PATTERN RECOGNITION USING CONTEXTDEPENDENT MEMORY MODEL (CDMM) IN MULTIMODAL AUTHENTICATION SYSTEM

International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.1, January 2015
DOI:10.5121/ijfcst.2015.5105 47
PATTERN RECOGNITION USING CONTEXT-
DEPENDENT MEMORY MODEL (CDMM) IN
MULTIMODAL AUTHENTICATION SYSTEM
Y Suresh1
, K Pavan Kumar2
, PESN Krishna Prasad3
, BDCN Prasad4
Prasad V. Potluri Siddhartha Institute of Technology,
Kanuru, Vijayawada-7, India
ABSTRACT
Pattern recognition is one of the prime concepts in current technologies in both private and public sectors.
The analysis and recognition of two or more patterns is a complex task due to several factors. The
consideration of two or more patterns requires huge space for keeping the storage media as well as
computational aspect. Vector logic gives very good strategy for recognition of patterns. This paper
proposes pattern recognition in multimodal authentication system with the use of vector logic and makes
the computation model hard and less error rate. Using PCA two to three biometric patterns will be fusion
and then various key sizes will be extracted using LU factorization approach. The selected keys will be
combined using vector logic, which introduces a memory model often called Context Dependent Memory
Model (CDMM) as computational model in multimodal authentication system that gives very accurate and
very effective outcome for authentication as well as verification. In the verification step, Mean Square
Error (MSE) and Normalized Correlation (NC) as metrics to minimize the error rate for the proposed
model and the performance analysis will be presented.
KEYWORDS
CDMM, Vector logic, Feature extraction, LU factorization, Biometric pattern, Mean Square Error (MSE)
and Normalized Correlation (NC).
1. INTRODUCTION
The mathematical analysis of unique characteristics of human patterns such as face, fingerprint,
palm, Iris, Vein and so on has been adopted worldwide on a large scale and used in different
sectors, including Educational Institutions, banking, Airports, public and private organizations
and also to facilitate the person identification cards (Smart User Cards) such as Aadhaar Cards,
Driving License, Debit/Credit cards, etc. However, the successful use of biometric technologies
in these areas does not imply ideal security systems.
In general, most of the pattern recognition agents are unimodal, i.e., they use only single
biometric patterns to recognize whether the user is authorize or not, but even the best unimodals
usually use iris, fingerprint and face patterns are far from perfect. They have many inherent
problems with the use of unimodal technologies suffer from noisy sensor data, lack of
individuality, non-universality, spoof attacks and error rates included in the use.
In the logical formalism, the combination of multiple patterns gets the best systems and
overcomes some of the inherent problems of unimodal systems. The multimodal systems use two
or more patterns as a combination either in fusion based or non-fusion based

48
authentication/verification from the same person in order to identify him/her. These models are
expected to be more accurate and less problematic because of consideration of more independent
pieces of evidence into account, prior to the decision.
The relentless advance in the accuracy of biometric pattern recognition technology, increase in
the availability of digital media and mass availability of cheap computing power now provide
unique opportunities to meet challenging budgets by drastically enhancing the operational
efficiency of forensic investigators while even further enhancing public safety. Digital media can
be bulk-ingested in an automated fashion to be processed to identify and extract potential
actionable intelligence. Processing is continuous, consistent and predictable. Multiple
identification technologies can be deployed and the most suitable algorithms integrated to meet
evolving requirements.
In the development of human patterns based authentication system, the process can be done in
two to three stages: In stage one, the selection of appropriate patterns of a human being with
reasonably invariant and having different degradations is to be done and that can be processed for
the extraction of features with different pattern sizes. Before extracting the features, the patterns
can be fusion using some geometrical algorithms such as PCA, LDA, ICA and its variants. In the
next stage, the features can be extracted using factorization methods to generate key components
for the computational process. In the computational stage, the usage of vector logic to make the
system as complex and produce effective and accurate outcome in the testing criteria based on
decision strategy for the test sample by using threshold. In the decision strategy, the error rate will
be computed for the training and testing computational patterns using either by Mean Square
Error (MSE), Normalized Correlation (NC) or any other error models for pair of vectors. Finally,
based on threshold criteria the system gets the acceptance of rejection of the person as a result.
In this paper, an intuitive strategy is considered from the vector logic as Context-Dependent
Memory Model (CDMM). This model works with the concept Kronecker product in the vector
logic, which makes the authentication systems very complex and less error rate and is very
effective for verification. This paper presents the computational model in the multimodal
authentication system using vector logic and will be discussed in section 1. The empirical study
of this model and methodology approach of PCA and LU will be presented in section 3. Finally
the derived computational model along with experimental results and analysis will be presented in
section 5.
1.1. Vector Logic
Vector logic is a mathematical representation of matrix algebra [2] inspired from the models of
image processing and recognition systems. In this formalism, the image data is represented as
matrix models to compute the best features in terms of Eigen values and Eigen vectors as feature
vectors. Vector logic is one of prime concept of vector algebra in the matrix calculus. In the
algebraic formalism, the operations on vector logic can be done in numerous ways for a square
matrices and rectangular matrices. In the theory of rectangular matrices, kronecker product is
one of the prime products in the process of tensor products which compute in the region of image
segmentation, de-noising, convolution filters, authentication security models and medical
applications too.
In the process of image computation models, matrices and vectors have been used to represent the
logical formalism of feature extraction and selection, sub spaces, sub regions, pattern filtering,
pattern classification, compression, restoration, digitization, enhancement. The representation of
pattern data in matrices extend to the complex domain; that actualizes the biometric
authentication and verification as vectors that can be used to construct geometrical representation

