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International Journal of Neural Systems, Vol. 13, No. 4 (2003) 263271c World Scientic Publishing Company

FINGERPRINT MATCHING USING RECURRENTAUTOASSOCIATIVE MEMORY

B. POORNADepartment of Computer Applications, Dr. M.G.R. Engineering College,

Maduravoyal, Chennai 602 102, India [email protected]

K. S. EASWARAKUMAR

School of Computer Science and Engineering, Anna University, Chennai 600 025, India [email protected]

Received 14 November 2002Revised 28 May 2003

Accepted 28 May 2003

An efficient method for ngerprint searching using recurrent autoassociative memory is proposed. Thisalgorithm uses recurrent autoassociative memory, which uses a connectivity matrix to nd if the patternbeing searched is already stored in the database. The advantage of this memory is that a big database isto be searched only if there is a matching pattern. Fingerprint comparison is usually based on minutiaematching, and its efficiency depends on the extraction of minutiae. This process may reduce the speed,when large amount of data is involved. So, in the proposed method, a simple approach has been adopted,wherein rst determines the closely matched ngerprint images, and then determines the minutiae of only those images for nding the more appropriate one. The gray level value of pixels along with itsneighboring ones are considered for the extraction of minutiae, which is more easier than using ridgeinformation. This approach is best suitable when database size is large.

Keywords : Minutiae extraction; pattern matching; recurrent autoassociative memory.

1. Introduction

Fingerprint classication and identication has beenaddressed by many researchers in the past. Theyare the most widely used biometric feature for auto-matic personal identication. Law enforcement agen-cies use it routinely for criminal identication. It isalso being used in several other applications such asaccess control for high security installations, creditcard usage verication and employee identication.The main reason of ngerprints as a form of identi-cation is that the ngerprint of a person is uniqueand remains invariant through age.

Several pattern recognition algorithms exist nowfor ngerprint classication, including early syntactic

approach, methods based on detection of singularpoints; and connectionist algorithms such as self-organizing feature maps and neural networks. Thesingular points, namely the core and delta points, actas registration point for comparing the ngerprints.A structure based approach using the estimatedorientation eld in a ngerprint image to classifythe ngerprint was given in Ref. 6. A topologicalapproach to detect core point was proposed inRef. 18. A Fourier transform method to reach thecore point was given in Ref. 2. A method for theautomatic detection of these points using syntactictree grammar has been reported in Ref. 16. Thesemethods rely on the accuracy of the immediate

Corresponding author.

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264 B. Poorna & K. S. Easwarakumar

neighborhood information. In syntactic approachusing stochastic grammars, probabilities associatedwith the production rules have been considered forthe ngerprint classication problem. 13 The mainstumbling block of this approach is that mecha-nisms for the inference of grammars from training

samples have not been well understood. Here, themajor drawback is the noise and the requirementof numerous matching lters. Optical techniques inngerprint classication using stochastic grammarshas been attempted in Ref. 11. The method proposedin Ref. 10 primarily deals with the description of ngerprint impressions by determining the locationof ridge endings, bifurcations and enclosures. Thematching techniques of most of the ngerprint iden-tication systems presume a high level of accuracyof singular points.

In Ref. 5, the ngerprint features are combinedusing two types of Neural networks, one to clas-sify the ngerprints and the other to train thematching networks. The network training is stronglydependent on regularization and pruning for accu-rate generalization. Some of the methods requireridge width, ridge length, ridge direction and minu-tiae direction to decide spurious minutiae. Most of the approaches use local ridge directions and a locallyadaptive threshold method. To improve ngerprintimage quality, directional ridge enhancement is used.The adaptive ow orientation based feature extrac-tion method proposed in Ref. 17, involves tremen-dous execution time. Direct optical correlations andhybrid optical neural network correlations are usedin the matching system for inked ngerprints. 5 Theimages in both binary and gray level forms are testedfor cross correlation and auto correlation sensitivity.Results are found to be strongly inuenced byplastic distortion of the nger. One of the problems

encountered by many existing systems for ngerprintmatching is that they are highly sensitive to imper-fections introduced during ngerprinting.

