optimum uniform piecewise linear approximation of planar curves

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-8, NO. 1, JANUARY 1986

Optimum Uniform Piecewise Linear Approximationof Planar Curves

JAMES GEORGE DUNHAM, MEMBER, IEEE

Abstract-Two-dimensional digital curves are often uniformly ap-proximated by polygons or piecewise linear curves. Several algorithmshave been proposed in the literature to find such curves. We presentan algorithm that finds a piecewise linear curve with the minimal num-ber of segments required to approximate a curve within a uniform er-ror with fixed initial and final points. We compare our optimal algo-rithm to several suboptimal algorithms with respect to the number oflinear segments required in the approximation and the execution timeof the algorithm.

Index Terms-Optimal approximation, piecewise linear approxima-tion, planar curve segmentation, polygonal approximation, scan-along.

I. INTRODUCTIONpIECEWISE linear approximation of a two-dimen-

sional digital curve is often a compact and effectiverepresentation of the curve for shape analysis and patternclassification [1], [2]. Such representations facilitate theextraction of numerical "features" for the description orclassification of the given curve.The problem of piecewise linear approximation for two-

dimensional curves has received considerable attention inthe literature [3]-[17]. These approximation problems areclassified according to the type of norm used in the ap-proximation, to whether the knots of the piecewise linearcurve are fixed or variable, and to the continuity or dis-continuity of the approximation. We shall consider the casewhere the knots are constrained to lie on the digital curveto be approximated, the approximations are continuous,and the norm is uniform where the maximum of the ab-solute value of the pointwise error between the point andthe approximating line segment is taken.Ramer [3] in 1972 and Duda and Hart [4] in 1973 pro-

posed the first such algorithm which iteratively splits theapproximation curve into finer and finer linear segmentsuntil the error criterion is satisfied on each segment. In1974, Pavlidis and Horowitz [5] proposed an algorithmbased upon splitting and merging segments as well as ad-justing segment end points until the desired criterion issatisfied, while Reumann and Witkam [15] proposed astrip algorithm that approximates the tangent to the curveof the fixed endpoint. Williams [6] in 1978 proposed analgorithm based upon the idea of finding the longest seg-ment such that all subsegments with the same initial point

Manuscript received March 25, 1985; revised June 24, 1985.The author is with the Department of Electrical Engineering, Southern

Methodist University, Dallas, TX 75275.IEEE Log Number 8405793.

satisfy the error criterion. Davis [9] in 1979 proposed analgorithm based upon the smoothed k-curvature of theplanar curve. In 19-80, Sklansky and Gonzalez [7] pro-posed an algorithm based upon finding the longest possiblesegment to satisfy the error criterion. Both Williams, andSklansky and Gonzalez found scan-along implementationsof their algorithms which made them run more efficiently.Badi'i and Peikari [8] in 1982 proposed an algorithm thatfinds a segment satisfying the error criterion up to a max-imal possible length, while Pavlidis [10] developed an al-gorithm based upon the notion of the points in a line seg-ment being almost collinear. Recently, in 1985, Roberge[11] proposed an enhanced version of Reumann and Wit-kam's algorithm which was designed to run very quickly.

Since the time for processing features of a polygon isoften proportional to the number of linear segments, it isdesirable to find approximations with the fewest numberof linear segments that satisfy the uniform error criterion.In Section II, we present an algorithm that achieves suchan approximation with fixed initial and final points. Wecall this the optimal algorithm since it seeks a represen-tation with a minimal number of linear segments. Theprinciple of optimality [18] is used and the algorithm issomewhat similar to Bellman's [19] dynamic program-ming solution of an approximation problem using a squarederror norm.

In Section III, an efficient scan-along implementation ofthe optimal algorithm is developed along the lines of Wil-liams [6] and Sklansky and Gonzalez [7]. We point outthat neither Williams nor Sklansky and Gonzalez suffi-ciently take into account how the distance between a po-tential endpoint and the intitial point of a segment affectsthe scan-along procedure. We clarify this point and showits effect on the scan-along procedure.

