Sensors 2010, 10, 4825-4837; doi:10 3390/s100504825

O PEN ACCESS

sensors

ISSN 1424-8220 www m dpicom jurnal/sensors

Articé

GPCA vs. PCA in Recognition and 3-D Localization of Ultrasound Reflectors

Carlos A . Luna *,José A . Jiménez, Daniel Pizarro, Cristina Losada and José M . Rodriguez

Electronics Deparment,High Polytechnic School, Alcalá University,Alcalá de Henares, M adrid, Spain; E-M ails:: jimenez@ depecauahjes (TA J); pjzarro@ depecauahes (D PP.); cristina® depecauahes (C L.); jmra@ depecajuahes (TM R .)

* Autthortowhom correspondence should be addressed; E-M ail: caluna® depecauahjes; Tel: +34-918-856-551; Fax: +34-91-885-6591.

Received: 25 February 2010; in revised form : 20 April 2010 /Accepted: 7 M ay 2010 / Published: 11 M ay 2010

Abstract: in this paper, a new m etthod of classification and localization of reflectors, using the tme-of-flight (TOF) data obtained fiom ultrasonic transducers, is presented. The method of classification and localization is based on Generalized Principal Component Analysis (GPCA) applied to the TOF values obtained from a sensor that contains four ultrasound em iters and 16 receivers. Since PCA works with vectorized representations of TOF, it does not take into account the gpatial locality of receivers. The GPCA works with two-dim ensional representations of TO F, taking into account inform aton on the spatial position of the receivers. This report. includes a detailed description of the m ethod of classification and localization and the results of achieved testswith three types of reflectors in 3-D environments: planes, edges, and comers. The results in term s of processing tim e, classification and localization were very satisfactory for the reflectors located in the range of 50-350 cm .

K eyw ords: principal com ponent analysis PCA ) ; generalized principal com ponent analysis (GPCA) ; reflector classification ; tim es-f-flight TO Fs) ; ultrasonic sensois

1. Introduction

The classification and localization of reflectors constitutes a fundam entaltask in the field of m obile robotics, since this inform atincontrbutes ina decisive way to other higher level tasks, such as the generation of environment maps and the robot's localization. With respect to the process of reflector classification, the techniques m ore broadly used are basad on geom etric considerations obtained from the TOFs for every reflector type [12]. An im portant inconvenience of the system s based on geom etric considerations is their high dependence on the precision with which the m easurem ents of the TOFs are carried out, and consequently, the classification results are strongly influenced by noise.

Principal Com ponentAnalysis (PCA) has been used to reduce the dim ension of data sets and object recognitinindifiPTentworks related to image processing [3-6]. The classiificatinandlccalizatbn technique for 3-D reflectors based on PCA using an ultrasonic sensor is explicitly discussed in [7], in which it is applied to 18 TO F values originated from a sensor that contains two em itter/receiver transducers and 12 receivers, (see Figure 1). The pulses em itted by E0/R0 are processed by itelf (Eo/R 0) and by transducers E1/R1; R2, R3; R4; R5; R6; R7; and R8. The pulses earn itted by E1/R1 are processed by transducers E1/R1, Eo/R o, R5,R7, R9,R10,R11,R12/ and R13. In [7-11] to reduce the number of transducers, the sim ultaneous em issdon of com plem entary sequences by tw o or m ore em iters is proposed.

Figure 1. View of sensor structure proposed in [7].

The PCA is applicable only for data in vectorized representation. Therefore, the data obtained from a m atrix sensor m ust have been previously converted to a vector form . A typical way to do this is the ^-called "matrix to vector alignment", which consists of concatenating all the rows in the matrix together to get a single vector. Figure 1 Shows two matrix sensors of nine transducers. The TOFs t_2, t_6,t)_3,t_8, t_o/te_5,t0_4, t0_7, t0_1 obtained from reception of sequence em itted by E0/R0 and the TOFs t_o / t1_5 / t_7 / tt_9, ti_10, ti_11, ti_12, ti_13 obtained from reception of sequence em itted by E1/R1 are aligned to get a single vector (1) . R eceiver R 2 andR 8 are neighbours inthe sensor, w hile they are far aw ay ftom each other in the vectorized representation. The same observations hold for positions R0

and R6, etc.Due to the vectoralignmentthe spatial information ismissed. Also it can be remarked that the original 3 x 3 m atrixes are converted to an 18 x 18 scatterm atrix in PCA , which leads to higher tm e and m em ory space costs;:

