Progressive Transmission of Textured Graphic Model Over IP Networks

Transcription

1 Progressive Trnsmission of Textured Grphic Model Over IP Networks. Reserch Tem Project Leder: Other Fculty: Post Doc(s): Grdute Students: Undergrdute Students: Industril Prtner(s): Prof. C.-C. Jy Kuo, Electricl Engineering Prof. Mthieu Desbrun, Computer Science Prof. Antonio Orteg, Electricl Engineering Qing Li, Meiyin Shen, Di Yng, Chi-Hung Yeh, Cho-Hu Lee, Ming-Sui Lee, Jingling Peng, Sheng Yng Hilton Chu Institute of Informtion Industry, Microsoft 2. Sttement of Project Gols 3D grphic trnsmission hs recently ttrcted gret ttention in both cdemic reserch nd commercil pplictions, such s rchitecturl presenttions, virtul mrketing, nd network gming, etc. In these pplictions, it is desirble to progressively encode nd trnsmit grphic models nd their ssocited textures in order to reduce the lengthy trnsmission nd rendering time. The textured model trnsmission consists of three steps: () grphic nd imge compression, (2) multiplexing, nd (3) rendering. To reduce the initil witing time, it is desirble to represent both the mesh nd texture dt by levels of detils. The lyered representtion enbles progressive trnsmission nd rendering. Mesh compression nd rendering hve been well studied. However, multiplexing is still n open problem nd will be studied in this work. For n intermedite mesh of compressed grphic model, the distortion comes from three spects: the geometric error, surfce wrping, nd texture distortion. The geometric error is introduced becuse of the inccurcy of vertex positions. Surfce wrping is devition of texture mpping due to the error of texture coordintes. Texture distortion is distortion of textured imge due to its lossy representtion. The purpose of progressive mesh trnsmission is to mximize the visul qulity for every intermedite mesh. Most existing methods [.2] put the focus on the geometric error. However, the joint optimiztion of mesh nd texture dt for progressive trnsmission is seldom ddressed. As result, the optimlity of rendered qulity t ech stge of progressive trnsmission cnnot be gurnteed. In this reserch, we propose generl rte-distortion model tht tkes both mesh nd texture dt into ccount so tht the bit lloction between them cn be optimized to chieve the best disply qulity t every stge of progressive trnsmission. 75

2 3. Project Role in Support of IMSC Strtegic Pln Techniques of grphic streming ply n importnt role in IMSC strtegic pln, which ims t fcilitting the browsing of grphic models vi wired or wireless chnnels. It is desirble to progressively compress nd trnsmit grphic models to reduce the initil wit time. However, it is importnt to mximize the visul qulity with limited network resources. The development nd understnding to the qulity mesurement of levels of detils cn be used to fcilitte controlled qulity of service of wlkthrough virtul environments. Our work is crried out jointly with quite few other fculty members in IMSC due to the multi-disciplinry nture of the proposed reserch. 4. Discussion of Methodology Used An optimized scheme of multiplexing coded mesh nd texture dt to fcilitte progressive trnsmission of 3D textured models is proposed in this work. The mesh nd texture dt of 3D textured model re fed into their respective compression modules nd represented by series of levels of detils. Then, for given viewpoint, rte-distortion surfce cn be generted bsed on the multiplexing of mesh nd texture dt in different detils. The distortion is clculted by mesuring the visul qulity of rendered imges. Furthermore, n optiml pth over the rtedistortion surfce is determined by the steepest descent lgorithm. When the mesh nd texture dt strems re trnsmitted in certin rtio long the optiml pth, it is gurnteed tht the rendered imges hve the best visul qulity perceived from the given viewpoint. To del with n rbitrry viewpoint, we propose lyered smpling lgorithm so tht the optiml pth cn be interpolted from finite smpling points. Experimentl results demonstrte tht the proposed method cn provide the best visul qulity of textured 3D model for ny observtion point t ny bit rte. 5. Short Description of Achievements in Previous Yers We studied progressive trnsmission technique of textured grphic models. This is chllenging problem, since there is no simple metric to mesure the distortion of compressed textured grphic models. Therefore, we proposed novel lgorithm bsed on rte-distortion surfce to solve this problem. 5. Detil of Accomplishments during the Pst Yer Problem Definition Although the mesh nd texture dt cn be esily progressively compressed by the respective codecs, the multiplexing of mesh nd texture dt into one bit strem hs not yet been ddressed before. This issue is criticl to progressive textured model delivery. After compression, the rte-distortion functions of mesh nd texture cn be written s D = m f m(rm ), Dt = f t(rt ), 76

