OPTIMAL CONFIGURATION OF HASH TABLE BASED MULTIMEDIA FINGERPRINTDATABASES USING WEAK BITS
Claus Bauer, Regunathan Radhakrishnan and Wenyu Jiang
Dolby Laboratories, San Francisco, CA, 94103, USA
Email: cb@dolby.com,regu.r@dolby.com,wzj@dolby.com
ABSTRACTThe increasingly large amount of digital multimedia content hascreated a need for technologies to search and identify multimedia files. Multimedia fingerprinting has been widely researchedand successfully commercialized as a technology to trace andrecognize both audio and video content.
Most published research on multimedia fingerprinting focuseson the design of the fingerprinting extraction algorithm. The architecture of the fingerprinting database and the database querysearch algorithm has only been investigated in very few publications. In particular, no comprehensive mathematical models
describing the accuracy of database query algorithms have beendeveloped. Therefore, most performance evaluations of fingerprint system rely on experimental data.
In this paper, we develop an analytic model to describe the operation of hash-table based multimedia fingerprinting databases.
It allows to predict the accuracy of a fingerprinting query algorithm as a function of the statistical distribution of the fingerprints and the specific design of the fingerprint database.
Keywords— Multimedia fingerprints, database configuration, analytic model
1. INTRODUCTION
The explosive growth of digital multimedia content and its rapid
distribution over the Internet have created a need for technologies to classify, trace, and identify multimedia files. Multimedia
fingerprinting is considered as one of the most promising technologies that can provide these functionalities. For both audio[1] - [5] and video [6] - [10], substantial research on the designof multimedia fingerprinting systems has been performed andvarious companies have developed complete fingerprint systems[11] - [14].
Multimedia fingerprint extraction algorithms capture perceptualfeatures of the multimedia object and store them in a compact
bit sequence called the fingerprint. The accuracy of a fingerprint system is defined via the notions of robustness and sensitivity. Depending on the specific application of multimedia
fingerprinting, the fingerprints are required to be robust againstchanges of the multimedia object from which they are derived.
For an audio file, a modification could be a transcoding of thecontent. For a video file, a modification could be a rotation orc
978-1-4244-7493-6/10/$26.00 2010
IEEE

978-1-4244-7492-9/10/$26.00 ©2010 IEEEa cropping of the video. A modification of the multimedia object that does not prevent a human being from recognizing thatthe modified and the unmodified multimedia object contain thesame content should only lead to a relatively small change ofthe fingerprint. Fingerprints should be sensitive to content, i.e.,fingerprints derived from different multimedia content must differ significantly. Indeed, fingerprint should not only be able touniquely identify a multimedia file, but should moreover allow
to identify a certain point in time in a multimedia file with hightimely accuracy. This characteristic is required to identify partsof multimedia files illegally posted on Web 2.0 web pages likeYouTube, MySpace, etc.
Most applications of fingerprints rely on the following fingerprint system architecture. Initially, a reference database of multimedia fingerprints is built. It contains the fingerprints of all
multimedia objects the application shall recognize. During operation, a fingerprint of a query multimedia object is comparedagainst the reference database using a specific database queryalgorithm. The algorithm decides if the fingerprint of the query
multimedia object corresponds to a fingerprint in the reference
database. Fingerprint matching algorithms use techniques fromstatistics and fuzzy logic.
Large scale multimedia fingerprint applications require highly
scalable database design and query techniques. Most fingerprinting databases use either a tree structure [15] or a hash-tablebased design [16]. For both types of databases, the database
design requires the configuration of specific parameters determining the exact structure of the database. In this paper, weinvestigate a way to model and optimally design a hash-tablebased multimedia fingerprint database. In particular, we develop a model to predict the performance of a query algorithm
searching through a hash-based database as a function of boththe statistical distribution of the fingerprints and the actual values taken by the database design parameters. The architecture of
the database and the choice of the query algorithm determine theaccuracy of the fingerprint matching algorithm as well the storage requirements of the database and the speed of the databasesearch. The model proposed in this paper helps the system designer to methodically evaluate the different design trade-offs.
In sec. 2, we define some terminology used in this paper. The
concept of hash-based databases using weak bits is explained insec. 3. In sec. 4, we develop an analytic model of the performance of the database query algorithm. In sec. 5, we describe