^ ~ [t0 2 t0 6 t0 3 t0 8 t0 0 t0 5 t0 4 t0 7

t1 0 t1 5 t"! 10 t1

From the em iite E0 t-X 1 t1 12

t1 11 t1 13 t1 9]

Fromtheem itterE.

in [8], the sensorial structure is fcrm edbyfour transducers which can smultaneouslyobtain16 TO F values at every scanning process. These 16 TO F values are aligned in a vector that is usad in the classification algorithm based on PCA .

In [12] the generalized pca (GPCA) algorithm is used for image compression.. The GPCA algorithm deals with the data in its original m atrix representation and considers the projection onto a space, which isthe tensorproductof two vectorspaces. In thispaper, we use the GPCA fiorrecognition and localization of ultrasound reflectors which aim to overcom e the drawbacks in the traditional PCA . W e used a m atrix of 16 transducers (using four of the centre sensors as em ¿teis/receivers and the other 12 as receivers). Thus, we can obtain up to four 4 x 4 matrices, each of which perform s a classification procedure independently. The results of the four cJassifkatbnprxesses m erge to give the final result.

The restof this paper is organized as follow s. Sect;bn2 illustrates the proposed sensor structure. Section 3 describes the GPCA algorithm . Experimental results are presented in Section 4. And, conclusions are offered in Section 5.

2. Sensor Structure

The sensor structure used to extract inform ation in a 3-D environm ent and to subsequently classify and locate ultrasonic reflectors with the obtained data (TOF sets) from echoes isshown in Figure 2.

Figure 2. Sensor structure proposed in this paper.

dh 13) (£4)

<R5) EcR6 E4R7 <e)

<°) tof lo)"

© <RD ©

The physical structure of the ultrasonic ssnEor consists of 16 transduce]:s, four of them working as emittersand reœivers E1/R6 , E4/R7 , E3/R10 and E2/R11) and the othersonJy working as reœivers. AU the tr^^uc^rs are located in a plane plane xz), witth the axial axis in the y-direction. Furtherm ore, the œpaaation between ttan^^uc^isdet^imined by the distance a (a = 017 m).

W ith the proposed œnscr structure it is po^ible tto obtain up to 64 TOFs atevery em i^don^/scann^ing process. To do this, it is necessary to assign a différent m acrosaquence to every transducer to encode their em irions. These m acrc^sequences, obtained from com plem entary sets of sequences, allow sm uHtaneous em i^aonandr^ep^tionto be carried out w ithalL the tran^Juc^ for the saim e ^anned environm ent [8,11].

3. G PCA Algorithm .

In this paper, the usage of G PC A is pioposed to cary out the reflector classification using the m easurem ents of 16, 32 and 64 TO Fs provided by the uLtrasonic sensors. This m ethod m aintains the gpatial distribution of sensordata, which is represented by the TOFs in a m atrx form at (2) :

t t t t

t t t t

t tl0 tl t2

t3 tl4 tl5 ts

In 2), t.... t16 are the TOFs asEo^da^teEl to each receiver and te e lRr<c is the TOF matrix obtainedfor each em itt^r (r and c are the num ber of row s andcolum ns of the sensor, ie^spectiivelyr).

■ ■ ■E1E2E3E4 ■

Therefore, in ourcase we can obtaon up to fburmatrœs (t,, ,Te ,t e and ^ for the em itters E1, E2, E3 and E4).

InGPCA we compote an optimal 11, ^-dinensbnal gpace, such thatthe projections of the data points (subtracted by the m ean) onto this axis system have the m axim um variance am ong alll possible 0l, 1)-dmensiona1 axes sysems. Unie PCA, the projections of the data points onto the 11, l)-dinensionalaxis^stem in GPCA are matrice, instead ofvэctcls.

