DTU Informatics Introduction to Medical Image Analysis Rasmus R. Paulsen DTU Informatics rrp@imm.dtu.dk TexPoint fonts.
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DTU Informatics
Introduction to Medical Image AnalysisRasmus R. PaulsenDTU Informatics
rrp@imm.dtu.dk
http://www.imm.dtu.dk/courses/02511
DTU Informatics
1/4/2009Introduction to Medical Imaging2 DTU Informatics, Technical University of Denmark
Lecture 9 – Pixel Classification
9.00 Lecture
Exercises
12.00 – 13.00 Lunch break
13.00 Lecture ?
Exercises
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What can you do after today? Describe the concept of pixel classification Use Matlab to select pixel training data Compute the pixel value ranges in a minimum distance
classifier Implement and use a minimum distance classifier Describe how a pixel value histogram can be
approximated using a Gaussian distribution Describe how the pixel value ranges can be selected in a
parametric classifier Implement and use a parametric classifier Explain the concept of Bayesian classification
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Classification Take a measurement and put it into a class
?Wheels: 2
HP: 50
Weight: 200
• Bike
• Truck
• Car
• Motorbike
• Train
• Bus
Measurement Classes
Classifier
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General Classification Multi-dimensional measurement Pre-defined classes
– Can also be found automatically – can be very difficult!
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Pixel Classification Classify each pixel
– Independent of neighbours Also called labelling
– Put a label on each pixel We look at the pixel value
and assign them a label Labels already definedCT scan of human head
Background
Soft-Tissue
Trabecular Bone
Hard Bone
Quiz: Two class classification with background and object?
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Pixel Classification – formal definition
Pixel value (the measurement) v 2 R
C = c1; : : : ;ckk classes
Classification rule c : R ! f c1; : : :;ckg
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Pixel Classification – example
Pixel value
v 2 [0;255]
C = fbackground;soft-tissue;trabeculae;bonegSet of 4 classes
Classification rule
c : v ! fbackground;soft-tissue;trabeculae;boneg
How do we construct a classification rule?
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Pixel classification rule
background
soft-tissue
trabeculae
bone
How do we do this?
c : v ! fbackground;soft-tissue;trabeculae;boneg
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Pixel classification rule – manual inspection
background soft-tissue trabeculae bone
c : v ! fbackground;soft-tissue;trabeculae;boneg
Looking at some few pixels
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Pixel classification rule – manual inspection
background soft-tissue trabeculae bone
c : v ! fbackground;soft-tissue;trabeculae;boneg
Looking at some few pixels New pixel – where do we put it?
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Pixel classification rule – manual inspection
background soft-tissue trabeculae bone
c : v ! fbackground;soft-tissue;trabeculae;boneg
Looking at some few pixels New pixel – where do we put it?
• Measure the “distance” to the other classes
• Select the closest class
db dst
Minimum distance classification
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Pixel classification ruleMinimum Distance Classification
background soft-tissue trabeculae bone
The possible pixel values are divided into ranges
Here the distance to “background” is equal to “soft-tissue”
Background
range
soft-tissue
range
Trabecular
range
Bone
range
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Pixel classification ruleMinimum Distance Classification
background soft-tissue trabeculae bone
Background
range
soft-tissue
range
Trabecular
range
Bone
range
c(v) =
8>><
>>:
background if v · (4+ 67)=2soft-tissue if (4+ 67)=2< v · (67+ 95)=2trabeculae if (67+ 95)=2< v · (95+ 150)=2bone if v > (95+ 150)=2
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Pixel classification rule For all pixel in the image do
c(v) =
8>><
>>:
background if v · (4+ 67)=2soft-tissue if (4+ 67)=2< v · (67+ 95)=2trabeculae if (67+ 95)=2< v · (95+ 150)=2bone if v > (95+ 150)=2
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Pixel Classification example
Background
Soft-Tissue
Trabecular Bone
Hard Bone
CT scan of human head
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Better range selection Guessing range values is not a
good idea Better to use “training data” Start by selecting representative
regions from an image Annotation
– To mark points, regions, lines or other significant structures
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Classifier training - annotation An “expert” is asked how
many different tissue types that are possible
Then the expert is asked to mark representative regions of the selected tissue types
Background
Soft-Tissue
Trabecular Bone
Hard Bone
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Classifier training – region selection Many tools exist Matlab tool roipoly
– Select closed regions using a piecewise polygon
Training is only done once!
Optimally, the training can be used on many pictures that contains the same
tissue types
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Initial analysis - histograms
Gaussian
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Initial analysis - histograms
Class separation
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Simple pixel statistics Calculate the average (mean) and the standard
deviation of each class
Average
Standard deviation
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Minimum distance classification
Any objections?
The pixel value ranges are not always in good correspondence with the histograms?
