Geometric Transformations of BDD Encoded Image

Transcription

1 IAENG Interntionl Journl of Applie Mthemtics, 36:1, IJAM_36_1_1 Geometric Trnsformtions of BDD Encoe Imge Wtis Leelptr Knchn Knchnsut Chichnok Lursinsp Astrct Binr Decision Digrm (BDD) hs een shown to e pplicle in imge coing. A technique presente in this pper etens the functionlit of BDD in imge coing to cover mjor geometric trnsformtions of imge such s trnsltion, rottions, scling n shering. B introucing the concept of trnsition rnch, nonuniform n uniform trnsltion of imge given isplcement coul e otine using onl eclusive-or opertion. Bse on the trnsltion, we show tht imge rottion, scling n shering coul e chieve. Kewors: BDD, imge coing, trnsltion, rottion, scling, shering 1 Introuction Binr ecision igrm (BDD), which cn e use for representing Boolen function [1], is roote cclic grph tht contins nonterminl n terminl noes. Ech nonterminl noe of BDD is ssocite with Boolen vrile, n two terminl noes re ssigne to constnt n 1. Ech nonterminl noe hs two outgoing eges, zero-ege n one-ege, tht link to its chilren. Either one of these eges is tken epening on the vlue of the vrile ssociting with the nonterminl noe. The represente Boolen function cn e evlute trversing through ll noes long the pth from the root to the terminl noe. The interesting propert of the BDD is tht it cn e me cnonicl representtion of Boolen function ppling reuction rules presente in []. The results from ppling reuction rules re tht the size of BDD is reuce n is uniquel representing Boolen function. This ecomes vntge for imge coing when consier lck-n-white imge s Krnugh mp of log M + log N vriles, where M n N re with n height of imge, respectivel [3], [6]. Detil of representing imge using BDD is epline in Section. Imge Representtion In this pper, rectngulr imge of size W H piel is consiere s Krnugh mp of w+h vriles where Computer Science n Informtion Mngement Deprtment, Asin Institute of Technolog, Klong Lung, Pthumthni, Thiln 11. Emil: {wtis,kk}@cs.it.c.th Deprtment of Mthemtics, Chullongkorn Universit Bngkok, Thiln 133. Emil: chichnok.l@chul.c.th w = log W, h = log H n the vrile orering is:, 1,..., w 1,, 1,..., h 1. The h-it n w-it Gr coes re use to represent the rows n columns of the Krnugh mp, respectivel s shown in Figure 1. We will refer to the rows n columns of the Krnugh mp s the Gr coe coorintes throughout this pper..1 Definitions of BDD Components The efinitions of term escriing components of BDD use in our pproch for imge representtion re s follows. Definition 1 A BDD is set of noes V n eges E tht forms n unirecte roote tree. A nonterminl noe v i of BDD is ssocite with Boolen vrile i. The evlution of Boolen vrile i iels outgoing ege e i whose ttriute is when the vrile is evlute s, n is 1 when the vrile is evlute s 1. There is onl one terminl noe T which is ssigne to constnt 1. Definition A pth of BDD is set of noes n eges tht ppers in n lternting sequence {v 1, e 1, v, e,..., e n 1, v n }, where n = w + h, with restriction tht ech noe n ech ege must pper onl once. Definition 3 A rnch B i of BDD is pth whose v 1 is the root of BDD n v n, where n = w + h, is the terminl noe. Using our efinitions, rnch of BDD shown in Figure 1 cn e written s B = {, e 1, 1, e,..., e w 1, w 1, e w,, e w+1, 1,..., e w+h 1, h 1, e w+h, T }. A rnch B of BDD encoes the coorinte (, ) of piel n cn e ecompose into two prts, surnch B which contins informtion of coorinte n surnch B contins informtion of coorinte written s B = {, e 1, 1, e,..., e w 1, w 1, e w, }, n B = {, e w+1, 1, e w+,..., e w+h 1, h 1, e w+h, T }.. Definitions of BDD Opertions Given rnch B i = {v i 1, e i 1,..., e i n 1, v i n} n rnch B j = {v j 1, ej 1,..., ej n 1, vj n} where n = w + h. Definition 4 An eclusive-or opertion etween rnch B i n B j iels rnch B k if n onl if sequence (Avnce online puliction: 1 Ferur 7)

