1. Introduction. 2. Existing Shape Distortion Measurement Algorithms

Transcription

1 A Moifie Distortion Measurement Algorithm for Shape Coing Ferous Ahme Sohel, Laurence S Doole an Gour C Karmaar Gippslan School of Computing an Information an Technolog Monash Universit, Churchill, Victoria, Australia. Ferous.Sohel@infotech.monash.eu.au Abstract: Efficient encoing of object bounaries has become increasingl prominent in areas such as content-base storage an retrieval, stuio an television post-prouction facilities, mobile communications an other real-time multimeia applications. The wa istortion between the actual an approimate shapes is measure however, has a major impact upon the qualit of the shape coing algorithms. In eisting shape coing methos, the istortion measure o not generate an actual istortion value, so this paper proposes a new istortion measure, calle a moifie istortion measure for shape coing (DMSC) which incorporates an actual perceptual istance. The performance of the Operational Rate Distortion optimal algorithm [1] incorporating DMSC has been empiricall evaluate upon a number of ifferent natural an snthetic arbitrar shapes. Both qualitative an quantitative results confirm the superior results in comparison with the ORD algorithm for all test shapes, without an increase in computational compleit. Kewors Distortion measure, shortest absolute istance, perceptual istance, shape coing, object base coing. 1. Introuction Object-oriente vieo coing base upon shape information is a ver challenging topic in the literature currentl as it facilitates retrieval, interactive eiting an manipulation of both natural an snthetic vieos. Due to the inherent banwith limitations of eisting communication technolog, vieo applications such as vieo over the Internet, vieo on eman, wire an mobile vieo transmission for han-hel evices will all be benefite immensel from fast an efficient shape coing techniques. Within the object-oriente framewor, a vieo sequence is represente b the evolution of vieo object planes (VOPs), with each frame comprise of one or more VOPs, which in turn are escribe in terms of shape, teture an motion information. In MPEG- [] for eample, shape or bounar encoing is wiel use. In a classical verte-base shape coing sstem, a shape is encoe to an optimal number of bits within some prescribe istortion value. The total number of bits require to encoe a particular shape is nown as the rate (cost), an the istortion is usuall measure as the maimum istance between the original an approimate shape. Distance measurement is an important issue in rate-istortion algorithms. For instance, the shortest absolute istance is use in [1- ], where the vertical istance from a point on a shape to a line or etene line segment of the approimate shape is efine as the istortion for that point. This measure however oes not prouce the actual istortion for those points where the vertical istance is measure from the etene line segment. To aress this issue, a istortion measure algorithm, namel a moifie istortion measurement algorithm for shape coing (DMSC) is propose on the beroc of Operational Rare Distortion (ORD) optimal algorithms in [1-]. The performance, in terms of subjective assessment of the ecoe shape an the actual istortion with respect to the constraine maimum istortion, of the DMSC has been teste for a number of ifferent natural an snthetic arbitrar shapes, an will be shown that it can mae guarantee on the maimum istortion value for the ecoing. The remainer of this paper is organize as follows: Section provies a brief review of eisting shape measurement techniques, while Section presents the new propose istortion measure algorithm. Section gives a escription of the ORD optimal algorithms in [1-], while Section provies the eperimental results, analsis an performance comparison of the various ORD techniques. Finall, some concluing remars are mae in Section.. Eisting Shape Distortion Measurement Algorithms A shape is generall approimate as a polgon or a higher orer parametric curve. Usuall parametric curves are comprise of a sequence of iscrete points that ultimatel form a polgon. Amongst the istance measurement algorithms, the Eucliian istance is etensivel use to etermine the istance between two points. The Eucliian istance ( p 1, p ), between two points p 1 an p in a plane, having Cartesian coorinates ( ) (, ) respectivel, is efine as:- 1, p) ( ) + ( ) 1 1 1, 1 an ( p = (1)

