Image Processing on Tensor-Product Basis

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1 Óbuda Uversty e Bullet Vol. 2 o. 20 mage Processg o Tesor-Product Bass Adrás Rövd László Szedl 2 mre J. Rudas Péter Várla 2 Óbuda Uversty Joh vo euma Faculty of formatcs Bécs út 96/b 04 Budapest Hugary rovd.adras@.u-obuda.hu rudas@u-obuda.hu 2 Széchey stvá Uversty System Theory Laboratory Egyetem tér 9026 Győr Hugary szedl@sze.hu varla@sze.hu Abstract: The paper troduces a tesor-product based represetato of dgtal mages ad shows how ther processg ca be performed. The mage fucto ths case s expressed by oe-varable smooth fuctos formg a orthoormal bass whch are specfc to the expressed fucto ad therefore less umber of compoets s eeded to acheve the same approxmato accuracy tha case of trgoometrc fuctos or orthoormal polyomals. t wll also be show how these oe-varable fuctos ca be determed usg the hgher order sgular value decomposto (HOSVD). The proposed techques wor well eve cases whe besde the color further attrbutes are assged to the pxels e.g. temperature varous type of labels etc. Fally the results are compared to the techques worg the frequecy doma. t ca be observed that may cases the usage of the proposed doma s more advatageous. Keywords: scalg; smoothg; compresso; tesor-product; approxmato; HOSVD troducto There are umerous dgtal mage processg tass e.g. mage smoothg edge detecto etc. whch effcecy strogly depeds o the doma they are worg o e.g. mage resoluto ehacemet flterg [] mage compresso [2] etc. The two-dmesoal array of pxels s the most atural way to represet dscrete mages applcable for example for hstogram modfcato pxel ad eghbor operatos etc. [0]. O the other had some applcatos the data are actually collected the frequecy doma specfcally the form of Fourer coeffcets e.g. MR or CT magg ad spectral methods for PDEs []. Aother type of represetato s based o Wavelet trasforms whch are mult-resoluto represetatos of sgals ad mages decomposg them to mult-scale detals [2]. eural etwor based represetato techques stad for a further group of mage represetatos e.g o-lear mage represetatos based o pyramdal 247

2 A. Rövd et al. mage Processg o Tesor Product Bass decomposto wth eural etwor [] etc. The ma am of ths paper s to propose a tesor-product based represetato doma whch the mage ca be expressed by less umber of compoets tha for example the frequecy based represetato requres by supportg the effcet multdmesoal flterg compresso ad rescalg. There are umerous mage processg tass whch ca be performed more effcetly whe swtchg to aother doma e.g. the well ow frequecy doma the mage flterg or compresso. O the other had to represet the mage frequecy doma wthout meagful qualty decle relatvely large umber of compoets s eeded cotrast to the proposed tesor-product based represetato where the umber of compoets to acheve the same approxmato accuracy tha case of trgoometrc oes s much more less. As show the upcomg sectos ay -varable smooth fucto ca be expressed wth the help of a system of orthoormal oe-varable smooth fuctos o hgher order sgular value decomposto (HOSVD) bass. The ma am of the paper s to umercally recostruct these specally determed oe-varable fuctos usg the HOSVD ad to show how ths approach ca gve support for certa mage processg tass ad problems. The paper s orgazed as follows: Secto 2 deals wth the recostructo of the oe-varable fuctos detal Secto shows how ths represetato ca be appled mage processg for resoluto ehacemet ad flterg whle Secto 4 the propertes of the proposed method are compared to the well ow Fourer trasformato. Fally Secto 5 expermetal results ad coclusos are reported. 2 The HOSVD-based Represetato of Fuctos The approxmato methods of mathematcs are wdely used theory ad practce for several problems. f we cosder a -varable smooth fucto [ ] T f ( x) x= ( x... x ) x a b the we ca approxmate the fucto f ( x) wth a seres f ( x ) =... α p ( x )... p ( x ). ()... = = where the system of orthoormal fuctos p ( x ) ca be chose classcal way by orthoormal polyomals or trgoometrc fuctos separate varables ad the umbers of fuctos playg role () are large eough. Wth the help of Hgher Order Sgular Value Decomposto (HOSVD) a ew approxmato method was developed [7] [5] whch a specally determed system of orthoormal fuctos ca be used depedg o fucto f ( x ) stead of some other systems of orthoormal polyomals or trgoometrc fuctos. 248