3 where Dm nd Dt re the mesh nd texture distortion functions, nd Rm nd R t re the mesh nd the texture bit rtes, respectively. Let us tke the 'Bunny' model s n exmple, its rte-distortion curves of mesh nd texture re illustrted in Figure. Typiclly, the mesh distortion is mesured in terms of the displcement of 3D mesh vertex points while the texture distortion is mesured in the difference of the gry-level vlues defined on 2D grid. When we wnt to multiplex these two progressive bit strems into one, it is nontrivil problem to define distortion mesure for the textured grphic model. + = + =? Figure The difficulty of defining distortion metric for textured grphic model. Rte-Distortion Surfce To solve the mesh-texture multiplexing problem, we propose new concept clled rte-distortion surfce to describe how the disply qulity of textured models vries with rte. Without loss of generlity, we compress the mesh dt using the edge collpse/vertex split method [8.9] nd the texture dt using the JPEG2 imge compression stndrd [] in this reserch. However, it is worthwhile to point out tht the frmework dopted in this study is generlly pplicble to ny progressive mesh nd texture codecs. The dopted specific codecs serve s concrete exmple for the ese of discussion. It is ssumed tht the mesh nd texture dt re, respectively, represented by m nd n lyers s M Æ M Æ L Æ M m -, T Æ T Æ L Æ Tn -. Thus, there re m n combintions of different mesh nd texture resolutions: 77

4 mmnnmnmtmtmt M T M T L M m- T M T M T L M m- T M M O M M T n- M T n- L M m- T n- For the distortion mesure, we first consider the cse where viewer exmines the model from fixed viewpoint. Then, to mesure the joint distortion ssocited with M i T j with i < m nd j < n, we my render the model from the given viewpoint, nd compre the difference of this imge with tht rendered in the full resolution of both mesh nd texture, i.e. M m- T n-. For ny combintion, we cn obtin such distortion mesure. Finlly, the distortion vlues for ll possible combintions form surfce, clled rte-distortion surfce, which is illustrted in Figure 2. Figure 2 The rte-distortion surfce. We implemented four distortion metrics in the experiments; nmely, MSE, TM, Mnnos- Skrison's filter, nd Dly's filter. The former two re the sptil domin metrics nd the ltter two re the sptil-frequency domin metrics. We only present the results of Mnnos-Skrison's filter [] nd MSE since the two in the pir hve similr performnce. The rte-distortion surfce cn be clculted off-line to sve viewer's witing time. It is worthwhile to point out tht illumintion could chnge the distortion function lso. However, since it is generted on rendered imges, it is impossible to pre-clculte the rte-distortion surfce with respect to the illumintion effect. Bsed on Phong's lighting model, the luminnce of one given point P on the model surfce cn be expressed by I = k i + M Â i= f È iili k ÍÎ d n ( N Li) + k s( N H i), () 78

5 where k, k d, nd k s re the coefficients of the mbient, diffuse, nd speculr lights. i is the intensity of the mbient light. i di nd i si re the intensities of the diffuse nd speculr components of the ith light source. f i is the ttenution coefficient of the ith light source. N is the norml vector t P. L i is the vector pointing from P to the ith light source. We hve H i Li + V = 2 where V is the viewing vector. For clrity, we denote the three prts of the bove eqution s I = k i, I di = f i i li k d ( N L i ), ( ) n. I si = f i i li k s N H i Since k is constnt, the luminnce is proportionl to the intensity of the mbient light. Therefore, the influence of the mbient light on the rte-distortion surfce cn be esily obtined. For diffuse nd speculr components, the clcultion is much more complicted. We discuss them with two cses below. First, only the intensities of light sources vry while the viewing positions remin unchnged. Since N Li remins invrint, the diffuse component of the rte-distortion surfce only depends on the intensity of the diffuse light nd cn be solved in similr wy to the mbient light. Furthermore, considering tht the clcultion of rte-distortion surfce implies fixed viewpoint, we cn clim tht N Hi is constnt s well. Therefore, the speculr component cn be derived similrly. The rte-distortion surfce under fixed illumintion is clculted vi () () () () D = D + D + D, j d s () () () where D, D d nd D s denote the mbient, diffuse, nd speculr components, respectively. When the intensities of light sources vry, the rte-distortion surfce cn be clculted vi () () () () D = c D + c D + c D, j d d s s () where D j is the joint rte-distortion surfce under the new illumintion scenrio. The coefficients re clculted vi () c = i i, () c d = Â Â c s = c d. M i= M i= f () () i i li f i () i li (), For the second cse, both the intensities nd positions vry. It is difficult to directly derive the rte-distortion surfce under ny illumintion in this cse. We leve this to our future work. In this work, we only py ttention to the first cse. 79