Let consider S = (t0, zi,t2, ...Tn-1} be a training œt of n sampfes of TOF matrix. The mean TOF m atrix of the set is defined by :

= 1 Z - i

M atrices with m ean zero are represented as:

= Ti - T (4)

Then the variance of the projections of Oionto the 1, 1) dimensional axis^stem isdefined as:

varL ,R ) = — ^ LT ^ R

where | | |F is the Frobeniusnorm andL e IR 1X1 andR e IR^1 are two matrices with orthonormal columns, such that the variance var(L R) ism aximum . The m aximum value of (5) cannot be found in closed form and thus an iterative approach is needed:

♦ For a given R , the optimalmatrix L consists of the l eigenvectors of the matrix M L which correspond to the largest l eigenvalues, where:

M L =X ^iRRT ^iT (6)

i— 0

♦ In the sam e w ay, given L, the m atrix R consists of the l eigenvectors of the m atrix M R, corresponding to the largest l eigenvalues, where:

Mr °iTLLT ^i 7)

i— 0

In the realized experiments we u^d i = l = 2 (L e IR4x2 and R e IR4x2) and l = l = 1 (L e IR4x1 and R e ]R4x1).

To calculate the L and R m aSies that m axim ize Equation 5, it is necessary to initially fix one of them . Fixing L, we can calculate R by com puting the eigenvectors of the m atrix M r , and then., with the calculated R we can then update L by com puting the eigenvectors of the m atrix M l . It is necessary to repeat the procedure until the result converges. The solution depends on the initial choice of L (L0). As itis rrecommendedin [12], we use L0 = (I, 0)T, where I is the idenStym atrix. To measure the convergence of the GPCA procedure we use the rootm ean square error Q, defined as follow s:

/1 n~1

£ = J-Z p i - llT ° iRRT

V nl-011

In the realized experim ente for<f = 108 the procedure converges within five iterations. O nce the transform ationm atrices L and R are deteim inedandgivena new TO F m atrix t± tobe classified. its zero-m ean version Oi is transform ed into the future ^ace as:

= LT 0±R 9)

Then we can reconstructOi as:

®± « Ln±RT (10)

The reconstruction error (?R) for O ican be com puted as:

£R = i - <J>J| = ||Oi - LLT® iRR^ (11)

31. O ffline Generation of the Classes

The objective of this work is to clarify one of the Sues reflector types (plane, corner, and edge) and its approximate direction azimuth angle y, elevation angle d) and distance (r) with respect So the frontal ^ace of the sensor. Before beginning the classification process, is is necessary So create different classes, depending on the reflector type and its gpatal location. Every class has two

■ ■ ■ ■ ■ P P

transformation matures L, R associated to it These matrces are referred as L and R for the plane,

C C EE

L and R for the comer, and L and R for the edge.

As it is stated in [781, we assume that the frontal space of the sensorial structure is formed byQ directions defined by 0q),wih q e {1, 2,.., Q } .Along every direction q, there are D discrete distances referred to as k, with d e {1, 2,.., D}. To generate the tran^sfbrmaton matrices associated to every direction class and every reflector type, the reflectors {P,CE} have been located at every direction q and for all the d distances, obtaining the TO F vectors. In G PCA, to generate the transform atonm atrces associated w ith everydirection and everyrefector type (LPq, R p , LCq, R q , LEq, rE ), the reflectors {P, C, E} have been lxatedateverydirection q and forallthe D distances. W hen more than one transduce is fitting, we use a matrix of TOF for each em iter', ie.:

E1 e 2 E 3 E4 ■

T e ,x e ,T e and T e for the em iters E1, E2, E3andE4. Therefore, itds necessary to compute the

■ ■ ■ ■ ■ P P c C E

transform aton m atrces associated to every dllececn and every reflector type (Lq, Rq, Lq, Rq, Lq, R q ) foreach em itte^r'.

3 2. Classifi:ation and Position Estrnaton of the Reflector

The strategy proposed in this paper t carry out the online classification process is to first classify the type and approxim ate direction in which the reflector is located and then to estim ate its distance with respect to the frontal space of the sensor.

To classify the type and approximate direction of the reflector, we calculate the square reconstructioner-rpr fc), using the transform ationm atrices associated to everydirection q and every reflector type.

