A Fast Distortion Measurement Using Chord

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A Fast Distortion Measurement Using Chord-Length Parameterisation within the Vertex-. Based Operational Rate-Distortion Optimal Shape Coding Framework.
A Fast Distortion Measurement Using Chord-Length Parameterisation within the VertexBased Operational Rate-Distortion Optimal Shape Coding Framework Ferdous A. Sohel1, Gour C. Karmakar, and Laurence S. Dooley Gippsland School of Information Technology Monash University, Churchill, Victoria – 3842, Australia.

ABSTRACT Existing vertex-based operational rate-distortion (ORD) optimal shape coding algorithms use the shortest absolute distance (SAD) or alternatively either the distortion band (DB) or tolerance band (TB) as their distortion measuring technique. Each approach however can lead to inaccurate distortion measurements, though these can be avoided by employing the accurate distortion measurement technique for shape coding (ADMSC). From a computational time perspective, an N point contour requires O N 2 time for DB and TB for both polygon and B-spline based encoding,

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while SAD and ADMSC incur order O(N ) time for polygonal encoding, but O N 2 complexity for B-spline based encoding, thereby rendering ORD optimal algorithms computationally inefficient. This paper presents a novel distortion measurement strategy based on chord-length parameterisation (DMCLP) of a boundary that incurs order O(N ) complexity for both polygon and B-spline based encoding, while preserving an analogous rate-distortion performance to the original ORD optimal shape coding algorithms, when it is embedded within the ORD framework. I.

INTRODUCTION

Despite facilitating increasingly effective retrieval, manipulation and interactive editing functionality for both natural and synthetic video sequences, object-oriented video coding using shape information remains a challenging research topic [1]-[6]. The universal pursuit for efficiency in existing communication technologies mean that applications such as video-on-demand, video streaming over the Internet, real-time applications and mobile video transmissions for handheld devices will derive significant benefits from strategies enabling fast shape coding. In [1], a rigorous review of shape coding algorithms was presented with the conclusion that the classical vertexbased shape coding framework was optimal in an operational rate-distortion (ORD) sense. With both polygonal and quadratic B-spline based shape encoding strategies being deployed in [1], these have become the kernel for several other shape coding algorithms [1]-[5] within the ORD framework. The general aim of all these algorithms is that for some prescribed distortion, a shape contour is optimally encoded in terms of the number of bits, by selecting the set of control points (CP) that incurs the lowest bit rate and vice versa. Distortion measures thus play a critical role in shape coding algorithms, as evidenced in the fixed admissible distortion framework proposed by Katsaggelos et al. [1], which applied either the shortest absolute distance (SAD) or distortion band (DB) technique. The DB was subsequently generalised as a tolerance band (TB) to support variable admissible distortions [5], though it inherited the drawbacks of DB including the propensity for trivial solutions, a problem that can be partially solved by using a sliding window (SW). The SW constrains the search space for the next CP to only points within the window so it improves the computational time requirements, but crucially compromises optimality in a bit-rate sense. Moreover, both SAD and DB suffer from distortion measurement inaccuracies particularly at sharp corners and edges, a limitation resolved with the introduction of the accurate distortion measurement technique for shape coding (ADMSC) [6]. Since distortion measurement must be seamlessly integrated into the core of the ORD optimal shape coding framework [1]-[5], the overall computational complexity of these algorithms is highly dependent on the distortion measuring overhead and so to ensure computationally efficient encoding, it is essential to employ a fast technique. For both polygonal and B-spline based encoding, DB and TB incur order O N 2 computational complexity, where N is the number of shape boundary points. In addition, the computational impost for both DB and TB proportionally increases with admissible distortion as the width of the respective admissible bands widens. In contrast, the SAD and ADMSC distortion measures require O(N ) time for polygonal encoding, but still mandate

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order O N 2 time for B-spline based shape encoding. B-spline encoding affords a twofold superiority over its polygonal counterpart, in firstly requiring a lower bit-rate because higher-order degree curves are used and secondly 1

