Parallel Polygon Scan Conversion on RMESH Multiprocessors
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1 Parallel Polygon Scan Converion on RMESH Multiproceor John Jenq Department of Computer Science Montclair State Univerity Wing Ning Li Department of Computer Science and Computer Engineering Univerity of Arkana ABSTRACT Polygon can converion i an important operation in computer graphic application. Speeding up thi operation make the animation more realitic. In thi paper, we preent a parallel algorithm to can convert a polygon from it object pace repreentation into it image pace repreentation uing RMESH multiproceor. The total number of SIMD RMESH proceor i N N, where each proceor element correpond to a picture element in the image plane. For convex polygon, an algorithm with time and pace complexitie of O(1) wa developed, which i aymptotically optimal for RMESH computer. Keyword Polygon can converion, computer graphic, RMESH algorithm, parallel algorithm. INTRODUCTION The polygon can converion problem refer to the contour filling or hading of the interior of a region preented by one or more polygon and ha found it application in computer graphic, pattern recognition, cene analyi, and computer aided deign. The input of the problem i a cloed contour, either in the form of a lit of edge that repreent a cloed polygon or a collection of pixel that form a polygon contour. In either cae, a polygon may be with or without hole. Several algorithm exit to olve thi problem. The algorithm may be claified a polygon baed contour filling or pixel baed contour filling depending on whether a polygon i repreented by a lit of edge (or vertice) or a collection of pixel. The output i either a collection of line egment that tand for the whole region encloed by the polygon, or imply the interior pixel of the polygon which are filled with deired color or pattern. If the output i line egment one might include or exclude the contour point. Although polygon baed contour filling can be apply to both vector and rater graphic device, we will conider polygon can converion on rater device only in thi paper becaue the mot popular diplay device in ue today are rater can device. Rater can device are now ued in a wide range of application from peronal computer, ATM machine, CAD/CAM deign, to computer imulation. The image created by a graphic ytem will be written onto a frame buffer (video memory) which can then be ued by diplay hardware to diplay the image on a rater output device uch a a monitor creen. Each time when a creen i to be updated, the polygon can converion algorithm i invoked. For certain application, fat interaction and quick repone are eential. To achieve rapid creen update, fat polygon can converion become very important becaue can converion i the bottleneck of creen update. Therefore peeding up thi operation will ignificantly improve the ytem performance and make better human machine interaction. The parallel implementation of the converion operation i one way of peeding up the whole proce of image creation and recreation. It i epecially true when complex 3D object have to be dealt with. Many erial implementation of contour filling exit, ee for example [1] [][8][9][1][13][14]. The popular one, a decribed in [3], ue bucket to tore the edge information. An edge table and an active edge lit i created. Every table entry i proceed if it i not null. The image i generated one horizontal line by one horizontal line until the object i can converted. Odd parity rule i ued to determine if
2 a pixel i inide or outide the polygon. See Figure 1 for an example. Special care ha to be taken to handle horizontal polygon edge and boundary condition. We will omit thee detail here. polygon, uing RMESH multiproceor. The vertice are organized counterclockwie. The algorithm aume that the input polygon are convex. In ection, the RMESH i introduced. In ection 3 baic operation on RMESH are defined. In ection 4, the polygon can converion algorithm i developed. The concluion i in ection Preliminarie The particular reconfigurable meh architecture that we ue in thi paper i called RMESH. It employ a reconfigurable bu to connect together all proceor. Figure how a 4 4 RMESH. By opening ome of the witche, the bu may be reconfigured into maller bue that connect only a ubet of the proceor. (0,0 Figure 1 Odd Parity rule in action; Segment 1-, 3-4, 5-6 will be filled In [4], Ghoal and Patnaik propoed three parallel polygon can converion algorithm on a hared bu architecture. The parallel algorithm that doe not ue edge coherence outperform thoe that ue edge coherence. In [15], Theohari howed how polygon rendering can be efficiently ditributed among dual-paradigm parallel proceor called diputer which can be configured by uing SIMD DiArray and MIMD Tranputer. In [7], Kaufman preented three dimenional can converion algorithm which convert 3D polygon into their dicrete voxel-map repreentation within a Cubic Frame Buffer. In [6], Jenq preented two can converion algorithm to convert a polygon into image pace uing Hypercube multiproceor. The algorithm that handle general hape polygon ha time complexity of O(log N ). When polygon are convex the time complexity drop to O (log N) in the other algorithm and i optimal for hypercube multiproceor In thi paper, we develop a parallel algorithm for can converting an M edge polygon, repreented by the vertice of the Figure A 4 x 4 RMESH : Proceor : Switch : Link The important feature of a RMESH are: 1. An N M RMESH i a -dimenional meh connected array of proceing element (PE). Each PE in the RMESH i connected to a broadcat bu which i itelf contructed a an N M grid. The PE are connected to the bu at the interection of the grid. Each proceor ha up to four bu witche (Figure ) that are oftware controlled and that can be ued to reconfigure the bu into (3,3