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Exam question 09.15
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Parametric classification Describe the histogram using
a few parameters Gaussian/Normal distribution
– Average– Standard deviation
Trabecular bone
f (x) =1
¾p
2¼exp
µ¡
(x ¡ ¹ )2
2¾2
¶
¹¾
Only two values needed
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Parametric classification
Trabecular bone f (x) =1
¾p
2¼exp
µ¡
(x ¡ ¹ )2
2¾2
¶
¹̂ =1n
nX
i=1
vi
¾̂2 =1
n ¡ 1
nX
i=1
(vi ¡ ¹̂ )
v1;v2; : : : ;vn ;Training pixel values
Estimated average
Estimated
standard
deviation
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Parametric classification Fit a Gaussian to the training pixels for all classes
What do we see here?
What is the difference between the two classes?
Trabeculae has much higher variation in the pixel values
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Parametric classification New pixel with value
78– Is it soft-tissue or
trabecular bone? Minimum distance
classifier?– Soft-tissue
Is that fair?– Soft-tissue Gaussian
says “Extremely low probability that this pixel is soft-tissue”
v = 78
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Parametric classification – repeat the question
New pixel with value 78– Is it soft-tissue or
trabecular bone?– Most probably
trabecular bone Where should we set
the limit?– Where the two
Gaussians cross!
v = 78
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Parametric classification – ranges The pixel value ranges
depends on– The average– The standard
deviation Compared to the
minimum distance classifier– Only the average
Soft-tissue Trabecular bone
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Parametric classification – how to Select training pixels for each class Fit Gaussians to each class Use Gaussians to determine pixel value ranges Little bit difficult with the Gaussians
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Parametric classifier - ranges We want to compute
where they cross
f 1(x) =1
¾1p
2¼exp
µ¡
(x ¡ ¹ 1)2
2¾21
¶
f 2(x) =1
¾2p
2¼exp
µ¡
(x ¡ ¹ 2)2
2¾22
¶
f 1(x) > f 2(x)
f 1(x) < f 2(x)Create a lookup table:
• Run through all 256 possible pixel values
• Check which Gaussian is the highest
• Store the [value, class] in the table
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Alternatively – analytic solution
v =¾1
2¹ 2 ¡ ¾22¹ 1 §
r
¡ ¾12¾2
2³2¹ 2 ¹ 1 ¡ ¹ 2
2 ¡ 2¾22 ln
³¾2¾1
´¡ ¹ 1
2 + 2¾12 ln
³¾2¾1
´´
¡ ¾22 + ¾1
2
1
¾1p
2¼exp
µ¡
(v ¡ ¹ 1)2
2¾21
¶=
1
¾2p
2¼exp
µ¡
(v ¡ ¹ 2)2
2¾22
¶The two Gaussians
Intercept at
Piece of cake!
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Exam Question 09.2 and 09.19
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Exercises
?
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Bayesian Classification
Area = 1
Pure parametric classifier assumes equal amount of
different tissue types
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Bayesian Classification
Area = 1
Much more soft-tissue than
trabecular bone
How do we handle that?
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Bayesian Classification An expert tells us that a CT
scan of a head contains– 20% Trabecular bone– 50% Soft-tissue
Picking a random pixel in the image– 20% Chance that it is
trabecular bone– 50% Chance that it is soft-
tissue How do use that?
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Bayesian Classification – histogram scaling
Parametric classifier Bayesian classifier
Scaled with 0.50
Scaled with 0.20
Little change in class border
(sometimes significant changes)
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Formal definition Given a pixel value What is the probability that the pixel belongs to class
P (ci jv) =P (vjci )P (ci )
P (v)
vci
Example: If the pixel value is 78, what is the probability that the pixel is bone
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Formal definition
P (ci jv) =P (vjci )P (ci )
P (v)
Constant – ignored from now on
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Formal definition
P (ci jv) =P (vjci )P (ci )
P (v)
The a priori probability (what is known from before)
Example: From general biology it is known that 20% of a brain CT scan is trabecular bone. Therefore P(trabecular) = 0.20
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Formal definition
P (ci jv) =P (vjci )P (ci )
P (v)
The class conditional probability Given a class, what is the probability of a pixel with value v
Example: If we consider class = soft-tissue. What is the probability that the pixel value is 78?
Very low
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Formal definition – sum up
P (ci jv) =P (vjci )P (ci )
P (v)ci = trabeculae
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Parametric classification – how to Select training pixels for each class Fit Gaussians to each class Ask an expert for the prior probabilities (how much
there normally is in total of each type) For each pixel in the image
– Compute for each class (the a posterior probability)– Select the class with the highest
P (ci jv)P (ci jv)
P (ci jv) =P (vjci )P (ci )
P (v)
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When to use Bayesian classification The parametric classifier is good when there are
approximately the same amount of all type of tissues Use Bayesian classification if there are very little or
very much of some types
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Exercises
?
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