Corresponding author: E-mail: [email protected]; Phone: +61 3 990 26133; Fax.: +61 3 990 26842. Postal address: 7N 106, GSIT, Monash University, Churchill, Victoria – 3842, Australia.

smoothness due to their inherent parametric continuity, B-spline approximating curves possess greater smoothness. It is for these reasons that most ORD optimal algorithms [4]-[5] have been developed primarily within a B-spline framework, with the TB usually employed as the distortion measuring technique. To secure computationally fast distortion measurements within the ORD optimal shape coding framework while concomitantly preserving comparable quality, this paper presents a novel distortion measurement technique using chord-length parameterisation (DMCLP) of the associated boundary to determine the approximating curve points and hence the distortion. It will be proven that for both polygon and B-spline based encoding, DMCLP entails only O(N ) complexity, with the rate-distortion (RD) performance of embedding DMCLP in the vertex-based ORD optimal shape coding framework being analysed and tested upon a large number of arbitrary shapes. The results corroborate that DMCLP ensures computationally faster encoding, while maintaining comparable RD performance. The rest of this paper is organised as follows: Section II provides a short overview of the existing vertex-based ORD optimal shape coding framework together with a brief description of both the TB and ADMSC measuring techniques from a computational complexity perspective. Section III details the new DMCLP measurement strategy for both polygon and B-spline based encoding, with Section IV providing a comprehensive discussion on the empirical results and performance comparison with DMCLP embedded into the original ORD framework. Finally, some concluding remarks are given in Section V. II.

EXISTING VERTEX-BASED ORD OPTIMAL SHAPE CODING FRAMEWORK

Existing ORD optimal shape coding algorithms seek to determine and encode a set of CP representing a particular shape. Let the boundary B = {b0 , b1, L , bN B −1} be an ordered set of points, where N B is the total number of boundary points and b0 = bN B −1 for a closed boundary. P is an ordered set of CP used to approximate B with P ⊆ C , where C is the set of vertices in the admissible CP band, which is normally a fixed width band around the boundary of the shape. The ORD optimal shape coding technique summarised in Algorithm 1 for quadratic B-spline based encoding (detailed in [1], [2] and [4]) then determines the optimal P for boundary B within the RD constraints.

Algorithm 1: The quadratic B-spline based ORD optimal shape coding algorithm [5]. Inputs: B , Tmax and Tmin – peak admissible distortions. Variables: MinR (ci,l , c j , m ) – minimum bit-rate to encode up to c j ,m from b0 including edge ci ,l c j ,m , pred (ci ,l , ck , n ) –

preceding CP of state ci , c k , T [i ] – admissible distortion at bi , L[i ] – the number of vertices associated with bi in C . Output: P – ordered set of CP approximating B . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Determine T [i ] , 1 < i < N B − 1 ; Initialise MinR (c0,0 , c1,0 ) with the total bits required to encode the first shape point b0 ;

Set MinR (ci ,l , c j ,m ) ,

0 < i < N C − 1, 0 ≤ l < L[i ],

FOR each vertex ci ,l , 0 ≤ i < N C − 2, 0 ≤ l < L[i ]

i < j ≤ N C − 1,0 ≤ m < L[ j ] to infinity;

FOR each vertex c j ,m , i < j < N C − 1, 0 ≤ m < L[ j ] FOR each vertex ck ,n ,

j < k ≤ N C − 1, 0 ≤ n < L[k ]

Check the distortion d (ci,l , c j , m , ck , n , T ) ;

Determine edge-weight w(ci,l , c j ,m , ck , n ) ;

IF (MinR (ci ,l , c j , m ) + w(ci,l , c j , m , ck , n )) < MinR (c j , m , ck , n ) THEN

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MinR c j , m , ck , n = MinR ci ,l , c j , m + w ci ,l , c j , m , ck , n ; pred c j , m , ck , n = c i ,l ;

11. Find P with properly indexed values from pred ; Tolerance band (TB): As alluded in Section I, the variable-width TB is a generalisation of the fixed-width DB, and works as follows [4]: Draw a circle around each boundary point bi of radius T [i ] , TB consists of the set of all points that lie inside the circles; check the distortion and if all points on a candidate edge (curve) lie inside the TB, it is considered that the candidate edge (curve) maintains the distortion criterion. As every curve point is searched within the TB, the complete checking process for a candidate curve entails O N B 2 computational time and since the distortion of each individual associated boundary point is not tested to see if it lies within the TB, it thus inherits the same accuracy measurement problems as the DB [6].