3 ubbue. The ID of each PE i a pair (i, j ) where i i the row index and j i the column index. The ID of the upper left corner PE i (0,0) and that of the lower right one i (N-1,M-1). Aociated with a PE there are up to four witche that are labeled E (eat), W (wet), S (outh) and N (north). Notice that the eat (wet, north, outh) witch of a PE i alo the wet (eat, outh, north) witch of the PE (if any ) on it right (left, top, bottom). Two PE can imultaneouly et (connect, cloe) or unet (diconnect, open) a particular witch a long a the etting do not conflict. The broadcat bu can be ubdivided into ubbue by opening (diconnecting) ome o f the witche. 3 Only one proceor can put data onto a given ub bu at any time. 4 In unit time, data put on a ubbu can be read by every PE connected to it. If a PE i to broadcat a value in regiter I to all of the PE on it ubbu, then it ue the command broadcat(i). 5 To read the content of the broadcat bu into a regiter R, the tatement R = content(bu) i ued. 6 Row bue are formed if each proceor diconnect (open) it S witch and connect (cloe) it E witch. Column bue are formed by diconnecting the E witche and connecting the S witche. Baic Data Manipulation Operation In thi ection we define everal baic data manipulation algorithm for RMESH multiproceor. Thee operation are ued a building block in t he next ection to develop the parallel algorithm for polygon can converion. Data Diagonalization In thi, a pecific row(or column) of element i moved to the diagonal potion fo the window which contain that row(or column). Thi can be done by broadcating the row(column) along column(row) bue and having the diagonal proceor read the data from their bu Row(Column) Bu Broadcat Thi operation will broadcat a datum that i initially in a PE to all other PE that are connected to the ame ub bu(either row or column). Since it i a pecial cae of window broadcat operation a defined in [ 5], thi operation can be done in O(1) time. Shift Each PE ha data in it A variable that i to be hifted to the B variable of a proceor that i, > 0, unit to the right but on the ame row. One of the variant of the hift i the operation of circular hift which perform hift with wrap around feature. Thee operation can be done in O() time. If =1 then the time become O(1). Sub MESH Bu Broadcat Thi operation will broadcat a datum that i initially in a PE to all other PE that are connected to the ame ub MESH. It i imilar to Row bu broadcat operation RAR and RAW The random acce read(rar) and random acce write(raw) operation are defined in [11]. In a RAR each PE ha a read addre aociated with it. Thi i the addre of the PE from which information i to be read. In a RAW each PE ha a write addre which i the addre of the PE to which it want to end information. Conflict can be reolved arbitrarily. Miller et al. [10] have developed RMESH algorithm for RAR and RAW when k data item are to be moved in the RAR or RAW, their algorithm take O ( k + log N) time when k N. If the number of ource and detination proceor in each k k block of PE i O (k), 1 k N then their algorithm take O (log N ) time. Ranking Conider the linear ordering of the N PE defined by row major order. Aume that each PE ha a Boolean variable elected. If elected(i,j) i true then rank(i,j) i the number of PE with elected(i,j) true that precede it in the defined linear ordering. If elected(i,j) i fale, then rank(i,j) i undefined. N element can be ranked uing N proceor in O (log N ) time. However, N element on a ingle column or row can be ranked in O(1) time on an N N RMESH [5]. If number of element in a row i contant then it can be eaily done in O(1) time. We omit the detail here.
4 POLYGON SCAN CONVERSION ON RMESH When a polygon i convex, then the edge lit can be partitioned into four egment chain. The firt chain ha x coor dinate in non -decreaing order and y in non -increaing order. The econd chain ha x coordinate in non -decreaing order and y in non-decreaing order. The third chain ha x coordinate in non -increaing order and y in non - decreaing order. The lat chain ha x coordinate in non -increaing order and y in non -increaing order. For implicity reaon, let u conider the chain having non -decreaing x and non - increaing y. The other three chain can be dealt with imilarly. When x i in non -decreaing order, it i impoible for three egment fall into the ame x at the ame time, while it i poible for two egment to fall into the ame x a hown in Figure 3(a). If there are three egment that have ame tarting x then the ituation have to be thoe hown in Figure 3(b). In Figure 3(b), egment S and egment T can be merged together. The ituation uch a that in Figure 3(c) violate our aumption that thee egment are in non -decreaing x. The validity of our aumption i due to the fact that the polyg on i convex. S T (a) (b) (c) Figure 3 egment can overlap in x T S Another obervation of a convex polygon i that the total number of line egment that it ha i in the order of O(N). Therefore, intead of uing the regular random acce write operation, a