The Recurrent autoassociative memory is usefulfor application dealing with large sized databases.As large amount of data is to be stored, com-pact representation and efficient access mechanismis essential. A lot of work on neural network andother learning machines was stopped by the needfor adequate representation. 12 Normally sequentialand non-sequential data structures, which are used

for data representations, are simple. A number of connectionist models capable of representing datawith compositional structure have already appeared.Distributed representations have been the focusof much research, especially for a connectionistnetwork. The systematic patterns developed by

recurrent autoassociative memory are a differentkind of representation, called recursive distributedrepresentation. 15 This was used for performing holis-tic structure-sensitive computations with distributedrepresentations in Ref. 4. Also non-monotonic rea-soning is a core problem in articial intelligence.A connectionist structure with exceptions repre-sented using recurrent autoassociative memory is inRef. 3. In general, variable sized recursive data struc-tures and compositional structures use recursive dis-tributed representations. The proposed method hereis focused particularly on handling huge amount of data each time the recognition process takes place,the recurrent autoassociative memory is used.

The most important features for ngerprintmatching is accuracy and speed of retrieval. Finger-print identication is very crucial as it is used forcriminal identication and high security installa-tions. Hence, accuracy in the matching process isvery essential. Also, the applications that use thisbiometric feature for identication demands imme-diate response and so speed of retrieval is a must.Since, the size of ngerprint database is usuallyhuge, an efficient method for storage and retrieval ismandatory, which is attempted in this paper. Also,the accuracy of matching is improved by comparingthe number and type of minutiae of the searchpattern and the retrieved pattern. Some existingmethods tried to reduce the size of database by clas-sifying the data. However, this alone is not sufficientfor improving the speed of recognition.

The proposed method determines the core pointsusing the algorithm given in Ref. 9. The algorithmgiven in Ref. 9 does not depend on a particular dataset and it can be tested on the entire database. Also,smoothing for nding the singularities and classi-fying them are substantially faster, as in this methodthe given image is reduced to a size of 64 64which is relatively a small image. After determiningthe core points, the intensity of the pixels aroundthe core points are choosen for further processing.Generally, the intensity changes relatively. Due to


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Fingerprint Matching Using Recurrent Autoassociative Memory 265

this fact, our method determines the matching evenif the image under consideration is of low quality.

2. Minutiae Extraction

Minutiae are local discontinuities(ridge anomalies)in the ngerprint pattern. Fingerprint identica-tion is mainly based on the detection of the minu-tiae. There are generally four types of minutiae. 7

They are terminations, bifurcations, crossovers andundetermined. Of these only the rst two types of minutiae are considered for the identication. Theefficiency of ngerprint identication system dependson the method used for the extraction of minutiae.Most of the minutiae extraction methods transformngerprint images into binary images using someadhoc algorithms. The images obtained are subject

to a thinning algorithm.The method proposed in this paper for ngerprintmatching uses a simple process to detect the minu-tiae. The core point is the top most point of theinner most ridge, and a delta point is the triradialpoint with three ridges radiating from it, and areknown as singular points. The two singular pointsof interest are identied using already existingmethods. 9 In order to do the ngerprint matchingaccurately, the images are normalized. The normali-zation must account for translation, rotation andscaling. The singular points are good candidatesfor registration points, and for core classicationof the ngerprint pattern. An arch ngerprint doesnot contain any singular point. Tented arch , right loop and left loop contains one core and one deltapoint, whereas whorls and twin loops contain twocore and two delta points. By connecting the coreand delta point, it is possible to decide on the type of the ngerprint. This classication reduces the searchspace when matching is done. The area of the nger-prints containing the singular points and the ridgesis called a pattern area .

The method proposed here rst reduces the ridgethickness by applying a thinning algorithm. Then,a circular pattern area, with the registration pointas the centre of the circle and a particular radiusr is formed. Whenever the ngerprint classicationhas two core points as in the case of whorl and twinloop, the mid-point of those two core points are con-sidered to be the point of registration. In the case of an arch type ngerprint classication, as there are

no singular points, the following step is performed todetermine the centre of the pattern area. The archtype ngerprint has some ridges that are nearlly hori-zontal at the bottom, above which arches are formed.The local maxima of each of the arcs are considered,and if more than one local maxima exists for an arc,

then the centroid of such local maxima is consideredas the local maxima of that arc for further processing.The registration point is then determined as the cen-troid of the local maxima of all arcs exit in theimage. Now, a circular pattern area is determinedwith radius r and the center as the registration point,as shown in Fig. 1.