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ICME 2010

• PW Probability of correct weak bit prediction: Assuming that a query multimedia fingerprint corresponds to afingerprint in the reference database, this probability denotes the probability that the reliable bits of the index of
the fingerprint of the query multimedia object are identical with the corresponding bits of the index of the fingerprint of the reference multimedia object. In other words,
PW denotes the probability that the weak bit prediction iscorrect. Using common notation, we set PW c = 1 − PW .
For any event A, we denote by PA|W (PA|W c ) the conditional probability of A given that the weak bit prediction
is correct (wrong).

how this model can be used to dynamically configure a fingerprint database.
2. TERMINOLOGY
A multimedia fingerprint h is derived from a multimedia object
x using a hash function H(x), i.e., h = H(x). The fingerprint
h is a sequence of bits of length L. We divide the fingerprint
into an index of length I < L and a remainder of length L − I.
We define the following probabilities to analytically describe
the notions of robustness and sensitivity:• Pf Probability of false positive: A query fingerprint
which does not correspond to any fingerprint in the reference fingerprinting database is erroneously declared tocorrespond to a fingerprint in the database.

• p : p defines the probability that an individual bit of the
L − I bits of the remainder of the fingerprint changes
when the multimedia object, from which the fingerprint isderived, is modified. The value of the variable p depends
on the design of the fingerprint extraction algorithm andthe considered signal modifications.

• Pc Probability of correct identification: A query fingerprint which corresponds to a fingerprint in the fingerprintreference database is correctly declared to correspond to
this fingerprint in the reference database.
• Pm Probability of misclassification: A query fingerprint
which corresponds to a fingerprint in the fingerprint reference database is erroneously declared to correspond toa different fingerprint in the reference database.

Note: In practice, the quantities p and PW tend to
be related. In this paper, we do not consider this possible
correlation, but assume that both quantities can be
independently measured and used as input parameters
for the system model described in sect. 4.

• Pd Probability of detection: A query fingerprint whichcorresponds to a fingerprint in the fingerprint reference
database is declared to correspond to a fingerprint in thereference database. Obviously,

3. MULTIMEDIA FINGERPRINT DATABASES USINGWEAK BITS

P d = P c + Pm .

(1)

Further, we introduce the following terminology:• Collision: Depending on the fingerprint extraction algorithm and the way the index is chosen as a part of thefingerprint, the fingerprints of different multimedia objects might be indexed by the same database index. This
scenario, where indexes of fingerprints of different multimedia objects collide, is called a collision.
• In order to simplify the calculations in this paper, we assume that for each index the expected number of collisions is identical. We denote the average number of collisions per fingerprint as D.
• Weak bits: Fingerprints are typically designed such thatthe fingerprint of an original multimedia object and thefingerprint of a modified version of the multimedia object
differ only by few bits. When extracting the fingerprintof the query multimedia object, each of the I bits of theindex of the fingerprint of the query multimedia object
is assigned a certain probability. Using techniques from[17], this probability describes the likelihood with which
the respective bit would change if the query multimedia
object would be modified. The bits which are assigned ahigh(low) probability of change are called weak(reliable)bits.

We first summarize the concept of multimedia fingerprint
databases using weak bits described in [17]. When a query multimedia object is queried against the database, the fingerprint of
the multimedia object is extracted. According to methods as described in [17], the I bits of the query fingerprint that constitutethe index are divided into W ≤ I weak and I − W reliable bits.
We then consider all indexes that have the same I − W reliable
bits as the index of the fingerprint of the query multimedia object. The number of these index equals 2W as we consider all
possible permutations of the W unreliable bits. We recall the
assumption in sec. 2 that the average number of collisions foreach of the 2W indexes is D. Thus, the 2W indices index
N = D2W .

