Segmentation of Multi spectral Magnetic Resonance Image using Penalized Fuzzy competitive Learning N

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Segmentation of Multi spectral Magnetic Resonance
Image using Penalized Fuzzy competitive Learning
Network

• Jzau-Sheng Lin, Kuo-Sheng Cheng Chi-Wu Mao

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Some Background

• Segmentation of Images
• Multi Spectral Images
• MR Scanner
• Magnetic Resonance Imaging

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Segmentation

• Segmentation of an image is a process of finding
  a region which is connected to a set of pixels
  that have one or more similar properties that are
  different from other surrounding pixels.
• Can be done by Thresholding, Clustering , etc.

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Multi Spectral Images

• More than one image of the same object or scene
  taken in different bands of optic or infra
  wavelength.
• Ex
• Green Red
• Photo IR False IR

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MRI- Magnetic Resonance Imaging

• Its a scanning technique
• MRI does not use radiation
• It combines the use of a large magnet and radio
  waves.
• The hydrogen atoms in the patient's body react to
  the magnetic field, and a computer analyzes the
  results and makes pictures of the inside of your
  body.

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MR Scanner
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MR Scanner
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MR Images
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Critical Parameters in MRI

• Spin-Spin Relaxation time (T2)
• Spin-Lattice Relaxation time (T1)
• Proton density (PD)
• Spin- is a fundamental property of nature like
  electric charge or mass. Protons, electrons and
  neutrons have spin(1/2).

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Spin-Lattice Relaxation time T1

• External Magnetic field is along Z axis.
• If we move a magnetic field from its equilibrium
  then its net Magnetization vector is 0.
• It tries to return back to its original position,
  the time it takes to do so is called Spin Lattice
  Relaxation time T1

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Spin-Spin Relaxation time T2

• If magnetization is along XY
• then time taken to return it to equilibrium is
  called spin-spin relaxation time or T2
• T2 images are used to distinguish healthy from
  pathological tissues or for classification of
  tissues

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The Problem

• To segment the multi spectral magnetic resonance
  images.
• Get a good segmentation method for the
  classification of the tissues.

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Artificial Neural Networks

• Powerful technique in pattern recognition and
  decision making
• ANNs are Supervised ANN
• Unsupervised ANN
• This paper uses an unsupervised scheme based on
  least squares criterion.

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Methods

• Fuzzy Clustering Algorithms
• Fuzzy C-Means, Penalized C-Means
• Architecture between conventional competitive and
  penalized fuzzy CNN
• Experimental results obtained from PFCM

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FUZZY CLUSTERING TECHNIQUES
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Fuzzy Clustering Techniques

• Overview on Clustering
• Basic Notions
• -Data Sets
• -Clusters Cluster Prototypes
• Overview on Clustering Methods
• Fuzzy C-Means Objective Function
• Fuzzy C-Means Algorithm
• Penalized Fuzzy C-Means Algorithm

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Clustering Techniques

• Features
• - Unsupervised Methods
• - Organization of Data based on Similarities
• - Useful in Absence of Priori Knowledge
• -Applications
• - Image Processing Segmentation
• - Speech Recognition
• - Data Compression
• - Modeling Identification
• - Pattern Recognition

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Basic Notions in Clustering

• The Data Set
• Clusters Prototypes

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Overview on Clustering Methods

• Crisp or Hard Clustering
• Soft or Fuzzy Clustering

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Fuzzy Clustering Medical Science ?
Easy Versatile Valuable Supportive Tool
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Most widely used
FCM Algorithm
Well known
Powerful
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Goal of the FCM

Detect similarity between members of a
collection
Minimize the criteria in the least squared
error sense
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Basis of the FCM Algorithm

• Fuzzy C Means Objective Function / Functional

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The Fuzzy C-Means Functional

• c n
• JFCM 1/2 ?j1 ? i1 (?i,j)m xi-wj2
• where,
• xi i th sample, i1,2,3,.n
• c number of clusters, c? 2
• n number of data sample vectors
• ?i,j grade of membership of xi in cluster j
• wj cluster centroids,w1,w2,w3wjwc
• m fuzziness parameter

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Parameters of the FCM algorithm
Termination Criterion, ?
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Number of clusters,c

• Most Important
• Assumed
• Or Chosen Equal to the Number of Groups Existing
  in the Data.

• - Other parameters
• have less influence
• on the resulting
• partition
• - When clustering real data if no priori
  information available
• - In case of brain images 4 clusters to
  represent the CSF, the white matter, the gray
  matter, and the skull bone

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Fuzziness Parameter, m

• Influences Fuzziness of Resulting Partition
• Reduces the Noise Sensitivity
• Effect for ? i,j Depends on m
• - larger the value of m, higher is the
  dependence
• Usually, m2

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Termination Criterion, ?

