Clustering Algorithms for Perceptual Image Hashing

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Clustering Algorithms for Perceptual Image
Hashing
IEEE Eleventh DSP Workshop, August 3rd 2004
Vishal Monga, Arindam Banerjee, and Brian L.
Evans
vishal, abanerje, bevans_at_ece.utexas.edu
Embedded Signal Processing Laboratory Dept. of
Electrical and Computer Engineering The
University of Texas at Austin http//signal.ece.ut
exas.edu
Research supported by a gift from the Xerox
Foundation
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Hash Example

• Hash function Projects value from set with large
  (possibly infinite) number of members to set with
  fixed number of (fewer) members
• Irreversible
• Provides short, simple representationof large
  digital message
• Example sum of ASCII codes forcharacters in
  name modulo N,a prime number (N 7)

Database name search example
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Perceptual Hash Desirable Properties

• Perceptual robustness
• Fragility to distinct inputs
• Randomization
• Necessary in security applicationsto minimize
  vulnerability againstmalicious attacks

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Hashing Framework

• Two-stage hash algorithm
• Goal Retain perceptual significance
• Let (li, lj) denote vectors in metric space of
  feature vectors V and 0 lt e lt d, then it is
  desired
• Minimizing average distance between clusters
  inappropriate

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Cost Function for Feature Vector Compression

• Define joint cost matrices C1 and C2 (n x n)
• n total number of vectors be clustered, C(li),
  C(lj) denote the clusters that these vectors are
  mapped to
• Exponential cost
• Ensures severe penalty associated if feature
  vectors far apart
• Perceptually distinct clustered together

a gt 0, ? gt 1 are algorithm parameters
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Cost Function for Feature Vector Compression

• Define S1 as
• S2 is defined similarly
• Normalize to get ,
• Then, minimize expected cost
• p(i) p(li), p(j) p(lj)

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Basic Clustering Algorithm

• Obtain e, d, set k 1. Select the data point
  associated with highest probability mass, label
  it l1
• Make the first cluster by including all
  unclustered points lj such that D(l1, lj) lt e/2
• 3. k k 1. Select the highest probability data
  point lk among the unclustered points such that
• where
• S is any cluster, C set of clusters formed
  till this step
• Form the kth cluster Sk by including all
  unclustered points lj such that D(lk, lj) lt e/2
• 5. Repeat steps 3-4 until no more clusters can be
  formed

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Observations

• For any (li, lj) in cluster Sk
• No errors up to this stage of algorithm
• Each cluster is at least e away from any other
  cluster
• Within each cluster, maximum distance between
  any two points is at most e

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Approach 1

• Select data point l among unclustered data
  points that has highest probability mass
• For each existing cluster Si, i 1,2,, k
  compute
• Let S(d)
  Si such that di d
• IF S(d) F THEN k k 1. Sk l is a
  cluster of its own
• ELSE for each Si in S(d) define
• where denotes the complement of Si i.e.
  all clusters in S(d) except Si. Then, l is
  assigned to the cluster S arg min F(Si)
• 4. Repeat steps 1 through 3 until all data points
  are exhausted

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Approach 2

• Select data point l among unclustered data
  points that has highest probability mass
• For each existing cluster Si, i 1, 2,, k,
  define
• and ß lies in 1/2, 1
• Here, denotes the complement of Si i.e.
  all existing clusters except Si. Then, l is
  assigned to the cluster S arg min F(Si)
• 3. Repeat steps 1 and 2 until all data points are
  exhausted

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Summary

• Approach 1
• Tries to minimize conditioned on
  0
• Approach 2
• Smoothly trades off the minimization of
  vs.
• via the parameter ß
• ß ½ ? joint minimization
• ß 1 ? exclusive minimization of
• Final hash length determined automatically!
• Given by bits, where k is number
  of clusters formed
• Proposed clustering can compress feature vectors
  in any metric space, e.g. Euclidean, Hamming, and
  Levenshtein

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Clustering Results

• Compress binary feature vector of L 240 bits
• Final hash length 46 bits, with Approach 2, ß
  1/2
• Value of cost function is orders of magnitude
  lower for proposed clustering

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Conclusion Future Work

• Two-stage framework for image hashing
• Feature extraction followed by feature vector
  compression
• Second stage is media independent
• Clustering algorithms for compression
• Novel cost function for hashing applications
• Applicable to feature vectors in any metric space
• Trade-offs facilitated between robustness and
  fragility
• Final hash length determined automatically
• Future work
• Randomized clustering for secure hashing
• Information theoretically secure hashing