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João Marques
nº58513
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Outline
Medical Imaging
Clustering Overview
Segmentation Methods
Kmeans
Riddler
Otsu
Kittler
ImageJ
Results obtained
Flow Simulation Analysis
João Marques
58513
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Medical Imaging
A set of techniques and procedures used to create
images of the human body for clinical purposes
(diagnosis) or medical science.
Examples of widely used techniques:
•MRI
•Radiography
•Tomography
•Ultrasound
João Marques
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Medical Imaging
João Marques
58513
Example: MRI
•Characterizes and discriminates among tissues using their
physical and biochemical properties
•Produces sectional images of equivalent resolution in any
projection without moving the patient
•Patient acceptability is high
•MRI contrast agents are very well tolerated
Despite all advantages, it is still difficult to look at the image and
identify the artery
Segmentation methods are important!!!
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Medical Imaging
João Marques
58513
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Medical Imaging
João Marques
58513
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Medical Imaging
João Marques
58513
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Clustering Overview
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In a general way, Clustering is a process of grouping a set of objects
into sets of similar objects.
Two basic principles in clustering: separation and homogeneity.
• Separation: Elements in different clusters are far from each other
• Homogeneity: Elements within a cluster are close to each other
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Clustering Overview
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Bad Clustering – Violates both the Separation and Homogeneity
principles!
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Clustering Overview
João Marques
58513
Good Clustering – Satisfies both the Separation and Homogeneity
principles!
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Clustering Techniques
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• Agglomerative: Start with every element in its own cluster, and
iteratively join clusters together, until there is only one cluster or
some condition is satisfied
• Divisive: Start with one cluster and iteratively divide it into smaller
clusters, until each element is in its own cluster or some condition
is satisfied
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Clustering Techniques
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• Hierarchical: Organize elements into a tree, where leaves
represent the objects and the length of the paths between leaves
represents the distances between objects. Similar objects lie
within the same sub-trees. This type of clustering results in a
dendogram.
• Partitional: Partitions the elements into a specified number of
groups. This is the most used type of clustering used when we
want to differentiate between image and background, where the
number of groups is two.
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Kmeans
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This method requires a set V of points (pixel values) and the number k
of clusters to be formed.
In the end, the algorithm returns a set X of k points that minimizes the
squared error distortion over all possible choices of X.
One popular heuristic that is efficient is the Lloyd Algorithm.
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Kmeans – Lloyd Algorithm
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1. Arbitrarily assign 2 cluster centers (0 and 255 for example).
2. While the cluster centers keep changing,
3. Set each pixel to the cluster that it is closest to.
4. At the end, calculate the centers of the 2 clusters as the mean
of all the values of that cluster.
This algorithm may lead to a merely local optimal solution!
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Riddler1
João Marques
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Algorithm that selects a statistically optimal threshold, based on the
contents of the image.
The main principle is to divide the pixels according to the average value
of the whole signal.
1Riddler, T. W., “Picture Thresholding Using an Iterative Selection Method”, 1978
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Riddler
João Marques
58513
1. Calculate the average value of all pixels
2. Divide the pixels into two groups: those below the average value
computed and those above it.
3. Calculate the average value of both groups.
4. Calculate the mean of both average values.
5. While the mean keeps changing,
6. Redivide the pixels into two groups.
7. Compute the average value of both groups.
8. Calculate the mean of both average values.
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Otsu2
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This algorithm makes use of the probability distribution of the pixel
values.
An optimal threshold is selected as to maximize the separability of the
resultant classes in gray levels, using the zero-th and first-order
cumulative moments of the gray level histogram.
2Otsu, N., “A Threshold Selection Method from Gray-Level histograms”, 1979
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Otsu
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1. Calculate μ = average value of the set of pixels.
2. Compute the probability histogram.
3. Create two clusters: one with the first histogram level, and another
with the rest.
4. Assing σmax = 0;
5. While there is more than one element in the second cluster,
6. Calculate the variance between the two clusters.
7. If this variance is greater than σmax ,keep it.
8. Remove one element from cluster 2 and add it to cluster 1.
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Kittler3
João Marques
58513
In this algorithm, it is assumed that the pixel values distribution will be
Gaussian for each gray level.
Knowing the probability distribution a priori allows certain deductions
to be made, transforming this threshold-finding problem into a
minimization problem of a criterion function, which is much simpler.
3Kittler, J., Illingworth, J., “Minimum Error Thresholding”, 1986
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Kittler – Criterion Function
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The criterion function results from algebraic manipulation of the relation
P1p(g|1)=P2p(g|2),
where Pi represents probability of occurrence of the class i and p(g|i)
represents the normal distribution of the class i. Here, g represents the
optimal threshold that separates image and background.
The function to be minimized will be the following:
J(T)=1+2[P1(T)log(σ1(T))+P2(T)log(σ2(T))]
-2[P1(T)log(P1(T))+P2(T)log(P2(T))]
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Kittler
João Marques
58513
1. Compute the probability histogram.
2. Create two clusters: one with the first histogram level, and another
with the rest.
3. Assign Jmin = ∞.
4. While there is more than one element in the second cluster,
5. Calculate J(g) for the specific gray level separation.
6. If this value is smaller than Jmin,keep it.
7. Remove one element from cluster 2 and add it to cluster 1.
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ImageJ4
João Marques
58513
Java application available on the Internet.
Provides tools for image processing.
ImageJ provides with a set of Java packages that allow
(relatively) easy plugin creation.
Anyone is free to add their own plugin to the ImageJ database.
4For more information please visit http://rsbweb.nih.gov/ij/
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Example 1
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Image Background
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Example 1
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Image + Background
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Example 1
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Results:
• Kmeans: 77
• Riddler: 77
• Otsu: 76
• Kittler: 77
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Example 2
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Image Background
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Example 2
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Image + Background
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Example 2
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Results:
• Kmeans: 121
• Riddler: 120
• Otsu: 119
• Kittler: 129
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Results Obtained
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All these algorithms were tested in a real set of 35 medical images, and
produced threshold values as follows:
40
50
60
70
80
90
100
Th
resh
old Otsu
Kittler
Riddler
Kmeans
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Results Obtained
João Marques
58513
All results agree with each other, except for the Kittler algorithm.
This can be explained by the fact the distributions are not exactly
normal, and as such there are errors inherent to this approach. One
of the main limitations is that the background histogram is in fact a
normal distribution, but finishes suddenly at 0 (incomplete normal).
This will affect the calculations of mean value and standard deviation.
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Flow Simulation Analysis
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58513
It is interesting to study applications of this kind of algorithms.
This can be done with Computational Flow Dynamics.
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Flow Simulation Analysis
João Marques
58513
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Flow Simulation Analysis
João Marques
58513
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Flow Simulation Analysis
João Marques
58513
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Flow Simulation Analysis
João Marques
58513
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Flow Simulation Analysis
João Marques
58513
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Flow Simulation Analysis
João Marques
58513
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Final Remarks
João Marques
58513
The segmentation method used to model the vessel is determinant in
CFD results. Different methods may lead to very different results!
There are still other methods that can be implemented.
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João Marques
58513
The End