In Section IV, we compare the optimal algorithm withthe algorithms of Ramer [3], Pavlidis and Horowitz [5],Williams [6], Sklansky and Gonzalez [7], Badi'i andPeikari [8], Davis [9], Pavlidis [10], and Roberge [11].We compare the performance of these algorithms on fourclosed digital curves with respect to the number of linearsegments required in the approximation and the executiontime of a Fortran 77 implementation on a TI professionalcomputer. Finally, conclusions are drawn in Section V.

II. OPTIMAL ALGORITHMA discrete planar curve parameterized by 0 c i c I is

described by

0162-8828/86/0100-0067$01.00 © 1986 IEEE

67


Y-AXI S Y-AXIS

PK

Pu]

puUJ-1

X-AXIS

Fig. 1. Computing the approximation error.

Cd = {(Xi, Y)|O < i < I} -

For simplicity, we sometimes denote the point (xi, yi) byPi.

The discrete planar curve is to be approximated by acontinuous piecewise linear planar curve C0 whose break-points are on Cd and satisfy

0 =-UO < U I < U2 < . . . < UN-I

Thus, the jth interval of Ca consists of the straight linethat connects the points p1j and Pu,, Plij ,IPa> and it ap-proximates the points Pk of Cd where uj 1 < k < uj.The error between a point of Cd and the approximating

planar curve C, is the Euclidean distance of the point tothe nearest point in the approximation interval. For ex-ample, in Fig. 1 the approximation interval is the straightline connecting pNj and pa,,p , , pu. The error betweenpoint Pk and the line pj1 ,1I pUJ is found by the line whichends at Pk and is perpendicular to the line pj 1, pj. Theerror between the point P,U and the line Pu] 1' pj is thedistance of the line between pk, and Puj, Pk', Puj We denotethis error by e(uj,, luj, k) and e(uj-, uj, k'), respec-tively.We next give a formal procedure for computing the error

between a point and the approximating line segment. Ifwe translate the coordinates so that p16 is at the originof the x-y axis and then rotate by the amount -0 where 0is the angle between the line puj ,, pu] and the x-axis ofFig. 1, we get the situation shown in Fig. 2 where all Eu-clidean distances are preserved. Now if the x-coordinateof p' lies between 0 and the x-coordinate of pa>, then theerror is simply the absolute value of the y-coordinate ofpk. If the x-coordinate of p, is less than 0, then the erroris the distance between p, and the origin. If the x-coor-dinate of p' is greater than p' , then the error is the dis-tance between p, and p, . Using elementary trignometry,we get

cos 0x -xu

(Xu,I - XU] )2 + (YuJ - YU]-I)

P U_

sin 0 =

PI K

P K

PI X-AX I S

Fig. 2. Translated and rotated axis.

YUj - Yuj I

(Xu1 -xj_ )2 + (y - y )2 '(lb)

The coordinate translation and rotation of the point Pk isaccomplished by

Xk = (Xk - Xuj ) COS 0 + ( Yk Yu] ) sin 6 (2a)

Yk =-(Xk Xu_ I) sin 0 + (Yk - Yuj-) cos. (2b)

Thus, for uj c k <

e (u- 1, uj, k) =

Uj the error is

V/(Xk - XU]- I) + (Yk - Yu] )2

if Xk < 0

V(Xk - Xu])2 + (yk_y

ifxI > XI(3)

IYk,

else.

We note that when the transformed x-coordinate of thepoint lies between the transformed end points of the line,the distance formula is equivalent to that given by Pavlidisand Horowitz [5].Having defined the error between a point and its ap-

proximating line segment, we define the error over a seg-ment as

e(uj 1, uj) = max e(uj 1uUj, k).Uj-]1k<uj

Finally, we define the error between Cd and its approxi-mation Ca as

e(Cd, C) = max e(uj_I uj)I <j<N

and thus we have a uniform norm.For a given planar curve Cd, fix the maximum allowable

approximation error to be e. Our goal is to find a contin-

68

DUNHAM: PIECEWISE LINEAR APPROXIMATION OF PLANAR CURVES

uous piecewise linear planar curve Ca with breakpoints onCd and with a minimal number of segments N where e (Cd,Ca) < e. Given an index u, let V(u) denote all the pointswith v c u where e(v, u) ' c and let F(u) denote theminimal number of segments needed to approximate thecurve Cd from points po to pu within c. Then F(u) satisfies