49
of logical rules. The data elements could be represented as Monadic, Dyadic and so on. In the
monadic data the data is a one-dimensional vector, whereas dyadic data is a two dimensional data
matrix.
For the dyadic data in various biometric applications, both key dimensions of the data matrix
simultaneously is often more desirable than traditional one-way decomposition. The prime key
benefit of exploiting the duality between rows and columns to effectively deal with the high-
dimensional and sparse data that are typical in many applications. The key idea is that the block
structure in a two-dimensional dyadic data matrix A can be explored by its triple decomposition.
The dyadic data matrix is factorized into three components: PA, Left Triangulation and Right
Triangulation .
1.2. Kronecker Product as Context Dependent Memory Model (CDMM)
The kroneckerproduct [25] of two matrices denoted by has been researched since the
19thcentury. Many properties and other decomposition techniques have been discover during this
time and are a part of classical linear algebraic models and it has been applied in many classical
applications and produces new theories. In the recent research the vector logic as tensor products
are used in the fields of Signal and Image processing to eliminate noise, error rate and improves
the accuracy and efficiency of the samples. One nice product is Kronecker product, which has
several properties; some of the properties are [3, 4, 7]
(1)
(2)
(3)
The kroneckerproduct can be applied to vectors (monadic) and matrices (dyadic). The
representaion of kronecker product for a set of dyadic matrices is
(4)
Where is the tensor product of . The elements of this tensor product [7] are the
Kernel products of the vectors belonging to The product is
essential in Vector Logic and also in the current application, and is defiend as follows. Given two
matrices , the Kronecker product is given by
Each is a block of size . is of size .
Now-a-days, tensor algebra is playing a key role in the pattern analysis and recognition systems
with the use of vector logic. The kronecker product can be used in the analysis of image patterns
and numerous representations for segmentation, sub region analysis, de-noising, filtering and also

