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Abstract Soft hashing, also known as robust hashing or perceptual hashing, consists of summarising multimedia data, so as to obtain a concise representation called a hash value. There has been an increasing interest in the soft hashing problem

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A FRAMEWORK FOR SOFT HASHING AND ITS APPLICATION TO ROBUSTIMAGE HASHING
E. McCarthy, F. Balado, G.C.M. Silvestre and N.J. Hurley
University College Dublin, Belﬁeld, Dublin 4 – Ireland
ABSTRACT
An increasing interest in the soft hashing problem has beenwitnessed in recent times. Techniques implementing softhashing intend to mirror the behaviour of cryptographichashing when the information to be hashed can be subjectto diﬀerent kinds of distortions. Many heuristic techniquesfor undertaking soft hashing of images and other multime-dia data have been devised. Except for some attempts, aframework giving solid guidelines to solve the problem islargely lacking. In this extended summary we provide onepossible approach to undertake the modelling of robust softhashing, detailing the basic problems involved. We showhow some prior schemes partly ﬁt inside our model, andwe provide an example of soft image hashing following thegiven scheme.
1. INTRODUCTION
Soft hashing, also known as robust hashing or perceptualhashing, consists of summarising multimedia data, so as toobtain a concise representation called a hash value (also,ﬁngerprint, message digest, or label). The hash generationprocedure should be such that perceptually similar datayield the same hash value. The soft hashing problem isinteresting for a number of scenarios which in most casesinvolve indexing of multimedia databases and/or authenti-cation, where the hash provides a compact representationwhich can be used to identify the data eﬃciently.Diﬀerent applications impose diﬀerent requirements, butusually soft hashing should signiﬁcantly reduce the dimen-sionality of the data. This has to be done without substan-tially increasing the probability of collision (or false positiverate), i.e., the probability of having the same hash value forany two perceptually diﬀerent data objects. In the case of authentication applications, the dimensionality reductionshould be made using a one-way key-dependent function.Last, a relevant requirement in indexing applications is therobustness to (usually unintentional) distortions which donot aﬀect the perceptual similarity of the multimedia data.Up to now, many diﬀerent robust hashing schemes havebeen proposed for image hashing (see for instance [1, 2, 3,4]), not forgetting those for other kinds of multimedia sig-nals. Nevertheless, a more general approach allowing theproblem to be addressed in a systematic way is largely lack-ing. Previous proposals by Johnson and Ramchandran [5]
Enterprise Ireland is kindly acknowledged for supporting thiswork under the Informatics Research Initiative.
and by Mıh¸cak and Venkatesan [6] have partially tried toﬁll this gap already.In this extended summary we propose a soft-hashingframework to gain insight into the main design lines of these types of systems. We emphasize its robustness as-pect, as required by their application to database index-ing. We identify the central blocks of the problem, which,although already hinted at by diﬀerent researchers in oneway or another, are presented here in a uniﬁed way. Last,we propose an application of the developed methodology tothe problem of image hashing.
2. SOFT HASHING FOR DATABASEINDEXING
In the following, we will consider a soft hashing system fordatabase indexing. Due to this, key-related security issueswill not be discussed, and we will focus our eﬀorts on thedesign of distortion-resistant soft hashing methods. Themultimedia signal to be hashed will be denoted without lossof generality by a continuous-valued
n
-dimensional vector
x
= (
x
1
,...,x
n
). In the general case, this vector may un-dergo some possibly random distortion function that we canwrite as
f
(
·
) :
R
n
→
R
n
. Our objective is to map the signal
x
to an index belonging to a ﬁnite set
H
, and as indepen-dent as possible of the distortion function applied. A work-ing hypothesis is that distortions have to be constrainedso that the distorted signal
˜x
=
f
(
x
) is not too diﬀerentfrom
x
under some perceptually meaningful criterion (seeSect. 3.1).As depicted in Fig. 1, it is possible to divide databaseindexing systems using soft hashing into three quite inde-pendent blocks, namely:
•
Synchronisation. As in communications problems itis necessary that the signal
y
= (
y
1
,...,y
m
) pre-sented to the hashing function always matches thesame indices, in spite of possible desynchronisationsundergone by
x
. Distortions that aﬀect synchronismcan be of very diﬀerent nature, such as warpings,croppings, rotations, etc. It is not possible to insertsynchronisation pilots in
x
, as is common practice incommunications systems. Most existing soft hashingsystems try to solve this issue through the use of fea-ture mappings. These mappings are just functions
s
(
·
) :
R
n
→
R
m
that exploit geometrical invariancesof
x
, and that are therefore very dependent on thenature of the multimedia signal. Examples of suchmappings for images may be moments of diﬀerent or-ders, or the Fourier-Mellin transform [7], and hence
x
Synchronisation
y
Hashing
c
Search
Fig. 1
. A Model for a Database Indexing System using Soft Hashing .we have that
m
≤
n
.