F(u) = min 1 + F(v) (4)veV(U)

with initial conditions

F(1) = 1 and F(O) = 0

since u > 2, there is one line segment PV, PU and then aminimal number of possible segments from 0 to v. This isa particular application of the principle of optimality [18].We can recursively compute F(u) using dynamic program-ming with the given initial conditions and F(I) will be theminimal number of segments needed for Ca to approxi-mate Cd within e.We note that there have been other efforts to implement

optimal algorithms based upon dynamic programming.The first work to consider the approximation of curves byline segments using dynamic programming was Bellman[19] in 1961. Gluss wrote a series of papers [20]-[23] ex-panding upon Bellman's work. Lawson [24] in 1964 useddynamic programming to establish the existence of a bal-anced error property. Bellman, Gluss, and Roth [25] ex-tended the notion to quasilinearization in 1964. In 1971,Cox [26], apparently unaware of Bellman's work, redis-covered the usefulness of dynamic programming. Vande-walle [27] in 1975 mentions adapting Bellman's solutionto the uniform error norm, but gives no details or descrip-tions of the algorithm.

There are several significant differences between ouroptimal algorithm and Bellman's algorithm. First, Bell-man uses the mean square error or 12 norm while we usethe uniform or 1lo norm. Second, Bellman considers a dis-continuous solution while ours is continuous. Third, froma philosophical point of view, Bellman fixes the number ofline segments and minimizes the error while we fix theerror and minimize the number of line segments. We notethat our recursion is easier to implement and can result insubstantially less computational effort, especially for smallvalues of error.By modifying the recursion slightly, we can also gen-

erate the breakpoints for Ca. With the calculations of F(u)store one of the v E V(u) that achieves the minimum in(4). Then from F(I) = F(UN) we get UN- 1, from F(UN- 1)we get UN 2, and so forth, thus generating an optimal con-tinuous piecewise linear planar curve that approximatesCd with fixed endpoints UN and uo within error e.When the planar curve Cd is closed, then the endpoints

are the same and so po = pi. In this case the optimal al-gorithm may not find the approximation Ca with the small-est number of segments. However, we observe that thepoint po must be contained in some segment of the curve.By applying the optimal algorithm with endpoints Pi, P2,*..until we have exhausted all possible segments

PUN- I' PUN containing po with e (UN- 1, UN) < -, one of thesewill have the overall optimal approximation to the closedcurve. In practice, the optimal approximation derived fromendpoints po = pi should tend to be very close to the over-all optimal.

III. SCAN-ALONG IMPLEMENTATIONThe major difficulty in an efficient implementation of

the optimal algorithm is in the evaluation of the set V(u).We now present a scan-along approach for determiningV(u).Let po be the origin of a line segment in Fig. 3. For the

first point PI draw a circle around it of radius c. Then theset S1 represents all lines through po which pass withindistance e of PI. For the point P2 draw a circle around itof radius c and the set S2 represents all lines through powhich pass within e Of P2. Further, the set T2 = S1 n s2represents all lines through po which pass within e of bothPI and P2. Similarly, for the point P3 the set T3 = S1 n s2n S3 represents all lines through po which pass within eof points P1, P2, and p3.

Mathematically, let

Oi = arctan ( i12)Xi - xO

(5)

and

j = arcsin ((x-x)2 + (y-YO)2) (6)

Then

Si = {e °i - fi C 06 Oi + ij}

and

T= n s .j=1

We can recursively compute Ti as follows. Let

i min(i) IGmax(i)]. (7)

Then

Omin(i + 1) = min ((max (lmin(i),Xi +I

- i +I )), 6max(i) + 6)

Omax(i + 1) = max ((min (0max(i), Oi +I

(8)

+ b±+ 1)), 6min(i) - 6). (9)

If Ti is empty, there can be no line passing within e ofpoints po, P', - * , Pi and, hence, no more points inV(pO). If Ti is not empty, then if /i E Ti, all the points po,Pl, * * *, pi are potentially within e of the linepo, Pi whileif qi t Ti, some points in po, PI, * * *, pi is greater thane away from the line po, pi and pi L V(pO). For example,in Fig. 3 the points po, Pi, P2 are within e of the linePo, P2 while the point P2 is more than e away from the linePO, P3-

69

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-8. NO. 1. JANUARY 1986

Fig. 3. Scan-along procedure.Fig. 4. Small test figure.

