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Segmentation Techniques Luis E. Tirado PhD qualifying exam presentation Northeastern University. Segmentation. Spectral Clustering Graph-cut Normalized graph-cut Expectation Maximization (EM) clustering. Segmentation. Spectral Clustering Graph-cut Normalized graph-cut

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Segmentation techniques luis e tirado phd qualifying exam presentation northeastern university

SegmentationTechniquesLuis E. TiradoPhD qualifying exam presentation Northeastern University


Segmentation
Segmentation

  • Spectral Clustering

    • Graph-cut

    • Normalized graph-cut

  • Expectation Maximization (EM) clustering


Segmentation1
Segmentation

  • Spectral Clustering

    • Graph-cut

    • Normalized graph-cut

  • Expectation Maximization (EM) clustering

9/15/2014


Graph theory terminology
Graph Theory Terminology

A

B

  • Graph G(V,E)

    • Set of vertices and edges

    • Numbers represent weights

  • Graphs for Clustering

    • Points are vertices

    • Weights reduced with distance

    • Segmentation: look for minimum cut in graph

9/15/2014


Spectral clustering
Spectral Clustering

5

9

4

2

6

1

8

1

1

3

7

from Forsyth & Ponce

  • Graph-cut

    • Undirected, weighted graph G = (V,E) as affinity matrix A

    • Use eigenvectors for segmentation

      • Assume k elements and c clusters

      • Represent cluster n with vector w of k components

        • Values represent cluster association; normalize so that

      • Extract good clusters

        • Select wn which maximizes

        • Solution is

        • wn is an eigenvector of A; select eigenvector with largest eigenvalue

9/15/2014


Spectral clustering1
Spectral Clustering

  • Normalized Cut

    • Address

      drawbacks of

      graph-cut

    • Define association

      between vertex subset A and full set V as:

    • Previously maximized assoc(A,A); now also wish to minimize assoc(A,V). Define normalized cut as:

9/15/2014


Spectral clustering2
Spectral Clustering

  • Normalized Cuts Algorithm

    • Define where A is affinity matrix.

    • Define vector x depicting cluster membership

      • xi = 1 if point i is in A, and -1, otherwise

    • Define real approximation to x:

    • We now wish to minimize objective function:

    • This constitutes solving:

    • Solution is eigenvector with second smallest eigenvalue

    • If normcut is over some threshold, re-partition graph.

9/15/2014


Probabilistic mixture resolving approach to clustering
Probabilistic Mixture Resolving Approach to Clustering

  • Expectation Maximization (EM) Algorithm

    • Density estimation of data points in unsupervised setting

    • Finds ML estimates when data depends on latent variables

      • E step – likelihood expectation including latent variables as observed

      • M step – computes ML estimates of parameters by maximizing above

    • Start with Gaussian Mixture Model:

    • Segmentation: reformulate as missing data problem

      • Latent variable Z provides labeling

    • Gaussian bivariate PDF:

9/15/2014


Probabilistic mixture resolving approach to clustering1
Probabilistic Mixture Resolving Approach to Clustering

  • EM Process

    • Maximize log-likelihood function:

    • Not trivial; introduce Z, & denote complete data Y = [XTZT]T:

    • We know above data; ML is easy:

9/15/2014






Conclusions
Conclusions

  • For simple case like example of four Gaussians, both algorithms perform well, as can be seen from results

  • From literature: (k = # of clusters)

    • EM is good for small k; coarse segmentation for large k

      • Needs to know number of components to cluster

      • Initial conditions are essential; prior knowledge helpful to accelerate convergence and achieving a local/global maximum of likelihood

    • Ncut gives good results for large k

      • For fully connected graph, intensive space & computation time requirements

    • Graph cut’s first eigenvector approach finds points in the ‘dominant’ cluster

      • Not very consistent; literature advocates for normalized approach

    • In end, tradeoff depending on source data


References for slide images
References (for slide images)

  • J. Shi & J. Malik “Normalized Cuts and Image Segmentation”

    • http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdf

  • C. Bishop “Latent Variables, Mixture Models and EM”