2 If Eucliian istance is use as the istortion measure, the istance of each point on the original shape from all points on the approimate polgon are neee to be calculate to fin the minimum istance, as shown in Figure 1. In Figure 1, the shortest istance of q from line a = qc while from p it is pa. p a b c Figure 1: Shortest istance using Eucliian Distance. This requires a computational compleit of O ( n), where n is the number of points in the approimate shape (polgon), to measure the istortion for each point of the original shape. The shortest absolute istance ha been introuce to conquer the high computational compleit, where the perpenicular istance is measure from each point on the original shape to the polgon. Let ( a,, m) be the shortest absolute (perpenicular) istance from a point m on a line passing through points a an. This is formulate as:- ( m a)( b a) ( m a)( b a) ( a,, m) = () ( b a) + ( b a) where subscripts an represent the an coorinate values respectivel. This is clearl computationall more efficient, as it taes onl O ( 1) to calculate the istance from a point to a line segment; however, the perceive shape istortion is not properl reflecte in (). To unerstan wh, consier the eample in Figure : g b c a h f e Figure : The minimum istance eample using (). Eq. () measures the shortest geometric istance between a point an a line segment. The minimum istance point of the line segment ma either be on the line or its etension in Figure. The minimum istance point is on the line segment ( gh is the istance of point g from a ), the minimum istance point is on the etene line segment ( b is the istance of point b from a, which lies on the etension a ). This leas to an inaccurac in the istortion measure from a perceptual point of view, since the actual perceptual istance of b from line q a is ba, while accoring to () it is b. This algorithm therefore oes not tae account of human visual perception with respect to shape an essentiall introuces further istortion in sharp eges, especiall in concave or conve areas of a shape. This was the primar motivation behin the evelopment of the new shape istortion measure that is escribe in Section.. Propose Distortion Measurement Algorithm for Shape Coing This Section introuces a novel istortion measurement algorithm, namel a moifie istortion measurement algorithm for shape coing (DMSC) which consiers the minimum istance of a shape point from a respective line segment connecting two approimate polgon points. Note that, the etene portion of the line segment ( a in Figure ) is not taen into account because of the nee to measure the true shape istortion. b Figure : Verte positioning strateg. Let the upper part of the shape abcefa, abc be approimate b the polgon segment a. The istance of ever point of shape abc from segment a is then calculate. Two perpenicular lines are rawn at a an (Figure ) an it is assume that the intersect the original bounar at points p, q respectivel. When the shape points lie within or on the perpenicular lines, the istortion is measure using (). For all other shape points, the istances are measure either from a or epening on the caniate shape point position with respect to the perpenicular lines. In Figure for eample, if the caniate shape point lies before the vertical line at a or after the vertical line at, the istortion is measure from a or respectivel. If the equation of a straight line is nown, using elementar geometr, the position of an point can be easil etermine whether it lies on the straight line or to which sie of it. For a given straight line ( α + β + δ = 0 ), the position etermining function is f (, ) = α + β + δ. The position of a point ( c, ) with respect to the straight line α + β + δ = 0 is then etermine using the following functions: f p s a q r c e

3 < 0, = 0, for below ( c, ) = > 0, for above ( after) the line () f on the line ( before) the line The perpenicular lines on a at a an can respectivel be efine as, a + a + a + a a a = an ( ) ( ) ( ) 0 ( a ) + a ( ) + ( + a a ) = 0 () The complete DMSC algorithm is summarize in the following Algorithm: Algorithm 1: Moifie istortion measurement algorithm for shape coing: Input: A shape point m an a line segment boune b points a an inclusive. Output: ( a,, m), the minimum istortion between the shape point m an the line segment a. Step 1: Construct two perpenicular lines on the current caniate segment enpoints. Step : Locate the position of an shape point m within a with respect to the