3 Óbuda Uversty e Bullet Vol. 2 o. 20 Assume that the fucto f ( x) [ ] w ( x) x a b the form ca be gve wth some fuctos f ( x)=... α w ( x )... w ( x ). (2)... = =... Deote by A R the -dmesoal tesor determed by the elemets α ad let us use the followg otatos (see : [4]).... A U = : the -mode tesor-matrx product A U : the multple product as A U... 2 U2 U. The -mode tesor-matrx product s defed by the followg way. Let U be a K M -matrx the A U s a M... M K M+... M -tesor for whch the relato ( A U) def = m... m m +... m m... m... m m m M a holds. Detaled dscusso of tesor otatos ad operatos s gve [4]. We also ote that we use the sg stead the sg gve [4]. Usg ths defto the fucto (2) ca be rewrtte as a tesor product form f ( x)= A w ( x ) () = where w ( ) = ( ( )... ( )) T x w x w x. Based o HOSVD t was proved [6] that uder mld codtos the () ca be represeted the form f ( x)= D w ( x ) (4) where = r... r D R s a specal (so called core) tesor wth the propertes: (a) r = ra( A ) s the -mode ra of the tesor A.e. ra of the lear space spaed by the -mode vectors of A : {( a... a ) : j } T j (b) all-orthogoalty of tesor D : two subtesors D = α ad D = β (the th dces = α ad = β of the elemets of the tesor D eepg fx) orthogoal for all possble values of α ad β : D = α D = β =0 U 249

4 A. Rövd et al. mage Processg o Tesor Product Bass whe α β. Here the scalar product D = α D = β deotes the sum of products of the approprate elemets of subtesors D = α ad D = β (c) orderg: D = D =2 D = >0 for all possble values of r ( D = α = D = α D = α deotes the Kroecer-orm of the tesor D ). = α Compoets w ( x ) of the vector valued fuctos r T w ( x )=( w ( x )... w ( x )) are orthoormal L2 -sese o the terval [ a b ].e. b δ a j j : w ( x ) w ( x ) dx = j r where δ j s a Kroecer-fucto ( δ j= f = j ad δ j=0 f j) The form (4) was called [6] HOSVD caocal form of the fucto (2). Let us decompose the tervals [ a b ] =.. to M umber of dsjuct subtervals Δ m m M as follows: ξ = a < ξ < < ξ = b Δ =[ ξ ξ ). 0 M m m m w ( x) x a b the equato (2) are pece-wse cotuously dfferetable ad assume also that we ca observe the values of the fucto f ( x ) the pots Assume that the fuctos [ ] y = ( x... x ) M (5)... where x m Δm m M Based o the HOSVD a ew method was developed [6] for umercal recostructo of the caocal form of the fucto f ( x ) usg the values f ( y ).... M We dscretze fucto f ( x ) for all grd pots as: b = f( y ). m.. m m.. m 250