    • http://cmp.felk.cvut.cz/cmp/courses/recognition/Resources/_EM/Bishop-EM.ppt

  • R. Nugent & L. Stanberry “Spectral Clustering”

    • http://www.stat.washington.edu/wxs/Stat593-s03/Student-presentations/SpectralClustering2.ppt

  • S. Candemir “Graph-based Algorithms for Segmentation”

    • http://www.bilmuh.gyte.edu.tr/BIL629/special%20section-%20graphs/GraphBasedAlgorithmsForComputerVision.ppt

  • W. H. Liao “Segmentation: Graph-Theoretic Clustering”

    • http://www.cs.nccu.edu.tw/~whliao/acv2008/segmentation_by_graph.ppt

  • D. Forsyth & J. Ponce “Computer Vision: A Modern Approach”



K means used by some clustering algorithms
K-means(used by some clustering algorithms)

  • Determine Euclidean distance of each object in data set to (randomly picked) center points

  • Construct K clusters by assigning all points to closest cluster

  • Move the center points to the real centers of the resulting clusters


Responsibilities
Responsibilities

  • Responsibilities assign data points to clusterssuch that

  • Example: 5 data points and 3 clusters


K means cost function

data

prototypes

responsibilities

K-means Cost Function


Minimizing the cost function
Minimizing the Cost Function

  • E-step: minimize w.r.t.

    • assigns each data point to nearest prototype

  • M-step: minimize w.r.t

    • gives

    • each prototype set to the mean of points in that cluster

  • Convergence guaranteed since there is a finite number of possible settings for the responsibilities


Limitations of k means
Limitations of K-means

  • Hard assignments of data points to clusters – small shift of a data point can flip it to a different cluster

  • Not clear how to choose the value of K – and value must be chosen beforehand.

    • Solution: replace ‘hard’ clustering of K-means with ‘soft’ probabilistic assignments of EM

  • Not robust to outliers – Far data from centroid may pull centroid away from real one.




Em algorithm informal derivation
EM Algorithm – Informal Derivation

  • Let us proceed by simply differentiating the log likelihood

  • Setting derivative with respect to equal to zero givesgivingwhich is simply the weighted mean of the data


Ng jordan weiss algorithm
Ng, Jordan, Weiss Algorithm

  • Form the matrix

  • Find , the k largest eigenvectors of L

  • These form the columns of the new matrix X

    • Note: have reduced dimension from nxn to nxk


Ng jordan weiss algorithm1
Ng, Jordan, Weiss Algorithm

  • Form the matrix Y

    • Renormalize each of X’s rows to have unit length

    • Y

  • Treat each row of Y as a point in

  • Cluster into k clusters via K-means

  • Final Cluster Assignment

    • Assign point to cluster j iff row i of Y was assigned to cluster j


Reasoning for ng
Reasoning for Ng

  • If we eventually use K-means, why not just apply K-means to the original data?

  • This method allows us to cluster non-convex regions


User s prerogative
User’s Prerogative

  • Choice of k, the number of clusters

  • Choice of scaling factor

    • Realistically, search over and pick value that gives the tightest clusters

  • Choice of clustering method



Advantages disadvantages
Advantages/Disadvantages

  • Perona & Freeman

    • For block diagonal affinity matrices, the first eigenvector finds points in the “dominant” cluster; not very consistent

  • Shi & Malik

    • 2nd generalized eigenvector minimizes affinity between groups by affinity within each group; no guarantee, constraints

  • Ng, Jordan, Weiss

    • Again depends on choice of k

    • Claim: effectively handles clusters whose overlap or connectedness varies across clusters


Segmentation techniques luis e tirado phd qualifying exam presentation northeastern university

Affinity Matrix Perona/Freeman Shi/Malik

1st eigenv. 2nd gen. eigenv.

Affinity Matrix Perona/Freeman Shi/Malik

1st eigenv. 2nd gen. eigenv.

Affinity Matrix Perona/Freeman Shi/Malik

1st eigenv. 2nd gen. eigenv.