relative orientation of the perpenicular lines erive in Step 1 using () an (). Step : IF position of m is within perpenicular line inclusive, THEN calculate istortion using (). Step : ELSE IF position of m is outsie of the perpenicular line at a, on the further sie of perpenicular line at, THEN the istortion is : ( a, m) = ( a m ) + ( a ), m Step : ELSE position of m is outsie of the perpenicular line at, on the farther sie of the perpenicular line at a, the istortion is : Step : Stop. ( a, m) = ( m ) + ( ), m The computational compleit of this algorithm is same as the shortest absolute istance measure algorithm, which is O () 1, once is onl require to be. This algorithm DMSC is mathematicall moelle an the potentialit of DMSC has teste using the ORD optimal algorithms escribe in [1-], which ha introuce shape coing as ORD optimal, for ifferent tpes of shapes with ifferent size an ifferent orientation. The following Section will provie a brief escription of the ORD algorithm.. The ORD Optimal Technique Incorporating the Propose Distortion Measurement Algorithm for Shape Coing The ORD optimal techniques [1-] approimate a bounar of a given shape b a set of significant points an encoe them instea of all the actual shape B = b, b,, be an orere set of points. Let { } 1 L bounar points, where points. Note Let S { s, s,, } b NB N B is the total bounar b 1 = b NB represents a close bounar. = 1 L s NS be an orere set of significant points use to approimate the bounar B, where N is the total number of points. The th ( ) S ege starts from s 1 an ens at s. Since S is an orere set, it uniquel represents the approimate polgon an is ifferentiall encoe. If r( s 1, ) is the requisite bit rate for ifferentiall encoing of significant point s, given that s 1 has alrea R require for set S is then:- encoe, the bit rate ( S ) N s R S) = r( s 1, s ) = 1 ( () where ( ) s 0,s 1 r is the number of bits require to encoe the absolute position of the first point s 1. For a close bounar r ( s N 1, s ) = 0 S N as the last point S oes not nee to be encoe. The - th ege which connects two consecutive significant points, 1 an s is an approimation to a portion of shape j = 1 j+ L, j+ l consisting of + 1 { b s, b 1, b = s } points. (, ) l shape s 1 s enotes the ege istortion between the shape an its approimate bounar efine using 1 maimum of all istortion measures. Let ( s s q) an s. The istortion measure is then the 1,, be the shortest absolute istance of q from the line connecting points s 1 an s. This istance is measure in the ORD algorithm b (). Note that this will also be measure using the new algorithm escribe in Algorithm 1 an will be compare an contraste with () in orer to test its potential. The maimum absolute istance between b s, b 1, b = s the portion of a shape { j = 1 j+ L, j+ l } an the ege of the approimate polgon ( s, ) 1 s is then given b: s, s ) = ma ( s, s, q) () ( 1 1 q { bj = 1, bj+ 1,..., bj+ l = } Hence, the maimum polgon istortion is :- D S) = ma ( s, s ) () ( 1 [1,..., NS ] s0, s1 =. where ( ) 0

4 Hence the problem becomes a solution to the following constraine optimization problem: min D( S), subject to R( S) R () s1,..., s N S ma where Rma is the maimum allowable rate. An also its complementar problem, min R( S), subject to: D( S) D (9) s1,..., s NS ma Dma is the maimum allowable istortion. This can be recursivel efine as a namic problem formulation: R ( ) = R 1( 1 ) + w( 1, ) (10) where : ( 1, ) > Dma w( 1, ) = (11) r( 1, ) : ( 1, ) Dma A etaile escription of rate measure r (, ) s 1 s is provie in [] which uses both linear an logarithmic schemes for a combine chain coe an run-length coing along with orientation encoing scheme for all significant points. The recursion nees to be initialize b setting ( ) 0 s 0 R equal to the number of bits require to coe the first point; so RN ( s ) R( S ) S N S =, the rate for the entire set of significant points for the complete bounar. The Directe Acclic Graph (DAG) [] algorithm has the least time compleit of currentl nown shortest path algorithms for a graph. The selection of a starting point plas an important role in such algorithms an [1] an [] have presente a clear efinition on how to select this point. Thus a weighte DAG is forme with ege list E = {( bi, b j ) B, i < j} an the weighting value is etermine b (11) to fin the shortest path. A DAG formation eample for an arbitrar shape point 9 is shown in Figure , Figure : Formation of DAG for point 9, i.e., (b 9, b