implified random acce write m ay be developed. The implified operation reduce the time complexity from O(N) down to O(1) a hown in the following algorithm. T S Rather than decribing the implified RAW operation eparately, we incorporate the idea directly into the polygon can conver ion algorithm. A high level peudo code of the polygon can converion algorithm on RMESH i outlined in Figure 4. Step1 Diagonalization with edge record ( x, y, color, xd, yd ) Step Set up row bu Step3 Broadcat the edge record, for PE received non -null record and i == j Step4 Read bu Step5 If the row index, i, of PE matche the x of the record read Then ( mark[i,j] = true; Iteration1[i,j] = true) ele mark[i,j] = fale Step6 form column ubu by open outh witch if hold a record Step7 broadcat the record if mark[i,j] == true Step8 Shift the non null record received from Step7 to it north neighbor PE Step9 If (mark[i,j] && received a record) then iteration[i,j] = true; Step10 For PE with index[i] matche the x of the record received then keep the record Step11 Form column bu Step1 If (mark[i,j] && iteration1[i,j] ) Then broadcat a ignal to all PE on the ame column Step13 For PE received the ignal open wet witch Step14 If (mark[i,j] && iteration1[i,j] ) Then broadcat x, y, color, x, y ) ( d d Step15 For PE between the line (( x, y ),( xd, yd )) keep the record of x, y, color, x, y ) ( d d Step16 repeat Step1 to Step 15 for iteration [i,j] ==true Step17 Ranking on row bu Step18 odd number PE open their W witch and even number PE open their E witch o to form ub row bu. Step19 odd number PE end the color and normal information to it right PE through row bu Figure Step1 4 ditribute Scan convert a the convex edge polygon on record RMESH x, y, color, x, y ) to the ( d d
5 diagonal PE. Step, 3, and 4 et up the ub bue and then broadcat the edge record from the diagonal PE to the PE on the ame row. By comparing the x field of the record with the row index, i, of the PE, flag mark[i,j] ca n then be et. The major function of thi tep i to align the x. The following tep will align y. At the ame time, a flag iteration1[i,j] alo get et. Thi i to take care of the ituation when two edge record may have the ame x. From thi tep to Step 9 i a etup mechanim to determine the order for the write edge record to be written to their proper detination PE when there are more than one edge record having the ame x. Figure 5 how Step 6, 7 and 8 in action. Step 11 to Step 16 are required o that edge record are properly ditributed to their proper PE. Step 17 i a ranking proce. Becaue the number of PE with the elected flag et i contant in convex polygon o the operation i not O (log N) but O(1). Step 18 and 19 are to be done after all four egment chain are proceed. They carry out the rendering proce to determine the color on the polygon. Special care need to be taken t o handle the ituation when an edge i a turning point of the polygon. I.e., it hall be counted twice. We omit the detail here for implicity. Since all the operation are in O(1) time, therefore the run time i contant and i optimal for RMESH. CONCLUDING REMARKS A parallel polygon can converion algorithm on RMESH multiproceor to can convert a polygon from it object pace repreentation into it image pace repreentation ha been developed. The time and pace complexitie of the algorithm are both O(1) and are aymptotically optimal for RMESH. REFERENCES [1] M. Dunlavey, "Efficient polygon -filling algorithm for rater diplay", ACM Tran. on Graphic, vol., no. 4, pp 64-73, 1983 [] K. Fihkin, and B. Barky, "A family of new algorith m for oft filling", Computer Graphic, vol. 18, no. 3, pp 35-44, 1984 Figure 5 (a) (b) (C) (a)when two record have the ame ource x (b) Step6 and [3] J. Foley, et. al., "Computer Graphic: Principle and Practice", nd ed. Addion-Weley, 1990 [4] D. Ghoal and L. M. Patnaik, Parallel polygon can converio n algorithm: performance evaluation on a hared bu architecture, Computer & Graphic, vol. 10, no. 1, pp 7-5, 1986 [5] J. Jenq and S. Sahni, Reconfigurable Meh Algorithm for Fundamental Data Manipulation Operation Computing on Ditributed Memory Multiproceor, NATO Serie F, ed. F. Ozguner, Spring Verlag, 1993, pp 7-46 [6] J. Jenq, Parallel Polygon Scan Converion on Hypercube Multiproceor, Proceeding of Sympoium on Applied Computing, pp , 1999 [7] A. Kaufman, Efficient algor ithm for can-converting 3D polygon, Computer & Graphic, vol. 1, no., pp 13-18, 1988 [8] G. Knott, "Computing polygon fill - line", Computer and Graphic, vol. 11, no. 1, pp 1-5, 1987 [9] J. Lane, "An algorithm for filling region on graphic d iplay device", ACM Tran. on Graphic, vol., no. 3, pp , 1983 [10] R. Miller, V. K. Praanna Kumar, D. Rei and Q. Stout, Data movement operation and application on reconfigurable VLSI array, Proceeding of the 1988 International Step7, (c) Step8 of Figure 4
6 Conference on Parallel Proceing, pp05-08 [11] D. Naimi and S. Sahni, "Data broadcating in SIMD computer," IEEE Tran. on Computer, pp , [1] T. Pavldi, "Filling algorithm for rater graphic", Computer Graphic and Image Proceing, 1 0, pp , 1979 [13] T. Pavldi, "Contour filling in Rater graphic", Computer Graphic, vol. 15, no. 3, pp 9-36, 1981 [14] A. R. Smith, "Tint fill", SIGGRAPH '79 Proceedding, pp 76-83, 1979 [15] T. Theohari and I. Page, "Polygon Rendering on a Dual-Paradigm Parallel Proceor", Computer & Graphic, 13(), pp , 1989
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