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Accurate distortion measurement technique for shape coding (ADMSC) [6]: This was introduced to resolve the measurement inaccuracies originating from the DB. In polygonal encoding, the edge-distortion for all associated boundary points are calculated from the candidate edge and checked against the corresponding admissible distortion which takes O (N B ) computational time. B-spline based encoding however, requires O N B 2 time to check the distortion because the B-spline curve is in fact a piecewise polygon-edge and so for each boundary point associated with a candidate curve, the individual distortion has to be measured from all edges that form the approximating curve. The minimum edge-distortion value is then taken as the distortion for that particular boundary point and compared with the corresponding admissible distortion, so while the ADMSC approach takes O (N B ) for polygonal

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encoding, it still requires O N B 2 for B-spline encoding. In the next section, a novel distortion measuring technique is presented that is computationally efficient, incurring only order O(N B ) time complexity for both polygon and Bspline based encoding. III.

DISTORTION MEASUREMENT TECHNIQUE USING CHORD-LENGTH PARAMETERISATION

This proposed chord-length parameterisation based distortion measurement method has been developed to be embedded within both B-spline and polygon based shape coding framework, which will now be respectively considered. B-spline based framework: Before detailing the actual distortion measurement technique, a short overview of the B-spline curve is presented. B-splines are members of a family of parametric curves, where a 2-dimensional ⎧[x (t ), y (t )], ⎩0

curve segment is defined as: Q(CP set , t ) = ⎨

for 0 ≤ t ≤ 1 otherwise.

(1)

A quadratic B-spline curve for a CP set ( pu −1, pu , pu +1 ) is defined as: ⎡ 0.5 − 1.0 Qu ( pu −1, pu , pu +1 ) = t t 1 ⋅ ⎢⎢− 1.0 1.0 ⎢⎣ 0.5 0.5

[

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0.5 ⎤ ⎡ pu −1, x ⎥ ⎢ 0.0 ⎥ ⋅ ⎢ pu , x 0.0 ⎥⎦ ⎢⎢ pu +1, x ⎣

pu −1, y ⎤ ⎥ pu , y ⎥ , 0 ≤ t ≤ 1 ⎥ pu +1, y ⎦⎥

(2)

So for each value of parameter t , one B-spline point is generated and the number of t values thereby determines the number of points on the B-spline curve. As mentioned in [1], each curve segment approximating the boundary is associated with an ordered set of continuous boundary points. The philosophy in this paper is that if each boundary point associated with a B-spline curve has its corresponding point on the curve then the distortion can be measured as the Euclidean distance between these two points. This can be achieved provided every associated boundary point for a candidate curve has a corresponding value of t , so there is one point on the candidate curve for each associated boundary point. If the distortion between respective boundary points (associated with the curve segment) and their corresponding curve points is less than or equal to the admissible distortion of the respective boundary points, the curve upholds the permitted distortion and is also considered a candidate curve segment in the RD optimisation process. To determine this value of t, chord-length parameterisation is applied since it produces a smooth curve as well as being extensively used in the development of parametric curve algorithms [7]. For an arbitrary curve segment having start and end indices k1 and k 2 respectively of the associated boundary points in B , the values of t can be determined as: , if r = k1 ⎧0 ⎪⎪ t r = ⎨ bk1 bk1 +1 + bk1 +1bk1 +2 + L+ br −1br otherwise. ⎪ ⎪⎩ bk1 bk1 +1 + bk1 +1bk1 +2 + L+ bk2 −1bk2 where tr is associated to br and br −1br is the Euclidean distance between points br −1 and br .