(2)fingerprints. The fingerprint matching algorithm calculates the
N Hamming distances between the fingerprint of the multimedia query object and each of the N indexed fingerprints. Subsequently, the fingerprint among the N fingerprints with the lowest Hamming distance to the fingerprint of the multimedia query
object is chosen. If the lowest Hamming distance is achievedby more than one of the S fingerprints, then one of these fingerprints is picked randomly. If this Hamming distance is notbigger than a pre-defined threshold τ, then the corresponding
indexed fingerprint is declared to present the query multimedia
object. If this Hamming distance is bigger than a pre-defined
threshold τ, then the query multimedia object is declared not to1668

correspond to any multimedia object the fingerprint of which isincluded in the reference database.
4. ANALYTIC MODEL OF THE DATABASE QUERYALGORITHM
In sec. 4.1, we describe a framework to analyze fingerprint systems from [18]. In sec. 4.2 we derive an analytic model modelof a multimedia fingerprint database using weak bits.
4.1. Decision theoretic framework for analyzing binary
hash-based content identification systems
In [18], a general model to analyze the performance of binary
hash-based content identification systems is considered. The
reference database consists of U copyrighted multimedia objects. However, as outlined in sec. 3, for each datatbase query
we only consider N ≤ U multimedia objects V1 , .., VN with fingerprints X1 , ..., XN . In order to simplify our presentation, we
assume that no two fingerprints Xi are identical. When a user
uploads a query multimedia object Z the fingerprint Y of Z is
computed. The fingerprint matching algorithm decides if themultimedia object Z corresponds to any of the N multimedia
objects V1 , ..., VN . The input to the fingerprint matching algorithm are the N Hamming distances di = d(Vi , Y ), i = 1, .., N
which respectively denote the N Hamming distances between
the fingerprints of the query multimedia object Y and the N
multimedia objects V1 , .., VN . The database query algorithm
uses a threshold parameter τ as follows:• If i = argj=1,2,...N min dj and di ≤ τ, the algorithm
decides that the multimedia object Y corresponds to themultimedia object Vi . If the minimum value of all N
Hamming distances di is attained by more than one index i, one of them is chosen randomly.
• Else, the algorithm decides that the multimedia object Y
does not correspond to any of the N multimedia objects
VN .

We define Amin as the number of j for which dj =min di . Further, we will make use of the parameter Bj , 0 ≤i=1,...Nj ≤ I, which denotes the number of di satisfying dI,i = j,i.e., Bj = number{i|1 ≤ i ≤ N, dI,i = j.}. According to[18], [19] signal modifications lead to changes in the fingerprint
space which can be modeled as additive noise. Here, the effect of signal modifications on fingerprint bits is modeled by
the probability p defined in sec. 2.

4.2. Calculation of Pf , Pc , Pm , and Pd
We first introduce some auxiliary notation similar to the oneused in [18]. We abbreviate
 the multinomial coefficients asN
a1 ....ah, where a1 + ..ah = M. Further,CM:=
a1 ....ahkk, f1 (k) = CL−I
pk (1 −
we define: f0 (k) = 2−L+I CL−I
L
L−k
, Fu (k) = j=k fu (j) for 0 ≤ k ≤ L − I, u = 1, 2.
p)4.2.1. Calculation of Pf

Pf= P (at least one of the d1 , d2 , .., dN ≤ τ |H0 )= 1 − P (none one of the d1 , d2 , .., dN ≤ τ |H0 )
=

1−

N


G0 (τ + 1 − dI,m )

m=1

=

1−

W


u
DCW

G0

(τ + 1 − u),u=0

where G0 (·) is defined as follows: We first note that under the
hypothesis Hs for any i ∈ {1, .., N }, i = s satisfying dI,i =≤j + 1 for a fixed j, we have P (di > j) = F0 (j + 1 − dI,i ),whereas for dI,i > j + 1, P (di > j) = 1. Thus
P (di > j) = G0 (j + 1 − dI,i )
F0 (j + 1 − dI,i ) , if dI,i ≤ j + 1,.
=
1,
, if dI,i > j + 1.