50
in biometric technologies. One advantage of kronecker product [4] is their compact representation
and approximation reasoning.
2. RELATED WORK
In the computation of multimodal systems, user has to enroll two to three patterns through
different sensors for registration as well as verification/authentication. Such patterns may be noise
due to capturing of patterns or environment or sensing devices. Such noise may lead to more
error prone.
Various kinds of multimodal systems have been discussed in [8-18] among them user
authentication based on face and speech patterns, is one of the first multimodal biometric system
[9]. Multimodal system using face and fingerprint features is then proposed [10] by taking two
acoustic patterns from speech and three visual patterns. Clustering techniques have been used for
the fusion of decisions from speech and face modalities are explored in [11]. Security can be
improved by considering dynamic pattern like lip movement, a practical multimodal system by
considering face, voice and lip movement is developed [12]. A frame work proposed by A. K.
Jain et al, for multimodal biometric person authentication discussed in [8]. Details about desirable
characteristics of a physiological and behavioral of human, different levels of fusion, security and
privacy concerns are also discussed.
In order to consider the level of authentication, the extraction of features for different patterns is
completely different and can be uniqueness. In the combination of such patterns, the computation
modal and decision modal must produce accurate and less error rate and complexity should be
high. For such kind of computation modals vector logic gives better outcome to minimize the
error rate with good accuracy.
2. EMPIRICAL STUDY
We demonstrate the effectiveness of our biometric system using the two level multimodal
authentication problems. We conduct a detail study of authentication system into three levels of
fusion, for encoding and decoding of biometric patterns. At level one, the normalized person
patterns fusion through PCA, at two using computational model the generated keys through LU
factorization, and at level three the decision process can be computed. An overview of the
computational process of the model is provided in Fig. 1 and Fig. 2 respectively. For empirical
evaluation, we demonstrate the performance of authentication system on different bench mark
data sets Yale and AT&T. We compare the obtained results with various recent distance metrics.
In this paper, we focus on the computational model often called Kronecker logic as associative
model often called Context-Dependent Memory Model(CDMM) and also various pair wise vector
distance can be considered for evaluating the chosen model to minimize the error rate and to
increase the efficiency and accuracy of the person patterns and we present False Accepting Rate
(FAR) and False Not Accepting Rate (FNAR) are summarized in Table. 1

51
Figure 1: Frame work of the proposed model
Figure 2. Block Diagram of three level computation models
3.1 Context-Dependent Memory Model (CDMM)
Any linear system can be characterized by context dependency memory model [1,2]. If
is a linear system which maps the input signal to the output , it is described by a function
such that
(5)
The symbol represents the Context dependency operation operation (also often called
Kronecker Product). The theory generalizes the above system can be mapped into the
characterization of nonlinear translation. If is a non-linear system which maps the input
signal to the output , it is described by an infinite sequence of functions indexed by
n as
(6)
Where represents the order kernel operation in the vector logic and
(7)
Here, are called as Taylor series exponential of the function. Note that (1) is a
special case of (3) for n=1. Another intuitive way to think of series is to see it as the Taylor series
expansion of the functional above. Switching to a discrete formulation, (3) can be written as
(8)
The infinite series in (4) does not lend itself well to practical implementations. Further, for a
given application, only the first few terms may be able to give the desired approximation of the
functional. Thus, we need a truncated form of Taylor series, which is denoted in this paper as
(9)
Where p denotes the order kernel logic. Note that is a place holder for all different
orders of kernels. The above equations can be defined for one-dimensional signals. All these

52
equations can be seamlessly generalized for two-dimensional image data, i.e. . The
approach of this model can be mapped to 2-Dimage processing in the field of biometrics. Thus,
this operation is an approximate operator that can be applied to image restoration and noise
removal.
4. FUSION OF BIOMETRIC PATTERNS USING PCA
Image fusion process can be defined as the integration of information from a number of registered
images without the introduction of distortion [19]. One of the goals of image fusion is to create a
single enhanced image more suitable for the purpose of human visual perception, object detection
and target recognition. This fusion of images obtained from different modalities is of great
importance in applications such as medical imaging, remote sensing, Multimodal Biometrics and
machine vision [20, 21].
In the process of image fusion, multi-sensor outputs can be combined to give a better quality
image of the scene. Image fusion is generally divided into three categories; pixel level, feature
level and decision level fusion [22, 23]. Pixel level fusion works directly on the pixels obtained at
imaging sensor outputs while feature level fusion algorithms operate on features extracted from
the source images. Decision level fusion uses the outputs of initial object detection and
classification as inputs to the fusion algorithm to perform the data integration. From these three
levels, the model considers feature level fusion patterns. The principle component analysis (PCA)
is a mathematical way of determining the linear transformation of a sample of points in a N-
dimensional space exhibiting the properties of the sample along the coordinate axes. It enables a
decrease in the number of channels (or images) by reducing the inter-channel dependencies. This
analysis is based on the computation of the covariance matrix and its diagonalization by finding
corresponding Eigen values and eigenvectors. The first eigenvector (or principle component) is a
linear combination of the initial images and usually contains more than 90% of the information
contained in all the images. Two sets of data can then fused by concatenating all channels and
performing a PCA transformation on all the channels to keep only the non-redundant information.
5. FEATURE EXTRACTION USING LU FACTORIZATION
There have been many research studies that perform decomposition based on Singular Value
Decomposition (SVD) or eigenvector-based decomposition, each data vector into the singular
vector space through the SVD, and then conducts the decomposition using traditional algorithms
in the transformed space. Since, the computed singular vectors or eigen vectors do not correspond
directly to individual ones, the decompositions from SVD- or eigenvector-based methods are
difficult to interpret and to map to the final segments; as a result, traditional data segmentation
methods must be applied in the transformed space. Another matrix decomposition formulation,
LU factorization has been used for key extraction and generation from the fusion patterns. LU
factorization has the intuitive interpretation for the result. However, it focuses on triangulation of
the data matrix and does not take advantage of the duality between the rows and the columns of a
matrix, where as SVD or other eigenvector decomposition methods focus on one dimensional
data matrix, i.e. vector representation.
5.1 LU Factorization
The matrices can be factorized into various ways with the use vector logic [3,24], some of the
factorization methods are LU, Cholesky, QR, Schur, SVD and Block matrix factorization and
many more. Among these methods LU factorization is chosen since it works on Gaussian