j ) B, j>9 Algorithm : The ORD algorithm incorporating DMSC. 1. R*(s 1 ) = r(s 0, s 1 );. for i =,, N B ;. R*(b i ) = ;. for i = 1,, N B 1;. for j = i + 1,, N B ;. calculate ege istortion (b i, b j ) using algorithm (1);. loo up ege rate r(b i, b j );. assign w(b i, b j ); 9. if (R* (b i ) + w (b i, b j ) < R* (b j )); 10. R* (b j ) = (R* (b i ) + w (b i, b j ); 11. prev(b j ) = b i ; In Algorithm, R * ( b i ) represents the minimum rate to reach the shape point b i from the source via a polgon approimation. ( ) b NB p 1 = bi R * is the solution to () or (9). Prev escribe in step 11 of algorithm will contain all the significant points in a recursive fashion.. Results an Analsis Both the original ORD optimal algorithm [1-] an the ORD algorithm incorporating DMSC were implemente using Matlab.1 (The Mathwors Inc.) for a number of ifferent natural an snthetic arbitrar shapes, some of which were selecte b collecting the objects from IMSI 1. All object shapes were manuall segmente an the results were erive using the coe shape information generate b each respective algorithm. The eperimental results prouce b the original ORD an ORD with DMSC algorithms for the butterfl object in Figure (a), are shown in Figures (b)-(), with a maimum istortion value D ma = 10 pel. Note the two front antenna of the butterfl object were not separate b manual segmentation, because it was ifficult to intuitivel separate. The results in Table 1 show that the ecoe shape (Figure (c)) encoe using ORD with DMSC has a maimum istortion of 10 pel i.e. equal to D ma, while the corresponing istortion value for Figure (b) using the original ORD algorithm is pel. This improvement is ue to the fact that the original ORD algorithm oes not consier istance in its istortion 1 IMSI s Master Photo Collection, 19 Francisco Blv. East, San Rafael, CA , USA.

5 measure. To further illustrate the superiorit of ORD incorporating the new DMSC over the original, two results are superimpose in ifferent colours in Figure (). If the results in Figures (b)-() are compare with the original shape shown in Figure (a), it is visibl apparent that ORD with DMSC prouces a much more accurate shape representation i.e. has a lower shape istortion. (a) (b) (c) As in [1] With DMSC (a) (b) Figure : Decoe shapes using shortest absolute istance measure with D ma = 1 pel (a) Original ORD [1-], note highlighte area has a istortion of pel, b) shape coing with DMSC has Dma = 1 pel [1 - ] DMS () Figure : a) Original butterfl object, b) The ecoe shape using the original ORD algorithm, c) The ecoe shape using ORD with DMSC, ) A comparative plot showing the shapes prouce b the original ORD an ORD incorporating DMSC. A secon series of eperiments were conucte using the 1 st frame of the Miss America vieo sequence (Figure ) which was etensivel use in [1-] an the fish image (Figure (a)). The results for both are shown in Figures an (b)-() respectivel. The corresponing istortion values for the original ORD an ORD incorporating DMSC for prescribe maimum istortion values are given in Table 1. Results for the highlighte nec region in the 1 st frame of the Miss America sequence are summarize in Figure, coe with a istortion of 1 pel. The maimum istortion for the ecoe shape using ORD [1] is pel (inicate b the ellipse in Figure (a)), while DMSC maintaine a Dma = 1 pel (Figure (b)), so confirming the superiorit of this new approach. Figure : The 1 st frame of the Miss America sequence with the nec region highlighte in white. (a) (b) (c) [1 - ] DMS () Figure : a) The original fish object, b) The ecoe shape using the original ORD algorithm, c) The ecoe shape using ORD with DMSC, ) A comparative plot of the original ORD an ORD incorporating DMSC. The algorithm in [1-] prouces more istortion that the stipulate maimum istortion at the sharp corners as inicate with otte ellipses, whereas the new algorithm shows the minimum rate istortion an obes the tolerable istortion. Because, DMSC is consiering all perceptual istances. A final series of eperiments were performe upon the snthetic shape in Figure 9(a) to confirm the superiorit of the new DMSC algorithm. The results are presente in Figures 9(b c) an Figure 10 for D ma = 1 an D ma = pel respectivel. In Figure 9(b), when the top soli line is consiere to approimate the upper bounar, the shortest absolute istance will have a istortion of 1 pel, but in realit ( 1) =. 1 + pel will be perceive b a human. The same thing occurs in Figure 10(a).