(3)

Once a B-spline curve point is generated using tr , the peak distortion at boundary point br is determined as the Euclidean distance between these two points. It is then necessary only to ensure whether the boundary point distortion is maintained. The complete DMCLP measurement procedure is summarised in Algorithm 2. Polygon based framework: In polygonal encoding, two CP ( pu −1, p u ) are required for each candidate edge rather than the three in quadratic B-spline based encoding. The value of tr is again determined by chord-length parameterisation using (2) though in this particular case the corresponding edge point q is found by linear interpolation in (4), rather than a B-spline curve point. q = tr ⋅ pu + (1 − tr ) ⋅ p u −1 (4)

This means Algorithm 2 can therefore also be applied for polygon-based encoding with appropriate inputs and using (4) instead of (2) in Step 7. Algorithm 2: DMCLP for B-spline based shape encoding. Inputs: B - boundary, ( pu −1 , pu , pu +1 ) - control points, T – admissible distortion, k1 , k 2 indices of the associated boundary points. Output: Flag – 1 means distortion criteria upheld; 0 means not upheld. 1. Flag=1; 2. chord _ lenr = 0 for r = k1 ; 3. FOR ∀r , k1 < r ≤ k 2 4. 5.

chord _ lenr = chord _ lenr −1 + br −1br ;

FOR ∀r , k1 ≤ r ≤ k2 chord _ lenr chord _ lenk 2

6.

tr =

7.

Calculate B-spline point q using tr in (2); dist = qbr ;

8. 9.

;

IF dist > T [r ] Flag =0;

It will now be proven in Lemma 1 that the proposed DMCLP technique maintains the admissible distortion for all object shapes. Lemma 1: If d [ j ] and T [ j ] are respectively the generated and peak admissible distortions for an arbitrary boundary point b j , then the DMCLP technique always upholds the admissible distortion so that d [ j ] ≤ T [ j ] . Proof: By embedding DMCLP into the ORD optimal shape coding framework, every boundary point will have a respective point on the encoded shape. If the distortion defined in Step 8 of Algorithm 2 at boundary point b j is dist j while the actual distortion at the same point is d [ j ] , then: d [ j ] ≤ dist j

(5)

From the distortion test in Step 9, for a curve segment included in the encoded shape the following must be true: dist j ≤ T [ j ]

(6)

therefore, from (5) and (6) d [ j ] ≤ T [ j ] . Computational complexity analysis: In terms of the order of complexity for Algorithm 2, Steps 3 and 4 calculate the cumulative chord-length of the boundary at a cost, in the worst case of O (N B ) , while Steps 5-9 firstly determine the value of t and then generate the B-spline point using (2) before checking to see whether the admissible distortion is maintained which also incurs O(N B ) . The overall complexity for Algorithm 2 in the worst case is therefore O(N B ) , which is one degree less than all existing distortion measurement techniques for B-spline based encoding, while for polygonal based encoding, DMCLP exhibits order of computational complexity O (N B ) , which is exactly the same as for SAD and ADMSC, but one degree lower than either the DB or TB. IV.

RESULTS AND ANALYSIS

To substantiate that ORD optimal algorithms employing the new DMCLP distortion measure produce faster encoding with comparable RD results to existing distortion measurement techniques, a number of experiments were performed upon a range of arbitrarily object shapes, though in this section the analysis focuses upon the Neck region of the 31st frame of the Miss America video sequence. This was selected as it possesses both sharp and gradually shaped parts and was also extensively used in the literature [1], [2], while experimental results on some other test object shapes have been included in Appendix A. For presentational clarity, the following nomenclature is adopted: Encoding type—-Distortion measurement technique, with the former defining whether either B-spline or polygon based encoding was used, and the latter the distortion measuring technique employed. The first series of experiments was conducted to compare the computational time requirements incurred by algorithm combinations for various admissible distortion pairs. Table 1 displays the total CPU time for various

algorithm implementations running on a 2.8GHz Pentium-4 processor, with 512MB RAM under a Windows XP operating system. It should be noted that to ensure trivial solutions did not occur in the TB measure, a sliding window was used, which for the purposes of equity in experiments for all algorithms, had a length 21 pel as this was the maximum feasible length with a 15 codeword based logarithmic code [2]. From the results in Table 1, it is evident those algorithms using DMCLP as their distortion measurement technique were computationally faster compared with the respective TB and ADMSC counterparts, to underscore the theoretical time complexity analysis given in Sections II and III. For instance, with Tmax = 3 pel , Tmin = 1 pel B-spline-TB, B-spline-ADMSC and B-spline-

