3D Image Representation through Hierarchical Tensor Decomposition, Based on SVD with Elementary Tensor of size 2 2 2

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1 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev 3D Imge Represenion roug Hierrcicl ensor Decomposiion, Bse on SVD wi Elemenry ensor of size ROUME KOUCHEV Rio Communicions n Vieo ecnologies ecnicl Universiy of Sofi Bul. Kl. Orisky 8, Sofi BULGARIA rkounc@u-sofi.g; p:// ROUMIAA KOUCHEVA K Engineering Druj, Bl. 44/ BULGARIA kouncev_r@yoo.com Asrc: - As i is known, groups of correle D imges of vrious kin coul e represene s 3D imges, wic re memiclly escrie s 3 r orer ensors. Vrious generlizions of e Singulr Vlue Decomposiion (SVD exis, ime e ensor escripion reucion. In is work, new pproc is presene for 3 r orer ensor ecomposiion, were unlike e fmous meos for ecomposiion componens efiniion, ierive clculions re no use. e sic srucure uni of e new ecomposiion is n elemenry ensor (E of size, wic uils e 3D ensors of size, were = n. e ecomposiion of e single ЕТ is execue y using Hierrcicl -level SVD, were (in ec level e SVD of size (SVD is pplie on ll su-mrices oine fer e elemenry ensor unfoling. e so clcule new su-mrices of e SVD in ec ierrcicl level, re rerrnge in ccornce wi e lessening of eir corresponing singulr vlues. e compuionl complexiy of e new ensor ecomposiion is lower n of e ecomposiions, se on ierive meos, n permis prllel clculions for ll SVD for e su-mrices in given ierrcicl level. Key-Wors: - 3D imges, ensor ecomposiion, Hierrcicl SVD (HSVD, elemenry ensor of size. Inroucion In e ls yers, e scienific ineres ime e processing, nlysis n recogniion of 3D imges, represene roug ensor ecomposiion, ws significnly increse. e sic meos for ensor ecomposiion [,,3,4] re ifferen muliliner exensions of e mrix SVD, clle Muliliner SVD (MSVD, or generlizions of e SVD mrix for iger-orer ensors, clle Higer- Orer SVD (HOSVD. Suc re e fmous meos: CADECOM/ARAFAC (C or Cnonicl olyic Decomposiion, were e ensor is represene s sum of rnk-one ensors; e ucker ecomposiion, wic is iger-orer form of e rincipl Componen Anlysis (CA; e Kruskl ecomposiion, ec. e componens of e ensor ecomposiion re clcule y using vrious meos, suc s, for exmple: e ensor power ierion; e QR-fcorizion, e Higer- Orer Eigenvlue Decomposiion (HOEVD; e Jcoi lgorim, ec. o ennce e ecomposiion of e 3 r orer ensors, wic represen sequences of D correle imges, in is pper is propose non-ierive meo for Hierrcicl SVD (HSVD, se on e SVD of size (SVD [5]. e sic ie is o represen e ensor ecomposiion of size (for =n roug ierrcicl ree-like srucure of n levels. In e firs ierrcicl level e ensor is ivie in elemenry ensors (ЕТ of size, rnsforme ino mrix form, n processe wi SVD. e ecomposiion componens of ll mrices of size re rerrnge in ccornce wi e lessening of eir singulr vlues n rnsforme ck ino corresponing E. In ec consecuive ierrcicl level e eges of e cues, wic represen ll Es, re increse wice, in resul of wic eir voxels re inerlce. ese increse Es re rnsforme gin ino mrices, eir su-mrices re ecompose roug SVD, n rerrnge following e lessening of eir singulr vlues. en, e so oine mrices re rnsforme ino ensors gin, ec. e essence of e new meo is given in e following secions of is pper. 3D imge represenion roug ensor ecomposiion formulion e ensor form is suile for e represenion of 3D imges, wic correspon o sequences of correle D imges of e kin: compuer omogrpy imges, ulrsoun imges, mulispecrl n yper specrl imges of sill ojecs, moving ojecs, ec. For illusrion only, E-ISS: Volume, 6