2 ,1,...,h 1 H row X X w X w 1 Y,1,...,w 1 Y h Y h 1 Root Noe e 1 e w 1 e w e w+1 e w+h 1 e w+h Terminl Noe W column 1 X X 1 X Y Y 1 Y root 1 Terminl Noe (c) rnch 1 rnch Figure 1: Binr imge. BDD representtion. (c) Emple of inr imge n its BDD representtion. of noes from oth rnches re orerl ienticl, tht is v i 1 = v j 1, vi = v j,..., vi n = v j n n the rnch B k is etermine B k = {v i 1, e i 1 e j 1,..., ei n 1 e j n 1, vi n} Definition 5 Let p e the numer of eges whose ttriute is one. The prit of rnch is even if p is even, otherwise the prit is si to e o. Definition 6 Left shifting of rnch replces the ttriute of ege e i the ttriute of ege e i+1 n ssigns the ttriute of e n 1 to. Definition 7 Right shifting of rnch replces the ttriute of ege e i+1 the ttriute of ege e i n ssigns the ttriute of e 1 to. 3 Trnsition Brnch Suppose tht B p is rnch of BDD representing piel p t its originl loction. Let p e the piel p fter trnsltion n p is represente rnch B p. The rnch B p cn e otine ppling eclusive-or etween rnch B p n trnsition rnch Bt. 3.1 Trnsition Brnch of Unit Displcement A trnsition rnch Bt = {, e t 1, 1, e t,..., et w 1, w 1, e t w,, e t w+1,..., e t w+h 1, h 1, e t w+h, T } cn e ecompose into two prts, Bt n Bt where Bt = {, e t 1, 1, e t,..., et w 1, w 1, e t w, } n Bt = {, e t w+1,..., e t w+h 1, h 1, e t w+h, T }. The trnsition rnch Bt is constructe epening on the prit of B s follows: Cse 1.1 Prit of B is even. Set the ttriute of e t w of Bt to 1. Set the ttriutes of other eges to. Cse 1. Prit of B is o. Let e e n ege in B n e t e n ege in Bt. Locte the first ege e w j, j w 1, such tht e w j = 1. Set ttriute of e t w j 1 to 1, n ttriute of the remining eges re set to. Similrl, the trnsition rnch Bt is constructe epening on the prit of B s follows: Cse.1 Prit of B is even. Set the ttriute of e t w+h to 1. Set the ttriutes of other eges to. Cse. Prit of B is o. Let e e n ege in B n e t e n ege in Bt. Locte the first ege e w+h k, w + 1 k w + h 1 such tht e w+h k = 1. Set ttriute of e t w+h k 1 to 1, n ttriute of the remining eges re set to. After Bt n Bt re constructe, we merge them to form Bt. A trnsition rnch representing negtive cn e constructe simpl echnging the cse of prit from even to o n vice vers. An emple of trnsltion is shown in Figure. A given isplcement of trnsltion is (1,-1). To clrif the steps of the lgorithm, we consier these rnches iniviull s shown in Figure 3. The lgorithm strts constructing trnsition rnch Bt 1, Bt n Bt 3 corresponing to B 1, B n B 3, respectivel. To construct Bt 1, we consier the prit of B1. Since the prit of B1 is o, we follow the proceure s stte in cse 1.1. Bt 1 is otine consiering the prit of B 1. Since the prit of B 1 is o n the isplcement is -1, we follow proceures stte in cse.. But the prit conition hs to e switche. Merging Bt 1 n Bt 1 iels Bt 1 s shown in Figure 3(). We repet the process to construct Bt n Bt 3. The rnches of BDD fter the imge is trnslte is otine B 1 = B 1 Bt 1, B = B Bt n B 3 = B 3 Bt Trnsition Brnch of n Displcement Given B = {, e 1, 1, e,..., e w 1, w 1, e w, } n isplcement of trnsltion = n, n = 1,, 3,..., log W where W is the with of the imge. The trnsition rnch Bt of B cn e constructe the following proceures: step 1 Shift B to the right n positions, then ssign to the temporr rnch B, tht is B = {,, 1,...,, n, e 1,..., e w, }. step Using proceures escrie in Section 3.1 to etermine the trnsition rnch Bt of B. Suppose tht Bt = {, e t 1, 1, e t,..., et w 1, w 1, e t w, }. step 3 Shift eges of the rnch Bt from step to

