Gray level image enhancement using the Bernstein polynomials

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Spencer Norton
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1 Buletiul Ştiiţiic l Uiersităţii "Politehic" di Timişor Seri ELECTRONICĂ şi TELECOMUNICAŢII TRANSACTIONS o ELECTRONICS d COMMUNICATIONS Tom 47(6), Fscicol -, 00 Gry leel imge ehcemet usig the Berstei polyomils Abstrct This pper presets method or ehcig the gry leel imges. This preseted method tes prt rom the ctegory o poit opertios d it is bsed o piecewise lier uctios. The iterpoltio odes o these uctios re clculted usig the Berstei polyomils. Keywords: imge ehcemet, piecewise lier uctio, Berstei polyomils. I. INTROUCTION Imge ehcemet is importt ts i the ield o imge processig. Legthwys time, there were built my methods i this purpose [], [], [6], [7]. The gret umber o existig methods is determied by the gret riety o imges, which eed speciic methods [4]. This pper describes method o imge ehcemet, which tes prt i the group o poit trsorms [3]. The piecewise lier uctios re used or gry leel trsorms. These uctios he grphic represeted by polygol lies [5], while the iterpoltio odes re determied usig the Berstei polyomils. The ollowig prt o the rticle is orgized thus: sectio comprises the mthemticl theory presettio or the piecewise lier uctios while the sectio 3 itroduce the - mes, the -histogrms. The piecewise lier pproximtio or the gry leel trsorm d the experimetl results re preseted i sectio 4. Filly the sectio 5 oers seerl coclusios. Vsile P. Pătrşcu 0 with M 0, E,M. Becuse o tht or imge processig oe cosiders the ollowig prticulrly mily o the PL: : 0, M 0, M, i i () i This uctio depeds o the rel prmeters i i, d i i,. Let be,,...,, set o lues rom 0,M. The elemets i i, re the uctio lues i the brepoits i i, Oe determies the coeiciets coditios: i i.,,..., rom the with i,,..., () The system () hs the subsequet solutio [5]: i i i i i, i,..., (3) i i i i The reltios (3) supply the coeiciets lues i i, i determiig the uctio whe owig the lues i i, i the poits i i,. II. THE PIECEWISE LINEAR FUNCTIONS Gie the rel umbers... let be cotiuous uctio : R R tht is lier o ech iterl i, i, i,,...,. Such uctio is clled piecewise lier d the umbers i re clled brepoits. Piecewise lier uctios (PL) re ote used to pproximte (cotiuous) olier uctios. The PL re lso ow s lier splies with rible ots. The grphic o PL uctio is represeted with polygol lie [5]. Usully the gry leel set is rel bouded iterl III. -MEANS AN -HISTOGRAM FOR GRAY LEVEL IMAGES A. The Berstei polyomils Polyomils re icredibly useul mthemticl tools s they re simply deied d c be clculted quicly o computer systems. The most used orm is: P t 0 t... t t which represets polyomil o degree s lier combitio o certi elemetry polyomils, t, t,..., t. The set o polyomils o degree less TAROM Compy, eprtmet o Iormtics Techology, e-mil: ptrscu@trom.ro

The iterpoltio odes o these uctios re clculted usig the Berstei polyomils. Keywords: imge ehcemet, piecewise lier uctio, Berstei polyomils. I.

2 the or equl to orms ector spce. The set o uctios, t, t,..., t orms bsis or this ector spce. I this pproch it use the Berstei bsis, which re dieret to the commo bses or the spce o polyomils. The Berstei polyomils o degree B i : 0, M 0,, re deied by: i i i t t Bit C or i 0,,..., (4) M M! where C i. The Berstei i! i! polyomils o degree orm prtitio o uity: B i t (5) i0 B. The -mes or gry leel imges A gry leel imge is deied o compct sptil domi R by gry leel uctio l : E E 0,M is the gry leel set. Usully where M 55. Assume tht the uctio l is cotiuous oe. The Berstei mes b0 ( l), b ( l),..., b ( l) re deied s ollowig: or i 0,,..., B i lx, y lx, y bi ( l) (6) B x, y dxdy l dxdy i It c use tuig prmeter to obti more lexibility. Let be the uctios: 0, M 0, F i :, B i t F i t or i 0,,..., (7) B i t i0 where,. From (7) it results similrly property lie (5): F i t (8) i0 With the uctios (7) oe c deie the -mes: or i 0,,..., lx, y lx, y Fi dxdy b, Fi lx, ydxdy (9) C. The -histogrms or gry leel imges Let be gry leel imge l : E where l is cotiuous uctio. The Berstei histogrm is discrete oe. Ech bi h i (l) is gie by: l Bi x, y h l i or i 0,,..., (0) re Usig the uctios F i deied by (7) oe c deie the -histogrms: l Fi x, y h l, i or i,..., re 0, () which belogs lso to the discrete histogrms mily. It is ery simple to proe tht the ollowig reltios result rom (0) d (): h i l, i0 h, i l i0 Usig the bis o -histogrm () we c deie the ccumulted -histogrm s ollows: H, 0 ( l) h, 0 ( l) () H, H, i( l) h, h, i( l) (b) or i,..., IV. THE ENHANCEMENT PROCEURE A. The gry leel trsorm or imge ehcemet Let be l : E the imge tht must be ehced d u : E imge with uiorm distributio o gry leels. The uctio tht deies the gry leel trsorm is PL oe with odes. For the iterpoltio odes re used the -mes d the bis o the ccumulted -histogrm computed with reltios (9) d () or 3. The brepoits d re stedy i the lues: mi lx, y, lx y x, y mx,. The others x, y brepoits, 3,..., will be the -mes b, 0( l), b,( l),..., b, ( l). So, i b,, or i 0,,...,. To deie the PL uctio or imge ehcemet we eed the prmeters i i 0, d tht re deied by: H, i or i 0,,..., (3) H, i ( u) d 0 b, 0 ( u) i b, i ( u) b, i( u) i (4) M The lues i i, o the iterpoltio uctio re computed with the ollowig reltios:

The -mes or gry leel imges A gry leel imge is deied o compct sptil domi R by gry leel uctio l : E E 0,M is the gry leel set. Usully where M 55. Assume tht the uctio l is cotiuous oe.