•
Hashing. Once the synchronisation of
y
is assured wecan safely design a hashing function
h
(
·
) :
R
m
→ H
.
(1)A mapping from an
m
-dimensional continuous vec-tor to an index belonging to a ﬁnite set
H
can beseen as quantization [5] or clustering. It is clear thatonly the knowledge of the statistics of
y
can lead toan optimal design of this multidimensional quantizer.Nevertheless, quantization is not the only ingredientof the problem, as, even if the vector at the input of
h
(
·
) is perfectly synchronised, the
amplitudes
of itssamples could be modiﬁed by the distortion function
f
(
·
), having a
˜y
=
y
+
n
. In this situation, codes maybe required to recover from the errors caused by thisdistortion. This block constitutes the main subjectof this extended summary, and we discuss it in moredetail in the following sections.
•
Search. Searching takes place when we need to com-pare a hash value with those precomputed and storedin a multimedia database. This search can be veryexpensive for huge databases, and for this reason wewould like our hash size to be the shortest possible.In any case, the minimum size can be quite large,depending on the nature of the hashed signals, andsmart strategies such as Viterbi search might be suit-able in order to deal with complexity issues.Notice that two dimensionality reductions are involved:one potentially due to the synchronisation block, and theother one due to the hashing function itself. One would likethe dimensionality reduction due to synchronisation to bethe less important one, so that more degrees of freedom arepossible for the design of the hashing function itself.
3. MODELLING THE HASHING BLOCK
In [5] guidelines are given for designing a perfectly
secure
hashing block, whose hash values can only be computed bythe owner of a secret key. The hashing block is proposedto be source coding with side information at the decoder,where this side information is a key-dependent dither whichis added to
y
before source coding.Alternatively, we focus here on the robustness of thehash values to further distortions that aﬀect the srcinalmultimedia signal. Still, we will see that source and channelcodes are also involved. Notice that we cannot rely here ona dither key-dependent vector for each database signal
1
.This would imply that we know the database signal we are
1
Using the same key for every signal would break the securityassumptions in [5].
dealing with, but that is precisely what we are trying toascertain with the hashing method.We will assume in the following perfect synchronisationat the input of the hashing block, such that the indices of
y
and its amplitude-distorted version˜
y
match. Also, weassume that the samples
y
i
,
i
= 1
,...,m
, are i.i.d. andthat they follow a certain distribution
f
Y
(
y
) with variance
σ
2
y
. Fig. 2 shows the main building blocks of our proposal.
3.1. Quantization
The ﬁrst objective is to design the quantizer (1) that maps
y
to an integer index
k
∈ H
=
{
1
,...,
2
mR
}
. This index
k
may be represented using
mR
bits (assumed integer), and,hence, the dimensionality reduction is given by the rate
R
.The quantizer should reach a compromise between the twofollowing conﬂicting properties:1. Two “similar” realisations of the stochastic process
Y
should map to the same index
k
.2.
|H|
= 2
mR
must be large enough to distinguish a“suﬃcient” number of instances of
Y
, i.e., to decreasethe probability of collision.Both properties require a distortion criterion for their ob- jective evaluation. Note that we are proposing to measuredistortion before the hashing block, and not afterwards asin [6]. Although the latter option is also coherent, we willsee below that our deﬁnition has several advantages. Wemay deﬁne a distortion criterion between any two vectors
y
and
ˆy
using a sample-by-sample error function such as thesquare error:
d
(
y
,
ˆy
) =1
m
m
i
=1
(
y
i
−
ˆ
y
i
)
2
.
(2)Other more accurate criteria using perceptual models arealso possible, leading to diﬀerent results. Using (2), theoverall distortion is measured in expectation as
D
E
{
d
(
y
,
ˆy
)
}
.