Next, we must consider the fact that the line po, pi doesnot go on forever. Let the distance between points po andp, be

di (x;- + ( -

and the maximum distance is

dmax i - max dI cjc5

If di + > dmax, i, then the Xi, coordinate of the translatedand rotated point given by (2a & b) will lie between 0 and

x' for 0 c j c i. Hence, all the points po, Pi,* Piare within e of the line po, pc + I and so pi t I e V(p0). Ifdi, I < dax, j, then we only need to check all the pointswhose distance is greater than di ± using (3) to see if theyare all within 6 of the line po, pi + If these points arewithin 6 of po, pi+ 1, then pi+ E V(po), otherwise pi+ I tV(po). Finally, if di < i, we don't need to worry aboutthis point as it is always within 6 of any line through po.

IV. EXPERIMENTAL RESULTS

In this section we discuss some experimental results.We chose to compare nine algorithms which are dividedinto two classes. In the first class are algorithms in whichthe approximation error can be directly controlled: the Ra-mer [3] and Duda and Hart [4] algorithm using the errormeasure (3); the Pavlidis and Horowitz [5] algorithmwhere the R algorithm for adjusting segment endpoints istaken from Pavlidis [28] with M = 1; the Williams [6]algorithm using exact calculations and taking into accountdistance as discussed at the end of Section III; the Sklan-sky and Gonzalez [7] algorithm taking into account dis-tance discussed at the end of Section III; the Badi'i andPeikari [8] algorithm neglecting the parts of the algorithmassociated with the 11 norm and using maximal extensionlengths of 5, 10, 15, 20, 25, 30, 40, and 50; and the op-timal algorithm discussed in Section II and III. The sec-

ond class of algorithms consists of those in which theapproximation error is indirectly controlled by some pa-rameter of the algorithm: the Davis [9] algorithm wherethe smoothed k-curvature was increased in increments of1 until the maximum allowable error was surpassed; thePavlidis [10] algorithm where the shaded region of Fig.12.5 of [10] was determined by the equation

T = 0.4583 + 0.25 C

and ko took on the values 1, 2, 3, 4, 6, 8, and 10; and theRoberge [11] algorithm where the extension factor took onthe values 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0.

Four closed curves were chosen to compare the algo-rithms with regard to the number of vertices in the ap-proximation and the total time to process the planar curve.Fig. 4 is a small polygon with 20 vertices, Fig. 5 is theoutline of a chromosome taken from Montanari [12] with148 vertices, Fig. 6 is the outline of a right lung consistingof 402 vertices, and Fig. 7 is the outine of Lake Lewisvillenear Dallas, Texas, and has 527 vertices. All algorithmsfind continuous approximations with breakpoints along Cdwith the same initial and final vertices for allowable errorsof 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, and 4.0. Allalgorithms were implemented in Fortran 77 and executedon a TI professional computer with an 8087 numeric co-processor for floating point calculations. All algorithmsconsider the planar curve points as real numbers, inputdata in the same way, and write out results using the sameformats. For those algorithms where the approximation er-ror was indirectly controlled or there were several optionsfor a fixed error, we first chose a result with a minimalnumber of vertices to satisfy a given approximation errorand then, among all those with the same number of ver-tices, chose one with the minimal program execution time.

Tables I-IV show the number of vertices required in theapproximation of Figs. 4-7, respectively. Several obser-

70


Fig. 7. Lake Lewisville near Dallas, TX.

Fig. 5. Chromosome outline taken from Montanari [9].

Fig. 6. Right lung figure.

vations can be made from the data. First, the optimal al-gorithm always required the least number of vertices as

expected. Considering the suboptimal algorithms in whichthe approximation error is directly controlled, we see thatthe Sklansky and Gonzalez algorithm tended to outper-form all the rest followed by the Badi'i and Peikari algo-rithm. When the error was small, the Ramer, and Pavlidisand Horowitz algorithms performed about the same fol-lowed by the Williams algorithm. When the error was largethe Williams algorithm tended to outperform the Pavlidisand Horowitz algorithm, which in turn tended to outper-form the Ramer algorithm. For the suboptimal algorithms

in which the approximation error is indirectly controlled,for small errors the Pavlidis algorithm tended to outper-form the Roberge algorithm while for large errors the Ro-berge algorithm tended to outperform the Davis algo-rithm, which in turn tended to outperform the Pavlidisalgorithm. Comparing the two classes of algorithms, thealgorithms with direct control over the error tended to uni-formly outperform the algorithms with indirect controlover the error.

