Generalized Phi Number System and its Applications for Image Decomposition and Enhancement

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1 Generalzed Ph Number System and ts Applcatons for Image Decomposton and Enhancement Sarks Agaan a and Ycong Zhou b, * a Department of Electrcal Engneerng, Stanford Unversty, Stanford, CA b Interdscplnary Solutons, LLC, San Antono, TX 7815; ABSTRACT Technologes and applcatons of the feld-programmable gate array (FPGAs) and dgtal sgnal processng (DSP) requre both new customzable number systems and new data formats. Ths paper ntroduces a new class of parameterzed number systems, namely the generalzed Ph number system (GPNS). By selectng approprate parameters, the new system derves the tradtonal Ph number system, bnary number system, beta encoder, and other commonly used number systems. GPNS also creates new opportuntes for developng customzed number systems, multmeda securty systems, and mage decomposton and enhancement systems. A new mage enhancement algorthm s also developed by ntegratng the GPNS-based bt-plane decomposton wth Parameterzed Logarthmc Image Processng (PLIP) models. Smulaton results are gven to demonstrate the GPNS s performance. Keywords: Number system, Ratonal number system, Ph number system, Parameterzed Logarthmc Image Processng, Bt-plane decomposton, Image enhancement 1. INTRODUCTION The development of computer technologes s how to utlze the hardware and software to perform computaton. Dgtal computer arthmetc s based on data or numbers represented as a strng of zeroes and ones. The hardware of dgtal computers can execute the smple and prmtve formats of Boolean operatons. The sze of the dgts s called radx r: k X ar a r... ar a (1) k ( k1) k1 1 0 The radx r could be an nteger, postve or a constant. A very useful radx case s r=, also called the base- number system. It represents numbers by a combnaton of two numerals: ak {0,1}, zero (0) and one (1). All modern computers are based on the bnary system. The computer archtecture s from embedded processors to supercomputers. The bnary system s also used for parallel programmng and parallel archtectures, computer arthmetc, custom computng, algorthms and programmng methodologes as well. Bnary arthmetc operatons are mplemented by Boolean logcs whch are easly realzed wth dgtal electroncs. In other words, bnary number system was selected for computer systems because of ts straghtforward and one-to-one mappng n logc crcuts. Due to the long carry/borrow propagaton paths extendng from the least-sgnfcant bt to the most-sgnfcant bt poston, the bnary number systems are lmted n the computng speed [1, ]. Ths motvates researchers to develop alternatve approaches to desgn hgh-speed arthmetc unts. Usng unconventonal number representatons has brought much attenton n recent years. Usng non-bnary number representatons, several schemes have been presented n order to obtan effcent arthmetc operatons. Examples nclude the multple-valued fxed radx-number resdue number [3], Logarthmc Number System (LNS) [4, 5], Resdue Number System (RNS) [6, 7], sgned-dgt [8], and some hybrd number systems [9, 10]. An approprate number representaton s very mportant for dfferent FPGA applcatons snce t sgnfcantly affects the performance and accuracy of the FPGA systems. Recently several encoders have been developed for dfferent number systems such as ratonal number system [5], beta or Golden rato encoders [11, 1] whch are used on analog to dgtal (A/D) converson system [13]. The golden rato ( (1 5) / ) sometmes refers to base- (the Greek letter ph ). It has been used as the base of a number system called golden mean base, ph number system (PNS), or phnary. Ths number system was developed by Bergman n It s a non-nteger postonal number system n whch the base- s an rratonal number. * Ycong.Zhou@yahoo.com Multmeda on Moble Devces 011; and Multmeda Content Access: Algorthms and Systems V, edted by Davd Akopan, Rener Creutzburg, Cees G. M. Snoek, Ncu Sebe, Lyndon Kennedy, Proc. of SPIE-IS&T Electronc Imagng, SPIE Vol. 7881, 78810M 011 SPIE-IS&T CCC code: X/11/$18 do: / SPIE-IS&T/ Vol M-1

2 Every non-negatve real number can be represented as a base- sequence usng dgts 0 and 1, and avodng the dgt sequence 11 ths called a standard form [14-16]. In ths number system, every non-negatve nteger s represented by a fnte base- sequence of dstnct ntegers k1, k,..., k m such that [15, 17-19]. X 1 m k k... k () Every non-negatve nteger has a base- representaton usng two dgts 0 and 1. Ths representaton s unque f the followng condton s satsfed [14, 15]. 1) It contans no consecutve 1s or has a standard form. ) It does not become after some fnte number of dgts. Usng the arthmetc propertes of the base- that +1 =, a base- numeral contanng the dgt sequence "11" can always be rewrtten as a standard form. For example, 11 = 100. Due to the fact of that all ntegers have a unque representaton as a fnte expanson of the rratonal base-, the ph number system s a remarkable number system. Other numbers have unque standard representatons n base- as well. Ratonal numbers have recurrng expansons. The numbers wth a termnatng expanson also have a non-termnatng representaton [0]. Smlar cases could be found n base-10 system, for example, 1= [1]. Ths property can be extended nto the negatve of a base- representaton by negatng each dgt, standardzng the result, and then markng t as negatve. For example, use a mnus sgn, or some other sgnfcance to denote negatve numbers. If the arthmetc s beng performed on a computer, an error message may be returned. Detal dscussons of the ph number system can be found n [16, 19, ]. Ternary -representaton [3, 4] s another rratonal base number system whch the base s the square of the golden rato, (3 5) /, wth the ternary numerals { 1,0,1}. Ths number system have been used n computers for arthmetcal operaton control [3, 4] FPGA technologes make applcaton-specfc number representatons avalable wth customzable bt-wdths. However, they requre a new customzed number system and new data treatment [10, 5]. In ths paper, we ntroduce a new generalzed Ph number system (GPNS). By selectng approprate parameters, The GPNS can be specfed to the tradtonal PNS, the bnary number system (base-), and other nteger base number systems. We nvestgate the applcatons of the new GPNS n mage processng. We ntroduce a new parameter bt-plane decomposton method usng the new GPNS. Integratng ths new decomposton method wth the chaotc logstc map, a new mage encrypton algorthm s ntroduced. Expermental results are gven to demonstrate that the GPNS shows excellent performance n mage decomposton and encrypton. The rest of the paper s organzed as followed. Secton ntroduces the new GPNS and dscusses ts propertes. Secton 3 ntroduces an mage bt-plane decomposton method usng the GPNS. Secton 4 draws a concluson.. GENERALIED PHI NUMBER SYSTEM In ths secton, we ntroduce a new class of parameterzed number system, namely the generalzed ph number system (GPNS). We show that, by selectng approprate parameters, the new system can derve other commonly used number systems ncludng the Ph number system, bnary number system, and beta encoder. Some GPNS propertes are also presented. By extendng the concept of the base- representaton, we ntroduce a new number system defned n the defnton.1. b b4bc Defnton.1: If s a root of x bxc 0 (where bc, 0 ),, each non-negatve nteger X can be b represented by a fnte sum, X a a a... a a a a a a... (3) km ( k 1) m ( k ) m m m m m 3m where a, m, kare ntegers and m 0, 0,1,,3,.... a (, ). Ths s called the generalzed ph number system (GPNS). SPIE-IS&T/ Vol M-

3 The new GPNS has followng propertes: 1) The sequence/coeffcent ak can be calculated by usng the followng teraton v1 X; a1 Q( v1); 1: v 1 ( v a); for a 1 Q( v 1) 1 v 0 whle Qv ( ) 1 v 0 ) The new GPNS s ether a ratonal or rratonal number system. The coeffcents of x bxc 0 determne that the base- s a ratonal or rratonal number. 3) It can be shown that X a a f 1, and m0 N ( N1) m km km k k km k1 kn1 1 (4) 1 f 1, and m km k k0 k0 1 (5) (6) k k 0 1 f 4) By varyng ts coeffcents, bcma,the,,, GPNS be specfed nto many well-known number systems, ncludng tradtonal ph number system, bnary numeral system, ternary numeral systems and many other arbtrary base systems. Bnary number system If b1, c, m1, and a {0,1}, then x x0,. The GPNS turns nto the tradtonal bnary numeral system defned by, where a {0,1}, and 0,1,,3,..., k. X a a a... a a a a a a... (7) The bnary numeral system s a base- number system. It s wdely used n many felds such as dgtal electroncs, dgtal communcatons, and all modern computers. Ph number system If b1, c 1, m1, and a {0,1}, then 1 5 x x10,, the equaton (3) changes to X a a a... a a a a a a... (8) k k 1 k 1 3 The GPNS becomes the tradtonal ph number system. The equaton (8) s another format of the equaton (). Ternary -representaton If b1, c 1, m, and a { 1,0,1}, then 4] defned by, 3 5 x x1 0,.618. The GPNS s the ternary -representaton [3, X a a a... a a a a a a... (9) k ( k 1) ( k ) SPIE-IS&T/ Vol M-3

4 where a { 1,0,1}, and 0,1,,3,..., k. The ternary -representaton s the ternary symmetrcal number system wth the ternary numerals{ 1,0,1}. It has rratonal base the square of the golden rato. It can be used n computers for arthmetcal operaton control [3, 4]. Ternary numeral system If b1, c 6, m1, and a {0,1,}, then system [6, 7]. It s defned by, where a {0,1,}, and 0,1,,3,..., k. x x 60, 3. The GPNS s the ternary numeral system a base-3 number X a 3 a 3 a 3... a 3 a 3 a 3 a 3 a 3 a 3... (10) Balanced ternary numeral system If b1, c 6, m1, and a { 1, 0,1}, then defned by, where a { 1,0,1}, and 0,1,,3,..., k. x x 60, 3. The GPNS s the balanced ternary numeral system [8], It s X a 3 a 3 a 3... a 3 a 3 a 3 a 3 a 3 a 3... (11) The balanced ternary numeral system s also called the ternary symmetrcal number system [3, 9]. It s used n comparson logc and ternary computers. Table 1. GPNS representaton of ntegers. GPNS b1, c, m1, a {0,1} b1, c1, m1, a {0,1} b1, c1, m, a { 1,0,1} b1, c6, m1, a {0,1, } b1, c6, m1, a { 1,0,1} Bnary number Ternary representaton system numeral system Ternary numeral Balanced ternary Ph number system system Note: ndcates -1 n the table. SPIE-IS&T/ Vol M-4

5 Examples of new number systems Case #1: Case #: b1, c, 1 7 b3, c 1, 1 11 Case #3: b7, c1, Table I shows the dfferent representatons of ntegers from 1 to GPNS BIT-PLANE DECOMPOSITION Benefted from the GPNS propertes, we ntroduce a new GPNS-based mage bt-plane decomposton. The tradtonal bnary bt-plane decomposton [30] ntends to decompose mage nto several bnary bt-planes. Each btplane contans the correspondng bts of the bnary representaton of all mage pxels. For example, a grayscale mage can be decomposed nto eght bnary bt-planes. The 4th bt-plane conssts of the 4th bts of all mage pxels. The tradtonal bnary bt-plane decomposton has been used n mage processng such as edge detecton [31], mage codng and compresson [3-34], as well as for securty applcatons such as mage encrypton [35-37], data hdng [38, 39], watermarkng [40] and steganography [41, 4]. Bt-plane 5 Bt-plane 4 Bt-plane 3 Bt-plane Bt-plane 1 Bt-plane 0 Fg. 1. Image bt-plane decomposton usng the balance ternary numeral system (base-3) wth numerals { 1,0,1}. SPIE-IS&T/ Vol M-5

6 Bt-plane 11 Bt-plane 10 Bt-plane 9 Bt-plane 8 Bt-plane 7 Bt-plane 6 Bt-plane 5 Bt-plane 4 Bt-plane 3 Bt-plane Bt-plane 1 Bt-plane 0 Bt-plane -1 Bt-plane - Bt-plane -3 Bt-plane -4 Bt-plane -5 Bt-plane -6 Bt-plane -7 Bt-plane -8 SPIE-IS&T/ Vol M-6

7 Bt-plane -9 Bt-plane -10 Bt-plane -11 Bt-plane -1 Fg.. Image bt-plane decomposton usng the Ph number system (base-) wth numerals {0,1}. Bt-plane 6 Bt-plane 5 Bt-plane 4 Bt-plane 3 Bt-plane Bt-plane 1 Bt-plane 0 Bt-plane -1 Bt-plane - Bt-plane -3 Bt-plane -4 Bt-plane -5 Bt-plane -6 Fg. 3. Image bt-plane decomposton usng the ternary -representaton (base- ) wth numerals { 1,0,1}. SPIE-IS&T/ Vol M-7

8 The pxel ntensty values of a dgtal mage are non-negatve ntegers. In the same manner of bnary bt-plane decomposton, a dgtal mage can also be composed nto several GPNS bt-planes. Ths s called GPNS bt-plane decomposton. Snce the base of the GPNS could be an rratonal number, a ratonal number, or an nteger. Therefore the GPNS bt-planes may consst of bnary bts or arbtrary-base bts. Moreover, the tradtonal bnary bt-plane decomposton s a specal case of the GPNS bt-plane decomposton when b1, c, m1, and a {0,1}. Fg.1-3 provdes decomposton results of a grayscale Lena mage usng the GPNS bt-plane decomposton. Fg. 1 shows mage decomposton usng the balance ternary numeral system (base-3) wth numerals { 1,0,1}. Usng the Ph number system (base-) wth numerals {0,1}, the mage s decomposed nto twenty-fve bt-planes shown n Fg.. Usng the ternary -representaton (base- ) wth numerals { 1,0,1}, Fg. 3 demonstrates that an mage s decomposed nto thrteen bt-planes. 4. IMAGE ENHANCEMENT USING GPNS-BASED BIT-PLANE DECOMPOSITION For a specfc base of the GPNS, an mage can be decomposed nto a certan number of bt-planes wth specfc numerals. Ths decomposton result could be used for other applcatons n mage processng. Ths secton nvestgates the applcaton of the GPNS bt-plane decomposton for mage enhancement. A new mage enhancement algorthm s ntroduced. Some results of mage enhancement are also presented. 4.1 New Image Enhancement Algorthm The Parameterzed Logarthmc Image Processng (PLIP) model s a mathematcal framework [43]. Its operatons make use of a parameterzed gray-tone functon (, ) g jdefned by, where f (, j) s the orgnal mage ntensty. The multplcaton of two gray-tone mages, g1, g n the PLIP model s defned by, where transform and ts nverse transform are defned by g(, j) ( M) f(, j) (1) 1 g * g ( g ) ( g ) (13) 1 1 1/ 1 g ( ) ( ) ln 1 and g g M ( g) ( M) 1exp ( M) ( M) The PLIP multplcaton has been demonstrated to be able to yeld vsually appealng mages [43]. Integratng the PLIP multplcaton wth the GPNS bt-plane decomposton, we ntroduce a new algorthm for mage enhancement, called. The algorthm s shown n Fg. 4. U bsug OL!d!L EUIJSU( IWSdE cowpue bisue!wsag Fg. 4. Image Enhancement Algorthm usng the GPNS bt-plane decomposton and PLIP Multplcaton SPIE-IS&T/ Vol M-8

9 The new enhancement algorthm frst decomposes the nput mage nto n GPNS bt-planes wth a specfc base-. Usng the defnton.1, the algorthm combnes the frst k GPNS bt-planes to obtan the sub-mage #1. The sub-mage # s generated from the rest GPNS bt-planes by the same manner. The enhanced mage s obtaned by mergng these two sub-mages usng the PLIP multplcaton. As shown n equatons (8) and (9), the varaton of parameters, μ(m), (M), and, wll change the enhancement results of the presented algorthm by modfyng the propertes of the PLIP multplcaton. When (M) s negatve, the parameter wll play mportant role for the PLIP multplcaton. The new algorthm has potental for enhancng dfferent types of mages such as grayscale mages, bometrcs and medcal mages. 4. Smulaton Results The presented algorthm has been appled to more than 6 mages. Fg. 5 shows the enhanced results of four dfferent types of mages. The enhanced mages are more vsually pleasurng and recognzable n detals than the orgnal ones. Ths demonstrates that the new algorthm shows excellent performance for enhancng the contrast and detals regons n dfferent types of mages. (a) Grayscale mage (b) Satelte mage (c) MRI mage (d) X-ray mage (e) (f) (g) (h) Fg. 5. Image enhancement usng the presented algorthm. (a)-(d) shows the orgnal mages; (e)-(h) shows the correspondng enhanced mages. Ths demonstrates that the presented algorthm can enhance dfferent types of mages. 5. CONCLUSION Ths paper has ntroduced a new generalzed ph number system, whch has shown the ablty of drvng several exstng rratonal-, ratonal- and nteger-base number systems. Its applcatons for mage decomposton and enhancement have been nvestgated. We ntroduced a new parametrc mage bt-plane decomposton method usng the new GPNS. It can decompose an mage nto dfferent number of bt-planes accordng the specfcaton by parameters. Combnng ths decomposton method wth the PLIP multplcaton, we have ntroduced a new mage enhancement algorthm. The algorthm has shown excellent performance for mprovng vsual qualty of dfferent types of mages. SPIE-IS&T/ Vol M-9