3 X = 5 Y = X = 1 Y = rnch 1 rnch rnch 1 rnch 1 B B B 1 3 B 3 B B Figure 4: BDD representtion of imge t originl loction. BDD n imge fter trnsltion. Figure : BDD representtion of imge t originl loction. BDD fter trnsltion. the left one position, then ssign to the temporr rnch Bt. step 4 Set the ttriute of the rightmost ege of Bt to 1. The content of Bt ecomes Bt = {, e t, 1, e t 3,..., et w, w 1, 1, }. step 5 Continue shifting the eges of Bt to the left n - 1 positions. The content of Bt fter shifting ecomes Bt = {, e t n+1, 1, e t n+,..., et w, w n 1, 1,,...,, }. 1 1 B 1 B B 3 (c) Bt 1 () Bt (e) rnch 1 rnch Bt B 3 1 (f) (g) Bt 1 B Bt (h) B 3 (i) Bt 3 Figure 3:, n (c) re rnches of the originl BDD. (), (e) n (f) re constructe trnsition rnches. (g), (h) n (i) re results fter trnsltion. After the process termintes, the content of Bt ecomes the esire trnsition rnch Bt. Similrl, Bt coul e constructe repeting step 1-5. After Bt n Bt re otine, we merge them to form Bt. 3.3 Aritrr Trnsltion The generl rules of trnsltion n ritrr isplcement, m 1 where m is the numer of its of the Gr coe coorintes, re s follows. First, ivie the trnsltion of isplcement into k steps such tht = k 1 k 1 + k k where k 1, k,..., 1, {, 1}. Then successivel ppl the trnsltion of nonzero isplcement i i for k times. For emple, trnsltion of isplcement = 13 cn e chieve iviing trnsltion into k = 3 steps since = = which mens we trnslte the oject 8 piels first then follow 4 piels n then 1 piel. We choose the rnch B 1 of the BDD from Figure 4 for emonstrtion s shown in Figure 5. Note tht we use the prime ( ) to inicte the temporr storge use in our lgorithm. In Figure 5, the lgorithm is pplie to the surnch B1 n B 1, seprtel. The rnch B 1 n B 1, s shown in Figure 5, re the result from right shifting of B1 two positions n B 1 one position,

4 B1 1 1 B1 step 1 B 1 -rnch 1-rnch step Bt1 step 3 Bt 1 step 4 trnsition rnch Bt 1 of B1 for = 4, = c = c c c = B1 B 1 Bt1 Bt 1 = + = + = = + c = c = (c) () (e) = + = + B 1 Bt 1 B 1 1 Figure 6: Imge flipping. verticl flipping horizontl flipping. 1 B1 fter trnsltion = 4, = B 1 = B1 Bt 1 trnsition rnch Bt 1 for further trnsltion = 1, = 1 rnch B1 fter trnsltion = 5, = 3 B 1 = B 1 Bt 1 (, ) (, ) ( c, c) c (, ) ( c, c) (, ) c (, ) (, ) (f) (g) (h) Figure 5: Detil of the rnch from Figure 4 when the lgorithm is pplie. respectivel. Net, we etermine the trnsition rnch Bt 1 of B 1 n Bt 1 of B 1 using proceure escrie in Section 3.1. The result is shown in Figure 5(c). B shifting Bt 1 n Bt 1 to the left one position n then setting the ttriute of ege e 3 n e 6 to 1, we hve Bt 1 n Bt 1 s shown in Figure 5(). Since n = for, we continue shifting Bt 1 further one position to the left. The lgorithm stops processing the Bt 1 ecuse n = 1. Therefore, we hve the trnsition rnch of B 1 for isplcement (4, -) s shown in Figure 5(e). Net, we eclusive-or the rnch B 1 to the trnsition rnch we otine. The result is rnch B 1 fter trnsltion (4,- ) s shown in Figure 5(f). We repet the lgorithm gin for the successive trnsltion (1, -1). The lgorithm strts with the rnch B 1, which is B 1 fter trnsltion (-4, ). The trnsition rnch Bt 1 of B 1 is shown in Figure 5(g). The rnch B 1, which is the complete trnsltion of B 1, cn e otine from eclusive-or the trnsition rnch Bt 1 to the rnch B 1 s shown in Figure 5(h). 4 Orthogonl Imge Rottion 4.1 Imge Flipping Definition 8 Imge flipping long n is =, > is n opertion tht moves ever piel i of the Figure 7: Coorinte Swpping. imge from ( i, i ) to the new coorinte ( i, i ) efine i = i + i, i = o i n i = i. Definition 9 Imge flipping long n is =, > is n opertion tht moves ever piel i of the imge from ( i, i ) to the new coorinte ( i, j ) efine i = i + i, i = i n i = i. Flipping opertion is equivlent to trnsltion of ech piel i. Figure 6 shows the concept of imge flipping when consiere s nonuniform trnsltion. 4. Coorinte Swpping Definition 1 Coorinte swpping is n opertion tht moves ever piel i of the imge from ( i, i ) to the new coorinte ( i, j ) efine i = i n i = i. Coorinte swpping is equivlent to echnging of eges from surnch B n B of rnch B. Figure 7 shows the concept of coorinte swpping. Imge rottion of 9 o cn e chieve coorinte swpping, n then follow flipping long = is. The rottion of 18 o is chieve flipping long = is, n then follow flipping long = is. The rottion of 7 o is chieve flipping long = is, n then follow coorinte swpping. Figure 8 shows