53
transformations and generates triangular system with the complexity . The computation of
LU factorization can be described for the matrices A and B of 3x3 matrices as:
(10)
6. COMPUTATION MODEL (CONVOLUTION KERNEL PRODUCT)
We describe a model where the input-output associations of a matrix are dependent on the context
of multiplicative vector logic. In this context a remarkable operation that transforms the real
matrix over the truly computing, into a triangular matrix. This transformation promoted by the
context leads to the LU Decomposition of a matrix representation, which is a mathematical
method in the field of Image Processing and signal processing.
The meaningful event in the representation is the activity of the large group of data patterns,
naturally represented by high dimensional vectors or dyadic matrices. These representations are
processed by associative memories (The association input-output pattern pairs) which store the
information distributed and superposed over coefficients. In this paper, the associative memory is
computed, the keys which are generated from the fusion patterns by using LU factorization,
(11)
(12)
Where A and B are fusion patterns of two pair biometric samples ( face and finger print). The
fusion is done by using PCA. The importance of this factorization is here to extract the best
features by eliminating noise and lighting effect using Gaussian-distribution models. In the
process of computation the CDMM uses the two keys from LU factorization and generates a
memory representation over k, where k is the order that uses Kronecker product as
CDMM[5,6].
(13)
In general, represents the Kronecker product.
(14)
Where A is the one key pattern of one fusion set and B is another key pattern of the second fusion
set, with order kernel logic. In this expression an association can also be weighted by a scalar
coefficient that represents frequency of the given association.
(15)
Where, W is the weighted frequency of the association. In the linear combination we assume that
the memory model has equal weights. Thus,
(16)
Where, and W is identity matrix. In this associative model, the memories are based on a
matrix over a multiplicity of vector associations are superimposed. Another representation of the
memory is,

54
(17)
which is a structural representation of M for the context . In this representation, the singular
values of matrix correspond to the scalar product . The potential flexibility of this
model emerges from a compromise between the structured rigidity and selective capabilities of
the information.
7. DECISION STRATEGY
In this paper, we considered two kinds of decision strategy approaches for
verification/authentication or the recognition of chosen patterns with the use of CDMM at the
training and tested levels M and M’. Mean Square Error (MSE) and Normalization Correlation
(NC) are considered as two approaches for verification of the proposed model shown in figure(1).
7.1 Mean Square Error
The Mean Square Error (MSE) of an estimator of a parameter is the function of defined
by and it is denoted as ̂. This is also called the risk function of an estimator,
with called the quadratic loss function. MSE measures the average squared difference
between the estimator and the parameter , a somewhat reasonable measure of performance
for an estimator. MSE has at least two advantages over other distances:
1. It is analytically tractable 2. It has the interpretation
7.2 Normalization Correlation
A popular approach of the pathway construction is based on the sample collection coefficient are
mutual information measures to characterize the interaction between the two patterns via the
computation task at the decision level. These measures of the interaction are computed from the
computational task of observed patterns across various experimental conditions.
The present paper focuses on the correlation between test statistics associated with the memory
signals produced by the trained and tested memory patterns from the effect of normalization
procedures on these correlations.
Normalization is intended to mitigate the effect of the noise (error signal) i.e., inherent in the
verification process. Normalization produces tend to reduce the variability of original data
pattern.
(18)
Where A and B are two congruent patterns, is the maximum possible distance between
them.
8. EXPERIMENTAL RESULTS AND ANALYSIS
By considering benchmark Yale and AT&T, FERET data sets [27, 28] number of experiments
made using the proposed framework with the computational task CDMM. The framework is
divided into three levels of process. At stage one, using LU factorization the best features will be
extracted as block matrix (dyadic matrix) from fusion patterns as keys. In stage two, the generated
keys will be passed as inputs to the CDMM for computation to increase the complexity. In stage

