Otsu binarization, Image Processing

2. Otsu binarization
Presented by
Seyede Yalda Akbarzadeh
Under the Guidance of
Dr. S. Z. Azimifar

3. Thresholding
 It is the simplest and powerful method
of image segmentation
 It is useful in discriminating foreground from
the background.
 Thresholding operation is used to convert a
gray scale image into binary image
 The advantage of obtaining first a binary
image is that it reduces the complexity of the
data and simplifies the process of recognition
and classifiction.

4. Where (x,y) represents a gray value/
are the coordinates of the threshold value p
T represent threshold value
g(x,y) represents threshold image
f(x,y) represents gray level image pixels/
input image
Thresholding
 The most common way to convert a gray-level image
into a binary image is to select a single threshold
value(T).Then all the gray level values below T will be
classified as black(0) i.e. background and those above
T will be white(1) i.e. objects.
 The Thresholding operation is a grey value remapping
operation g defined by:
0 if f(x,y) < T
g(x,y)=
1 if f(x,y) ≥ T,

5. 5
0.00
500.00
1000.00
1500.00
2000.00
2500.00
0.00 50.00 100.00 150.00 200.00 250.00
i
h(i)
Background
Object
T

6. Detection of Similarities-
Thresholding

7. Otsu Binarization
 Otsu method is a popular thresholding method.
 that it works on the histogram of the image.
 It assumes the image contains two classes of
pixels - foreground and background.
 It also assumes the image has a bimodal
histogram displaying two peaks.
 then sets a threshold on the image such that the
intra class variance is minimized.
 The first paper in this field:
N. Otsu, "A Threshold Selection Method from Gray-Level
Histograms," Transaction on systems, man and cybernetics,
vol. 9, no. 1, pp. 62-66, 1979

8.  The aim is to find the threshold value where
the sum of foreground and background
spreads is at its minimum.
 Based on a very simple idea: Find the
threshold that minimizes the weighted within-
class variance.
 This turns out to be the same as maximizing
the between-class variance.
 Operates directly on the gray level histogram
[e.g. 256 numbers, P(i)], so it’s fast (once the
histogram is computed).
Otsu Binarization…

9. Simple example
 The calculations for finding the foreground and background variances (the
measure of spread) for a single threshold are now shown in next slide.

10. Example…

11. Example…
 The next step is to calculate the 'Within-Class
Variance'. This is simply the sum of the two
variances multiplied by their associated
weights

12. Example…
 This final value is the 'sum of weighted variances' for the threshold
value 3. This same calculation needs to be performed for all the
possible threshold values 0 to 5. The table below shows the results
for these calculations. The highlighted column shows the values for
the threshold calculated above

13. Example…
 It can be seen that for the threshold equal to 3, as well as being
used for the example, also has the lowest sum of weighted
variances. Therefore, this is the final selected threshold. All pixels
with a level less than 3 are background, all those with a level equal
to or greater than 3 are foreground. As the images in the table
show, this threshold works well.

14. A Faster Approach
 By a bit of manipulation, you can calculate what is called
the between class variance, which is far quicker to calculate.
Luckily, the threshold with the maximum between class variance
also has the minimum within class variance. So it can also be used
for finding the best threshold and therefore due to being simpler is a
much better approach to use.

15. Formula:
 The weighted within-class variance is:
)()()()()( 2
22
2
11
2
ttwttwtw  
 Where the class probabilities are estimated as:
q1(t)  P(i)
i1
t
 q2 (t)  P(i)
i  t 1
I



t
i tw
iiP
t
1 1
1
)(
)(
)( 

I
ti tw
iiP
t
1 2
2
)(
)(
)(
 And the class means are given by:

16. Formula...
Within-class,
from before Between-class,
1
2
(t)  [i - 1
(t)]2 P(i)
w1
(t)i1
t
 2
2
(t)  [i - 2
(t)]2 P(i)
w2
(t)it1
I

𝜎2 = 𝜎 𝑤
2 t + 𝑤1 𝑡 1 − 𝑤1 𝑡 [𝜇1 𝑡 − 𝜇2 𝑡 ]2

17. Algorithm:

19. Source: Digital Image Processing by
Gonzalez and Woods