Second, for the very small errors all algorithms tendedto perform the same. The improvement offered by the op-timal algorithm was greatest over the middle range of er-ror. Finally, all algorithms performed about the same onthe small polygonal figure, but showed distinct differenceson the three larger figures.

Tables V-VIII show the total execution time for the al-gorithms for Figs. 4-7, respectively, in seconds. Severalobservations can be made. First, for the Ramer, Davis,and Badi'i and Peikari algorithms, the time tended to de-crease slightly as the error increased while for the Pavlidisalgorithm the time tended to increase slightly. The timefor the Pavlidis and Horowitz algorithm and the Robergealgorithm tended to increase to a maximum then decreaseas the error increased. The time for both the Williams,and Sklansky and Gonzalez algorithms tended to first de-crease to a minimum and then increase as the error in-creased. For the optimal algorithm, the running time in-creased as the error increased which is expected sinceincreasing the error increased the size of V(u).

Second, the Ramer, Williams, Sklansky and Gonzalez,Davis, Pavlidis, and Roberge algorithms tended to have arunning time proportional to the total number of verticesin the curve. This tended to be true for the Pavlidis andHorowitz algorithm and the Badi'i and Peikari algorithmand was definitely not true for the optimal algorithm.

Third, there was little difference in the running time of

71


TABLE INUMBER OF VERTICES FOR SMALL FIGURE

F_ F 1 1 ,1F ~~~~~Pavlidis Sklansky Badi'i Dvs Pwii oeg piaError Ramer Williams GzalDavis Pavlidis Roberge OptimalII ~~~~Horowitz Gozlz Peikari II

0.2 18 1 18 18 18 18 18 18

0.4 118 81 18 18 18 18 18 18I,

0.6 17 17 17 17 17 17 18 17

0.8 17 155 1 5 1 1 16 1 15

ii!i

1.0 15 18 14 14 14 14 15 14

1.5 11 10 12 10 1 14 12 10 1

2.0 17 79 79 17 7 10 12 17

2.5 1 16 17 6 10 6

3 .0 5 5 5 5 5 5 5 5 5

4.0 75 15 15 : 15 15 5 5 5 15

TABLE IINUMBER OF VERTICES FOR CHROMOSOME

Error RamerPavlidi Wilims 'klanszaky BPadili Davis Pavlidis Roberge Optimal

0.2 59 59 59 59 59 59 59 59

0.4 5 1 r 1 59 9 159 1

0.6 36 39 43 32 37 46 59 30

0.8 29 30 32 23 26 37 40 23

I,Ii

1.0 19 21 17 16 19 26 37 16

1.5 16 17 16 16 16 22 34 16

2.0 16 15 " 12 13 27 22 15 11

2.5 13 13 10 19 01 26 18. 141 91

I, I, I, ,!!

3.0 11 o 9 8 10 20 1 18 10 8

240 8 9 9 7 8 17 9 9 7

TABLE IIINUMBER OF VERTICES FOR LUNG

EYrror Ramer Palidroisz Williarns Sklanslkyz BPaedilai Davis Pavlidis Roberge Optimal

0.2 254 254 254 254 254 ,, 254 254 254

2 226 82156 6 1

0. 4 28 226 254 15 224 254 254 207

0.6 159 160 185 137 157 193 222 133

0.8 116 116 124 95 116 155 222 86

1.0 92 79 87 63 93 '' 153 161 56

1.5 5 5 51 37 50 129 70 35

2.0 40 33 36 26 35 105 129 51 24

_2.5 34 26 _27 22 2717 93 110 1 47 18

30 1 26 22 17 26 80 91 39 15ii Ii

4.0 20 19 18 12 18 78 53 22 10

72

73DUNHAM: PIECEWISE LINEAR APPROXIMATION OF PLANAR CURVES

TABLE IVNUMBER OF VERTICES FOR LAKE LEWIS VILLE

Error Ramer Pavlidis Sklanslky Pad' Davis Palds Rbre Otilrrr Raaar lirowitz Williams~ G,rzalez Peik'ari -lda obre Otn