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DMCLP respectively required 620.3, 587.8 and 312.5 second, so reflecting that while TB and ADMSC cost O N 2 for B-spline based encoding, the overhead for DMCLP is only O(N ) . For polygon based encoding, DMCLP algorithms required less time than their ADMSC based counterparts, while as expected TB algorithms consistently incurred a significantly greater time overhead because as detailed in Sections II and III, both DMCLP and ADMSC are of order O(N ) compared with O N 2 for the TB. For B-spline based algorithms using small admissible distortions, TB was faster than ADMSC, though for higher distortion values, the comparative time requirement for TB increased dramatically. For example, with Tmax = 2 pel , Tmin = 2 pel B-spline-TB and B-spline-ADMSC required 545.5 and 582.1secs respectively, while the corresponding values for Tmax = 3 pel , Tmin = 2 pel were respectively 680.4 and 591.6secs because at higher distortions, the TB width is extended so commensurately increasing the computational overhead for distortion checking. Similar observations can be made for the results of the other test shapes in Appendix A.

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Table 1: Computational time (in second) and bit-rate (in bit) requirements (with the obtained distortion in parenthesis whenever the peak distortion failed to be bounded for a prescribed admissible value) for different admissible distortion pairs ( Tmax , Tmin in pel) for various ORD optimal shape coding algorithms. Tmax = 1 , Tmin = 1

Algorithm Time Bit-rate Polygon-TB 4.26 115(2.24) Polygon-ADMSC 1.63 138 Polygon-DMCLP 1.61 146 B-spline-TB 390.60 133 (2.0) B-spline-ADMSC 554.20 127 B-spline-DMCLP 270.20 132

Tmax = 2 , Tmin = 1

Time 6.03 1.89 1.83 510.50 575.00 290.30

Bit-rate 95(2.24) 109 112 87 (3.6) 100 102

Tmax = 2 , Tmin = 2

Time 7.73 2.01 1.92 545.50 582.10 297.00

Tmax = 3 , Tmin = 1

Bit-rate Time Bit-rate Time Bit-rate 79 (4.0) 11.35 71 (5.0) 12.66 70 (5.0) 86 2.15 86 2.25 86 93 1.97 92 2.02 87 78 (2.8) 620.30 76 (6.0) 680.40 75 (6.0) 78 587.80 78 591.60 78 78 312.50 80 314.30 78

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d) B-spline-TB e) B-spline-ADMSC f) B-spline-DMCLP Figure 1: Results for the Neck region of the 31st frame of the Miss America sequence with Tmax = 2 pel and Tmin = 2 pel (legends – solid line: approximated boundary; dashed line: original boundary; asterisk: CP).

