolated point is detected when this sum is larger than a given
threshold

Discontinuity: Point Detection

Discontinuity: Line Detection
For line detection, the same idea is performed but with
different masks
Could detect horizontal, vertical, or ‘m’ slope lines
The preferred direction in the mask is weighted by a
different weight (larger value and opposite sign)
The main disadvantage in this method is that only
strong responses (lines having large differences than the
background) could be detected.

Discontinuity: Edge Detection
Edge detection is the most common approach for
detecting meaningful discontinuities in gray
level.
An edge: is the boundary between two regions
with relatively distinct gray level properties.
An edge could be ideal or blurred
Blurred edges need sophisticated algorithms to
be detected

Edge detection
• Goal: Identify sudden changes
(discontinuities) in an image
Intuitively, most semantic and shape
information from the image can be encoded in
the edges
More compact than pixels
• Ideal: artist’s line drawing (but artist is
also using object-level knowledge)

Characterizing edges
• An edge is a place of rapid change in the image intensity
function
image
intensity function
(along horizontal scanline) first derivative
edges correspond to
extrema of derivative

If the regions are sufficiently homogeneous, edge
detection could be performed by the computation of a
derivative operator, where the intensity variation is
plotted then the first derivative is used to detect the
presence of an edge
For non- homogeneous regions, a technique based
on gradient function should be used, as the maximum
rate of change of image intensity is in direction of
gradient.

The gradient points in the direction of most rapid increase in intensity
Image gradient
The gradient of an image:
The gradient direction is given by
• how does this relate to the direction of the edge?
The edge strength is given by the gradient magnitude
'( ) ( 1) ( )
f
f x f x f x
x

   


The magnitude and the direction of the gradient
should be calculated at each pixel:
𝑮 𝒇 𝒙, 𝒚 =
𝝏𝒇
𝝏𝒙
𝟐
+
𝝏𝒇
𝝏𝒚
𝟐
𝜶 𝒙, 𝒚 = 𝐭𝐚𝐧−𝟏
𝝏𝒇 𝝏𝒙
𝝏𝒇 𝝏𝒚
where G is the magnitude of the gradient and α is its direction. The
edge is then found by linking all points in the direction calculated.

Finite difference filters
• Other approximations of derivative filters exist:

Edge detection algorithms face several
problems:
Noise
Edge break due to illumination problems
Any other effect that causes discontinuity

Effects of noise
Consider a single row or column of the image
Plotting intensity as a function of position gives a signal
Where is the edge?

Effects of noise
• Finite difference filters respond strongly to noise
Image noise results in pixels that look very different from
their neighbors
Generally, the larger the noise the stronger the response
• What is to be done?
Smoothing the image should help, by forcing pixels
different to their neighbors (=noise pixels?) to look more like
neighbors

Solution: smooth first
• To find edges, look for peaks in )
( g
f
dx
d

f
g
f * g
)
( g
f
dx
d


• Differentiation is convolution, and convolution is associative:
• This saves us one operation:
g
dx
d
f
g
f
dx
d


 )
(
Derivative theorem of convolution
g
dx
d
f 
f
g
dx
d

Designing an edge detector
• Criteria for an “optimal” edge detector:
 Good detection: the optimal detector must minimize the probability of
false positives (detecting spurious edges caused by noise), as well as that
of false negatives (missing real edges)
 Good localization: the edges detected must be as close as possible to the
true edges
 Single response: the detector must return one point only for each true
edge point; that is, minimize the number of local maxima around the true
edge

Canny edge detector
• This is probably the most widely used edge
detector in computer vision
• Theoretical model: step-edges corrupted by
additive Gaussian noise
• Canny has shown that the first derivative of the
Gaussian closely approximates the operator that
optimizes the product of signal-to-noise ratio and
localization

Canny edge detector
1. Filter image (Smooth) with derivative of Gaussian
2. Find magnitude and orientation of gradient
3. Apply non-maximum suppression to the gradient magnitude image
 Thin multi-pixel wide “ridges” down to single pixel width
4. Use double thresholding and connectivity analysis to detect and
link edges (Linking and thresholding):
 Define two thresholds: low and high
 Use the high threshold to start edge curves and the low threshold to
continue them
 MATLAB: edge(image, ‘canny’)

Example
 original image (Lena)

Example
thinning
(non-maximum suppression)

Non-maximum suppression
At q, we have a
maximum if the value is
larger than those at
both p and at r.
Interpolate to get these
values.