The circular pattern area is divided into unitrectangular cells, each of one pixel size, and the valueof r is so determined that the circular pattern areahas atleast 512 pixels. The average gray value g oall the pixels in the pattern area is determined. Thecells are assigned a value zero if its intensity is belowthis value g, and 1 otherwise.

The minutiae points are then determined asfollows. Traverse the cells of circular area left toright within top to bottom order. For each cellthat has a value of 1, the eight neighbouring cellsthat surround it are considered to form a squarearea. Based on the gray value on these cells thetype of the minutiae is decided. If only one neigh-bour of the center cell has gray value 1 thenthat cell is a terminal minutiae. Similarly, if thecenter cell under consideration has only three neigh-bouring cells with gray value 1, then that point is

Registration point Axis

Quadrants

Fig. 1. Circular pattern area.


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Fig. 2. Some bifurcation minutiae.

considered to be a bifurcation minutiae. 8 Someexample patterns are shown in Fig. 2.

Now, by traversing the given circular area, it ispossible to determine the number and the type of

minutiae. The choice of circular area is for recog-nizing ngerprints that appears in different direc-tions. Moreover, the images to be matched are atthe same scale.

3. Recurrent Autoassociative Memory

Neural networks recognize patterns that is noteven dened, as the neural network characterizesmuch intelligent behaviour. An associative memory 1

belong to a class of neural networks that learns

according to a certain recording algorithm. Thememory is able to store data in a robust manner,so that local damage to its structure does not causetotal breakdown and inability to recall. It associatesor regenerates stored pattern vectors and do so bymeans of specic similarity criteria. The memorylocations have no addresses, and storage is distri-buted over a large interconnected neurons. Anefficient associative memory can store a large set of patterns as memories.

An associative memory performs an associativemapping of an input vector x into an output vectorv, by performing the transformation

v = M [x ] .

The operator M denotes a general nonlinear matrix-type operator. For the linear associative memory, aninput pattern x is presented and mapped to the out-put by simply performing the matrix multiplicationoperation,

v = W x (1)

where the matrix W is called the connectivity matrix ,which is an n n matrix containing network weights.The algorithm allowing the computation of W iscalled the Recording or Storage algorithm. 14 Themapping as in Eq. 1 performed on a key vector iscalled a Retrieval algorithm.

Updating the output of the ith neuron is donein an asynchronous fashion. Under asynchronousoperation of the network, each element of the out-put vector is updated separately, while taking intoaccount that the most recent values for the elementsthat have already been updated and remain sta-ble. The autoassociative recall of images uses theHopeld model to store and recall a set of bitmapimages. Images are stored by calculating a corre-sponding weight matrix. Thereafter, starting froman arbitrary conguration, the memory will settleon exactly that stored image, which is nearest to thestarting conguration. Thus, given an incomplete orcorrupted version of a stored image, the network isable to recall the corresponding original image. Thememory even shows a limited degree of fault tole-rance in case of corrupted input patterns.

Dynamic memory networks exhibits dynamicevolution in the sense that they converge to an equi-libirium state according to the recursive formula

vk +1 = M [x k , v k ] .

The operator M operates at the present instant kon the present input xk , and output vk to producethe output for the next instant k + 1. The memoryis essentially a single layer feedback network with nneurons and is a discrete time network. Under theasynchronous update mode as only one neuron isallowed to compute or change state at a time andthen all output are delayed by a time produced bythe unity delay element in the feedback loop. Thissymbolic delay allows for the time-stepping of theretrieval algorithm.

3.1. Encoding

Encoding of the information is a vital factor insolving a problem using neural network. The binaryencoding is used in this problem. The type of encod-ing is purely depending on the problem to be solved.The binary string is made of six parts. The length of each string is n , where n is the number of pixels thatmake up the circular area. The rst part is meant


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for the registration point, which consists of only onecell. Therefore, the binary value of the registrationpoint is considered for the rst part. Basically, forleft loop, right loop and whorl the bit value will bezero, and for tented arch the bit value will be one.However, in case of arch and twin loop it may be 0 or