2 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev on Fig. is sown e 3D silouee imge sequence, represene y 3-orer ensor []. Fig.. 3D silouee imge sequence, represene y ir-orer ensor e min vnge of e ensor represenion is, i reins o mximum egree e correlion eween e 3D-imge voxels in ll ree orogonl irecions. As resul of e C ecomposiion, ec 3D ensor is represene s sum of 3D ensors of sme size, wose vrinces (respecively, weigs ecrese rpily. Ec ensor in is sum is n ouer prouc of 3 muully orogonl vecors. e im of e ensor ecomposiion is o cieve full ecorrelion of e sum componens. Ec ensor is efine y relively smll numer of prmeers, n from is i follows, e generl represenion of e firs severl ensors in e ecomposiion, wic ve mximum weig, is significnly smller n of e ecompose ensor. e ecomposiion of e kin MSVD permis o reuce e ensor represenion roug cuing-off e ensors wi smlles weigs, reining minimum men-squre pproximion error. In is work, new pproc is propose for ensors clculion (i.e., e ecomposiion componens, y using e HSVD. For is, in ec ierrcicl level, on ll su-mrices of size, wic uil e rnsforme ensor, is pplie SVD. e su-mrices re rerrnge in ccornce wi eir singulr vlues, n e mrices, weige is wy, re rnsforme ck ino corresponing ensors (ecomposiion componens. In e nex secions of e pper is escrie e НSVD for 3-orer ensor, se on e mrix SVD. 3 SVD clculion for mrix of size Ec mrix [X] of size coul e ecompose roug SVD, s follows [6]: u u [X] c u u u UV U V u A 3 M 3 were v v v,v v,v 3 v v u u 3 [C] [C x =, x =, x =c, x = - elemens of mrix [X]; A A M /A; /A; ; ; R R U ; A R U ; A R Q Q V ; A Q V ; A Q R A ; A ; A ; A ; R Q Q RQ ; R Q ; 3 R Q ; 4 R Q ; A 4 ; c ; c ; c ; c. ( Here n re e singulr vlues of e mrix [X were ; U, U n V, V re is lef n rig eigen vecors, for wic [X][X] sus n [X] [X] svs for s=,. Here s s re e corresponing eigenvlues of e wo symmericl mrices: [ X][X] n [X] [X]. From Eq. ( i follows, mrices [C ] n [C ] coul e represene s follows: A (A (A (A (A [ C ], A (A (A (A (A A (A (A (A (A [ C ]. A (A (A (A (A In corresponence wi Eq. (, e elemens of [C ] n [C ] re efine y e four prmeers,,, ( E-ISS: Volume, 6

3 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev n, i.e., e ecomposiion, represene y Eq. (, is no overcomplee. 4 Clculion of e Hierrcicl SVD for ensor of size e ensor of size, noe s Т, is e kernel of e ecomposiion for e 3 r -orer ensor of size for = n. Afer moe- unfoling (or mricizion e elemenry ensor, is oine: e f unfol( [X ] [X] c g (3 In e firs level of e HSVD lgorim for e ensor Т (HSVD, on ec of e mrices [X ] n [X ] is pplie SVD of size (SVD, n in resul is go: [ X ] [C] [C c e f [ X] [C ] [C ] g (4 e so oine mrices [C i,j ] of size for i,j=, in Eq. (4 re rerrnge in new couples in corresponence o eir singulr vlues. Afer e rerrngemen, e firs couple of mrices [C ] n [C wic ve ig singulr vlue, efines e ensor ( y reverse mricizion, n e secon couple [C ] n [C ] wic ve lower singulr vlue - e ensor (. en: ( ( (5 Afer moe-3 unfoling of o ensors, is oine: unfol ( unfol( [X ] [X ] [X ] [X ] ( ( (6 In e secon level of HSVD, on ec mrix [X i,j ] of size is pplie SVD n in resul is go: [X [X ] [C ] [C ] [C ] [C [X [X ] [C ] [C ] [C ] [C ]. (7 e so clcule mrices [C i,j,k ] of size for i,j,k =, re rerrnge ino 4 new couples wi similr singulr vlues in orer, efine y is ecrese. Ec of ese 4 couples of mrices y reverse mricizion efines corresponing ensor of size. Afer eir moe- unfoling, is oine: unfol{ unfol{ (} unfol{ (} unfol{ [C ] [C ] [C ] [C ] [C ] [C ] [C ] [C ]. ( ( ( ( (} (} (8 In resul of e execuion of e wo levels of e HSVD, e ensor Т is represene s: i j ( j( ( (i. ( ( ( ( ( ( (9 On Fig. is sown e ecomposiion lgorim for e elemenry ensor Т. Afer e ecomposiion, e ensors in e sum re rrnge in corresponence o e lessening of e s vlues for e su-mrices, oine roug unfoling ll ensors in e sum. e voxels wi lrges ensor vlues re coloure in re, n e smlles - in lue. On Fig. 3 is sown e clculion grp of e -level HSVD lgorim for e elemenry ensor, se on e muliple clculion of SVD in ec ierrcicl level. Afer e execuion of e clculions for given level, e elemens of e su-mrices oine for e SVD re rerrnge in ccornce wi e lessening of e s vlues, n re rnsforme ck ino new elemenry ensors: wo ensors ( n ( - fer e en of e firs level, n four ensors (, (, ( n ( - fer e secon level. 5 Clculion of e HSVD for ensor of size n In e firs level of e HSVD lgorim e ensor Т (for =4, sown on Fig. 4, is ivie ino eig elemenry ensors (i for i=,,..,8 (cues of size. In ccornce wi is lgorim, ec elemenry ensor is ecompose ino 4 new ensors Т j( (i for j=,,3,4 of sme size. In e s level of HSVD, on ec ensor Т j( (i compose of 8 voxels of sme colour (yellow, re, green, lue, wie, purple, lig lue, n ornge, is pplie HSVD. Afer e rerrngemen of e elemenry ensors, n eir inegrion, re oine four new 4 ensors Т j( (i. E-ISS: Volume, 6