Assume the marked point
is an edge point. Then we
construct the tangent to
the edge curve (which is
normal to the gradient at
that point) and use this to
predict the next points
(here either r or s).
Edge linking

Discontinuity: Edge Linking 1
Edge Linking: becomes an important approach after edge
detection
How to link edges?
1. First define a small window (3x3) or (5x5)
2. The center of this window is a pixel defined as an edge (x0 ,y0 )
3. Check other pixels inside the window if considered as edges or not by
calculating the gradient at each
4. A pixel (x,y) is considered linked edge if:
1. |G(x,y)-G(x0 ,y0 )| < E
2. Θ(x,y) - Θ(x0 ,y0 ) < A
Where G is the magnitude of the gradient and Θ is the angle of the gradient
E and A are magnitude and phase thresholds respectively

Effect of  (Gaussian kernel spread/size)
Canny with Canny with
original
The choice of  depends on desired behavior
• large  detects large scale edges
• small  detects fine features

Edge detection is just the beginning…
image human segmentation gradient magnitude

35 Basic Edge Detection by Using First-Order Derivative
2 2
1
( )
The magnitude of
( , ) mag( )
The direction of
( , ) tan
The direction of the edge
-90
x
y
x y
x
y
f
g x
f grad f
f
g
y
f
M x y f g g
f
g
x y
g

 


 
 
  
 
   
  
 
 
 

 

   

 
  
 
 


36 Basic Edge Detection by Using First-Order Derivative
Edge normal: ( )
Edge unit normal: / mag( )
x
y
f
g x
f grad f
f
g
y
f f

 
 
  
 
   
  
 
 
 

 
 
In practice,sometimes the magnitude is approximated by
mag( )= + or mag( )=max | |,| |
f f f f
f f
x y x y
 
   
   
   
 

40
Advanced Techniques for Edge Detection
The Marr-Hildreth edge detector
2 2
2
2 2 2 2
2 2
2 2 2 2
2 2
2
2 2
2
2 2
2 2
2 2
2 2
2 2
4 2 4 2
2
( , ) , :space constant.
Laplacian of Gaussian (LoG)
( , ) ( , )
( , )
1 1
x y
x y x y
x y x y
G x y e
G x y G x y
G x y
x y
x y
e e
x y
x y
e e
x

 
 

 
   


 
 
 
 

 
  
 
   
   
 
   
 
   
   
   
   
   
   

2 2
2
2 2
2
4
x y
y
e 




 
 
 
 

42 Marr-Hildreth Algorithm
1. Filter the input image with an nxn Gaussian lowpass
filter. N is the smallest odd integer greater than or
equal to 6
2. Compute the Laplacian of the image resulting from
step1
3. Find the zero crossing of the image from step 2
 
2
( , ) ( , ) ( , )
g x y G x y f x y
 


Edge Linking Using Hough Transform
Effective method to solve linear segmentation problems
Assume the following segmentation problem:
A set of ‘n’ points are in a plane,
We want to determine the points that are linked together to form a
straight line
Solution1:
We will take a pair of points and form a straight line
Check the number of remaining points that are close to this line
This process is repeated for all possible pairs of points and each time the
rest of points are checked for being part of this proposed line
The line that has maximum number of points closer to it is the fittest
one

What is the problem concerned with solution 1 ?
Large number of computations
For ‘n’ points, we have nC2
= n
2
=
n(n+1)
2
possible pairs to
form the proposed lines
Each case, we have to compare (n-2) points, if the proposed
line passes closer to them or not
The number of computations is then:
n(n+1)(n+2)
2
Large mathematical computations

 Solution by means of Hough Method, Hough proposed the following model:
 Given that the equation of a straight line is: 𝒚 = 𝒂𝒙 + 𝒃 , “a” and “b” are the line parameters
 At any point (x1 ,y1), we can write the equation of all lines passing through it as:
𝑦1 = 𝑎𝑥1 + 𝑏 𝒐𝒓 𝑏 = −𝑎𝑥1 + 𝑦1
 We have got a linear equation between ‘a’ and ‘b’ that could be represented in another plane
called: “The Parameter Space”
 Thus each point (x,y) will be transformed into a straight line in the parameter space
 If two lines in the parameter space -representing the pair of points (x1 ,y1 ) and (x2 ,y2 ) -will
intersect at values (a, b)
 These values’ couple ‘a’ and ‘b’ are parameters of the straight line joining them in original space
 When all ‘n’ points are transformed into the parameter space domain, we will simply observe the
values of (a,b) where the most points’ cloud density occurs.
 These values for ‘a’ and ‘b’ are the required ones; and these points are linked to form the required
line

Hough Transform
All lines passing in this crowded area are then inversely
transformed to obtain the original points passing via the linked line
with parameters (a,b)

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