1. The given circular area is divided into four quad-rants. In the case of an arch type ngerprint, a per-pendicular line drawn from the centroid to the baseline is considered as one of the axis. For tented arch,left loop or right loop, the line joining the core anddelta point is considered as one of the axis. Whereasfor whorl and twin loop the line joining the middleof the two core points and the middle of the deltapoints is considered as one of the axis. The otheraxis in all the case is the one perpendicular to thisaxis. The bit information for the next four parts aredetermined based on the binary value exists in theclockwise order starting from the registration pointtowards the boundary of the circle correspondingto the four quadrants. The sixth part consists of the binary value of the pixels on the axis, startingfrom the positive y-axis and then in clockwise order.For each axis, cells are to be considered fromregistration point towards the boundary of the circle.Figure 3 illustrates the order of choosing cells for

1

21

31

41

42

32

22 33 44

43

(a)

chord 3

chord 1

(b)

Fig. 3. (a) Pixel ordering (b) Some chords in the

quadrant.

1 0 1 1 0 1

01010011 0

1 0 1 1 00 1010

1 0 0 1 0 1 01011011

1 0 1 11

Fig. 4. Encoding.

assigning bit values for each quarter, and that orderbe 1, 21 , 22 , 31 , 32 , . . . . In our approach, i th, 2 i 5, part corresponds to the ( i 1)th quadrant, afternormalization, which is explained in Sec. 3.2. Thus,the entire string is made up of sequence of zerosand ones.

For example, consider the pixel ordering shown

in Fig. 4. The circular pattern area consists of 45pixels. In this gure, the registration point is shadedblack. Let the value of the registration point be 0,and therefore the rst part of the correspondingstring is 0. The 2nd part of the binary string corre-sponds to the rst quadrant, and it is 00110001. Thepixels are considered here in clockwise order. Simi-larly, the 3rd, 4th and 5th parts of the binary stringrelate to the second, third and fourth quadrants,respectively. The respective substrings are 01100111,00011111 and 00010011. The nal part provide

the sequence corresponding to positive Y -axis,positive X -axis, negative Y -axis and negativeX -axis, in that order. For each axis (either positiveor negative), the sequence will be determined fromthe center point towards the boundary of the circle.Here, in our example, the 6th part of the sequence is111101110101. The entire sequence is thus

000110001011001110001111100010011111101110101

3.2. Normalization Let us assume that there are n cells in the circulararea. The value of n should atleast be 512. There-fore, the circular area is divided into four quadrants,(as shown in Fig. 1) each consists of 16 chords (seeFig. 3) as the rst one made of one pixel, the secondwith two pixels, the third with three pixels and soon. Let wt i be the weight assigned to a cell in thei th chord. The chord that is nearest to the core pointis assigned a weight wt 1 = 0 .95, followed by the nextsuccessive chords being assigned the weight starting


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0 0

1

1 0

00

1

1 0

1

0

0

0

1

(a)

.85 .8

.85

.9 .85

.8.9

.95

.8 .75

.75

.75

.8

.7

.7

(b)

Fig. 5. Weight determination.

from 0.9, and reducing the weight by 0.05 for eachchord. Here the value of wt 16 is 0.2. The rst valueis chosen randomly, which may be between 0 and 1.

The weight of each quadrant is determined as fol-lows. Let, Q i be the sum of the weight of chordsstarting from the one that is nearest to the core

point of quadrant i, then Q i = 16j =1 jk =1 (wt j ),for 1 i 4. For example, consider the pixelordering shown in Fig. 5. The quadrant here con-sists of ve arcs. The pixel values and weightsare respectively shown in Fig. 5(a) and Fig. 5(b).Thus, the sequence corresponding to this quadrantis 101010100001001, and the weight of this quadrantis (10.95+(0+1) 0.9+(0+1+0) 0.85+(1+0+0+0)0.8+(0+1+0) 0.75+(0+1) 0.7), which is equalto 4.95. After nding the values of Q i , 1 i 4, asequence of Q i s is formed as (Q 1)(Q 2 )(Q 3)(Q 4). Forexample if Q1 = 10, Q2 = 5, Q3 = 12 and Q4 = 9,then the initial sequence would be (10)(5)(12)(9).

The weight of the sequence is dened as concate-nating the values of the elements in the sequence, inthat order, with respect to the base as the maximumvalue exists in the sequence. For instance, supposeQ 3 is maximum among {Q i |1 i 4}, then theweight of the sequence is ( Q 1Q 2Q 3Q 4)Q 3 . The nor-malized sequence is the one having minimum weight.This is achieved by performing sequence of circularleft shifting on the values of elements in the sequence.