4 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev [X] c ( ( ( ( 3 e [X] g 5 7 HSVD xx ( 4 f voxel ( 3 SVD x [C ] [C ] 7 SVD x 6 c e f SVD x g SVD x [X ] Rerrengemen 4 Level 8 ( ( 5 [X ] [C ] [C ] Rerrengemen Rerrengemen X ] [C ] [C ] g e f [ X ] [C ] [C ] c [ 3 4 SVD x 7 8 SVD x 6 [X] c 3 e [X] g f6 8 [C ] [C ] Level [C ] [C ] [C ] [C ] [C ] [C ] ( + ( + ( + ( Fig.. wo-level HSVD lgorim for elemenry ensor of size se on e SVD ensor e f [X ] g c [X ] unfol [X ] c [X ] e g f X ] c [ [X ] c e [X] [X] [C] [C [C ] [C [X [X] ] [C] [C Level Rerrngemen Level [ C] [ C ] [ C] [ C ] [ C ] c(, c(, [ C ] c(, c(, SVD x [ C] c(, c(, c(, c(, c(, c(, [ C] c(, c(, c(, c(, [X ] c(, c(, [X ] c(, c(, c(, c(, SVD x SVD SVD SVD [C(] x 3/4 C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, [ C ] [ C ] [ C ] C(, [ C ] C(, C(, C(, C(, c(, c(, C(, f c(, c(, C(, SVD x C(, SVD SVD x SVD SVD SVD [C(] x [X ] [X ] C(, c(, c(, 5/6 C(, g c(, C(, c(, C(, [X ] C(, c(, c(, C(, c(, c(, C(, C(, [C] [C] Rerrngemen [ C ] [ C ] [ C] [ C ] [C] C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, [ C ] [ C ] [ C] C(, [ C C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, C(, [C] e f [ X ] [C] [C ]. g [X] [X] [X ] [C [X ] [C ] [C ] [C ]. [C] [C] [C] [C] Fig. 3. e flow grp of e -level HSVD for e elemenry ensor, se on e SVD E-ISS: Volume, 6

5 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev ensor Elemenry ensor In e secon level of e HSVD lgorim ec of e four ensors Т j( (i is ivie ino eig su-ensors i,k( (j for i=,, j=, n k=, in e wy, efine y e spil ne for voxel inerlcing s sown on Fig. 5. e colour of e voxels in ec cue correspons o from e firs level of e HSVD lgorim. On ec expne elemenry ensor (oule size ensor, sown on Fig. 5, is pplie once gin e НSVD lgorim. Fig. 4. e ensor Т is ivie ino eig elemenry ensors (i for i=,,..,8 in e firs HSVD level, were e HSVD is pplie on ec group of voxels of sme color (8 in ol ensor 4 ensor 3 ensor ensor Elemenry exene ensor Fig. 5. Division of ensors Т i( (j ino e elemenry ensors j( (i in e secon HSVD level, were e HSVD is pplie on ec group of voxels of sme color (3 in ol Afer e execuion of e s ecomposiion level, e ensor Т is represene s sum of 4 componens: j(444 (i i j ( e so clcule 4 ensors Т i( (j re rrnge in corresponence wi e lessening of e men singulr vlues (e energy of e elemenry ensors i,k( (j, for i=,, j=,, k=,. e ensors Т i( (j re rerrnge in ccornce wi e energy ecrese of e ЕТs i,k( (j, wic uil em. In ccornce wi Fig., on ec ЕТ is pplie e wo-level НSVD gin. Afer e execuion of e secon ecomposiion level, e ensor Т 444 is represene s sum of 6 componens: j(444 (i i j ( e compuionl grp of e -level HSVD ecomposiion is sown on Fig. 6. E-ISS: Volume, 6