55
three, this paper performs two kinds of decision strategy on pair wise data matrices from the
patterns of the computational model for registration and verification. Mean square Error (MSE)
and Normalized Correlation (NC) can be considered for verification of the biometric patterns
through the framework. Based on the error rate of MSE and NC, the acceptance rate will be
determined. From the observations of bench mark data sets for both similar and dissimilar patters
of different poses using the proposed framework, the thresholds of MSE and NC bounded as 0.10,
0.25 respectively. In this paper, we are presenting three kinds of patterns with various key sizes
The experimental results will be presented in the Table 1. Based on the
analysis, the key size 8X8 using MSE and NC have failed for recognition since less number of
features has been extracted. From the observations of MSE and NC thresholds the False
Acceptance Rate (FAR) and False Non Acceptance Rate (FNAR) have been measured for the
chosen key sizes of the patterns.
Key Size
Similar Dissimilar Dissimilar2
MSE NC MSE NC MSE NC
8x8 0.16989 0.45022 0.13558 0.408484 0.142142 0.423452
16x16 0.18351 0.2212 0.06473 0.470359 0.066315 0.478301
24x24 0.14647 0.21929 0.08831 0.306719 0.089309 0.346723
32x32 0.13255 0.24038 0.09124 0.26171 0.081218 0.279634
40x40 0.14347 0.1513 0.07946 0.390177 0.073142 0.385437
48x48 0.10454 0.15067 0.0962 0.282631 0.091011 0.232712
56x56 0.11772 0.14998 0.08888 0.291341 0.081913 0.285432
64x64 0.12101 0.14921 0.0755 0.311495 0.067432 0.334527
Table 1. MeanSquareError
Figure 3. Mean Square Error

56
Figure 4. Normalized Correlation
9. CONCLUSION
The proposed model from the vector logic as CDMM takes place an instrumental role in many
image restoration algorithms and removal of noisy with the use of LU factorization since
Gaussian transformation eliminates noise. And also the CDMMwill be used in various forms like
Approximate inverse pre-conditioners and Fast transformations such as Hartley transform,
Fourier Transforms and Wavelet transforms for Image and Signal systems, and also it fits the
problem of surface fitting with splines also can be used in stochastic network models. The
complexity of the CDMM is , assume the two matrices are of . It is NP-hard. Using
the proposed model of various samples, the key size 8X8 is failed due to less features from the
fusion pattern for recognition/verification.
REFERENCES
[1] David Haussler {1999}, “Convolution Kernels on Discrete Structures UCSC-CRL-99-10”, a technical
report, July 8.
[2] Eduardo Mizraji, “Modalities in Vector Logic”, Notre Dame Journal of Formal Logic Volume 35,
Number ,272-283 Spring 1994.
[3] H.V.Henderson, F.Pukelsheim and S.R.Searle. “On the history of the Kronecker product”. Linear and
Multilinear Algebra. 14:113-120, 1983.
[4] Lester Lipsky and Appie van deLiefvoort, “Transformations of the Kronecker Product of Identical
Servers to Reduced Product Space”, 1995.
[5] Alessandro Moschitti, “A Study on Convolution Kernels for Shallow Semantic Parsing”, proceedings
in ACL 2004.
[6] Thomas Gartner, “A survey of Kernels for Structured Data”, ACM SIGKDD, Volume 5(1), pages 49-
58, July 2003.
[7] Wolfgang Hackbusch, Boris N. Khoromskij, “Hierarchical Tensor-Product Approximations”, 1-26.
[8] A.K.Jain, A Ross and S. Prabhaker, “An introduction to biometric recognition”, IEEE Trans. Circuits
and Systems for Video Technology, vol. 14, no. 1, pp, 4-20, Jan. 2004.
[9] R.Bruneelli and D.Falavigna, “Person identification using multiple cues”, IEEE Trans. PAMI, vol. 17,
no.10, pp.955-966, oct.1995.
[10] L.Hong and A.K. Jain, “Integrating faces and fingerprints for person identification”, IEEE PAMI, vol.
20, no. 12, pp. 1295-1307, Dec. 1998.
[11] V.Ghattis, A.G. Bors and I. Pitas, “Multimodal decision level fusion for person authentication”, IEEE
Trans. Systems, Man and Cybernatics, vol. 29, no. 6, pp, 674-680, Nov. 1999.
[12] R.W..Frischholz and U. dieckmann, “Bioid: A multimodal biometric identification system”,IEEE
Computer Society, pp. 64-68, Feb.2000.