After each circular left shifting, the weight of thesequence will be determined. This gives four differentweights, after three shifting. Now, we select the mini-mal weight sequence as the normalized one. Whenmore than one sequence gives the minimal weight,any one can be chosen arbitrarily as normalized one.

Let, i be the number of shifting required to get thenormalized one, then ( i + 1)th quadrant becomesthe rst quadrant for the encoding process. In ourexample, one circular left shift leads to the sequence(5)(12)(9)(10). Also, the minimum weight sequenceis this particular one, which is obtained after onecircular left shift. Then, the quadrant correspondingto Q2 is now considered as the rst quadrant, fol-lowed by other three quadrants in clockwise orderfor encoding.

3.3. Algorithm

Recurrent networks are neural networks with one ormore feedback loops. There are two functional uses of recurrent networks, one associative memory and theother is input-output mapping. Biological memoryoperates according to the associative memory prin-ciples. Associative memory enables a parallel searchwithin a stored datale. An associative memory hasthe ability to retrieve a stored pattern, given a rea-sonable subset of the information content of that

pattern. The connectivity matrix, W , is used to per-form the associative mapping. The neural system issaid to have learned the association, when given theinput x, it identies the output v. Here, W is theouter product matrix, which is the generalization of Hebbs postulate of learning. 1 Suppose, each pairof association generates an associative matrix, thenthe overall connectivity matrix is the sum of all thematrices of the individual associations. Thus, theconnectivity matrix becomes

W =

p

m =1x (m ) x (m ) t pI

where p is the number of bipolar vectors stored, and Iis an identity matrix. The system here does not needthe individual vectors, however only the weights. Asthe Hebbs learning does not involve the presence of negative synaptic weight values, only bipolar vectorsare allowed for building the auto correlation matrix.

First, convert the normalized binary vectors B ( j )

into bipolar form using the formula x ( j )i = 2b( j )i 1

1 i n and 1 j p, where n is the number


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of pixels used for matching, and p is the number of bipolar vectors stored. In an autoassociative mem-ory, each x( j ) (when 1 j p), converts to v( j ) ,based on mapping equation given in 1. Here, vectorx ( j ) is the stored data, and the bipolar vector to bematched serves as a search argument. The following

algorithm is used to store the bipolar vectors x( j )

,1 j p, in the memory.

Algorithm: 1 [Storage Algorithm]

Input: p Bipolar vectors , x(1) , x (2) ,...,x ( p) .where x (m ) is (n 1), for 1 m p.

BEGIN

1. Initialize W as zero matrix and m as 1.2. Calculate W = W + x (m ) x(m ) t I , where I

is the identity matrix.

3. If (m < p ) then m = m + 1 and goto step 2.4. Store W.

END.

The associative memory used here is contentaddressable. Initializing v as a (a is the pattern whosematch is searched), the elements vi , 1 i n , of the given vector v can be calculated by using thediscrete-time recurrent network update rule

vk +1i

= sgnn

j =1

wij

vkj

where k denotes the index of recursive update, andi is the neuron number currently undergoing theupdate asynchronously. Here, the functions sgn() isapplied to each element of the vector. The functionsgn() returns either 1 or 1 depends on sgn( w ij vkj )is positive or not. Here, each neuron can updateits values simultaneously. Note that n neurons arerequired, as the number of pixels used for matchingis n . The method for retrieval process is as follows.

Algorithm: 2 [Retrieval Algorithm]

Input: (i) The bipolar vector a of the pattern to be matched

(ii) The connectivity matrix W .

BEGIN

1. Initialize k and i as 1, where k is the cycle counter , and i is the update counter .

2. Initialize v as a. (Note that a and v are thevectors of size n 1)

3. Update neuron i by computing vnew as

neti =n

j =1

wij vj

vnew = sgn (neti )

(Note that vnew is the vector of size n 1).

4. If (i n ) then update i = i + 1 , goto step 3.

5. If vnew = athen Output vnew and STOP .

6. If v = vnewthen Display Match not found and STOP .

7. Update k = k + 1 , v = vnew, goto step 3.

END.

The search argument a is a bipolar vector ofthe ngerprint whose match is required. Using thissearch argument, the closest stored vector is obtainedusing the retrieval algorithm. The network has theability to converge to the desired output, when acorrupted pattern is given.