0.2 330 ~~ 330 733( 7330 73730 33 330 7330

0.4 291 298 7329 274 :2973 32 730 266

0.6 29 211 227 198 219 7329 3730 192

0.8 168 174 175 140 : 157 219 73730 17130

1.0 145 1734 17134 105 1473 190 247 101

1.5 160 87 84 : 72 :95 177 198 673

:2.0 : 71 : 71 : 67 54 64 : 1735 : 152 : 1731 48

2.5 60 49 : : 40 : 50 : 1173 : 152 80 39

73.0 : 51 : 40 43 736 45 : 102 : 152 67 : 28

:4.0 37 2 731 22 0 93 151 737 21

TABLE VPROGRAm ExECUTION TiME FOR SMALL FIGURE IN SECONDS

Error Ramer :Pavlidia Wnilliasa Sklansky Badi'i :Davis Pavlidis Ro~berge Optimallhorowitz Gonzalez Peikari

0.2 :9.0 12. :8.9 :9.73 10.73: :1. 7.8 2.6

0.6 9 3 12.2 :8.8 9 .3 10.1 :10.1 7.8 9.8

:0.8 :9.73 :12.73 :9.2 :10.0: :10.1 : 8.1 : 9.8

1.0 :9.2 :12.6 8.7 :9.2 9.9: : 9.8 8.5

:1.5 :9.0 :12.7 :8.6 :9.1 : 9.4: 9.8 8.9 : 10.0:

2.0 8.8 12.6 : 8.5 8.9 9.1 : 10.1 8.9 : 10.2

2.5 : 8.8 12.7 : 8.5 8.8 9.0 8.6 10.73 8. : 10.73

73.0 : 8.7 11.4 : 8.73 : 8.8 9.0 8.2 : 10.2 8.1 10.4

4.0 8.8 : 11.4 : 8.73 : 8.9 : 9.0 : 8.2 10.2 : 8.1 10.6

TABLE VIPROGRAm EXECUTION TiME FOR CHROMOSOME IN SECONDS

Error Ramner :Pavlidis Willam Sklansky Badili :Davis :Pavlidia : oberge Optimalliorowitz lan Gonzalez :Peikari:

0.2 : 16.4 873.4 16.2 : 16.7 24.73 :: 15.1 173.1 24.2

0.4 16.2 84.6 16.2 17.1 i19.0 15.1 173.1 :27.0

0.6 15.7 89.1 15. 168 19.6 : 17.3 173.1 734.1

0.8 i15.5 7736 15.6 16.5 17ii:5 16.8 :173.8 :41.5

1.0 15.2 i66.4 15.2 j159 i17.8 i :18.8 14.0 44:

15 15. 66.s3 : 1.1 in16 .231 : : 19.5 13.9 49.4

2.0 : 15.0 58.5 15.1 : 16.2: 18.2 14.4 :12.5 13.8 :59.1

25 14.9 58.4 15.2 16.2 : 19.4 : 14.4 16.4 : 173.4 : 67.7

4.0 14.78 55.4 : 5. 16.73 7 18.5 : 14.1 1.4: 5. 173.2 7410 3ii I__ Ii

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-8, NO. 1. JANUARY 1986

TABLE VIIPROGRAM EXECUTION TIME FOR LUNG IN SECONDS

a1iIOlim, Crror 1, .erier ilaorowitz CWilli nzs.Glanle PBikai Davis Pavlidis Roberse Optimal