6 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev runce 3D HSVD Rejece componen Reine componen Rerrngemen ensor Rerrngemen Level of 3D HSVD Rerrngemen Rerrngemen ensor pproximion Level of 3D HSVD Fig.6.Srucure of e compuionl grps of e full n runce 4-level inry ree for e execuion of e -level 3D HSVD 444 lgorim se on e SVD Level Level Level 3 Fig.7. Secion for ensor of size in levels,, n 3 of e 3-level HSVD 888 lgorim, se on e SVD (ec ensor ere is represene s mrix of size 8 8; in ec level, e SVD is execue 6 imes e so clcule 6 ensors Т j( (i re rrnge in ccornce wi e ecresing vlues of e singulr vlues of kernels, i,k( (j, wic compose em, for i=,, j=, n k=,. In e s level of e full HSVD 444, e HSVD is execue 8 imes, n in e n level - 3 imes. In e cse, wen e ensor is of size 8 8 8, e corresponing HSVD lgorim if of 3 levels: in e firs level re oine 4 ensors of sme size; in e secon - 6 ensors, n in e ls, ir level - 64 ensors. On Fig. 7 re sown e corresponing secions of e iniil ensor in e firs ecomposiion level, n of one ensor in e secon n ir levels. In e secon level re sown only 8 of e sumrices of size, on wic is pplie e SVD, n in e ir level - only four of e sumrices of size. e elemens of ec sumrix re coloure in sme colour: re, green, lue, yellow. E-ISS: Volume, 6

7 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev 6 Generl HSVD cse for ensor of size e ecomposiion of ensors Т n Т 888 coul e generlize for e cse, wen e ensor is of size for = n. As resul of e use of e HSVD lgorim, e ensor n n n is represene s sum of = n singulr ensors: n n n i j n n n n n (i. ( j( e singulr ensors n n n (i of size j( n n n re rrnge in ccornce wi e ecresing vlues of e energies of i,k( (j for i, j, k =,,.., n, wic uil em. e numer of ierrcicl levels neee for e execuion of e HSVD lgorim for = n, is n. 7 Exmple for ensor ecomposiion of size Le e voxels of e elemenry ensor Т, represene in corresponence wi Eq. (3, ve e vlues: =, =, c=, e=, f=, j=, =. en Eq. (3 ecomes: unfol( [X ] [X] Afer pplying e SVD on ec of e mrices [X ] n [X is oine: X ] [C] [C [ X] [C ] [C ] [ were: [ C ], [ C ], [ C ], [ C ], Afer e rerrngemen of e voxels of e ensors ( n ( compose y e mrices [C [C [C [C followe y moe- 3 unfoling of o ensors, e nex four mrices re oine: [ X ] ; [ X] ; [ X ] ; [ X]. On ec of ese mrices is en pplie once more SVD, n s resul is oine: [X [C ] [C were ] [C ] [C [X [X ] [C ] [C ] [C [ C ], [ C ], [ C ], ] [C [X ] [ C] [C] [C ] [C ] [C ]. e vlues of e voxels of ensors (, (, (, n ( (ec of size, compose y e mrices [C [C [C [C [C [C [C [C re given on Fig. 8. In is cse, e voxels coloure in re only ( in ol, ve nonzero vlues. e remining voxels, coloure in lue (lf of ese, corresponing o e secon ensor, n ll voxels of e ir n four ensors in e ecomposiion ve nonzero vlues. 8 Evluion of e ecomposiion componens energy, n eir muul correlion 8.. ensor energy e energy of ec ensor, represene y e ecomposiion (, is efine y e relion: (i, j,k, (3 i j k s s were (i, j,k re e voxels of e ensor Т, n s is e energy of ec ensor s for s=,,.,, in ccornce wi Eq. (. In e cse for =, e energy of e ensors s for s=,,3,4 is efine y e relions: 3( ; ( ;. (4 3 4 E-ISS: Volume, 6