5 Coorinte swpping c Originl imge flipping c c o 9 rotte imge Originl imge c c flipping Coorinte swpping (c) c o 7 rotte imge Originl imge c c flipping flipping c o 18 rotte. imge Figure 8: Imge rottion. 9 o 18 o n (c) 7 o. process of rottion. 5 Aritrr Rottion Imge rottion n ritrr ngle α, α 9 o out the point ( o, o ) cn e consiere s the trnsltion s follows. Let ( i, i ) e the coorinte of the piel i t the originl loction n ( i, i ) e the coorinte of the piel i fter rottion. The coorintes i n i re etermine i = i cos(α) i sin(α) n i = i sin(α) + i cos(α). Then, we clculte the isplcement for ech piel iniviull, tht is i = i i n i = i i. Given the originl loction n the isplcement, we ppl the trnsltion proceures s escrie in Section 3. n Imge Shering Shering is trnsformtion of imge tht ever piel i of the imge is move from the loction ( i, i ) to the new loction ( i, i ) efine i = i + i n i = i+ i. The coefficients n re shering fctor which control shpe of the imge fter shering. If =, the result is clle shering long -is. If =, the result is clle shering long -is. After the new loction ( i, i ) is otine, we clculte the isplcement of piel i. Given the new loction n the isplcement, the trnsltion lgorithm cn e pplie to perform the imge shering. 7 Imge Scling Imge scling enlrges or reuces the imge the given scling fctor out the origin. Imge scling cn e consiere s the trnsltion tht moves the piel i from ( i, i ) to the new loction ( i, i ) efine i = S i n i = S i. S n S re clle scling fctor. After the new loction ( i, i ) is otine, we clculte the isplcement of piel i. Given the new loction n the isplcement, the trnsltion lgorithm cn e pplie to perform the imge scling. 8 Computtionl Requirements Since the propose lgorithm is se on the construction of trnsition rnch which involves prit checking n rnch serching. The est cse is when rnch serching is not require. In this cse, the construction of the trnsition rnch cn e one in O(1). If rnch serching tkes plce, the construction of the trnsition rnch cn e one in time linerl to the height of the rnch or O(w + h). If the BDD contins n rnches, then the require time is O(n(w + h)). The trnsition rnch for n ritrr trnsltion isplcement requires itionl time for ege shifting which is liner to the height of the BDD or O(w + h). The running time lso epens on the vlue of trnsltion isplcement. The est cse is when the isplcement of trnsltion is power of ecuse onl one trnsition rnch hs to e constructe. In prctice, the trnsltion of isplcement which is not power of two is one using the comintion of ecrementl n incrementl trnsltion. For emple, given = 55. If onl the incrementl trnsltion is use, then the trnsltion must e ivie into 7 steps. The smller numer of steps coul e chieve trnslting = 56 first, n, then = -1. For geometric trnsformtions which rel on nonuniform trnsltion, we nee to construct trnsition rnches for ech rnch of the BDD representtion of the imge inepenentl. 9 Eperimentl Results The performnce of the propose lgorithm is compre to the well-known stnr lossless imge coing lgorithms such s GIF, JPEG-LS, JBIG n PNG. The performnce of lgorithms uner test re mesure in term of eecution time when performing trnsformtions of the test imges encoe in their ntive formt. The lgorithms uner test re implemente using C lnguge n compile with GNU gcc compiler running on 1.8 GHz Pentium4 PC with 56MB of memor uner Linu operting sstem. We use the stnr test imge Len, which is converte to i-level 5656 n 6464 piels in our mesurement. Figure 9-13 n show the trnsformtion performnce of 6464 n 5656 piels imges, respectivel. Smple of imges otine from the eperiment re shown in Figure Conclusion We hve shown tht the geometric trnsformtions of n imge represente BDD cn e epresse using onl BDD mnipultion process. From the eperimentl results, the performnce of our propose lgorithm is comprle to those stnr lossless imge coing