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To confirm that ORD algorithms which integrate DMCLP produce analogous RD results to the original ORD framework, a set of experiments concentrating upon the required bit-rate for a prescribed set of admissible distortion values was undertaken. The respective results produced by different algorithm combinations are displayed in Figure 1(a)-(f) for the admissible distortion pair Tmax = 2 pel , Tmin = 2 pel while the numerical results together with some other distortion pairings are summarised in Table 1. The results in Figure 1 reveal that all algorithms produced very similar shapes with the notable exceptions of the Polygon-TB and B-spline-TB algorithms where, as indicated by the rectangular boxes, the distortion was greater than the prescribed peak admissible value. Tmax and Tmin were made equal so all boundary points had the same admissible distortions thereby manifesting the aforementioned measuring problems inherent in the TB. Furthermore, while TB-based algorithms failed to be constricted to the peak admissible distortion, in contrast both DMCLP and ADMSC consistently maintained a bounded distortion. The numerical results in Table 1 show the bit-rate requirement for different combinations of the same admissible distortion pairings are comparable between respective algorithms, as for example in Tmax = 2 pel , Tmin = 1 pel , Polygon-TB, PolygonADMSC and Polygon-DMCLP required 95, 109 and 112 bits respectively, while the corresponding B-spline-TB, Bspline-ADMSC and B-spline-DMCLP combinations required 87, 100 and 102 bits. The comparatively small number of additional bits incurred for DMCLP-based algorithms compared to ADMSC was due to it being a slightly more relaxed distortion measure, in the sense that ADMSC always guarantees to calculate the absolute minimum distortion. A cursory review of the TB results fallaciously reveals they consistently required a smaller number of bits for encoding, until it is realised these algorithms did not always uphold the maximum admissible distortion constraint as in the B-spline-TB case, which generated a maximum distortion of 3.6 pel despite being supposedly bound to a peak of 2 pel. This meant the TB erroneously ignored certain parts of the shape leading to a lower bit requirement than the ADMSC and DMCLP based algorithms, which both guaranteed the maximum admissible distortion. It is also worth mentioning that for the same admissible distortion pair, B-spline based encoding always required a lower number of bits than its polygon counterpart, provided the admissible distortion was maintained. For instance, with Tmax = 2 & Tmin = 2 pel , Polygon-ADMSC required 86 bits compared to only 78 bits for B-spline-ADMSC endorsing the earlier comment about higher degree curve B-splines requiring a lower bit rate than a lower degree polygon approximation. All these judgements are again corroborated for the other test shape results presented in Appendix A. V.

CONCLUSIONS

The computational speed of any algorithm is a vital benchmark of quality as many applications mandate fast and efficient throughput. This paper has presented a novel distortion measurement technique based on chord-length parameterisation (DMCLP) which can be seamlessly embedded into the classical vertex-based rate-distortion optimal shape coding framework to reduce the order of computational time complexity. It has been proven that for a N point boundary, DMCLP requires order O(N ) computational time for both B-spline and polygon based encoding

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in contrast to the conventional tolerance band approach which incurs O N 2 complexity in both cases. Experimental results have shown that algorithms embedding the new distortion measurement technique consistently provided both comparable rate-distortion performance to existing algorithms and faster encoding due to the improved order of complexity. VI.

REFERENCES

[1] A.K. Katsaggelos, L.P. Kondi, F.W. Meier, J. Ostermann, and G.M. Schuster, “MPEG-4 and rate-distortionbased shape-coding techniques,” Proc. IEEE, vol.86 (6), pp.1126-1154, 1998. [2] G.M. Schuster, and A.K. Katsaggelos, Rate-Distortion Based Video Compression: Optimal Video Frame Compression and Object Boundary Encoding, Kluwer Academic Publishers, 1997. [3] G.M. Schuster, G. Melnikov, and A.K. Katsaggelos, “Operationally optimal point-based shape coding,” IEEE Signal Processing Mag., vol.15 (6), pp.91-108, 1998. [4] L.P. Kondi, F.W. Meier, G.M. Schuster, and A.K. Katsaggelos, “Joint optimal object shape estimation and encoding,” in Proc. SPIE Conf. Visual Communications and Image Proc., pp.14-25, 1998. [5] L.P. Kondi, G. Melnikov, and A.K. Katsaggelos, “Joint optimal object shape estimation and encoding,” IEEE Trans. Circuits Systems. Video Tech., vol.14 (4), pp.528-533, 2004. [6] F.A. Sohel, L.S. Dooley, and G.C. Karmakar, “Accurate distortion measurement for generic shape coding,” Pattern Recognition Letters, vol.27 (2), pp.133-142, 2006.