57
[13] A.Kumar et. al., “Person verification using palmprint and handgeometry biometric”, proc. Fourth Int.
Conf. AVBPA, pp.668-678, 2003.
[14] S.Ribaric, D.Ribaric and N. Pavesic, “Multimodal biometric user identification system for network
based applications,” IEEE Proc. Vision, Image and Signal Processing, vol. 150, no.6, pp.409-416,
2003
[15] A.K.Jain and Ross, “Learning user specific parameters in multibiometric system”, Proc. Int. Conf.
Image Processing (ICIP), pp. 57-60,2002.
[16] A.K.Jain, L.Hong and Y. Kulkarni, “A multimodal biometric system using fingerprint, face and
speech”, Proc. Second Int. Conf. AVBPA, pp.182-187, 1999.
[17] S.Ribaric, I.Fratic and K. Kris, “A biometric verification system based on the fusion of palmprint and
face features”, Proc. Fourth Int. Symposium Image and Signal Processing, pp. 12-17, 2005.
[18] B.Ducet. al., “Fusion of audio and video information for multimodal person authentication”, Pattern
Recogntion Letters, vol. 18, pp.835-845, 1997.
[19] A.Haq, A.M.Mirza and S. Qamar, “An Optimized Image Fusion Algorithm For Night-Time
Surveillance And Navigation”, Proceedings of the International Conference on Emerging
Technologies, Islamabad, 2005, pp138-143.
[20] G.Pajares, J.M.de La Cruz, “A Wavelet Based Image Fusion Tutorial”,, Pattern Recognition, 37,
2004,pp.1855-1872
[21] C.Pohl, and J.L.VanGenderen, “Multisensor Image Fusion in Remote Sensing : Concepts, Methods
and Applications”, International Journal of Remote Sensing Vol. 19, No. 5, 2008, pp. 823-854.
[22] G.Piella, “A General Framework For Multi-Resolution Image Fusion: From Pixels To Regions”,
Information Fusion, 9, Dec 2003, pp. 259-280.
[23] Lawrence A.Klein, “Sensor and Data Fusion: A Tool for Information Assessment and Decision
Making”, SPIE, vol.PM138-H.Li.B.
[24] John W.Brewer, “Kronecker Products and Matrix Calculus in System Analysis”, IEEE Transactions
on Circuits and Systems, Vol. 25, No. 9, 1978.
[25] Kathrin Schacke, “On the Kronecker Product”, Master’s Thesis, University of Waterloo, 2004.
[26] Amy N.Langville, William J,Stewart, “The Kronecker Product and Stochastic automata networks”,
Journal of Computational and applied mathematics, Elsevier, 167,429-447,2004.
[27] http://www.advancedsourcecode.com/fingerprintdatabase.asp
[28] http://www.face-rec.org/databases/

PATTERN RECOGNITION USING CONTEXTDEPENDENT MEMORY MODEL (CDMM) IN MULTIMODAL AUTHENTICATION SYSTEM

More Related Content

Similar to PATTERN RECOGNITION USING CONTEXTDEPENDENT MEMORY MODEL (CDMM) IN MULTIMODAL AUTHENTICATION SYSTEM (20)

More from ijfcstjournal (20)

Recently uploaded (20)

PATTERN RECOGNITION USING CONTEXTDEPENDENT MEMORY MODEL (CDMM) IN MULTIMODAL AUTHENTICATION SYSTEM