6 Optiml Pth In prctice, we hve the integer constrint on the mesh lyer index i nd the texture lyer index j, s shown in Figure 3. Thus, if we strt from representtion M i T j, we only hve two choices for the next higher rte, i.e. M i+ T j nd M i T j+. Thus, t ech stge, we simply compre the rtedistortion slope for these two cses nd choose the one tht hs the steeper slope. The process is repeted until the mesh nd texture dt re completely delivered to rech the full resolution. The result is ccurte if the gird size is smll enough. The mesh nd texture grdient surfces re illustrted in Figure 4 () nd (b). To demonstrte the performnce of the proposed multiplexing scheme, we compre it with the following three strtegies. The constnt rtio scheme: The mesh nd texture dt re trnsmitted ccording to constnt rtio, which is determined by the totl sizes of the mesh nd the texture dt. The mesh-first scheme: The mesh dt re trnsmitted first with the minimum mount of the texture dt. Then, more texture refinement dt re trnsmitted fter ll mesh dt hve been completely delivered. The texture-first scheme: The texture dt re trnsmitted first with the minimum mount of the mesh dt. Then, more mesh refinement dt re trnsmitted fter ll texture dt hve been completely delivered. Figure 3 The mesh texture grid. The proposed multiplexing lgorithm works well due to the monotonicity of the rte-distortion surfce in most regions. However, it is worthwhile to mention tht, when the mesh is rendered in very low resolution, the joint distortion will sometime increse with the refinement of texture due to the fct tht the lrge texture coordinte error results in severe texture devition. In other words, the steepest descent method my not gurntee the globl optiml pth in non-convex region. However, this cn be treted s pthologicl cse, nd it only occurs t very low bit rtes. On the other hnd, the refinement of mesh will lwys decrese the joint distortion. Another wy to exclude the pthologicl cse is to demnd the bse rte for the simplest textured grphic model M T to be greter thn resonble threshold so tht the rte-distortion surfce possesses the monotonic decresing property fterwrds for lrger vlues of i nd j. 8

7 () Grdient due to mesh chnge (b)grdient due to texture chnge Figure 4 The grdient surfces. The curve shown in Figure 5 describes the mount of mesh dt tht should occupy mong the totl bit rte. This curve guides how to llocte bits for mesh nd texture dt representtion during progressive trnsmission to chieve the best disply qulity with respect to given viewpoint. To reduce the cost of storing the dt for this curve, we my pproximte the curve by low order polynomil. In prticulr, we py specil ttention to the intervl where both the mesh nd the texture dt hve not been completely delivered. It is observed tht the curve cn be well pproximted by third-order polynomil function. Plese note tht the performnce of the pproximtion cn be further improved by dopting weighted curve fitting, since the dt hve lrger influence on visul qulity in the beginning. The weights cn be determined ccording to the rte-distortion curve, which cn be good topic for future reserch. Continuity of Distortion Surfce Figure 5 The mesh percentge curve. Before presenting the smpling lgorithm, we first prove tht the imge-bsed distortion function is continuous with respect to viewer's motion. Bsed on this fct, we cn rgue tht, if the 8