)= oP - LPL V q / || q q

(-.E )= ^ ) =

_ TOPRPRPT

q q q q q

O E — LeLe O ERERE

qq C - C '

O C — LcLct O CRCRr,T

q q q q q q

2 || 2

P T O PR P q q q

LEqT ^ ERE

C T o CR C

The reflector is classified as a plane if £,Cmih > £qm±1 < , it is classified as an edge if

C e P . . . . , p C E ■

^qmin > ^qmn < ^qjmih and ± is classafed as a comer if > s^n < s^n . The rabe f q, whxh corresponds to the m inim um value ofsq determ ines the approxim ate direction of the reflector. W hen more than one em iter is being used, we determ ine the m inimum value of reconstruction error in each direction foreach em iter, and added to all these m inim um values. The m inim um values are taken with the sam e angle for all transm iter, otherwise the results w ill not olasErffyocrleotly. For four em iters that is:

qmin E

qmin C

fcp.)E1+fcp.)E2+frp.)E3 \ qmin/ \ qmin/ \ qmin/ \ qmin/

(^E.)E1 +tE.)E2 +tE.)E3 +(^E,)E

\ qmin/ \ qmin/ \ qmin/ \ qmin/

fcc.f + tc.)E2 +tc.)E3 +(^C,)E

\ qmin/ \ qmin/ \ qmin/ \ qmin/

Once the type of reflector and the diection inwhich it is posatbned are known, the approxim ate distance with respectto the sensor structure can be determ ined. The Frobenius norm in the transform ed space betw een the feature vector re for the object to be classified , and everyitieature vector of the training sam pites of the class to whih tthis reflector belongs, are calcuilated. For exam ple, if the object was classifid as a plane in the diection q, the TOF vector set used offline to generante the transfiormation matrix will have been {tP ,tp2 ,tp3 ,..,tpD } . Therefore, it is only necessary to com pute the Frobenius norm in the transform ed space am ong the featme vector corresponding to the TO F vectorTe, and the feature vectors conespond^ing to the training sam ptes, as is show n in:

^ qd. ^ q ' d _ 1' 2, •

The value corresponding to d, that provides the m inim um ed, w iH be the approxim ate reflector distance, in the direction q.

It has been proven em pirCaHy that the relationship am ong the distances of the reflectors to the sensor structure, and the Frobenius norm of their feature vectors in the transformed space, is approximaty linear::. In this way, considering the distance interval, where the reflector is, and the Frobenius norm inthe transform ed space, a correct estim attoncan be obtainedby m eans of a linear interpolation of the distance atwhich the reftector is positioned..

33. Processing Time Using PCA and GPCA.

To compare the processing time of the GPCA classffication method with the PCA method, we analyze the num ber of mulsplicaSbn operations required So classify the type of the reflector from the reconstruction error. To do this, we consider a generic sensor as proposed in Figure 2 with r row s and c columns. With TOF's obtained a r x c malrix and a vector of dimension rc are built for GPCA and PCA methods respectively. It's also considered that a number l of eigenvectors are ^Hected and that the sensor has m em iters.

In PCA , the reconstruction error is given by the follow ing expression:

Œ - <I>

Œ - LLT Œ

where L is the transform ation m atrix and O is the m easurements column vector with mean zero. The dimensions of the vector O and matrix L are rcm 0 <= ^tcm)) and (rcm) x l (L e ^(rcm)xl) respectively. W e can obtain the num ber of m ultplication operations required broking expression 13) down in different term s 14) and 15) :

LT® = E (14)

LE = F (15)

Therefore, taking the dim ensions of vector O and m atrX L into account, to calculate the term s E = LtO and F = LE l.(rcm) mi iltiplicationis are needed. To bbtainthe Euclidean distance, rem mulltplcaitions are needed. So, the tctal number of mulltplcaitions required to classify the type of the reflectorushg PCA is!:

n°mulilcatrns (PCA)=l- (r-c-m) +1- (r-c-m)+ (r-c-m) (16)

In GPCA the reconstruction error is given by.

e=\ 1^1 2f - |lt or| 2 (17)

where L and R are transform ation matrices andO is the measurementsmatrX: with m ean zero. In this case, the dim ensions of m atrices O, L and R are r x c ^ e ^rxc) ,rx 4 L e ^^) and c x l R e ^1x1 respectively. Fb1Lcwihg the same methodthae inPCA, we bbtainthe number of multiplicationsbroking expression (17) down in term s.

