Colorado School of Mines
Image and Multidimensional
Signal Processing
Professor William Hoff
Dept of Electrical Engineering &Computer Science
Colorado School of Mines
Department of Electrical Engineering and Computer Science
http://inside.mines.edu/~whoff/
Thresholding
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Thresholding
• Label pixel as belonging to one of two (or more)
classes
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Department of Electrical Engineering and Computer Science
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Basic Global Thresholding
•
A heuristic algorithm
1. Select an initial estimate for threshold T
2. Segment the image using T
•
•
G1 is all pixels with intensities > T
G2 is all pixels with intensities <= T
3. Compute averages m1 and m2 for the pixels in G1 and G2
4. Let T = (m1+m2)/2
5. Repeat steps 1-4 until no more change
•
Note – you can use the histogram to compute the
averages (very fast)
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
clear all, close all;
I = imread('cameraman.tif');
[H,W] = size(I);
figure, imshow(I);
% Get histogram. hist(i) is the count of pixels with value x(i).
[hist x] = imhist(I);
figure, imhist(I);
% Convert to probability
p = hist/(H*W);
t = 128;
% initial guess for threshold
while true
disp(t);
% Group 1 is all pixels with intensity > t
m1 = (x > t);
% create a logical mask; true where x>t
% Compute mean of group 1
u1 = sum( x(m1) .* p(m1) ) / sum(p(m1));
% Group 2 is all pixels with intensity <= t
m2 = (x <= t);
% create a logical mask; true where x<=t
% Compute mean of group 2
u2 = sum( x(m2) .* p(m2) ) / sum(p(m2));
tnew = (u1 + u2)/2;
if t==tnew
break;
% Quit if no further changes
else
t = tnew;
% Otherwise, this is new threshold
end
end
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Department of Electrical Engineering and Computer Science
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Optimal Global Thresholding
•
Want to pick a threshold that minimizes the error of miss-classifying a point
T
E1 (T )   p2 ( z ) dz and


E2 (T )   p1 ( z ) dz
T
Unfortunately, we usually don’t know the pdf’s p1(z), p2(z) in advance
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Otsu’s Method for Global Thresholding
• Choose threshold to minimize the
variance within groups
 W2  P1 12  P2  22
Used in Matlab’s
“graythresh” function
0.03
0.025
• Or equivalently, maximize the
variance between groups
 B2  P1  m1  mG   P2  m2  mG 
2
0.02
2
0.015
0.01
• where
0.005
k
P1   pi , P2 
i 0
L 1
p
i  k 1
i
0
0
50
100
150
200
250
300
• mG is the global mean; mk is the
mean of class k
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Matlab Examples
• Images
– cameraman.tif, eight.tif, pout.tif
• Functions
– t = graythresh(I) % Otsu algorithm
– BW = im2bw(I,t);
% performs thresholding
– or use imtool(I,[]) and use the “adjust contrast” tool
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example
Matlab image
“pout.tif”
The minimum
within-group
variance occurs at
threshold 114
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Multiple Thresholds
• Otsu’s method can be generalized to find multiple
thresholds
• In the case of K classes, we maximize the between-class
K
variance
2
 B   Pk mk  mG 2
k 1
where
Pk 
 pi
iCk
mk 
1
Pk
i p
iCk
i
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Variable Thresholding
•
•
Global thresholding may not work if illumination or reflectance varies across
image
Can partition image into subimages, compute threshold for each
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
“Split-and-merge” Segmentation
• Find a partitioning of connected regions such that each
region has a consistent property (eg, color or texture)
within it
• “Split and merge”:
– Split regions that are inconsistent
– Merge adjoining regions that are mutually consistent
– Repeat until no further change is possible
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Department of Electrical Engineering and Computer Science
Example
Consistency
property:
Q = TRUE if  >
a and m < b
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Department of Electrical Engineering and Computer Science
Quadtree Representation
• Recursively divide image into four quadrants
• Stop dividing when each quadrant is consistent
• A very concise and efficient representation
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Summary / Questions
• Thresholding is a way to segment an image, such that
each pixel is labeled as belonging to one of (typically)
two classes.
• A global thresholding algorithm chooses a single
threshold, based on the image histogram across the
entire image.
• What property does the Otsu thresholding algorithm
seek to minimize (or maximize)?
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Colorado School of Mines
Department of Electrical Engineering and Computer Science