8 smpling is fine enough, the rte-distortion surfce for n rbitrry viewpoint cn be reconstructed from its nerby smpling points. It is ssumed tht the textured grphic model is cptured by pinhole cmer C tht hs finite ngulr resolution s illustrted in Figure 6. Then, the pixel vlue of the cpturing cmer is n integrl of the illumintion of the light rriving t the cmer plne. The surfce of the given model is represented by m = f mesh ( s), s b, where nd b re the prmetric coordintes of A nd B, respectively. The sectors P CP nd P C P denote the fields of view of the cmer before nd fter the chnge of the viewing prmeters. Therefore, the illumintion of one point s in the surfce cn be clculted by (). C C ' B P ' P P P ' A Figure 6 The pixel vlue clcultion. The pixel vlue before nd fter cmer movement is clculted by P P ( s, s ) = s I( s) ds, Ús ' ( s ', s ') = s I( s) ds. Ú s ' (2) (3) Thus, the pixel vlue difference cn be clculted by subtrcting (3) from (2) p = P s ', s ' - P s, s d + ( ) ( ) Ú s ' = Á Ê - Ë s M Â i= Ú s ' s ˆI s ' s ' Á Ê Ú - s Ús Ë s ' s ' ( s) ds + Ê - ˆI ( s) ˆI si M Â i= ( s)ds Since k ( s), we hve k ( s) i i Á Ë Ú s Ú s di ds. Then, we cn obtin s ' ( s) ds i s -s, I ( s) ds i s ' s, s ' I ' s Ú - s Ú nd Ê -Ú Á Ë ( s) ds i ( s '-s + s '-s ) s ' s ' Ú I. s s ˆ 82

9 Let e = s' -s nd e = s '-s Ê Ú -Ú Á Ë s ' s ' s s ˆI ( s) ds i ( e + e ). Then, the bove inequlity cn be written s Similrly, since kd ( s), k s( s), N( s) L ( s) N( s) L ( s) = N ( s) H ( s) N( s) H ( s) = s ' s ' Á Ê Ú - s Ús Ë s ' s ' Á Ê Ú - s Ús Ë i ˆI ˆI di si i ( s) ds f i ( e + e ), i li ( s) ds f i ( e + e ). i li, we obtin i i nd Finlly, we hve the following inequlity: M Ê ˆ P s ', s' - P s, s Ái + 2Â f iili e + e Ë i= ( ) ( ) ( ) (4) The bove inequlity sys tht the pixel vlue difference is bounded when the viewer moves long t smll distnce. Accordingly, the distortion function is continuous with viewer's motion. Thus, the rte-distortion surfce cn be reconstructed within certin error bound from its nerby smpling points. Smpling Function With the method described bove, we successfully solve the problem of mesh-texture multiplexing when the viewer looks t the model from fixed viewpoint. However, it is typicl tht the viewer will move round nd observe the model from different viewpoints. We propose n dptive lyered smpling lgorithm to ddress this problem. Here, we only present the work when the viewing distnce between the viewer nd the model remins unchnged. The solution cn be generlized to the vrying distnce cse. As result of the constnt viewing distnce, the set of ll viewpoints forms sphere, which is clled viewing sphere. The ide of lyered smpling is to pproximte the viewing sphere by progressive tringle mesh, clled viewing mesh. We first ssign the four vertices of regulr tetrhedron internlly tngent with the sphere s the first lyer of the viewing mesh. Then, we continue to subdivide the viewing mesh until it is fine enough. The gol of viewing mesh construction is to itertively insert new viewpoints so tht the rte-distortion surfce of ny viewpoint cn be predicted from its nerby viewpoints within certin error bound. Given three vertices vp, vp, nd vp 2 of tringle in the viewing mesh, their rte-distortion surfce cn be denoted by 83

10 D D D j j j2 = = = f f f j j j2 ( Rm, Rt ), ( R, R ), m ( R, R ). m t t For ny viewpoint vp enclosed by the tringle, its ctul rte-distortion surfce, denoted by D j, cn be pproximted using the liner interpoltion of results observed from vp, vp, nd vp 2. Tht is, we hve D ' = vp - vp D + vp - vp D + vp - vp D vp - vp + vp - vp + vp - vp, j ( ) ( ) j j 2 where denotes the Eucliden norm in the 3D spce. The pproximtion error cn be written s 2 {[ D j ( Rm, Rt ) D j '( Rm, Rt )] drmdrt}( RM RT ) ÚÚ - e =, (5) where R M nd R T denote the totl size of mesh nd texture dt, respectively. One cn continue to do the refinement until the mximum pproximtion error, which usully occurs t the brycenter of the tringle, is less thn certin threshold. To reduce the pproximtion error s fst s possible, the viewing mesh construction process should meet the following two requirements. First, tringles in ech lyer should be pproximtely of the sme size nd similr shpe. Second, the sides of the tringle should be pproximtely of the sme length. In the proposed viewing mesh subdivision method, we first find the midpoint of ech edge nd extend it to intersect with the surfce of the viewing sphere. The points of intersection re viewed s new viewpoints. Then, we subdivide one tringle into four by connecting the newly inserted viewpoints. The viewing mesh subdivision process is shown in Figure 7. j2 2 Figure 7 The viewing mesh subdivision process. With the bove dptive lyered smpling lgorithm, we obtin the rte-distortion surfces t smpling points. By expressing the smpling points with 3D ngulr coordintes, we hve the following: D = g q,j, int ( ) 84