3 0, 0 b,0 ( u) (5) i i i b, i ( u) b, i3( u) or i 3,.., (5b) M b, ( u) (5c) Hig the lues uctio i i, i the poits i i, i i,, we c compute the coeiciets lues usig (3). The pplyig the PL uctio to the origil imge l, we obti the ehced imge l eh (l). B. The ehcemet procedure Let be l : E the imge tht must be ehced d u : E imge with uiorm distributio o gry leels. The ext procedure will be used or ehced imge clculus:. Iitiliztio: choose,, (the costt or stoppig the procedure) d m 0. We set l l d compute the -mes b i l (0), ( ), b i0,, i ( u) d the ccumulted i0, -histogrms H i l (0), ( ), H i0,, i ( u) or i0, the imges (0) l d u.. We compute the prmeters i, (m ) i0, d the lues i usig the reltios (3, i, 4, 5). Oe clculte mi l ( x,, ( x, mx l ( x, 3 d ( i b, i l ) ( x, or i 0,,...,. Oe compute the piecewise lier uctio : 0, M 0, M, 3 i i usig reltios (3), i 3 3 i i i i i or i,..., i i i i (0) We will compute l ( m) l b m i l ( ), ( ) i0, H m i l ( ), ( ) i0,, the -mes, the ccumulted -histogrm usig (9), () or this ew imge. To obti the illy gry leel trsorm eh we must compute the strig o uctios ( m ) (0) where E is the idetity uctio o gry leel set E. ( m) 3. I b, i ( l ) b, i ( u) pss to the step 4, otherwise m m d go to step. 4. Se the results ( ) l eh l m ( m), eh, h m i l ( ), ( ) d stop. i0, b m i l ( ), ( ) i0,, To exempliy, three imges were piced out: oe dr ( tire ) i Fig., oe bright ( cells ) i Fig.. d oe with low cotrst ( lx ) i Fig.3. Their - histogrms re i Fig.b, Fig.b d Fig.3b. The grphics o their gry leel trsorms eh re show i Fig.c, Fig.c d Fig.3c. The ehced imges c be see i Fig.d, Fig.d d Fig.3d with the -histogrms i Fig.e, Fig.e d Fig.3e. The -mes or the origil imges d or the ehced imges re represeted i Fig., Fig. d Fig.3. V. CONCLUSIONS The pper preseted method or ehcig the gry leel imges. The method is bsed o poit trsorms deied by piecewise lier uctios. These uctios re determied by simple ormule, which eed short clculus time. I estblishig the iterpoltio poits it ws chose lgorithm tht is similrly to the clssicl histogrm equliztio. Future perspecties or the show method could be the extesio or color imges. REFERENCES [] K.R. Cstlem, igitl Imge Processig, Pretice Hll, Eglewood Clis NJ, 996. [] R.C. Gozles, P. Witz, igitl Imge Processig, d Editio, Addiso-Wesley, New Yor, 987. [3] A.K. Ji, Fudmetls o igitl Imge Processig, Pretice Hll Itl., Eglewood Clis NJ, 989. [4] M. Jourli, J.C. Pioli, Imge dymic rge ehcemet d stbiliztio i the cotext o the logrithmic imge processig model, Sigl processig, Vol. 4, o., 995, pp [5] V. Pătrşcu, Gry leel imge ehcemet usig polygol uctios, IEEE-tttc Itertiol Coerece o Automtio, Qulity d Testig, Robotics, AQTR00, ol. Robotics, Imge d Sigl Processig, Cluj-Npoc, Romi, 3-5 My, 00, pp [6] W.K. Prtt, igitl Imge Processig, d Editio, Wiley / Itersciece, New Yor, 99. [7] A. Roseeld, A.C. K, igitl Picture Processig, Acdemic Press, New Yor, 98.

The ext procedure will be used or ehced imge clculus:. Iitiliztio: choose,, (the costt or stoppig the procedure) d m 0.

4 Fig.. ) The origil imge tire Fig.. d) The ehced imge Fig.. b) The - histogrm o origil imge Fig.. e) The - histogrm o ehced imge Fig.. c) The gry leel trsorm Fig.. ) The - mes or origil imge (circle) d or ehced imge (squre)

5 Fig.. ) The origil imge cells Fig.. d) The ehced imge Fig.. b) The - histogrm o origil imge Fig.. e) The - histogrm o ehced imge Fig.. c) The gry leel trsorm Fig.. ) The - mes or origil imge (circle) d or ehced imge (squre)

6 Fig. 3. ) The origil imge lx Fig. 3. d) The ehced imge Fig. 3. b) The - histogrm o origil imge Fig. 3. e) The - histogrm o ehced imge Fig. 3. c) The gry leel trsorm Fig. 3. ) The - mes or origil imge (circle) d or ehced imge (squre)

Chapter 04.05 System of Equations

Chapter 04.05 System of Equations hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee