Fused Number Representation Systems and their Barcode Applications

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1 Best Studet Paper Award Fused Nuber Represetatio Systes ad their Barcode Applicatios Sarkis Agaia* Departet of Electrical Egieerig, Staford Uiversity, 450 Serra Mall, Staford, Califoria ABSTRACT I this paper we focus o: a) ehacig the perforace of existig barcode systes ad b) buildig a barcode syste for obile applicatios. First we itroduce a ew cocept of geeratig a paraetric uber represetatio syste by fusig a uber of represetatio systes that use ultiplicatio, additio, ad other operatios. Secod we show how oe ca geerate a secure, reliable, ad high capacity color barcode by usig the fused syste. The represetatio, sybols, ad colors ay be used as ecryptio keys that ca be ecoded ito barcodes, thus eliiatig the direct depedece o cryptographic techiques. To supply a extra layer of security, the fused syste also allows oe to ecrypt give data usig differet types of ecryptio ethods. I additio, this fused syste ca be used to iprove iage processig applicatios ad cryptography. Keywords: Barcode, Multiple Base Represetatio, Fiboacci/Lucas p-codes, Double Base Represetatio, Cell/ obile phoes 1. INTRODUCTION Sice the 1960 s, barcodes have provided optical achie-readable represetatios of data. Iitially used to label railroad cars, barcodes are ow laid o alost every ediu, fro paper ad cardboard to plastics ad etals ad fro ski to DNA 1. Though other techologies are beig developed for cosuers to sca obects, icludig radio waves, coputer chips or satellite locatio systes, barcodes aitai their popularity due to their low cost, low rate of error, high iforatio capacity, ad portability 2. The potetial ad dead for barcode techology cotiues to icrease with costatly iprovig digital techologies. The recet itegratio of these techologies, aely through the pairig of obile caera phoes ad high desity barcodes, ca fulfill the dead for electroic data exchage. For istace, i Japa ad Korea, people ca save ad exchage Facebook iforatio, receive the utritioal iforatio about their McDoald s eal, board airplaes usig their phoes ad wireless payets 3. Despite its huge potetial, however, the use of caera phoes to sca barcodes is ot widespread. Resolutio liits, distortio, blurrig, ad oise iduced by the phoe caera hider the direct use of ost existig bar codes for obile phoes 4. These probles are exacerbated whe dealig with caera phoes that geerate low to ediu quality iages (less tha 640 x480 pixels) 5. Questios that have bee raised iclude: Are the existig barcodes suitable for scaig by obile phoes? Is it ecessary to desig a ew barcode tailored for obile phoes 4? As private wireless trasactios ay be proe to hackig ad piracy, a additioal questio is how ca ake barcodes ore secure? Copaies have developed their ow uique, proprietary barcode forats that are better geared toward obile applicatios 2. These developets have resulted i desigs that vary draatically fro those of traditioal barcodes, which represet data i the widths ad spacigs of black parallel lies. These barcodes, also kow as liear or 1D (1 diesioal) barcodes or sybologies carry a liited aout of iforatio with few security features. Other *sagaia@staford.edu Multiedia o Mobile Devices 2010, edited by Reier Creutzburg, David Akopia, Proc. of SPIE-IS&T Electroic Iagig, SPIE Vol. 7542, SPIE-IS&T CCC code: X/10/$18 doi: / SPIE-IS&T/ Vol

2 sybologies, such as 2-D ad color barcodes, have thus bee looked at i a effort to ake barcode systes ore efficiet, accessible, ad secure. May of these sybologies are siilar to their liear couterparts i that they are based o biary represetatios. They differ, however, i the aout of iforatio they ca hold. I exchage for the icreased iforatio cotet, 2D ad color barcodes have also icreased the coplexity of recogitio algoriths, as they require greater iage segetatio 6. Thus, due to the varyig resolutios of obile caera phoes, the iagig quality of a give caera phoe ay ot be satisfactory for decodig a high desity 2D or color barcode 4. I additio, iage capture by the phoe caera i a variety of lightig coditios ay ot retai eough discriiate iforatio of color ad grayscale. I the followig, we itroduce a ew represetatio syste to geerate secure, reliable, ad high capacity barcodes that are easier for cell phoes to decode. We call this represetatio syste the Multiple Base Nuber Syste (MNBS). MBNS fuses several uber represetatio systes icludig the double base uber syste, which has recetly bee itroduced to offer ore sparse ad efficiet eas of represetig data. MBNS ca be applied to barcode sybols with a variety of widths, colors, ad vertical arrays to represet ore iforatio. The rest of this paper is orgaized as follows: Sectio II itroduces soe ecessary backgroud iforatio. A cocept of geeratig fused uber represetatio syste is provided i Sectio III. I Sectio IV, the fused uber represetatio syste is used to costruct ore efficiet ad error-resistat barcode systes. Sectio V offers cocludig rearks ad suggestios for future work. 2. BACKGROUND: NUMBER REPRESENTATIONS Below we review soe uber represetatios, ost otably Fiboacci ad Lucas ubers ad their geeralizatios. We also itroduce ew Fiboacci ad Lucas p-uber based paraetric represetatio uber systes, which are later ipleeted i developig color barcodes. Fiboacci ad Lucas Sequeces: Fiboacci ad Lucas (F-L) sequeces iclude weighted F-L ubers, F-L p- ubers, -diesioal (Meta) F-L ubers, ad rado F-L ubers. I this article, we focus o sequeces that are geerated usig the first two classes of ubers. Fiboacci ad Lucas (F-L) Nubers 7 : Fiboacci (f ) ad Lucas (l ) ubers ca be preseted recursively as f f + f, l = l l, 2 (1) k = k 1 k 2 k k 1 + k where f 0 =0, f 1 =1, l 0 =2, l 1 =1, l 2 =3, k 2.Subsequet ters of F-L ubers are thus defied as the su of their two predecessors. Note that a Lucas uber l k ca be also expressed i ters of the Fiboacci ubers as: l = 2 f + f ; l = f + f ; f = ( l + l )/5, k k k 1 k k 1 k+ 1 k k 1 k+ 1 (2) Weighted Fiboacci ad Lucas (F-L) P-Nubers 7 : Nubers that are give weight based o their coefficiets ad are defied by the followig recurrece: 0 if 0, (p) F = 1 if 0< p + 1, (3) (p) (p) af -1 + bf-p-1 if >p+1 p+ 1 if = 0, (p) L = 1 if 0 < p, (4) (p) (p) al -1 + bl-p-1 if >p where a ad b are soe costats ad p is a o-egative iteger. I the followig, we cosider the case i which a = b = 1. Note that F-Lucas p-uber represetatios iclude a ifiite uber of various represetatios. For SPIE-IS&T/ Vol

3 values p=0 ad p=1, for istace, the F-L p-represetatios respectively reduce to the well-kow biary ad F-L systes previously discussed. As p, the F-L p-represetatios becoe the uitary code i which X = Other uber systes ca be geerated by varyig the value of p. The iitial sequeces for the X. first five p-values are give i the table below. Table of Fiboacci ad Lucas P-Nubers P Fiboacci p-ubers Lucas p-ubers 0 0,1,2,4,8,16,32,64,128,512,1024 1,1,2,4,8,16,32, 64,128,256,512,1024, ,1,1,2,3,5,8,13,21,34,55,89,143 2,1,3,4,7,11,18,29,47,76, 123, 199, ,1,1,1,2,3,4,6,9,13,19,28,41,60 3,1,1,4,5,6,10,15,21,31,46, 67, ,1,1,1,1,2,3,4,5,7,10,14,19,26,36,50,69,95.. 4,1,1,1,5,6,7,8,13,19,26, 34, ,1,1,1,1,1,2,3,4,5,6,8,11,15,20,26,34,45,60,80 5,1,1,1,1,6,7,8,9,10,16,23, 31 Paraetric Represetatios: Cosider the followig expasio of a iteger X X= arku (, ) + a Rk ( 1, u) + a Rk ( 2, u) ar(2, u) + ar(1, u) + a, a Z (5) k k 1 k where Rku (, ) is a paraetric base sequece of the uber syste. The table below gives soe saple represetatio systes we ca geerate fro (5). Table of Paraetric Represetatios Case R(k,u) Represetatio Siged Biary a 2 + a 2 + a a 2 + a 2 + a, a { 1,0,1} (6) 2 k 8 k k 1 k k k 1 k Siged Terary 3 k 9 k k 1 k bk 3 + bk 13 + bk b23 + b13 + b0, b { 1,0,1} or {0,1, 2} (7) Siged Geeral r k 10 k k 1 k ckr + ck 1r + ck 2r c2r + c1r + c0, c {0, ± 1, ± 2,..., ± ( r 1)} (8) P-Fiboacci Double Base F (p) 7 (p) (p) (p) (p) (p) k bk k k-1 k-1 b2f 2 b1f 1 b F + b F , {0,1}, F is a Fiboacci p-uber X (9) p k q u 11 k u k 1 u 1 k 2 u 2 d k, up q + d k 1, u 1p q + d k 2, u 2 p q d 0,0, d, {0,1}, (10) ( geeralized ) k < u, 0 d < k; ( specifically) k = 2, u = 3, Prie P k 12 ep k k + ek-1p k ep ep1, e {0,1}, P is a prie uber X (11) May such systes are sparse but redudat. We are iterested the coditios that lead to these systes uique represetatios ad iial expasios. For istace, it has bee show that the represetatio R(k,u) = q k, where q is a iteger 2 is uique ad iial if two cosecutive digits cc 1 = 0 for all k 8, or the coefficiets c, 0,1,2,..., k = satisfy the followig two coditios 13 : c + c + 1 < r, for all c < c + 1, if c c + 1 < 0 Case 1 ca also be exteded to fidig a uique Fiboacci represetatio (Zeckedorf s Theore). Such a represetatio for a iteger X exists if: X=F k -1, k=1,2,3,4 Both Fiboacci ad q k represetatios have bee applied i the optial desig of arithetical hardware 8, codig theory 6, ad cryptography 11,14. k k, SPIE-IS&T/ Vol

4 3. FUSED NUMBER REPRESENTATION SYSTEMS I this sectio we itroduce the Multiple Base Nuber Syste (MBNS), a collectio or fusio of differet uber systes. MBNS allows us to represet a iteger X with various systes{ S (p) (q) (l), Τ,...,Z k} i the for X = s S Τ *...* Z, p, q, l = 0,1,2,... (12) i,,.., k (p) (q) (l) i, k where * is a arbitrarily chose fusio operatio (i.e., additio, ultiplicatio, liear cobiatio, etc.) ad si, is either real or iagiary. Soe oteworthy s i values iclude s i, {0,1}, s i, {-1,0,1}, s i, {0,1,} where = 1. (p) (q) (1) Also to be oted is that S, Τ,..., Z k ca be both real ad coplex uber systes. Chagig the paraeter values i these systes provides greater represetatio possibilities. The ap below sus up the basic idea behid this ew uber syste. I the followig, we focus o a subset of MBNS, aely the two base represetatio syste which allows us to fuse two uber systes together. The ideas below, however, ca be exteded i fusig ultiple other uber systes. (p) (q) Two Base Represetatio Syste: A subset of MBNS i which two uber systes, S,Τ, are fused ito a sigle paraetric represetatio of the for: X = e S Τ, p, q = 0,1, 2... (13) i, (p) (q) i, e {0,1}, or e { 1,0,1}, e D = {0,..., q }, q = 1,2,... e {0,1,, + 1} = 1 etc. i, i, i, i, where, {0,1,2,3 } ad p ad q are paraeters. Exaples of (13), i which we let * be the ultiplicatio operatio, are provided below to illustrate the for s versatility. S = 2, T = 1, e {0,1}, gives the radix-2 represetatio (see 6); p q i, S = 3, T = 1, e {0,1,2}, gives the radix-3 represetatio (see 7); p q i, S 2 T, ei, {0,1} p q T ei, = =, gives the radix -4 represetatio; S = 2 or 3, = 1, { 1,0,1}, gives the siged digit uber syste (see 6,7); S p 2, q 3 T, ei, {0,1} S p p or p, q F L T 1, ei, {0,1}, = =, gives the double-base represetatio (see 10); = = gives the F-L p-represetatio(see 9); S = F or L, T = q, e { 1,0,1}, gives a so called siged-digit F-L p- represetatio. p p p q i, SPIE-IS&T/ Vol

5 S = F or L, T = F or L, e {0,1}, gives a ew, so called double F-L base p- represetatio. p p p q q q i, These exaples ephasize the ultiplicatio represetatio systes of Fiboacci ad Lucas ubers ad their variatios (i.e., o-weighted/ weighted F-L p-ubers), though they ca be expaded to iclude Golde Ratio ubers ad other systes. We ote that it is also possible to fuse these represetatios usig additio: X = e {as + bτ + d}, p, q= 0,1,..., k 1, (14) i, (p) (q) i, where a,b ad d are costats. This liear cobiatio of systes ay lead to iterestig represetatios such as X = ek{a kfk + bkl k} where a kad b kare costats: i.e., a k=, b k= or (15) k Defiitio: A uber represetatio of a iteger X is called a caoical or oral for if it is a stadard way of presetig X. For exaple, the double base represetatio is a caoical for whe d = e is i.15 Below we preset a approach for deteriig the ear caoical for of a certai two base represetatio. Two Base Represetatios ad the Greedy Algorith: Oe way to fid a certai two base represetatio of a iteger is to use the Greedy approach 11, which deteries the best approxiatio of, coputes the differece, ad reapplies the process. The geeral forula that describes this two base represetatio is 1 l = siaa i B i = 1 b, (16) where s i is a set of predefied coefficiets. Note this forula allows for the itroductio of otrivial coefficiets i b ai ai b b ai ultiple base expasios. To illustrate the forula, let = Settig A = 3 or F, B = 2 or F, we ca express as a su of double base ters or Fiboacci products usig the Greedy Algorith. Double Base Expasio: = = = Fiboacci Expasio: = = = = Note that we call ay syste of expressig a iteger with ixed Fiboacci ubers i the for Double Fiboacci Base Nuber Represetatio Syste (referred to as FBNS or Fiboacci for short)., d F F, The Greedy Algorith does ot always produce a caoic represetatio. For exaple, cosider the iteger X=49 expressed i DFNS. The represetatio produced by the Greedy Algorith is X=21*2+6*1+1*1. However, the caoic represetatio (oly oe such for exists for the iteger 49) is X=13*3+5*2. Deteriig the caoical for of such a two base represetatio is ofte difficult, especially for very large itegers. The Greedy Algorith is thus used as a straightforward eas of producig a ear-to-caoic for. Itegers represeted usig the Greedy Algorith ca have varyig represetatio efficiecies based o the chose two base represetatio syste. That is, the iial uber of ozero digits eeded to represet each iteger chages with respect to the chose represetatio syste of the for (16). I the followig, we copare two represetatio systes geerated fro (16), aely the double base ad the Fiboacci expasios, to see which syste is ore efficiet i represetig itegers. We begi our copariso betwee the double base ad Fiboacci expasios by aalyzig the possible direct products or ters geerated by each syste. the SPIE-IS&T/ Vol

6 These tables provide all double base ad Fiboacci products less tha 10,000. Note that ters <100, >100 ad <1000, >1000 ad <10,000 are represeted with gree, yellow, ad orage cells, respectively. Fro the data tables, we ca see that the Fiboacci ad double base expasios are both very sparse. Still, the Fiboacci syste sees to represet ay ubers with fewer ters. We ca attribute this to the fact that the Fiboacci syste icreases at a slower pace (though still quite quickly) tha the double base syste. As a result, ore itegers are represeted as direct products of Fiboacci ubers tha of biary ad terary ubers. For exaple, we ca see that there are 101 differet Fiboacci products less tha 10,000 (26 below 100, 32 below 1000 ad above 100, ad 43 below 10,000 ad above 1000). I copariso, there are oly 67 double base products less tha 10,000 (20 below 100, 20 below 1000 ad above 100, ad 27 below 10,000 ad above 1000). A iterestig observatio here is that as we allow the uber liit to icrease (say fro below 1000 to below 10000), the uber of Fiboacci products icreases ore tha the uber of double base products (double base: 20 27, Fiboacci: 32 43). Copariso of Double Base ad Fiboacci Products Less tha 10,000 Ters Below Double Base Fiboacci Ters I Coo , I other words, the differece i the uber of Fiboacci ad double base products icreases as we allow our uber rage to icrease (fro 12=32-20 to 16=43-27). Geerally, give its larger represetatio of ubers, which becoes ore proiet as we allow our uber rage to icrease, the Fiboacci product sequece is ore efficiet i represetig ubers while aitaiig a high degree of sparseess. We test this idea o various itegers. The first table below provides ear caoical Fiboacci ad double base expasios of itegers 100 calculated usig the Greedy Algorith. I early half of the itegers cosidered oe ethod of expasio fares better tha the other. That is, oe syste uses fewer ters to represet the respective iteger. I 30 cases (represeted by gree cells), the Fiboacci syste is ore efficiet i represetig the give iteger. I cotrast, the double base syste is ore efficiet i oly 19 cases (represeted by red cells). For the reaiig cases (represeted by o-shaded cells), both represetatio systes use the sae uber of ters to express the give iteger. We ca cotiue testig this idea. SPIE-IS&T/ Vol

7 Fro the table secod table below, we see that the Fiboacci syste is ore efficiet 37 ties (represeted by gree cells) while the double base is ore efficiet oly 21 ties (represeted by red cells). This data further affirs the idea that Fiboacci systes are ore ofte tha ot ore efficiet i represetig give itegers. That is, the Fiboacci systes represet ay ore ubers with fewer ters tha double base systes. As the uber of Fiboacci products icreases ore tha the uber of double base products, we expect the Fiboacci syste to geerally becoe eve ore efficiet as our give iteger value icreases (of course, there are exceptios to this expectatio: cosider, for istace, the above exaple X= discussed above).. SPIE-IS&T/ Vol

8 4. APPLICATIONS We will ow apply MBNS represetatios such as FBNS to costruct ore efficiet ad error resistat barcode systes. We cosider 2D barcodes as they expad traditioal 1D barcode iforatio capacities by addig the vertical array of bars ad spaces. By exaiatio of obile phoe caeras ad the oral use cases, the iiu requireets of the 2D codes for caera phoes are idetified 4. They are a) Matrix codes are preferred to Stacked codes. b) Code size should grow proportioally to the data ad have o sudde ups. c) Code should be easily detected ad read regardless of its size ad data aout. d) Code should have read flexibility so that it ca be read uder ay agle. 2D code should also be efficiet ad widely supported. For istace, although 2D barcodes ca be of ay shape, ost are ade up of squares ad rectagles due to pixel efficiecy. Moreover, 2D barcodes are ofte ot foud i color ad grayscale as black ad white barcodes are ore widely supported ad allow for faxig ad photocopyig without losig readability. Takig these criteria ito accout, we propose represetig the iput data with MBNS represetatios such as DBNS ad FBNS. The procedures for doig so are outlied i the flow chart below: Note that for ehaced security, both algoriths ay also use ecryptio techiques. As MBNS represetatios are sparse, they are efficiet i represetig give data ad ca lead to high perforace barcodes. I geeral, these systes are also well equipped for error correctio. I the followig, we test our algoriths agaist the traditioal base 2 represetatio used i QR code 3. Though our algoriths are explaied through a ueric exaple, they ca be exteded to differet types (ueric, alphabetic, alpha-ueric, etc.) ad laguages of iforatio. Algorith 1 (Illustrative Exaple): Usig the Double Base Nuber Syste to Represet Iput Data Separate the data ito oe bit groups. 2. Choose a represetatio syste: DBNS or FBNS. 3. Deterie each groups DBNS represetatio, for istace by usig the Greedy Algorith. 4. Fill i MNBS tables for each group represetatio ad alig the adacetly to oe aother. Though we ca select fro various represetatio sybols ad colors, we here costruct our barcode fro black ad white squares for pixel efficiecy. We also add a boarder to our barcode by a outliig row ad colu. This way we ca easily preserve the table forat that we used i our geeratio. DBNS Table ad Barcode SPIE-IS&T/ Vol

9 FBNS Table ad Barcode* *Note that because of repetitio, we ca costruct odified FBNS tables without losig represetatio possibilities. I this case, we ca igore the last two rows (or colus). We ca also represet the uber 9 as 8*1+ 1*1 istead of as 3*3, ad get rid of oe additioal colu (or row). The FBNS barcode above is geerated usig this idea. Algorith 2: We ow provide several versios of Algorith 2 ad apply the to codig the data Note that due to redudacy i systes such as FBNS, we are ofte ot liited to shadig i a particular cell ad have the possibility of choosig a cell we fid ore coducive to easy readig (i.e., a cell that is ot adacet to ay other filled cells). Coordiate Code Coordiate Code Nuber of Code Coordiate Code Nuber of Code Coordiates Coordiates Coordiate Code Versio A Versio B Versio C Versio D As the charts above illustrate, we ca vary ay of eleets of our algoriths icludig the uber of eleets i a group, the represetatio syste, the code for represetig coordiates, the atrix holdig the code represetatio, ad the barcode shape, color, ad boarder. We ca also vary the uber of represetatio systes to costruct higher diesioal barcodes. For istace, we ca us three base represetatios (istead of the two base used here) to costruct 3-D barcodes. Error Correctio ad Code Versios: The code versios are geerated usig Hadaard atrices 16, which ca be used to defie soe error correctio codes. Note that code versios ay also be geerated to use other error SPIE-IS&T/ Vol

10 correctio codes such as Reed-Soloo code. Though Versio A ay be the ost copact, it ay ot be used for all MBNS systes ad does ot provide the error correctio capabilities of the others. Hadaard code ca be used to correct 4 1 errors ad detect 4 errors i a -bit ecoded block. The table below illustrates soe of the various levels of error correctio usig versios of Algorith 2. Though code that ca reach a data correctio level of 15% is regarded as satisfactory 17, we ca reach a level of up to 25% Levels of Error Correctio Usig Algorith 2 # Bits/ Coordiate Error Detectio Capability Error Correctio Capability Versio Bits % Bits % A B,C,D E F Copariso to QR Code: We ow test our ethod agaist that used i QR Code. I order to ake this copariso, a brief overview of how QR code is geerated is provided below. Iput data i QR Code ca be represeted i several odes, as illustrated i the table below. Each ode has a correspodig four-bit-log biary represetatio as well as a separate bit cout for a group of iput characters. For istace, the biary represetatio ad reserved bit cout for ode ueric is 0001 ad 10, respectively; for alphaueric ode, 0010 ad 9, respectively. The biary represetatio correspodig to the iput data is placed at the begiig of the ecoded QR code. For istace, if we were to ecode the zip code 78249, we would begi the code with the bits 0001 ad reserve 10 bits for every group of ueric characters i the iput data. Below we fiish ecodig the zip code i order to specifically illustrate how QR ecodig works i ueric ode. I order to represet i ueric ode, we group the iput data with characters or bits of 3 or less. The data is separated ito oe 3 bit ad oe 2 bit group: 782 ad 49. Each full 3 bit group is reserved a 10 bit biary represetatio. However, if the group cotais fewer characters, say 2 or 1 bits, the a 7 or 4 bit biary represetatio is used respectively. Ecodig 782 ad 49 i 10 bit ad 7 bit biary represetatios, respectively, we get: Addig o the biary represetatio correspodig to the QR ueric ode gives: To fiish ecodig the exaples, we take the represetatio arrage it ito 8 bit groups. 0 s are added to groups at the ed of these represetatios that have less tha 8 characters: This is doe to fit the size of the QR code, which varies accordig to which versio is used. QR code has versios fro 1 to 40, where versio 1 (used here) is a 21x21 atrix ad each followig versio icreases i legth ad width by 4 cells so that versio 40 is a 177x177 atrix. It should be oted that QR code is utilized up to versio 10 for caera phoes as a greater desity exceeds the capabilities of a caera as a iager ad thus caot be successfully decoded (Kato, Ta). This fial code ca be apped to a barcode to for: QRcode has four levels of error correctio: L, M, Q, ad H. About 7%, 15%, 25%, ad 30% or less of errors ca be corrected with each respective level. It is ipleeted usig Reed- Soloo Code, which requires twice the aout of codewords to be corrected. I the preseted exaple, the total codewords that eed to be corrected are 24, requirig 48 additioal error correctig codewords. I copariso, Algorith 1 ad 2 use 50 ad 20 codewords, respectively, to ecode the sae data without error correctio. At this stage, both algoriths offer exceptioally easy reads ad security features. Error correctio features ca be added. For istace, we ca achieve a siilar error correctio level usig Versio E i Algorith 2 with oly 70 codewords. Note that the custo desig capability of both algoriths, icludig variatios i shape, size, color, ad codig represetatio, allows the sybol to provide additioal security. A ore SPIE-IS&T/ Vol

11 geeral copariso of how well QR code, Algorith 1, ad Algorith 2 satisfy the criteria that optiize 2D barcodes for obile phoes 17 is provided below. Copariso of QR Code, Algorith 1, ad Algorith 2 Data Capacity Error Correctio Scaability Desig Security Additioal Features Oi- Directioal Low (VGA) Resolutio High Speed Divisible Scalable Custo QR Code Sall pritout size Al Easy to read, especially at a distace. Legible uder varyig lightig coditios. Al Sall pritout size. Provides additioal error correctio ad ecryptio capabilities 5. CONCLUSION I this paper, we itroduced a cocept of fusig paraetric uber represetatio systes. We deteried that these systes provide sparse, efficiet, ad potetially secure eas of represetig data. Usig the features of these fused systes, we created two algoriths that geerate high data capacity barcodes. These barcodes are easy to read, scalable i size, potetially divisible, ad oi-directioal. Because the aout ad type of iforatio withi the sybol is user selectable, the user ca deterie the desity of the barcode ad thus solve resolutio proble of caera phoes. Moreover, by havig the optio of selectig various represetatio systes, the user is offered a secure eas of represetig data. Oe ca use the algoriths custo desig capabilities, i which sybol characteristics such as shape, size, ad color ca be varied, to provide additioal security ad carry ore iforatio. The geerated barcodes also provide high percetage error correctio ad detectio capabilities usig the Hadaard or other error correctio codes. By copariso, we foud that ay popular 2D barcodes, aely QR code, do ot exhibit all these features. Fially, the proposed algoriths provide a viable alterative to curret 2D barcodes ad show a ew directio i developig secure, reliable, ad high capacity color ad gray level barcodes. REFERENCES [1] Youssef, S.M. ad Sale, R.M., Autoated barcode recogitio for sart idetificatio ad ispectio autoatio, Expert systes with applicatios: A iteratioal oural 33(4), (2007). [2] Meg, J. ad Yag, Y., Applicatio of Mobile 2D Barcode i Chia, 4 th Iteratioal Coferece o Wireless Couicatios, Networkig ad Mobile Coputig, 1-4 (2008) [3] Ohbuchi, E. ad Haaizui, H. ad Hock, L., Barcode Readers usig the Caera Device i Mobile Phoes, Proceedigs of the 2004 Itl. Cof. o Cyberworlds, (2004) [4] Wag, H. ad Zou, Y., Caera Readable 2D Bar Codes Desig ad Decodig for Mobile Phoes, ICIP, (2006) [5] Rohs, M., Real-World Iteractio with Caera- Phoes, 2 d Iteratioal Syposiu o Ubiquitous Coputig Systes, (2004) SPIE-IS&T/ Vol

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