Signal Analysis Research (SAR) Group - RNet - Ryerson University

Fig 3: Two-dmensional mapping: (Left) Input pattern with 7 dstinct clusters, (Middle) 8 centres are generated using Ncut, and (fight) 7 centres are
generated using SONcut. Over-classification around the quev (triangle) will result in enoneous classification of the relevant class.
nodes in the input pattern,) and assoc(A, A) and where H(t) is defined similar to the hierarchy function used
assoc(B, B) measure the total intra-cluster similarity in the DSOTM algorithm, then increment maximum allowed
Ncut by 1;
(association) in A and B, assoc(A,V) is the Continuation: Continue with the Similarity Measurement
total connection from nodes in cluster A to all nodes in the ,tep until no noticeable changes in the feature map are
graph; assoc(B,V)is defined similarly. w, is a nonnegative observed;
weight function measuring degree of similarity between two Graph Generation: Given the input pattern, set up a
samples of input patterns and is defined as: weighted graph, G = (V, E), compute the weight, wm, on
('1
each edge, Em, using (8) and create affinity, W, and
diagonal, D, matrices;
where d(p,q) is a pre-defined distance metric (i.e.
Euclidean distance) and k is a user defined constant to
control decreasing rate of the weight function and is
empirically sets to 0.2. By using this function, the smallest
eigenvector remains constant and Ncut can find relatively
right partitions [2].
As Shi and Malik have also discussed, the optimal
partitioning (minimum possible Ncut) can be computed by
solving the generalized eigenvalue system. The second
smallest eigenvector of the generalized eigensystem is then
used to partition the graph.
In this paper we have used the Ncut algorithm [2] for the
purpose of unsupervised data clustering. The Ncut
partitioning method can be recursively applied on the input
pattern to generate more than two clusters. Decision on
maximum number of centres in the input pattern to stop the
clustering process is a challenging problem. In this work, we
Eigensystem Transformation: Solve (D- W)x = A- Dx for
eigenvector with the smallest eigenvalue;
Graph Bipartition: Use the eigenvector with the second
smallest eigenvalue to bipartition the graph;
Partitioning Continuation: Consider current partitions for
supplementary subdivision. Continue with repartitioning
until the Ncut value reaches to its maximum allowed.
In summary, we have proposed an unsupervised
hierarchical Ncut algorithm that is able to estimate the
maximum number of allowed Ncuts by training the
algorithm using the principles found in the DSOTM
architecture. Thus, by dynamically adapting the Ncut
algorithm to the nature of the input pattern, problem of overpartitioning
the relevant class can be prevented. Fig. 3
depicts importance of adapting such predictive mechanism
for the Ncut clustering algorithm and illustrates
effectiveness of employing such mechanism to avoid overclassification
around the query centre.
have integrated - the Ncut algorithm - with the principles found
in DSOTM to automatically estimate appropriate number of
clusters in the input pattern and set the maximum allowed
Ncut accordingly. We call this Ncut algorithm with self-
oriented centre detection, the Self-Organizing Normalized
cuts, SONcut.
The proposed SONcut algorithm is as follows:
Initialization: Choose a root node, {I-,)~=~,
from the
available set of input vectors, { x,)L, in a random manner.
N is the maximum allowed Ncut (initially set to 1) and K the
total number of inputs;
Similarity Measurement: Randomly select a new data point,
x, and find the winning centroid, n*, by minimizing a
predefined distance criterion in (1);
Maximum Allowed Ncut Estimation: If Ilx(t) - I-,, 11 > H (t)
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Previously in [7], we proposed an automatic CBIR engine
that was structured around an unsupervised learning
algorithm, the DSOTM. To reduce the gap between high-
level concepts (semantics) and low-level statistical features
and to evolve the search process according to what the
system believes to be the significant content within the
query, the above engine was integrated with a process of
feature weight detection using genetic algorithms (GA) as
illustrated in Fig. 4b. In this paper we use a relatively
simpler CBIR architecture (see Fig. 4a and Fig. 5) to solely
compare abilities of the proposed hierarchical clustering
algorithms for the purpose of data classification with other
three techniques, SOTM, SOFM, and Ncut.
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