The overall process of our ngerprint matching isbriefed through following steps.

1. Calculate the connectivity matrix W , asstated in the storage algorithm.

2. Store the input vectors in a separate databasecalled DB .

3. For the bipolar vector a , determine the vector(vnew) that matches (exactly or closely) ausing the retrieval algorithm.

4. Using the vector vnew, search DB to get allclosely matched stored vectors, any of whichlikely to match the pattern being searched.The closely matched patterns are determinedusing Hamming distance, which we call asan allowable error rate. The vector gives zeroHamming distance is the exactly matched one.

5. Now, compare the number and types of minu-tiae of the images, retrieved in the previousstep, with the number and types of minutiaeof the search pattern, for more accuracy. Thiscan be done only after converting the bipolarvectors to the respective binary vectors.

The Hamming distance is dened as an integerequal to the number of bit positions differing betweentwo binary vectors of same length. For two n-tuple


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binary vectors x and y, the Hamming distanceHD (x, y ) is

12

n

i =1

abs (x i yi ) .

Advantages

Some advantages of the proposed methods aredue to

1. The database is to be searched, vector byvector, only if there is atleast one closelymatched image exists in it. This is benecialwhen a large sized database is used.

2. The process of comparing the stored vectors ismuch more easier than extracting the minutiaeand then comparing.

3. Minutiae is determined only for closely relatedmatches, but not on the entire databaseimages. Also, this process is required only if exact match is not found.

4. The number of closely matched pattern isobviously far lesser than the number of patterns in the database. Thus, minutiaeshould be determined only for a limited setof ngerprints.

4. Results and Conclusion

The algorithm stated in Sec. 3.3 was implemented us-ing MATLAB, a technical computing language. Theimage processing toolbox of MATLAB is used forreading, thinning and rotation of the ngerprints.A database of 1000 ngerprints is used for testing.The test set is a mix of real and articial ngerprints.60% of the ngerprints were created using the syn-thetic ngerprint generator. Some of the ngerprintsgenerated are of different intensities and transforma-tions of the original one. The input gray scale imageis converted to binary using threshold value, say g.The value of g is specied within the range [0 , 1]. Theconversion is carried out as follows. When the lumi-nance of the pixel is less than the threshold value, itsvalue will be treated as 0. On the otherhand, it willbe 1. It is also observed that for a normal image withless noise, the threshold value within the range of [0.5, 0.7] yields correct result. Usually, the intensityof the image varies almost uniformly. So, an imagewith less quality must be used with small thresh-old value. Similarly, a dark picture is to be usedwith high threshold value. Therefore, we can modify

Table 1. Observation.

Intensity Thresholdvariation (%) range

020 0.80.92040 0.70.84060 0.50.76080 0.30.580100 0.10.3

the threshold value suitably by either increasingor decreasing, depending on quality of image. Theobservations that are made on threshold value, whilethere is change in intensity, is given in Table 1.

It is worth to note that the recurrent autoas-sociative memory neural network is found suitablefor matching, even if their intensities varies. Usu-ally, the quality of the image varies uniformly andrelatively on the gray scale values. Therefore, ouralgorithm is best suitable for carrying out detectionwith low quality images. Moreover, it is known thatthe performance and correctness depends on minu-tiae detection. As the minutiae extraction dependsonly on a particular unit cell along with eight neigh-bouring cells, the approach adopted in this paperis more effective, as this approach is not based ondirectional maps. Due to normalization, the algo-rithm determines the images even if it is rotated.The practical importance of recurrent autoassocia-tive memory is for large size networks. The networkis able to produce correct stored states when an in-complete or noisy version of the stored vector is ap-plied as the input. The update rule can reconstruct anoise corrupted or incomplete pattern. The approachadopted in this paper is more effective, as the sys-tem need not remember the individual vectors, butonly the weights. Hence, the algorithm works withthe same accuracy when applied to real world prob-lems of large data sets. This, denitely makes thealgorithm more effective than other methods. Theaccuracy of the algorithm is improved by the processof minutiae extraction when closely matching pat-terns are retrieved, that too when exact matching isnot found. The algorithm works substantially better,irrespective of intensities or of transformations.

Acknowledgment

We thank the referees for their valuable suggestionsfor the improvement of the quality of presentation.


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