6 As in [1] (a) DMSC (b) (c) Figure 9: a) An artificial object, b) results of original ORD algorithm in [1-] an c) results of ORD incorporating DMSC for D 1 pel. Distortion is ma = greater than 1 pel (actuall.1 pel) at an corner of the ecoe object for algorithms in [1-] As in [1] 10 0 DMSC (a) (b) Figure 10: Comparison of (a) Original ORD algorithm in [1-] an (b) ORD algorithms incorporating DMSC for D = ma pel. DMSC guarantees a boune maimum istortion, however in Figure (a) istortion is higher than pel (. pel) at an corner of the ecoe object. Table 1: Actual istortions in pel. Image D ma DMSC [1-] Butterfl Miss America 1 1 (Nec) Fish Artificial Object 1 1. Artificial Object. It can be conclue from the subjective assessment of the ecoe shape an actual istortion with respect to the stipulate limit, that the shortest absolute istance measure oes not maintain the constraint of maimum istortion. Table 1 confirms this because the perceptual istortion ecees the maimum allowable istortion in all eperiments, while the new DMSC algorithm taes cognisance of human visual perception b consiering ever istortion is alwas boune.. Conclusions Much research has alrea been conucte in respect of eveloping optimal shape coing algorithms, though none aress the ifference between geometric an perceptual istance in the shape coing technique. The original ORD algorithm is unable to ensure a boune shape istortion after ecoing, so to aress this issue, a new istortion measurement algorithm calle a moifie istortion measurement algorithm for shape coing (DMSC) has been presente in this paper. Eperimental results using a number of ifferent natural an snthetic arbitrar shapes confirme the improve performance of the DMSC algorithm, an prove it overcomes the limitation of the original ORD algorithm, without an increase in computational compleit.. References [1] A. K. Katsaggelos, L. P. Koni, F. W. Meier, J. Ostermann, an G. M. Schuster, MPEG- an Rate-Distortion-Base Shape-Coing Techniques, Proc. IEEE, vol., pp , June 199. [] G. M. Schuster, G. Melniov, an A. K. Katsaggelos, Operationall Optimal Point-Base Shape Coing, IEEE Signal Processing Mag., vol. 1, pp. -0, Nov [] G. M. Schuster an A. K. Katsaggelos, Rate- Distortion Base Vieo Compression: Optimal Vieo Frame Compression an Object Bounar Encoing, Kluwer Acaemic Publishers, 199. [] G. M. Schuster an A. K. Katsaggelos, An optimal bounar encoing scheme in the rate istortion sense, IEEE Trans. Image Processing, vol., pp. 1-, Jan [] H. Wang, G. M. Schuster, A. K. Katsaggelos, an T. N. Pappas, An Efficient Rate-Distortion Optimal Shape Coing Approach Utilizing a Seleton-Base Decomposition, IEEE Trans. Image Processing, vol. 1, pp , Oct. 00. [] L. P. Koni, G. Melniov, an A. K. Katsaggelos, Joint Optimal Object Shape Estimation an Encoing, IEEE Trans. Circuits Sst. Vieo Technol., vol. 1, pp. -, Apr. 00. [] R. Koenen, MPEG- multimeia for our time, IEEE Spectrum, vol., pp. -, Feb [] T. H. Cormen, C. h. Leiserson, R. L. Rivest, C. Stelin, Introuction to Algorithms, n eition, MIT Press, 001.