In order to model the database query algorithm as a multiple
hypothesis test, we define the following hypotheses:(3)• H0 : Z is not from the database V1 , .., VN

4.2.2. Calculation of Pc• Hi , 1 ≤ i ≤ N : Z corresponds to the multimedia object
Vi

We note that if the weak bit prediction is not correct, then thefingerprint in the reference database that corresponds to the fingerprint of the query multimedia object will not be identified.
Thus, Pc|W c = 0 which implies Pc = Pc|W PW . If the weak bit
prediction is correct, then a specific hypothesis Hs , 1 ≤ s ≤ N,
is correct, i.e., we need to calculate

As described in sec. 3, the L bits of the fingerprint consist of
the I index bits and the L − I remainder bits. Thus, di = dI,i +dR,i , 1 ≤ i ≤ N, where• dI,i : Hamming distance between the index of the fingerprint of multimedia object Vi and the index of the fingerprint of the query multimedia object Z• dR,i : Hamming distance between the remainder of thefingerprint of multimedia object Vi and the remainder ofthe fingerprint of the query multimedia object Z

Pc = Pc|W PW =N


Pc|Hs PHs .

(4)

s=1

The following calculations will show that the value of Pc|Hs
depends indeed on the actual hypothesis Hs and in particular

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The last probability can be written as

depends on the quantity dI,s . We first see
Pc|HS = P (deciding s|Hs )P (dR,s = j − dI,s ∧ min di = j ≤ ∧= P (ds ≤ τ ∧ ds < min di |Hs )

i=s

i=s

+ P (min di = ds ≤ τ ∧ s is decided|Hs ).

Amin = k + 1|Hs )

(5)

i=s

=
We now calculate the first probability on the RHS of (5):

=

i=s

by b1 , .., bk |Hs ).

P (ds = j ∧ j < min di |Hs )P (dR,s = j − dI,s ∧ j < min di |Hs ).

(6)

i=s

j=0

Arguing as in (7) - (9), we write the last probability in (11) as
P (dR,s = j − dI,s ∧ min di = j attained

Under the condition Hs , the events dR,s = j − dI,s and j <min di are independent, i.e.,

i=1,..Ni=s

by s, b1 , .., bk |Hs ) = g1 (j − dI,s )

i=s

i=s

It is easy to see that
P (dR,s = j − dI,s |Hs ∧ W ) = g1 (j − dI,s ),

(8)

N


×

P (dR,s = j − dI,s |Hs ∧)P (j < min di |Hs ∧). (7)g1 (j − dI,s ) =f1 (j − dI,s ), if j ≥ dI,s ,
0,
else.



Pc|Hs

i=s

=

τ


G0 (j + 1 − dI,m ). (9)m=1, m=s×G0 (j + 1 − dm )

τ N
−1



1 
...
k+1

g1 (j − dI,s )

N

N


b1 =1

bk =1
b1 <...bk , bi =s{bi |dI,b =dI,s }≤Bd
−1
i
I,s

k


We now calculate the second probability on the RHS of (5):×P (min di = ds ≤ τ ∧ s is decided|Hs )

=

i=s

=

k=1

m=1, m=s,b1 ,..,bk

=

k=1
N
−1

k=1

G0 (j + 1 − dm )τ

g1 (j − dI,s )G−1
0 (j + 1 − dI,s )×

P (min di = ds ≤ τ ∧ Amin = k + 1 ∧
i=s

+

1
g1 (j − dI,s )G−1
0 (j + 1 − dI,s )
k+1

0
CW
−δdI,s ,0



×

τ
1 
P (dR,s = j − dI,s ∧ min di = j ∧i=sk + 1 j=0

Amin = k + 1|Hs ).