[7] G. Farin, Curves and surfaces for computer aided geometric design – a practical guide, 4th edition, Academic Press, 1997. Appendix A: Supplementary Results To confirm the superior computational complexity performance of the ORD optimal shape coding algorithm with the proposed DMCLP technique embedded, a complete set of experimental results for two other popular test shapes is included in this Appendix. Table 2 summarises the computational time related results while Table 3 presents the numerical results from the ratedistortion perspective upon two popular test shapes – Lip and Fish. The computational time related results confirm that the algorithms using DMCLP as the distortion measurement was always computationally the fastest among the algorithms. Table 3 vindicates that the algorithms embedded DMCLP required an analogous number of bits to those embedded other distortion measuring algorithms and successfully bounded the admissible distortion. Moreover, TB failed to maintain the admissible distortion in all cases while both ADMSC and DMCLP successfully did this. Table 2: Computational time requirements for admissible distortion pairs ( Tmax , Tmin ) for various ORD optimal shape coding algorithm combinations. Shape Algorithm Lip

Fish

Polygon-TB Polygon-ADMSC Polygon-DMCLP B-spline-TB B-spline-ADMSC B-spline-DMCLP Polygon-TB Polygon-ADMSC Polygon-DMCLP B-spline-TB B-spline-ADMSC B-spline-DMCLP

Tmax = 1 , Tmin = 1

2.42 0.82 0.80 170.20 249.90 124.80 12.75 4.81 4.74 1220.35 1560.33 697.02

Time requirements (seconds) Tmax = 2 , Tmin = 2 Tmax = 3 , 3.48 4.35 0.97 1.05 0.83 0.94 220.20 246.30 253.60 257.30 136.00 143.20 19.73 22.18 5.35 5.48 5.16 5.24 1545.25 1610.05 1650.55 1669.52 960.25 967.50

Tmax = 2 , Tmin = 1

Tmin = 1

6.08 1.13 0.95 296.20 262.10 148.50 35.52 5.79 5.37 2172.22 1750.36 1040.52

Tmax = 3 , Tmin = 2

6.51 1.19 0.96 330.30 270.10 150.20 38.65 5.86 5.42 2260.50 1785.44 1065.32

* Lip shape was selected from the 31st frame of the Miss America video sequence. Table 3: Bit requirements for admissible distortion pairs ( Tmax , Tmin in pel) for various ORD optimal shape coding algorithms with the obtained distortion in parenthesis whenever the distortion failed to be bounded for a prescribed admissible value. Bit-rate requirements Shape Algorithm Lip Polygon-TB Polygon-ADMSC Polygon-DMCLP B-spline-TB B-spline-ADMSC B-spline-DMCLP Fish Polygon-TB Polygon-ADMSC Polygon-DMCLP B-spline-TB B-spline-ADMSC B-spline-DMCLP

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Tmax = 2 , Tmin = 1

Tmax = 2 , Tmin = 2

Tmax = 3 , Tmin = 1

Tmax = 3 , Tmin = 2

68 (2.24) 97 109 85 (4.24) 88 90 280 (8.0) 355 365 272 (7.0) 283 327

51 (5.0) 74 80 63 (4.24) 68 71 220 (8.0) 281 290 217 (7.0) 240 250

48 (5.0) 69 75 52 (4.24) 68 69 202 (9.0) 249 259 212 (7.0) 233 243

48 (5.0) 61 62 50 (6.0) 60 61 189 (9.0) 223 235 188 (8.0) 204 205

44 (5.0) 52 56 50 (6.0) 51 51 188 (9.0) 223 229 175 (8.0) 188 190

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d) B-spline-TB e) B-spline-ADMSC f) B-spline-DMCLP Figure 2: Results for the Lip region of the 31st frame of the Miss America sequence with Tmax = 2 pel and Tmin = 2 pel (legends – solid line: approximated boundary; dashed line: original boundary; asterisk: CP). 0

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Subjective results for given admissible distortion pair Tmax = 2 pel and Tmin = 2 pel using the different algorithm combinations are presented in Figure 2 and Figure 3 where both show that TB failed to maintain the admissible distortion while ADMSC and DMCLP were successful and moreover, the shapes were better preserved in both DMCLP and ADMSC than in TB.