10 REFERENCES [1] Johnson, E. L., Dgtal desgn: a pragmatc approach, PWS Publshng Co., Boston, MA, USA (1987). [] Hwang, K., Computer arthmetc: prncples, archtecture and desgn, John Wley & Sons, Inc., New York, NY, USA (1979). [3] Hurst, S. L., "Multple-Valued Logc--ts Status and ts Future," IEEE Trans. Comput., C-33(1), (1984). [4] Mrsaleh, M. M. and Gaylord, T. K., "Logcal mnmzaton of multlevel coded functons," Appl. Opt., 5(18), (1986). [5] Fu, H., et al., "Optmzng Logarthmc Arthmetc on FPGAs," Feld-Programmable Custom Computng Machnes, 007. FCCM th Annual IEEE Symposum on, (007). [6] Garner, H. L., "The resdue number system," n Papers presented at the the March 3-5, 1959, western jont computer conference ACM, San Francsco, Calforna (1959). [7] Soderstrand, M., et al., Resdue number system arthmetc: modern applcatons n dgtal sgnal processng, (1986). [8] Balakrshnan, W. and Burgess, N., "Very-hgh-speed VLSI s-complement multpler usng sgned bnary dgts," Computers and Dgtal Technques, IEE Proceedngs E, 139(1), 9-34(199). [9] Hasheman, R. and Sreedharan, B., "A hybrd number system and ts applcaton n FPGA-DSP technology," Informaton Technology: Codng and Computng, 004. Proceedngs. ITCC 004. Internatonal Conference on, (004). [10] Bajard, J.-C., et al., "Modular Number Systems: Beyond the Mersenne Famly," n Selected Areas n Cryptography, Handschuh and Hasan, Eds. Sprnger Berln / Hedelberg, (005). [11] Ward, R., "On Robustness Propertes of Beta Encoders and Golden Rato Encoders," IEEE Trans. Inf. Theory, 54(9), (008). [1] Daubeches, I., et al., "The Golden Rato Encoder," IEEE Trans. Inf. Theory, 56(10), (010). [13] Daubeches, I. and Ylmaz, O., "Robust and Practcal Analog-to-Dgtal Converson Wth Exponental Precson," IEEE Trans. Inf. Theory, 5(8), (006). [14] Bergman, G., "A Number System wth an Irratonal Base," Mathematcs Magazne, 31(), (1957). [15] Rousseau, C., "The Ph Number System Revsted," Mathematcs Magazne, 68(4), 83-84(1995). [16] Wkpeda. Golden rato base. Avalable from: [17] Knuth, D. E., Art of Computer Programmng, Volume I: Fundamental Algorthms, 3 ed. Addson-Wesley Professonal (1997). [18] Levasseur, K. The Ph Number System. Avalable from: [19] Wessten, E. W. Ph Number System. Avalable from: [0] wordq. Golden mean base - Defnton Avalable from: [1] Wkpeda Avalable from: [] Knott, R. Usng Powers of Ph to Represent Integers (Base Ph). Avalable from: [3] Stakhov, A., "Brousentsov's Ternary Prncple, Bergman's Number System and Ternary Mrror-symmetrcal Arthmetc," The Computer Journal, 45(), 1-36(00). [4] Stakhov, A. Ternary mrror-symmetrcal representaton. Avalable from: [5] Cet, M., et al., "Tradng Inversons for Multplcatons n Ellptc Curve Cryptography," Desgns, Codes and Cryptography, 39(), (006). [6] Hayes, B., "Thrd Base," Amercan Scentst, 3(4(1908). [7] Wkpeda. Ternary numeral system. Avalable from: [8] Wkpeda. Balanced ternary. Avalable from: [9] Museum, T. M. o. Nkolay Brusentsov - the Creator of the Trnary Computer. Avalable from: [30] Gonzalez, R. C. and Woods, R. E., Dgtal Image Processng, 3 ed. Pearson Prentce Hall (007). SPIE-IS&T/ Vol M-10

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