DC u
G0 W (j + 1 − u)
τ N
−1


j=0 k=1

1
P (min di = ds ≤ τ ∧ Amin = k + 1|Hs )k + 1 i=sW
CW
−δdI,s ,W

...

a0 =0

(10)

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×

W

u=0

j=0

s is decided|Hs )
N
−1


N


g0 (j − dI,bi )

i=1

N
−1


(12)

m=1, m=s

j=0 k=1NNg1 (j − dI,s )

j=0

+

=

G0 (j + 1 − dm ),where g0 (·) is defined via f0 (·) in the same way g1 (·) is defined
via f1 (·) above. From(5) - (8) and (9) - (12), we getUsing (3), we obtain
P (j < min di |Hs )g0 (j − dI,bi )

m=1, m=s,b1 ,..,bk

where


k

i=1

P (dR,s = j − dI,s ∧ j < min di |Hs ∧)
=

(11)

i=s

j=0
τ


bk =1
b1 <...bk , bi =s{bi |dI,b =dI,s }≤Bd
−1
i
I,s

P (dR,s = j − dI,s ∧ min di = j attained

i=s

=

N


...

b1 =1

P (ds ≤ τ ∧ ds < min di |Hs )
τ


N


W

i=0

g0ai (j − i)a0 ,..,aWCW

aW =0
a1 +..aW =k

W

u=0

i
DCW
−ai

G0

(j + 1 − u),(13)where G1 (·) is defined via F1 (·) in the same way G1 (·) is defined via F1 (·) in (3). The second probability in (16) is calculated as follows:

where δa,b is the Kronecker delta. Inserting (13) into (4),
τ
W

Cv 

Pc|W =×WW

2W j=0v=0

g1 (j − v)G−1
0 (j + 1 − v)

P (min di = ds ≤ τ ∧ i = s is decided|Hs )
i=s

DC u
G0 W (j + 1 − u)

=

u=0

+

τ N −1WCv  
W
2W

v=0

j=0 k=1

0
CW
−δv,0





...

×

W


a0 ,..,aWCW

aW =0
a1 +..aW =k
i
DCW
−ai

G0

j=0 k=1

1
g1 (j − v)G−1
0 (j + 1 − v)
k+1

W
CW
−δv,W

a0 =0

τ N
−1



W


Amin = k + 1|Hs ).

i=0

Pm|Hs =
τ N
−1 



(14)

i=0

k
G1 (j + 1 − dI,s ) + g1 (j − dI,s )k+1

j=0 k=1N

4.2.3. Calculation of Pm
We note that
(15)

bk =1
b1 <...bk , bi =s{bi |dI,b =dI,s }≤Bd
−1
i
I,s

i=1

=

G0 (j + 1 − dI,m )

τ N
−1 


j=0 k=1k
G1 (j + 1 − dI,s ) + g1 (j − dI,s )
k+1

i=s

+P (min di = ds ≤ τ ∧ i = s is decided|Hs ). (16)

×

i=s

G−1
0 (j + 1 − dI,s )0
CW−δdI,s ,0



i=s

τ N
−1



=

×

i=s

i=s

=

v=0

W
2W

(j + 1 − i).

G1 (j + 1 − v) + g1 (j − v)

j=0 k=10CW−δv,0i=1,..Nby s, b1 , .., bk |Hs ).

(17)
×Arguing as in (7) - (9), we write the last expression in (17) asi=1,..N

k


W

i=0

P (dR,s > j − dI,s ∧ min di = j attainedg0 (j − bi )

i=1
N


i
DCW
−ai

G0

i=0

×G−1
0 (j + 1 − v)

P (dR,s > j − dI,s ∧ min di = j attained×W

τ N −1 W
Cv  

bk =1
b1 <...bk , bi =s{bi |dI,b =dI,s }≤Bd
−1
i
I,s

by s, b1 , .., bk |Hs ) = G1 (j + 1 − dI,s )aW =0
a1 +..aW =k

Pm|W =

N

...

j=0 k=1 b1 =1

g0ai (j − i)...