• Terminaton condition
• for the FCM algorithm
• Usual choice for ?
• is 0.001.
• Choosing ? 0.01 drastically
• reduces the computing times

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The FCM Algorithm

• Step 1
• Given a data set,
• Select the number of clusters
• Initialise the cluster centroids, wj
• ( 2 ? j ? c )
• Initialise the fuzzification parameter, m
• (1 ? m ? ?)
• Initialise the termination tolerance, ? gt0
• Initialise the fuzzy membership matrix U to U(0),
• U(0) ? U

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The FCM Algorithm

• Step 2
• Calculate the Euclidean distance, di,j
• between each training sample xi and
• the class centroid wj
• Calculate the membership matrix Uµi,j,
• using the equation,
• (1/(d i,j )2)1/ (m-1)
• µi,j c
• ?j1 (1/(d i,j )2)1/(m-1)

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The FCM Algorithm
Step 3 Update the class centroids
n ? i1( ? i,j )m xi
wj c ?
i1( ? i,j )m

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The FCM Algorithm
Step 4 Compute ? max (
U(t1) - U(t) ). If ? gt ? , then go to Step
2 otherwise go to Step 5.
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The FCM Algorithm
Step 5 Find the results for the final class
centroids.
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Penalised FCM Algorithm
More Effective
Generalised type of FCM
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Penalized Fuzzy C-Means Functional

• c n
• J PFCM ½ ?j1 ? i1 (?i,j)m xi-wj2
  
• c n
• - ½ ? ?j1 ? i1 (?i,j)mln? j
• J FCM - ½ ? ?j1 ? i1 (?i,j)mln? j

The penalty term
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where, ? j proportional constant of class
j given by, n ? i1( ? i,j )m
? j c n
? i1 ? i1( ? i,j )m ? a constant, ? ?
0 J PFCM J FCM when ? 0
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and, wj same as for J FCM
c ((d i,j )2 - ?ln? j )1/ (m-1)
-1 µi,j ?l1 ((d
i,j )2 - ?ln? j )1/(m-1)
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The PFCM Algorithm

• Step 1
• Given a data set,
• randomly set the following
• - cluster centroids wj ( 2 ? j ? c )
• - fuzzification parameter, m, (1 ? m ? ?)
• - the termination tolerance, ? gt0
• Give a fuzzy c-partition U(0).

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The PFCM Algorithm

• Step 2
• Using the equations,
• compute the ? j (t) and wj(t) using U (t-1)
• Calculate the membership marix U ? i,j
• with ? j (t) and wj(t)

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The PFCM Algorithm

• Step 3
• Compute ? max ( U(t1) - U(t) )
• If ? gt ? , then go to Step 2
• otherwise go to Step

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The PFCM Algorithm

• Step 4
• Find the results for the final class centroids

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Neural Networks
Weights adapt thru unsupervised learning
rules

• Supervised
• Fixed Weight
• Unsupervised
• Competitive Neural Networks
• Learning rules
• Supervised (error based)
• unsupervised (output based)

No Teacher guidance
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Basic Competitive Learning Network

• One layer of input neurons and one layer of
  output neurons
• An input pattern x is a sample point in the
  n-dimensional real or binary vector space.
• Output nodes - Binary-valued (1 or 0) local
  representations
• output neurons number of classes

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Basic Competitive Learning Network

• comprises the
• Training Rules
• feed forward excitatory network(s)
• lateral inhibitory network(s).
• Minimal learning Model-Fixed output
  nodes(clusters)

Hebbian rules
Winner take all (WTA)
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Competitive learning network

• Input is multi spectral magnetic spin echo images
  X255, 240, 245
• Hebbian Learning Rules
• Single layer of neurons
• Fully connected to output nodes

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Competitive Learning Network

X1
? 1,1
X2
w1
X3
w2
X4
w3
X5
w4

.
wc
Xn-1
? n,c
Xn
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Conventional Competitive Learning Neural Network

• Least squares criterion
• Hebbian learning rules
• Modifies the winning unit to move them closer to
  the input
• Similar to c-means clustering, cluster centroids
  in multidimensional pattern space

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Parameters of FCM

• nj-number of pixels in a class cj
• wj be the mean of the class (i.e. centroids)
• wj?xj?cj xi/nj
• w0 be the global center of mass of X
• wj?xj?cj xi/nj

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Scatter Functions

• JT JWJB
• c n
• JT ? ? xi-w0 2
• j1 xj?cj
• c
• JW 1/2 ? ? xi-wj 2
• j1 xi?cj
• c
• JB 1/2 ? ? wi-w0 2
• j1 xi?cj

JT- Total Scatter Function
JW- Within Class
JB- Between-class
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Criterion

• Minimization of JW gt Maximization of JB
• So our criteria function could be
• c
• JW 1/2 ? ? xi-wj 2
• j1 xi?cj
• i.e. we want to minimize the sum squared error
  vector for each class xi-wj

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Cont.