(3)Let us assume ﬁrst that
˜y
=
y
after the synchronisationstage; we will deal with the general case in Sect. 3.2. Inthis case, the problem would come down to that of sourceencoding with a given ﬁdelity criterion. This means that
y
may be reconstructed from
k
=
h
(
y
)
∈ H
using a certainfunction
g
(
·
):
ˆy
=
g
(
k
) =
g
(
h
(
y
))
,
(4)under the constraint
D
≤
D
max
. The limits of the optimalsolution to this problem are given by Rate-Distortion (R-D) theory, which determines the minimum rate
R
min
of thequantizer (1) for the distortion constraint to hold.In our problem we may reinterpret
√
D
max
as the aver-age radius of the ball in
R
m
that maps to the same hash
y
Quantizer
h
(
·
)
k
BitAssignment
v
Mapping toa Codeword
c
Fig. 2
. A Model of the Hashing Blockvalue, i.e., as the
granularity
or resolution of the soft hash-ing function. The reconstruction function (4) would justgive the centroids or representatives of those regions. Inthis case, R-D theory establishes the theoretically minimumhash size for achieving the resolution target
D
max
(i.e., thereconstruction distortion). Interestingly, a quantizer builtalong these guidelines guarantees the property of
uniformity
of the hash values (quantization indices) over the multime-dia signal to be hashed.Moreover, once the quantizer is built we may computebeforehand the probability of collision between any two sim-ilar signals (
P
c
) using the statistical characterisation of
y
:
P
c
=
f
Y
(
y
)
I
(
y
)
f
Y
(
y
) d
y
d
y
,
(5)with
I
(
y
)
V
h
(
y
)
∩
S
(
y
), that is, the intersection of theVoronoi (quantization) region corresponding to
k
=
h
(
y
)with a ball of similar vectors
S
(
y
), centered at
y
.
3.2. Bit Assignment and Channel Code
In many practical cases we may not assume that˜
y
=
y
asdone in the preceding section, due to intentional or unin-tentional distortions such as lossy compression and others.Once the quantization function (1) is chosen, notice thata vector lying near the border of a Voronoi region couldbe easily moved to a neighbouring region (that maps toanother diﬀerent index) with just a small distortion.Consequently, let us assume next that, after the syn-chronisation stage, the input to the hash function is dis-torted by a zero-mean additive noise
n
, independent of
y
and with covariance matrix Γ
n
=
σ
2
n
I
, i.e.˜
y
=
y
+
n
.In order to cope with this situation several strategiesare possible. One possible approach would involve takinginto account the distortion
n
in the codebook design. In theliterature, this is known as noisy source coding [8], whoseusual design target is a distortion constraint between thenoiseless signal
y
and the reconstruction of the quantizednoisy signal,
g
(
h
(
˜y
)). In the hashing problem the errorprobability of the hash values has to be considered in thecodebook design too. To our knowledge, this approach con-stitutes an open line of research. An additional diﬃcultyis that we do not usually know the statistical distributionof
n
.As an alternative, we propose to use a binary channelcode
C
at the quantizer output, in order to ﬁght the errorsinduced by noise. A similar strategy was followed in [6].Notice that, under noise distortions, the optimal quantizerin Sect. 3.1 may no longer be useful. This is so becauseits optimality in the R-D sense implies that the more likelyvalues are more ﬁnely quantized, and then, due to the samereason, these more probable values are less robust to noise.We will give an example of this behaviour and discuss asuitable quantizer in Sect. 5.In general, for using a code we will need to convert thequantization index
k
to a binary vector representation
v
(bitassignment). This vector has to be mapped to the nearestcodeword vector
c
∈ C
, and so the length of both vectorsis assumed to be equal. Notice that the bit assignment is-sue is very important for the code to be useful. Imaginewe have a one-dimensional codebook with 16 centroids uni-formly placed. If we use a natural bit assignment we canhave neighbouring regions which diﬀer in as much as fourbits, such as 1000 and 0111. If we use instead a Gray bitassignment the number of bit diﬀerences between neigh-bouring regions will never be greater than one, allowing forbetter performance of the code
C
.A code rate
R
c
implies that the number of coded hashvalues is 2
mR
c
R
, instead of 2
mR
. Then, with a code we areactually trading resolution power for robustness of the hashsystem. We may denote the whole hashing block in Fig. 2as
h
c
(
·
) :
R
m
→ C
. The robustness of the scheme can bedetermined using the probability of error
P
e
=
P
{
h
c
(
˜y
)
=
h
c
(
y
)
}
,
(6)or the corresponding bit error rate (BER). This amount is afunction of a given granularity-to-distortion ratio, that maybe deﬁned asGDR
10log
10
D
max
σ
2
n
.