6 Figure 9: Performnce comprison of imge trnsltion (6464 piels). Figure 11: Performnce comprison of imge ritrr rottion (6464 piels). Figure 1: Performnce comprison of imge orthogonl rottion (6464 piels). Figure 1: Performnce comprison of imge shering (6464 piels).

7 Figure 13: Performnce comprison of imge scling (6464 piels). Figure 15: Performnce comprison of imge orthogonl rottion (5656 piels). Figure 14: Performnce comprison of imge trnsltion (5656 piels). Figure 16: Performnce comprison of imge ritrr rottion (5656 piels).

8 Figure 17: Performnce comprison of imge shering (5656 piels). Figure 19: Originl Len imge (6464 piels). Figure 18: Performnce comprison of imge scling (5656 piels). Figure : Imge Trnsltion (37, 3) piels.

9 Figure 1: Imge Orthogonl Rottion 18 egrees. Figure 3: Imge Scling Fctor (1.5, 1.5). Figure : Imge Aritrr Rottion 1 egrees. Figure 4: Imge Shering Fctor (.,.1).

10 lgorithms in term of eecution time. In trnsformtions of smll imge (6464 piels), our propose lgorithm outperforms ll other stnr lossless coing lgorithms. Normll, stnr lossless imge coing lgorithms o not support geometric trnsformtions, hence imges encoe using these lgorithms hve to e converte to rster or itmp representtion efore trnsformtions cn e pplie. Contrsting to our propose lgorithm, the trnsformtions cn e performe irectl on the BDD representtion of the imge which elimintes time for conversion ck n forth etween itmp n their ntive representtion. As we hve nlze, the propose lgorithm runs slower s the size of the imge increse. B incresing the test imge from 6464 piels to 5656 piels which is 16 times enlrging of re, the eecution time of the propose lgorithm increses the fctor of 3.4,.7, 3.9, 3. n.8 for imge trnsltion, orthogonl, ritrr rottion, shering n scling, respectivel. However, the performnce is still comprle to the other stnr lossless coing lgorithms. References [1] Brnt, R.E., Grph-Bse Algorithms for Boolen Function Mnipultion, IEEE Trns. Comput., Vol. C35, No. 8, pp , 8/86 [] Brnt, R.E., Smolic Boolen Mnipultion with Orere Binr Decision Digrms, ACM Computing Surves, Vol. 4, No. 3, pp , 9/9 [3] Lursinsp C., Knchnsut K., n Sirioon T., Bsic Binr Decision Digrm Opertions for Imge Processing, Proc. 3r Asin Computing Science Conf., Lecture Notes in Computer Science, Springer- Verlg, pp , 1997 [4] Srkr, D., Boolen function-se pproch for encoing of inr imges, Pttern Recognition Letters, 17(8), pp , 1996 [5] Srkr, D., Bnerjee, S., n Chttoph, S., Trnsltion n rottion of inr imges encoe s minimize Boolen functions, Pttern Recognition Letters, 18 pp , 1997 [6] Strke, M., Brnt, R.E., Using Orere Binr Decision Digrms for Compressing Imges n Imge Sequences, Technicl Reports, CMU-CS-95-15