11 q,. An exmple of the smpling surfce is illustrted in Figure 8. By using the smpling surfce, the rte-distortion function of n rbitrry viewpoint cn be clculted. where D int is the integrl over rte-distortion surfce in viewpoint ( j) Experimentl Results Figure 8 The smpling surfce. In this section, we test the proposed lgorithm in two stges. First, we test if the rte-distortion surfce cn correctly reflect the reltionship between the bit rte nd the disply qulity of textured models to offer the best multiplexing strtegy. Second, we test if the smpling function works well for the cse when the viewer looks t the model from n rbitrry viewpoint. As mentioned before, to test the performnce of the proposed lgorithm, we compre it with other three strtegies: the constnt rtio scheme, the mesh-first scheme, nd the texture-first scheme. The rte-distortion curves of 'Angelfish' by Mnnos-Skrison's filter nd MSE re illustrted in Figure 9. We see from the figures tht the proposed scheme lwys outperforms the other three schemes, no mtter wht distortion metrics re dopted. As mentioned bove, when the mesh resolution is very low, the incresing resolution of texture my sometimes increse the joint distortion. This is observed in the texture-first curve. Therefore, when the textured model is trnsmitted by the constnt rtio or the texture-first schemes, the joint distortion my not decrese monotoniclly. The imge rendered in the full resolution of both mesh nd texture is shown in Figure. When the ngelfish model is trnsmitted with % of the totl number of bits for the full resolution, the rendered imges nd residul imges under four multiplexing methods re shown in Figures nd 2, respectively. For the proposed method, the optiml pth selection is bsed on the distortion mesured with Mnnos-Skrison's filter. We observe tht the poor qulity of the mesh-first method minly lies in the texture component, while tht of the texture-first method mostly exists in the silhouette of the object, e.g. the til prt. The distortion of imges using the optimized multiplexing trnsmission is hrdly visible. We cn drw similr conclusion for the 85

12 bunny model. Although the imges of the mesh-first method look pprently worse thn those of the texture-first method, their MSE distortion vlues sometimes (for exmple, 'Angelfish' nd 'wolf' model) re smller. On the other hnd, the results of Mnnos-Skrison's filter re more consistent with humn perception in most cses. This is why people feel tht the sptilfrequency domin metrics perform better thn sptil domin metrics. () Mnnos-Skrison's filter (b) MSE Figure 9 Rte-distortion curves. Tble I. Distortions of imges obtined with \% of totl bits. Model Strtegy M-S's filter MSE Optiml Angelfish Constnt Rtio Mesh-First Texture-First Optiml Bunny Constnt Rtio Mesh-First Texture-First Optiml Wolf Constnt Rtio Mesh-First Texture-First To verify the dptive lyered smpling lgorithm, we rndomly choose viewpoint nd clculte its ctul nd predicted rte-distortion surfces in different lyers of viewing mesh. Bsed on the two surfces, we determine two bit lloction results. The prediction error of rtedistortion surfces nd curves re illustrted in Figure 3. The errors shown in the figures re clculted by (5). The error cn be esily controlled by viewing mesh subdivision ccording to user's requirements. 86

13 Figure. The Angelfish rendered in full resolution. () The proposed method (b) The constnt rtio method (c) The mesh-first method (d) The texture-first method Figure Rendered imges of the ngelfish model with % of the totl no. of bits. 87

14 () The proposed method (b) The constnt rtio method (c) The mesh-first method (d) The texture-first method Figure 2 Residul imges of the ngelfish model. () e=