H = ^R n°muleзLJceons > (r- c- Lf ) m (18)

G = lth nOmu1tiэLi0fens > (l- r- lF)m (19)

|g||F n°mu1epLioebns > (1. Lf).m 20)

ll^lF n°mu1eiPLjoebns > r c). m F1)

Then, the total number of mulltplcaitions required to classify the type of the reflector using GPCA is:

n°multp^lcatonB GPCA)=[(r-c-ll)+ lrD l +1)+ (r-c)

The number of m ultiplications for both methods particularized for r = 3, c = 3 and using г^iffprene numbers of emitters m) andeigenvalues l, i, ]2) are showninTable 1. One can see that the total num berof m ultiplications losing GPCA is slightly low er than using PCA . However, its m ain advantage is that because the C1assificaeon is perform ed independently with the TO FS obtained for each em itler and added to the obtained values, it is possible to m ake a parallel ization of the calculations when using a mulep)rcoesscr system . That is, for two em iters m = F) using a dual processor system , the processing tim e can be reduced by more than half using GPCA (160 using PCA and 68 using GPCA).

Table 1. Number of mulSplicaSons for PCA and GPCA , using different numbers of em itters and eigenvalues.

S = O - LL10

O G ^(r'c>m } L €= ^rom ><l

l-(r- c- m )+ (r- c- m )• l+ (r- c- m ) 64 + 64 +32 = 160

128 + 128 + 64 = 320

32 + 32 + 32 = 96

64 + 64 + 64 = 182

£ = \\o\\2 - |lt #r||f

O €= ^ rXc L e^1 R e^cxl2 (r- c1 +1 • r 1 + l1 • l2 + rc)"m 32 +16 + 4 + 16 = 68 x 2

32 +16 + 4 + 16 = 68 x 4

16 + 4 + 1 +16 = 37 x 2

16 + 4 + 1 +16 = 37 x 4

ReconsSucSon error. Dim ensims of m atrices.

Num berof m utplicatixL operations.

Number of mufcplicaSixL operations fori = l = l = 2 and m = 2. Number of mutplicatixL opersScms fori = l = l = 2 and m = 4. Number of mu opersScms

fori = 11 = 11 = 1 and m = 2. Number of mufcplicaSixL opersScms fori = 11 = 11 = 1 and m = 4.

4. Sim ulatin Results

A sim ulator has been used in order So carry out the sim u1ations; Shis allow s TOFs So be obtained in three-dim ensional environm ents( basэdomShe sensor m odel proposed byB arshan andK uc [13] and using She rays technique [14]. The system em ploys a frequency of 50 Khz. To this frequency we can suppose a specular m odel. This sim ulaSor is She sam e as that used in [7] and is is validated with real m easurem ems. The m easures used So com pare She two m ethods of classifica^lio^, PCA and G PCA , have been obtained under the sam e conditions.

To evaluate and com pare the GPCA c1assrfiratbn m ethod with the PCA m ethod, we generate TO Fs sim u1ating She sensor structure of Figure 2. To obtain the transform atson m atrices, the reflectors have been located as distances from 50 So 350 cm , with 30-cm intervals. The azim uth and elevation angles were from -12° So +12°, with intervalsof 2°.

In the simulations carried outs, we analyze the percentage of successful classificationis, using different- vai ies for the num berof em itters (m) and different- vaiies of the num berof used eigenvectors 1 = 11 = 11) corresponding So the largest eigenvalues. In all the cases, the results are obtained adding So the TOFs a zero mean, independent and identically distributed (ii.d.) Gaussian note with typical deviation of 15 ius. The distance forplane, edge, and corner-type reflectors placed atdistances from 50 So 350 cm , wifhintervaJs of 20 cm , and an azimuth angle of 7 5°. For each type of reflector aSeach distance 500 tests were conducted. W e also perform ed sim u1ations for different azim uth and elevation angles and She results are sim ilar So the ones ±low ed in this section.