0,2 55., .21'. 34.8 36.0 76.2 26.7 25.6 45.01 1 ! ~~~~~~~~~~~~I I I I i| .4 33 9I 230 .5 54 .5 6.1 6 6. 7 2. 7 50 .2

~ ~ ~~~I I --I

.22.1 56.) I 54Y 8~~~~~~~~5.1 69.1 2.7 28. 20

51 . :223. 32.8 35.5 95.0 32.5 68.7 :65.8

81̀ .S0.6 230.8F 31.- -,4.5 112.2 32-3 26-6 80-2I I I

30.1 241.1 5.1 141 1322947.4 35.4 26.6 96.8

15 ,28.1 224.9 29.7 32.4 j51.9 3 1.8 23.5 j1`32.6

2).0 28.3 199.3 30.7, 33.8 47.'3 26.4 31.8 ,23.9 ,182.1

~~~~~~I I I II II

12.5 027.7 175.8 31.1 35.2 31.5 26.3 32.0 23.4 253.0

3! .G 271.6 ! 176.5 1-9 36.2 31.2 26.0 42-1 22-2 340-9

4.0 26.9 151.0 .4 .4 522.8~~~~F11

14.0 26.9 24151.0 37.92 52.4 529125 . 9 834 22.2

TABLE VIIIPROGRAM EXECUTION TIME FOR LAKE LEWISVILLE IN SECONDS

- rror FRamer trliS|Williamrs kaak airiDavis Pavlidis |Roberge Optiinal

,I Ill

0.2 45.4 334.3 44.3 45.8 70.3 4.3 32.3 55.8

0.4 44.4 318.7 44.5 546.6 65.0 4.3 32.3 63.9

10.6 41.9 1 329.8 41.1 45.0 1 65 5234533 7259

19 .5 139 4 312-6 3.3.0 14 27 4210.0 323 86.6

41.0 179.3 322.2 38.4 42.0 56.3 52.6 33.5 8.8

1.5 37.6 335.6 38.1 43.3 j53.8 j ,39.4 33.2 1137.3

i! iii I_ i___!

2.0 36.6 308.6 3 9.2 44.8 65.8, 32.4 39.5 31.2 j178.4

2.536 . 1 3 1 2 . 8 4 7 48 *1 49 * 1 32 * 4 39* 5 3 1* 3227|31 TF

4.0 34.7 '284.6 49.3 60.9 43.1 32.0 82.9 29.1 48.1

the Ramer and Williams algorithm for most errors. Fourth,the running time of the optimal algorithm was comparableto that of the Ramer, Williams, Pavlidis, and Sklanskyand Gonzalez algorithms for small errors. Fifth, the run-ning time for the small polygonal figure was about thesame for all nine algorithms. Sixth, the Roberge had thefastest running time.

V. CONCLUSIONAn algorithm for optimal uniform piecewise linear ap-

proximation of planar curves was presented for fixed ini-tial and final vertices. A scan-along procedure was de-scribed to effectively implement the algorithm. Theperformance of the optimal algorithm was compared toeight other algorithms with regard to the number of ver-tices required and the total program execution time. Of allthe algorithms compared, the Roberge [11] algorithm con-sistently had the shortest program execution time, but

tended to require twice as many vertices as the optimalalgorithm. Of the eight suboptimal algorithms, the Sklan-sky and Gonzalez [7] algorithm, taking into account dis-tance as discussed at the end of Section III, consistentlyrequired the smallest number of vertices to meet a givenerror tolerance and was relatively close to the number ofvertices required by the optimal algorithm. The optimalalgorithm outperformed the Sklansky and Gonzalez algo-rithm with regard to the number of vertices required in theapproximation, but took more time. For small errors thetime of both algorithms was comparable.

REFERENCES

[1] T. Pavlidis, "A review of algorithms for shape analysis," Comput.Graphics Image Processing, vol. 7, pp. 243-258, 1978.

[2] , "Algorithms for shape analysis of contours and waveforms," IEEETrans. Pattern Anal. Mach. Intell., vol. PAMI-2, pp. 301-312, 1980.

[3] U. Ramer, "An iterative procedure for the polygonal approximation

7/4


of plane curves," Comput. Graphics Image Processing, vol. 1, pp.244-256, 1972.

[4] R. 0. Duda and P. E. Hart, Pattern Classification and Scene Analy-sis. New York: Wiley, 1973, pp. 328-339.

15] T. Pavlidis and S. L. Horowitz, "Segmentation of plane curves," IEEETrans. Comput., vol. C-23, pp. 860-870, Aug. 1974.

[6] C. M. Williams, "An efficient algorithm for the piecewise linear ap-proximation of planar curvers," Comput. Graphics Image Process-ing, vol. 8, pp. 286-293, 1978.

[7] J. Sklansky and V. Gonzalez, "Fast polygonal approximation of dig-itized curves," Pattern Recognition, vol. 12, pp. 327-331, 1980.