15 (b) e=.2283 (c) e= Figure 3 Rte-distortion curves. 6. Other Relevnt Work Being Conducted nd How this Project is Different Most previous work on progressive grphic coding delt with the mesh dt lone, where only the geometric error nd surfce wrping re relevnt [-5]. All the bove work considered progressive coding of the mesh dt under the ssumption tht the texture is represented by full resolution imge. Some work pid ttention to progressive coding of both geometry nd texture dt, which is closer to our objective, [6.7]. However, the metric used ws usully derived from user's preference bout the importnce of the geometry nd texture informtion. Thus, the userdependent pproch does not provide n utomtic solution nd my not be convenient in mny pplictions. Furthermore, it fils to chieve n optiml performnce, if there exists proper metric for performnce mesure. To solve the problem, we propose nd implement scheme for progressive trnsmission of textured grphic models. The min contributions of this work include the following. A new concept clled rte-distortion surfce is proposed to describe the reltionship between the bit rte nd the disply qulity of textured grphic model t given viewpoint. 89

16 An optiml strtegy is developed to multiplex the mesh nd the texture bit strems into one single strem for progressive trnsmission to present the best disply qulity subject to the bndwidth constrint t given viewpoint. A lyered smpling method is proposed so tht the multiplexing scheme cn be generlized to del with n rbitrry viewpoint. 7. Pln for the Next Yer Our pln for the next yer includes the following: Develop progressive rendering lgorithm, i.e. only updte the regions with newly incoming mesh or texture dt, for some client devices with limited computtionl cpbility. Develop more robust strtegy to determine the optiml pth over rte-distortion surfce. 8. Expected Milestones nd Deliverbles An optiml mesh-texture multiplexing codec hs been implemented by C++. It hs the following functions: () progressive mesh compression; (2) progressive texture compression; (3) rtedistortion surfce clcultion; (4) optiml pth determintion; nd (5) dptive lyered smpling. Severl conference ppers hve been published on this topic. There re severl long-term chllenging issues worth our further study. They include: progressive rendering, optiml pth determintion, etc. We will continue to investigte these problems in next yer nd seek some mjor brekthrough in engineering science. The milestone chrt is given below. 9. Member Compny Benefits Both III (Institute of Informtion Industry) nd Microsoft Reserch Lb show strong interested in the developed technologies. Demos hve been mde to them.. References [] H. Hoppe, New qudric metric for simplifying meshes with ppernce ttributes, in Visuliztion 99. Proceedings, Sn Frncisco, CA, USA, 999, pp , 5. [2] Michel Grlnd nd Pul S. Heckbert, Simplifying surfces with color nd texture using qudric error metrics, in Proceedings of Visuliztion 98, 998, pp [3] Jonthn Cohen, Mrc Olno, nd Dinesh Mnoch, Appernce-preserving simplifiction, in Proceedings of the 25th nnul conference on Computer grphics nd interctive techniques, 998, pp [4] Peter Lindstrom nd Greg Turk, Imge-driven simplifiction, ACM Trnsctions on Grphics, vol. 9, pp , July 2. [5] Pedro V. Snder, John Snyder, Steven J. Gortler, nd Hugues Hoppe, Texture mpping progressive meshes, in Proceedings of the 28th nnul conference on Computer grphics nd interctive techniques, August 2, pp [6] M. Okud nd T. Chen, Joint geometry/texture progressive coding of 3d models, in Proceedings of Int l Conf. on Imge Processing, 2, pp

17 [7] Mrí José Abásolo nd Frncisco Perles, Geometric-textured bitree: Trnsmission of multiresolution terrin cross the internet, Journl of Computer Science & Technology, vol. 2, no. 7, pp. 34 4, Oct. 22. [8] H. Hoppe, Progressive meshes, in Computer Grphics Proceedings, New Orlens, August 996, Annul Conference Series, pp [9] R. Pjrol nd J. Rossignc, Compressed progressive meshes, IEEE Trnsctions on Visuliztion nd Computer Grphics, vol. 6, pp , Jn. Mrch 2. [] M. D. Adms nd F. Kossentini, JsPer: softwre-bsed JPEG-2 codec implementtion, in Proceedings of IEEE Interntionl Conference on Imge Processing, October 2, pp [] J. L. Mnnos nd D. J. Skrison, The effects of visul fidelity criterion on the encoding of imges, IEEE Trnsctions on Informtion Theory, vol., no. 5, pp ,

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