]nFigu!e 3 the classil±at:bnpercentage for 1=2 andm = 2, using PCA andG PC A m ethiods, is shown. In this figule we can observe that the percentage of successful classifications using tthe PCA a1gbriihlm is greater than using tthe GPCA for distances greater than 200 cm . This is due to, the GPCA a1gbriihm need a am bunt of input data greater than PCA for a appropriate classification. In both cases we obtained a 100% successrate fbrdistances below 200 cm .

Figure 3. Percentage bf successful classifications using GPCA and PCA , fbrl = l = l = 2 and m = 2.

100* 90 -

I 80 "

E 70 -

- PCA Plane

—e- - PCA Corner

A - PCA Edge

- GPCA Plane

—• - GPCA Corner

-B— - GPCA Edge

| 20-Q.

0-1-1-1-1-1-

50 100 150 200 250 300 350

Distil nee (cm)

If we m ainttain the sam e num ber bf em itt^rs and use only the eigenvector corle£pbnding to the largest eigenvalue, the percentage bf success is greater than 95% up to 300 cm and then deceases very shaiply, as show n in Figure 4.

Figure 4. Percentage bf successful classifications using GPCA and PCA , fbrl = l = l = 1 and m = 2.

The sim ulatons carry out w ith other values of noise and a single eigenvector have show n that, using PCA and GPCA the percentages successful clarifications of comers and planes fall sharply for distances greaterthan 330 cm . This is due to the loss of dim ensionality in the transform ed space.

If one wants to increase the percentage of hits in the classification, we can increase the num ber of em iters. In Figure 5 the results obtained for m = 4 are shown. In this figure we can see that the percentage of hitsisover98% up to 350 cm . In thiscase the processing time forPCA is approximately twice thatwith two transm iters.However, using GPCA we can get a tme sim ilarto thatobtained fora single em iter, if we use parallel processing.

Figure 5. Percentage of successful classifications using GPCA and PCA , for l = l = l and m = 4.

In applications that do not require classifying objects at distances greaterthan 290 cm , you can use a single eigenvector to obtain 100% success. In Figure 6 the results obtained for m = 4 and l = 1 are shown. In this figure we can see that the percentage of success is 100% up to 290 cm and then decreases very sharply.

The G PC A approach better organizes the data inthe sense of adjacent com ponents inthe m atrix representation correspond to physically adjacent readings in the ^nsor array. This allow s, with m aller transformation matrixes to achieve sim ilar results to those obtained with PCA and the computational cost is low er.

The results obtained in estimating the distance are similar using PCA and GPCA methods. However, using GPCA has the advantage of a low ercom putational cost The com putational costin the estm ation of distance can be analyzed in the sam e way as is analyzed in the classification process.

Although all the results presented com e from sim ulations done with non-correlated noise, tests with correlated noise have been carried cut. It has been verified that the proposed algorithm is very robust against that kind of note. In these tests, the classification is successful even when the correlated note has higher standard deviations than the non-correlated noise.

Figure 6. Percentage of successful classifications using GPCA and PCA , fori = l = l = 1 and m = 4.

5. Conclusions

Inthis paper, a classifratinalgorithm based ontthe G PCA techniques have been proposed, w itth which three types of basic 3-D reflectors can be classified: planes, comers, and edges. The excellent behaviour of the GPCA proposed clarification algorithm for a wide range of distances between reflectors and sensor ^stem s has been dem onstrated bym eans of sim ulations, under extrem e noise conditions regarding measurements. Experimentsshow sim ilarperformance between GPCA and PCA , in term s of successful clarification percentage. However, GPCA uses transform ation m atrices that are m uch sm aller than PCA . This significantly reduces the space to store the transform ation m atrices and reduces the computational time inthe clarification procedure. Also, as the procering of the data is obtained for each em itter it is processed independently and processing tim e can be significantly reduced using parallel procering.

Acknow ledgem ents

This workhas been supported bytthe Spanish M inistryof Science and Innovationunder projects VISNU ref. TIN 2009-08984) and SDTEAM -UAH fcef. TIN2008-06856-C05-05).

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