5 Óbuda Uversty e Bullet Vol. 2 o. 20 The we costruct dmesoal tesor B =( b m m ) from the values b m.. m. Obvously the sze of ths tesor s M... M. Further we dscretze vector valued fuctos w ( x) over the dscretzato pots x m ad costruct matrces W from the dscretzed values as: w ( x ) w2 ( x ) w r ( x) w ( x2 ) w2 ( x2 ) w r ( x2) W = w ( x M ) w2 ( x M ) w r ( x M ) The tesor B ca smply be gve by (4) ad (6) as B = D W. (7) Matrces see [5]. = W ad tesor D ca be obtaed by HOSVD of B. For further detals (6) mage Scalg the HOSVD-Based Doma Let f ( x) x =( x x2 x) T represet the mage fucto where x ad x 2 correspod to the vertcal ad horzotal coordates of the pxel respectvely. x s related to the color compoets of the pxel.e. the red gree ad blue color compoets case of RGB mage. Fucto f ( x ) ca be approxmated (based o otes dscussed the prevous secto) the followg way: 2 f ( x )= w ( x ) w ( x ) w ( x ). (8) α = 2 = = The red gree ad blue color compoets of pxels ca be stored a m tesor where ad m correspod to the wdth ad heght of the mage respectvely. Let B deote ths tesor. The frst step s to recostruct the fuctos w based o the HOSVD of tesor B as follows: B = D W (9) ( ) = where D s the so called core tesor. Vectors correspodg to the colums of ( matrces W ) as descrbed the prevous secto are represetg the dscretzed form of fuctos w ( x ) correspodg to the approprate dmeso. 25

6 A. Rövd et al. mage Processg o Tesor Product Bass 2 W 2 B W r 2 r r D r r 2 r W Fgure llustrato of the hgher order sgular value decomposto for a -dmesoal array. Here D s the core tesor the W -s are the -mode sgular matrces. Our goal s to demostrate the effectveess of mage scalg the proposed doma. Let s {2...} deote the scalg factor of the mage. Frst let us () () cosder the frst colum W of matrx W. Based o the prevous sectos t ca be see that the value w () correspods to the st elemet of () W w (2) to the 2d elemet... w ( M ) to the M th elemet of () W. To elarge the mage by a factor s the () W =..2 matrces should be updated based o the scalg factor s as follows: The umber of colums remas the same the umber of les wll be exteded accordg to the factor s. For example let us cosder the () () () colum W of W. W () does ot chage () () () () () () W ( s) := W (2) W (2 s) := W ()... W (( M ) s) := W ( M ). () () () () () The mssg elemets W (2) W ()... W ( s ) W ( s+ )... W (2s ) () () W (2s+ )... W (( M ) s ) ca be determed by terpolato. the paper the cubc sple terpolato was appled. The remag colums should be processed smlarly. After every matrx elemet has bee determed the elarged mage ca be obtaed usg the equato (9). 4 HOSVD vs. Frequecy Doma Comparg the proposed represetato to the frequecy doma smlartes ca be observed ther behavour. As t s well ow the Fourer Trasformato s coected to trgoometrc fuctos whle case of HOSVD approach the fuctos w ( x ) are cosdered whch are specfc to the approxmated - varable fucto. both cases the fuctos are formg a orthoormal bass. Let us meto some commo wdely used applcatos of both approaches. case of the Fourer based smoothg some of the hgh frequecy compoets are dsmssed resultg a smoothed mage (low pass flter). case of the HOSVD 252

7 Óbuda Uversty e Bullet Vol. 2 o. 20 smlar effect ca be observed whe dsmssg fuctos correspodg to smaller sgular values. the opposte case.e. dsmssg small frequeces yelds a edge detector (hgh pass flter) whch HOSVD case s equvalet to dsmssg compoets correspodg to hgher sgular values []. 5 Examples 5. Part- (Approxmato) ths secto some approxmatos ca be observed performed by the proposed ad by the Fourer-based approach. As the umber of the used compoets decreases the observable dffereces qualty become more sgfcat. the examples below both the HOSVD-based ad Fourer-based cases the same umber of compoets have bee used order to show how the form of determed fuctos flueces the qualty. Fgure 2 Orgal mage (24bt RGB) 25

8 A. Rövd et al. mage Processg o Tesor Product Bass Fgure HOSVD-based approxmato usg 2700 compoets composed from polylear fuctos o HOSVD bass Fgure 4 Fourer-based approxmato usg 2700 compoets composed from trgoometrc fuctos 254

9 Óbuda Uversty e Bullet Vol. 2 o Part-2 (Scalg) The pctures are llustratg the effectveess of the mage scalg by applyg the proposed approach. The result s compared to the output obtaed by the blear ad bcubc mage terpolato methods. Fgure 5 The orgal mage Fgure 6 Elarged segmet usg blear terpolato 255