Arguing as in (14), we obtaini=sAmin = k + 1|Hs )
N
τ N
−1 



W

i=0

P (min di = j ∧ min j < ds ∧j=0 k=1

a0 ,..,aW
CW



W
CW
−δdI,s ,W

a0 =0

The first probability on the RHS of (16) is transformed as
P (min di ≤ τ ∧ min di < ds |Hs )

g0 (j − Ibi )m=1, m=s,b1 ,..,bk

Pm|Hs = P (deciding i = s = 0|Hs )= P (min di ≤ τ ∧ min di < ds |Hs )
i=s

k


N


×

We first calculate Pm|W . Similarly to the calculation Pc|W we
first consider the conditional probability Pm|Hs for a fixed s.

N


...

b1 =1

Pm = Pm|W PW + Pm|W c PW c .

(19)This probability has been calculated in (10) - (12). Summarizing (16) - (19), we obtaing0ai (j − i)(j + 1 − i).

k
P (min di = ds = j ∧k + 1 i=sG0 (j + 1 − dm ),m=1, m=s,b1 ,..,bk(18)

g0ai (j − i)

W




a0 =0



W
CW
−δv,W



...

a0 ,..,aWCW

aW =0
a1 +..aW =k

i
DCW
−ai

G0

k
k+1

(j + 1 − i).

(20)

i=0

Now, we calculate Pm|W c . The condition W c implies that thefingerprint of the query multimedia object does not correspondto any of the N fingerprints indexed as described in sec. 3.
Thus, the hypothesis H0 holds. The conditional probability ofthe erroneous identification of the query fingerprint with any of
the N fingerprints given that the weak bit prediction is wrong
is therefore equal to the probability of a false positive, i.e.,
Pm|W c = Pf .

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4.2.4. Calculation of Pd
We note that Pd = Pd|W PW + Pd|W c PW c . To calculate Pd|W ,we first consider Pd|Hs . We see
P (d|Hs ) = P ( min di ≤ τ |Hs )i=1,..,N

=

1−

N


P (di ≥ τ + 1|Hs )

i=1

=

N


1 − G1 (τ + 1 − dI,s )G0 (τ + 1 − dm ).

m=1, m=s

Averaging as in (14), we now derive Pd|W as follows:
Pd|W

= 1−
×

τ
W

Cv 
W

2W j=0
v=0

W


u
DCW

G0

G1 (j − v)G−1
0 (j + 1 − v)

(j + 1 − u).

(21)

u=0

Last, we conclude from (1) and Pc|W c = 0 and Pm|W c = Pf ,Pd|W c = Pm|W c = Pf . Alternatively, using (1), Pd can bederived from the formulas for Pc and Pm given above.
5. DYNAMIC DATABASE CONFIGURATION
When the system is initialized, a maximum fingerprint length
Lmax is defined. Using maximum values of N and p and a minimum value of PW envisioned for the target application, one uses
the probability formula derived in sec. 4 and a series of values
for L, I, τ, W to find the smallest value of L = Lmax for which
one obtains the target values for Pf , Pc ,Pm , and Pd . During
operation, one similarly uses the current values of N, p, PW todetermine the actual values of Lactual , Iactual , τactual , Wactual
such that the resulting values of Pf , Pc , Pm , and Pd best matchthe target values for the application. For each reference multimedia object, a fingerprint of length Lmax is extracted. The
Lmax bits are divided in two S1 and S2 containing Lactual andLM ax − Lactual bits, respectively. For any database query, only
the Lactual bits in S1 are considered. The sets S1 and S2 are
updated whenever one of the input parameters N, p, PW or oneof the target probabilities changes.

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