1. Clustering Algorithms for Perceptual Image Hashing
IEEE Eleventh DSP Workshop, August 3rd 2004
Clustering Algorithms for Perceptual Image Hashing
Vishal Monga, Arindam Banerjee, and Brian L. Evans
{vishal, abanerje,
Embedded Signal Processing Laboratory
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Research supported by a gift from the Xerox Foundation

2. Database name search example
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 representation of large digital message
Example: sum of ASCII codes for characters in name modulo N, a prime number (N = 7)
Name
Hash Value
Ghosh 1
Monga 2
Baldick 3
Vishwanath
Evans 5
Geisler
Gilbert 6
Database name search example

3. Perceptual Hash: Desirable Properties
Perceptual robustness
Fragility to distinct inputs
Randomization
Necessary in security applications to minimize vulnerability against malicious attacks
Symbol
Meaning
H(I)
Hash value extracted from image I
Iident
Image identical in appearance to I
Idiff
Image clearly distinct in appearance vs. I m
Length of hash (in bits)

4. Two-stage hash algorithm
Hashing Framework
Two-stage hash algorithm
Goal: Retain perceptual significance
Let (li, lj) denote vectors in metric space of feature vectors V and 0 < ε < δ, then it is desired
Minimizing average distance between clusters inappropriate
Feature Vector Extraction
Compress (or cluster) Feature Vectors
Final Hash
Input Image
Visually Robust Feature Vector

5. 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
α > 0, Г > 1
are algorithm parameters

6. 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)

7. Basic Clustering Algorithm
Obtain ε, δ, 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) < ε/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) < ε/2
5. Repeat steps 3-4 until no more clusters can be formed

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

9. 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(δ) = {Si such that di ≤ δ}
IF S(δ) = {Φ} THEN k = k + 1. Sk = l* is a cluster of its own
ELSE for each Si in S(δ) define
where denotes the complement of Si i.e. all clusters in S(δ) 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

10. 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 β 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

11. Summary Approach 1 Approach 2
Tries to minimize conditioned on = 0
Approach 2
Smoothly trades off the minimization of vs.
via the parameter β
β = ½  joint minimization
β = 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

12. Error correction decoding [Mihcak &Venkatesan, 2000]
Clustering Results
Compress binary feature vector of L = 240 bits
Final hash length = 46 bits, with Approach 2, β = 1/2
Value of cost function is orders of magnitude lower for proposed clustering
Clustering Algorithm
Approach 1
7.64 x 10-8
Approach 2, β = ½
7.43 x 10-9
7.464 x 10-10
Approach 2, β = 1
7.17 x 10-9
4.87 x 10-9
Error correction decoding [Mihcak &Venkatesan, 2000]
5.96 x 10-4
3.65 x 10-5

13. 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