10 A. Rövd et al. mage Processg o Tesor Product Bass Fgure 7 Elarged segmet usg bcubc terpolato Fgure 8 Elarged segmet usg the proposed HOSVD-based method. Smoother edges ca be observed 256

11 Óbuda Uversty e Bullet Vol. 2 o. 20 Coclusos the preset paper a ew mage represetato doma ad recostructo techque has bee troduced. The results show that how the effcecy of the certa tass depeds o the appled doma. mage rescalg has bee performed usg the proposed techque ad has bee compared to other well ow mage terpolato methods. Usg ths techque the resulted mage matas the edges more accurately the the other well-ow mage terpolato methods. Furthermore some propertes of the proposed represetato doma have bee compared to the correspodg propertes of the Fourer-based approxmato. The results show that the proposed doma some tass ca be performed more effcetly tha other domas. Acowledgemet The research was supported by the Jáos Bolya Research Scholarshp of the Hugara Academy of Sceces ad part by the Óbuda Uversty. Refereces [] P. Bojarcza ad S. Osows "Deosg of mages - A Comparso of Dfferet Flterg Approaches" WSEAS Trasactos o Computers ssue Vol. pp July 2004 [2] A. Khashma ad K. Dmller "mage Compresso usg eural etwors ad Haar Wavelet" WSEAS Trasactos o Sgal Processg Vol. 4 o pp. 0-9 [] Roume Koutchev Stuart Rub Marofaa Mlaova Vladmr Todorov Roumaa Koutcheva "o-lear mage Represetato Based o DP wth " WSEAS Trasactos o Sgal Processg SS: ssue 9 Volume 5 pp September 2009 [4] L. De Lathauwer B. De Moor ad J. Vadewalle "A multlear sgular value decomposto" SAM Joural o Matrx Aalyss ad Applcatos vol. 2 o. 4 pp [5] L. Szedl P. Várla "HOSVD Based Caocal Form for Polytopc Models of Dyamc Systems " Joural of Advaced Computatoal tellgece ad tellget formatcs SS : 4-00 Vol. o. pp [6] L. Szedl P. Baray Z. Petres ad P. Várla "umercal Recostructo of the HOSVD Based Caocal Form of Polytopc Dyamc Models" rd teratoal Symposum o Computatoal tellgece ad tellget formatcs Agadr Morocco 2007 pp. -6 [7] L. Szedl. Rudas A. Rövd P. Várla "HOSVD Based Method for Surface Data Approxmato ad Compresso " 2th teratoal Coferece o tellget Egeerg Systems Mam Florda February pp

12 A. Rövd et al. mage Processg o Tesor Product Bass [8] M. colaus L. Yue. Do Mh "mage terpolato usg multscale geometrc represetatos " Proceedgs of the SPE Volume 6498 pp [9] S. YUA M. ABE A. TAGUCH ad M. KAWAMATA "Hgh Accuracy Bcubc terpolato Usg mage Local Features " ECE Trasactos o Fudametals of Electrocs Commucatos ad Computer Sceces E90-A(8) pp [0] R. avarro A. Taberero ad G. Crstóbal "mage Represetato wth Gabor Wavelets ad ts Applcatos " Advaces magg ad Electro Physcs vol. 97 P. W. Hawes Academc Press Sa Dego CA pp [] Ae Gelb Taylor Hes "Detecto of Edges from ouform Fourer Data " Joural of Fourer Aalyss ad Applcatos DO 0.007/s pp [2] Yasu Xu Joh B. Weaver Des M. Healy Jr. ad Ja Lu "Wavelet Trasform Doma Flters: A Spatally Selectve ose Fltrato Techque " EEE Trasactos o mage Processg Vol. o. 6 pp [] A. Rövd. J. Rudas Sz. Sergyá L. Szedl "HOSVD Based mage Processg Techques" Proc. of the 0th WSEAS teratoal Coferece o Artfcal tellgece Kowledge Egeerg ad Data Bases Cambrdge UK SB: pp February