(7)
4. APPLICATION TO IMAGE HASHING
We describe next a proposal on how to apply this frame-work to image hashing. For rotation, scale and translationinvariance we may use the Fourier-Mellin transform as asynchronisation block, as in [7], for instance. Afterwards,the DCT may be applied, as its low-frequency coeﬃcientsare known to be well modelled by the generalised Gaussiandistribution. Other similar strategies yielding modelableinputs to the hashing block are also possible.As the input
y
to the quantization block can now bemodelled statistically, the design of a quantizer which is op-timal is possible. An approximation to it can be obtainedwith the generalised Lloyd algorithm using a suitable ini-tialisation, although, as discussed in the previous section,we have to take into account the potential weaknesses of anoptimal codebook if further distortions are present in oursystem.In the latter case, the hashing block requires the appli-cation of a suitable bit assignment method. An open prob-lem is that it is not obvious how to make this assignment
0 2 4 6 8 10 12 14 16 18 2010
−3
10
−2
10
−1
10
B E R
GDR
Lloyds NaturalUniform NaturalLloyds GrayUniform Gray
Fig. 3
. Performance of Lloyd and uniform quantizers with-out using a code, and using natural and Gray bit assign-ments.for an
m
-dimensional quantizer. As a compromise solu-tion, we may apply unidimensional quantizers separately toeach dimension. In one dimension, the Gray bit assignmentis arguably a good choice. The resulting bitstrings maybe concatenated in a systematic or possibly key-dependentpseudorandom manner, to give the resulting vector
v
. Thechoice of an appropriate code for the ﬁnal step is discussedin Sect. 5.
5. EXPERIMENTAL RESULTS
In order to illustrate the proposal above we assume a sce-nario where the image to be hashed is represented by azero-mean i.i.d. Laplacian vector
y
with
m
= 1000 sam-ples. All quantizers are built with 8 centroids. Fig. 3shows the performances of a quantizer designed with theLloyds method and the optimal uniform quantizer underi.i.d. Gaussian distortion with variance dependent on theGDR. Notice that the practical values of this parameterare high if we want to limit the perceptual impact of theadded noise, when using a coarse hash granularity. We ob-serve that the uniform quantizer shows better performance,due to the reasons explained in Sect. 3.2. In addition, theimportance of the bit assignment is seen.At the ﬁnal step, the use of a convolutional seems a suit-able choice for easily providing abitrarily long codewords.However, preliminary tests resulted in quite high BER’s,possibly due to the codewords being clustered in this typeof code. As decoding (i.e., mapping of
v
to a codeword) isnot preceded in this problem by encoding, clustered code-words may aﬀect negatively the decoding process. This sug-gests that the most suitable codes in this scenario shouldhave uniformly distributed codewords. In this sense Reed-Solomon codes proved better, but other codes may largelyimprove the performance results obtained with these codesand shown in Fig. 4.As a conclusion, in this extended summary we have pro-posed a framework giving guidelines to build robust softhashing methods, and pointed out several issues that needto be solved. An application of the methodology to image
0 2 4 6 8 10 12 14 16 18 2010
−3
10
−2
10
−
B E R
GDR
Lloyds codedLloyds uncodedUniform codedUniform uncoded
Fig. 4
. Hashing block performance for Lloyd and uniformquantizers, using Gray bit assignments and a R-S (255,55)code.hashing has been given, together with preliminary empiri-cal tests to validate the proposal. In the ﬁnal paper resultswill be given using a full system with real images, and testson collisions and their prediction will be provided.
6. REFERENCES
[1] R. Venkatesan, S.M. Koon, M.H. Jakubowski, andP. Moulin, “Robust image hashing,” in
Procs. of the IEEE International Conference on Image Processing
,Vancouver, Canada, 2000.[2] M. Mıh¸cak and R. Venkatesan, “New iterative geomet-ric methods for robust perceptual image hashing,” in
Procs. of ACM Workshop on Security and Privacy in Digital Rights Management
, Philadelphia, USA, 2001.[3] C. Kailasanathan, R. Safavi Naini, and P. Ogunbona,“Compression tolerant DCT based image hash,” in
Procs. of the 23rd Intnl. Conf. on Distributed Comput-ing Systems Workshops
, Rhode Island, USA, May 2003,pp. 562–567.[4] F. Lefebvre, J. Czyz, and B. Macq, “A robust softhash algorithm for digital image signature,” in
Procs.of the IEEE International Conf. on Image Processing
,Barcelona, Spain, September 2003, vol. 2, pp. 495–498.[5] M. Johnson and K. Ramchandran, “Dither-based se-cure image hashing using distributed coding,” in
Procs.of the IEEE International Conf. on Image Processing
,Barcelona, Spain, September 2003, vol. 2, pp. 751–754.[6] M. Mıh¸cak and R. Venkatesan, “A perceptual audiohashing algorithm: A tool for robust audio identiﬁca-tion and information hiding,” in
Procs. of the 4th In- formation Hiding Workshop
, Pittsburgh, USA, 2001.[7] J. Fridrich, “Visual hash for oblivious watermarking,”in
Procs. of SPIE: Security and Watermarking of Mul-timedia Contents
, San Jos´e, USA, 2000.[8] E. Ayanoglu, “Optimal quantization of noisy sources,”in
Procs. of ICASSP
, Seattle, USA, April 1988, vol. 1,pp. 569–572.

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