8 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev ( ( ( ( X ] X ] [ [ HSVD xx ( ( SVD x SVD x [X ] [X ] [ X] [C ] [C ] [ X] [C] [C ] Rerrengemen (+/ (+/ (-/ (-/ SVD x SVD x [ X ] Level [C ] [C ] [C ] [C ] (+/ (+/ (-/ (-/ SVD x ( SVD ( x [ X ] Rerrengemen Rerrengemen (+/ (+/ (-/ (-/ [C ] [C ] Level [C ] [C ] [C ] [C ] [C ] [C ] (-/ (-/ (+/ (+/ ( + ( + ( + ( Fig. 8. Exmple: wo-level HSVD for elemenry ensor Т o evlue e energy ( of e firs ensor in e ecomposiion, regring e ol energy ( coul e use e relion elow: 4 3( 3. (5 s 4( 4 ( / / s If =4 n = is oine =.95, i.e. 95% of e ol energy of e ensor Т re concenre in e firs ecomposiion ensor,. In priculr, if =, en = n in e firs ensor is concenre e ol energy (%, for ny vlue of. 8.. Muul correlion e muul correlion eween couples of ecomposiion ensors R( р, р+ is efine y eir sclr prouc: R(p,p p(i, j,k p(i, j,k. (6 i j k For e exmple from Fig. 7 e muul correlion R( р, р+ for р=,,3 eween ensor couples, is corresponingly: R(, ( ; R(,3 R(3,4, i.e., i rpily ecreses o zero. is resul sows e so oine ensor componens in e ecomposiion re significnly ecorrele. e ormlize Correlion Funcion (CF in respec of e firs one, is: for s ; R(, s ( (, for s ; (7 s R(, 3( for s,3. For e cse, wen =4 n =, on Fig. 9 is sown e grpic represenion of e funcion (s. CF (s,8,6,4, ormlize Correlion Funcion, 3 s Fig. 9. e ormlize Correlion Funcion of e ecomposiion componens E-ISS: Volume, 6

9 WSEAS RASACIOS on SIGAL ROCESSIG Roumen Kouncev, Roumin Kouncev 9 Compuionl complexiy of e new ierrcicl ensor ecomposiion In corresponence wi [6 e compuionl complexiy of new lgorim for ecomposiion of e ensor [ n n n ], is O( 4n. e comprison wi e H-ucker ensor ecomposiion О(3 3n +3 4n sows, for e new ecomposiion i is pproximely 3 imes lower. Bu, e neee memory soul e ou /3 lrger, n for e H-ucker ensor ecomposiion. Conclusions e sic vnges of e new pproc for clculion of e ierrcicl ensor ecomposiion, use o represen 3D imges of vrious kin, re s follows: e offere HSVD lgorim, unlike e H- ucker ensor ecomposiion, s lower compuionl complexiy n oes no nee ierive clculions; In ccornce wi Eqs. (3 n (9, in ec ierrcicl level is execue repeely sme ecomposiion of elemenry ensors of size, wic permis eir prllel clculion; e comprison wi e fmous ensor ecomposiions sows, e HSVD lgorim opens new n eer iliies o ennce e efficiency of e sysems for rel-ime processing of 3D imges, n lso in e 3D compuer vision sysems n 3D ojecs recogniion. References: [] H. Lu, K. lniois, A. Venesnopoulos, MCA: Muliliner rincipl Componen Anlysis of ensor Ojecs, IEEE rnscions on eurl eworks, Vol. 9, o, Jnury 8, pp [] G. Bergqvis, E. Lrsson, e Higer-Orer Singulr Vlue Decomposiion: eory n n Applicion, IEEE Signl rocessing Mgzine, Vol. 7, Iss. 3,, pp [3] A. Cicocki, D. Mnic, A. n, C. Cif, G. Zou, Q. Zo, L. De Luwer, ensor Decomposiions for Signl rocessing Applicions, IEEE Signl rocessing Mgzine, Vol. 3, Iss., 5, pp [4] L. De Luwer, B. De Moor, n J. Vnewlle, A Muliliner Singulr Vlue Decomposiion, SIAM Journl of Mrix Anlysis n Applicions, Vol. 4,, pp [5] R. Kouncev, R. Kouncev, Rix-( Hierrcicl SVD for muli-imensionl imges, roc. of e IEEE Inern. Conf. on elecommunicions in Moern Sellie, Cle n Brocsing Services (ELSIKS 5, is, Seri, Oc. 5, pp [6] R. Kouncev, R. Kouncev, Hierrcicl Decomposiion of D/3D Imges, se on SVD, Inernionl J. of Mulimei n Imge rocessing (IJMI, Vol. 5, Iss. 3/4, Sepemer/Decemer 5, pp E-ISS: Volume, 6