• JW represents the least sum of squares error
  between n samples within class.
• We get c class centers w1,w2,wc
• Objective function for the learning network is
  modified to
• c n
• JC 1/2 ? ? ? i,j xi-wj 2
• j1 i1
• ? i,j 1 if xi belongs to cj and ? i,j 0 for
  all other clusters

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Winner-take-all-neuron

• The neuron that wins is called winner-take-all-neu
  ron.
• ? i,j indicates whether the input sample xi
  activates neuron j to win.
• ? i,j 1 if xi - wj lt xi - wk , for
  all k
• 0 otherwise

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Gradient (Steepest) Descent

• Gradient descent on the objective function Jc
  gives
• n
• ?wj -??Jc -?? (xi-wj) ? i,j
• ?Wj i1
• The update rule is the sum over all samples, its
  is usually used incrementally, i.e., sample after
  sample
• ?wj ? (xi-wj) ? i,j valid for all j and
• wj(t1) wj(t) ?wj(t)
• ? - learning-rate parameter

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Algorithm

• Initialize the cluster centroids wj (2 lt j lt c)
• Initialize learning rate ?, and states of input
  samples U? i,j
• Update neuron states with competitive learning
• Compute synaptic weights
• Repeat 3, 4 for all input samples, record no of
  neurons with changes state. If no neuron state is
  changed go to 6
• Output final classification

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Penalized Fuzzy Competitive Learning Neural
Network

• Same as the conventional learning network.
• Unsupervised network with penalized fuzzy
  reasoning
• Objective function
• c n
  c n
• JC,PFCM 1/2 ? ? ? i,j xi-wj 2 -1/2 ? ? ? ?
  i,j ln aj
• j1 i1
  j1 i1
• c n
• JFCM -1/2 ? ? ? ? i,j ln aj
• j1 i1

aj -is a proportionality constant
? - is a constant gt0
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Why Penalty?

• c n
• JC,PFCM JFCM -1/2 ? ? ? ? i,j ln aj
• j1 i1
• since aj is the weighted average of ? over the
  entire data aj lt 1 ? ln aj lt 0
  
• Hence we are actually adding a penalty which is
  log of a proportionality constant times the
  membership of each sample summed over the entire
  data.
• This adds a term to the evaluation function,
  which increases it by a small constant, hence can
  give better weights.

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Gradient Descent

• Gradient descent on the objective function Jc
  gives
• ?wj
• n
• -??(Jc,PFCM)-???(JFCM) -1 ?m(? i,j)m-1(ln aj) ?
  ? i,j
• ?Wj i1 ?wj 2
  ?wj
• Simplifying this by obtaining the derivative of ?
  i,j and replacing in the above equation, the
  gradient descent on the objective function can be
  updated as
• n
• ?wj ? ? ? i,jm (xi-wj)1- m (1-? i,j )
• i1 m-1
• ? - learning-rate parameter

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Gradient descent

• ?wj ?? i,j (xi-wj)1- m (1-? i,j )
• m-1
• ? - learning-rate parameter

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Algorithm

• Initialize cluster centroids wj,
• fuzzification parameter m,
• learning rate ?,
• constant ?
• and the value egt0.
• Give a fuzzy c-partition U(0).

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Algorithm cont.

• Find aj (t) ,wj (t) with U(t-1). Calculate U?
  i,j .
• Sequentially update weights of a neuron using
  competitive learning
• Compute ?max( U(t1) -U(t)). If ? gt e, goto 2
  else goto 5.
• Execute a defuzzification process and output
  final results

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Defuzzification process

• A pixel is assigned to the cluster when its
  membership grade in that cluster is greater than
  0.5.
• If none of the grades satisfy then the class
  with maximum grade is chosen, provided that sum
  of two largest gt0.5

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Experimental Results Discussion

• Data Input
• Parameter Values Used
• Acquisition Parameters
• Network Associated Parameters
• Discussion
• Results
• Performance Evaluation
• Conclusion

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Data Input Used

• T2 weighted MR images
• Recorded from a man of 32 years
• Acquired on a Siemens 1.5 Tesla
  Magnetom MR scanner
• 5 mm Slice Thickness
• 1mm Interslice Space
• 256 X 256 Pixels Matrix Size
• 8-bit Gray Levels with Spin Echo
• Sequences

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Data UsedAcquired Brain Images

TR1/TE12500ms/75ms
TR2/TE22500ms/100ms
TR3/TE31500ms/59ms
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Data UsedExtracted Peritoneal Cavity Images
TR 2500 ms TE 158 ms
TR 2500 ms TE 130 ms
TR 2500ms TE 144 ms
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Parameter Values Used

• for Fig.2a-2c,
• TR1/TE12500 ms/75 ms
• TR2/TE22500 ms/100 ms
• TR3/TE31500 ms/59 ms
• for Fig.3a-3c ,
• TE130,144,158
• TR2500 ms
• Fuzzification parameter, m 1.5
• Learning rate,? 0.3
• The constant, ? 1.2

Acquisition Parameters Network
Associated Parameters
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Discussion

• Pixel Vectors
• Initial Centroid Values
• Iterations
• Errors Due to Noise
• Postclassification Filtering

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Results
Segmented Images with 4 Clusters with 5
Clusters
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Performance Evaluation

• Goals Achieved
• Successful in the Identification of Most
  Important
• Regions in the Image
• Results Produced are in Acceptable Visual
  Agreement with Human Expert Opinion

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Results Independent of the Initial Cluster
Centroids
No Operator Intervention
More Powerful Performance
Conclusion
Stable Results
More Efficient Mechanism
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QUESTIONS COMMENTS ?