[8] F. Badi'i and B. Peikari, "Functional approximation of planar curvesvia adaptive segmentation," Int. J. Syst. Sci., vol. 13, no. 6, pp. 667-674, 1982.

[9] L. S. Davis, "Shape matching using relaxation techniques," IEEETrans. Pattern Anal. Mach. Intell., vol. PAMI-1, pp. 60-72, Jan. 1979.

110] T. Pavlidis, Algorithms for Graphics and Image Processing. NewYork: Computer Science, 1982, pp. 281-297.

111] J. Roberge, "A data reduction algorithm for planar curves," Comput.Vision, Graphics, Image Processing, vol. 29, pp. 168-195, 1985.

[12] U. Montanari, "A note on minimal length polygonal approximationto a digitized contour," Commun. ACM, vol. 13, pp. 41-47, Jan. 1970.

113] J. Sklansky, R. L. Chazin, and B. J. Hansen, "Minimum perimeterpolygons of digitized silhouettes," IEEE Trans. Comput., vol. C-21,pp. 260-268, Mar. 1972.

[14] 1. Tomek, "Two algorithms for piecewise linear continuous approxi-mations of functions of one variable," IEEE Trans. Comput., vol. C-23, pp. 445-448, Apr. 1974.

[151 K. Reumann and A. P. M. Witkam, "Optimizing curve segmentationin computer graphics," in International Computer Symposium, A.Gunther, B. Levrat, and H. Lipps, Eds. New York: Elsevier, 1974,pp. 467-472.

[16] K. Ichida and T. Kiyono, "Segmentation of planar curves," Electron.Commun. Japan, vol. 58-D, no. 11, pp. 689-696, 1975.Press, 1957.

[17] Y. Kurozumi and W. A. Davis, "Polygonal approximation by the min-max method," Comput. Graphics Image Processing, vol. 19, pp. 248-264, 1982.

[18] R. Bellman, Dynamic Programming. Princeton, NJ: Princeton Univ.Press, 1957.

1191 R. Bellman, "On the approximation of curves by line segments usingdynamic programming," Commun. ACM, vol. 4, p. 284, 1961.

[201 B. Gluss, "Further remarks on line segment curve-fitting using dy-namic programming," Commun. ACM, vol. 5, pp. 441-443, Aug.1962.

[21] -, "A line segment curvc-fitting algorithm related to optimal en-coding of information," Inform. Contr., vol. 5, pp. 261-267, 1962.

122] -, "Least squares fitting of planes to surfaces using dynamic pro-gramming," Commun. ACM, vol. 6, pp. 172-175, Apr. 1963.

123]-, "An alternative method for continuous line segment curve-fit-ting," Inform. Contr., vol. 7, pp. 200-206, 1964.

124] C. L. Lawson, "Characteristic properties of the segmented rationalminimax approximation problem," Numer. Math., vol. 6, pp. 293-301, 1964.

[25] R. Bellman, B. Gluss, and R. Roth, "On the identification of systemsand the unscrambling of data: Some problems suggested by neuro-physiology," P NAS US, vol. 52, pp. 1239-1240, 1964.

[26] M. G. Cox, "Curve fitting with piecewise polynomials," J. Inst. Math.Appl., vol. 8, pp. 36-52, 1971.

[27] J. Vandewalle, "On the calculation of the piecewise linear approxi-mation to a discrete function," IEEE Trans. Comput., vol. C-24, pp.843-846, 1975.

128] T. Pavlidis, "Waveform segmentation through functional approxima-tion," IEEE Trans. Comput., vol. C-22, pp. 689-697, July 1973.

James George Dunham (S'72-M'77) was born inAkron, OH, on September 10, 1950. He receivedthe B.S., M.S., and Ph.D. degrees in electricalengineering from Stanford University, Stanford,CA, in 1973, 1973, and 1978, respectively.

From August of 1977 to August of 1984 he waswith the Department of Electrical Engineering,Washington University, St. Louis, MO, where hewas an Associate Professor. Since August of 1984he has been an Associate Professor of ElectricalEngineering at Southern Methodist University in

Dallas, TX. His research interests and activities are in the areas of com-munications, information theory, and cryptography.

Dr. Dunham is a member of Tau Beta Pi and Phi Beta Kappa.

75

optimum uniform piecewise linear approximation of planar curves

Documents