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Malik Zawwar Hussain, Samreen Abbas, Misbah Irshad
Quadratic trigonometric B-spline for image interpolation using GA
www.plosone.org
www.plosone.org
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5.9998 0 0 5.9998 275.1874 671.1307 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
5 0 0 5 279.6094 669.6 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 281.4803 671.1307 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
5 0 0 5 285.9023 669.6 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 294.1228 671.1307 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 296.2204 673.7952 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F13 1 Tf
0.8787 0 Td
()Tj
0.6576 0 Td
()Tj
/F10 1 Tf
1.0488 0 Td
[(\n)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 328.0251 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 330.1228 673.7952 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F13 1 Tf
0.8787 0 Td
()Tj
0.6576 0 Td
()Tj
/F16 1 Tf
1.0488 0.754 Td
(P)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 366.4629 679.011 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(3)Tj
/F10 1 Tf
0 -1.3323 Td
[()]TJ
/F13 1 Tf
0.2645 0 Td
()Tj
/F12 1 Tf
(0)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 376.1574 673.7952 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 380.7496 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F14 1 Tf
0.2645 0 Td
(;)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 386.0787 673.7952 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F14 1 Tf
(;)Tj
/F17 1 Tf
1.3946 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 405.0708 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 407.1685 673.7952 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F17 1 Tf
0.4365 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 417.0897 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 419.1874 673.7952 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 430.1291 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 434.7212 673.7952 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F13 1 Tf
0.9354 0 Td
(8)Tj
/F10 1 Tf
[()]TJ
/F13 1 Tf
1.3209 0 Td
(2)Tj
0.9354 0 Td
()Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 474.0094 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 476.1637 673.7952 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F10 1 Tf
0.4365 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 485.2346 671.3007 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 494.9858 673.7952 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
6.8484 0 Td
()Tj
/F5 1 Tf
(1)Tj
/F13 1 Tf
()Tj
/F5 1 Tf
-0.004 Tc
-36.3455 -2.3924 Td
(where)Tj
/F10 1 Tf
2.6476 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 233.5748 648.17 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 235.3322 649.8708 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(;)Tj
/F10 1 Tf
0.4422 0 Td
[()]TJ
/F5 1 Tf
0.4592 0 Td
(=)Tj
0.7993 0 Td
(0,1,2,3,)Tj
3.0217 0 Td
(are)Tj
1.4514 0 Td
(co-efficient)Tj
4.7281 0 Td
(of)Tj
1.0148 0 Td
(interpolation)Tj
5.4538 0 Td
(which)Tj
2.6476 0 Td
(are)Tj
1.4513 0 Td
(defined)Tj
3.2258 0 Td
(as)Tj
1.0148 0 Td
(follows:)Tj
/F10 1 Tf
0 Tc
-20.1372 -2.262 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 298.148 624.7559 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(0)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 304.4409 627.2503 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
1.0488 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 320.1448 624.7559 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 322.2425 627.2503 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F10 1 Tf
5.4424 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 383.7543 624.7559 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 389.9905 627.2503 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
1.0488 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 405.6944 624.7559 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 410.0598 627.2503 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
0.9921 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 425.1401 624.7559 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 427.2377 627.2503 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 434.211 624.7559 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 436.3653 627.2503 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(=)Tj
/F12 1 Tf
0.8844 0 Td
(1)Tj
/F13 1 Tf
0.7199 0 Td
()Tj
/F17 1 Tf
0.9978 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 467.3763 624.7559 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 441.3543 627.2503 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
2.8176 0 Td
()Tj
ET
Q
q
1 j
1 J
0 w
475.0299 629.9717 m
480.5291 629.9717 l
480.5291 629.5748 l
475.0299 629.5748 l
f*
10 0 0 10 475.0299 634.0535 cm
BT
/F17 1 Tf
1 TL
0 0 Td
(p)Tj
/F12 1 Tf
0.0283 -1.3662 Td
(2)Tj
/F14 1 Tf
0.6916 0.6859 Td
(;)Tj
/F10 1 Tf
-18.2889 -3.5829 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 306.4251 588.9259 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(2)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 312.7181 591.4204 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
1.0488 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 328.422 588.9259 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 340.4409 591.4204 cm
BT
/F13 1 Tf
1 TL
0 0 Td
( )Tj
/F10 1 Tf
0.9921 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 355.4645 588.9259 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 357.5622 591.4204 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 364.5921 588.9259 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 374.3433 591.4204 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(=)Tj
/F12 1 Tf
0.8844 0 Td
(1)Tj
/F13 1 Tf
0.7199 0 Td
()Tj
/F17 1 Tf
0.9978 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 405.8645 588.9259 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 379.3322 591.4204 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
2.8686 0 Td
()Tj
ET
Q
q
1 j
1 J
0 w
413.5181 594.085 m
419.074 594.085 l
419.074 593.6882 l
413.5181 593.6882 l
f*
10 0 0 10 413.5181 598.2236 cm
BT
/F17 1 Tf
1 TL
0 0 Td
(p)Tj
/F12 1 Tf
0.0283 -1.3662 Td
(2)Tj
/F14 1 Tf
0.6916 0.6859 Td
(;)Tj
/F10 1 Tf
1.44 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 442.2047 588.9259 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(3)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 448.4409 591.4204 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
1.0544 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 464.1448 588.9259 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 473.9527 591.4204 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F5 1 Tf
-0.004 Tc
-27.3938 -2.6305 Td
(with)Tj
/F10 1 Tf
1.9956 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 225.0141 563.4708 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 229.2094 565.1149 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(=)Tj
/F10 1 Tf
0.805 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 241.9086 563.4708 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
/F5 1 Tf
(+1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 253.9842 565.1149 cm
BT
/F15 1 Tf
1 TL
-0.004 Tc
0 0 Td
[()]TJ
/F10 1 Tf
0.788 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 266.5133 563.4708 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 268.4976 565.1149 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(.)Tj
/F10 1 Tf
0.4478 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 279.8929 563.4708 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 284.0881 565.1149 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(are)Tj
1.4513 0 Td
(the)Tj
1.4683 0 Td
(derivatives)Tj
4.5014 0 Td
(at)Tj
0.9524 0 Td
(the)Tj
1.474 0 Td
(knots)Tj
/F10 1 Tf
2.4265 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 411.5338 563.4708 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 415.6724 565.1149 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(which)Tj
2.6532 0 Td
(may)Tj
1.9275 0 Td
(be)Tj
1.1509 0 Td
(given)Tj
2.3811 0 Td
(or)Tj
1.0941 0 Td
(estimated)Tj
4.0762 0 Td
(using)Tj
-34.8488 -1.2982 Td
(any)Tj
1.6554 0 Td
(approximation)Tj
6.1738 0 Td
(schemes;)Tj
3.7871 0 Td
(geometric)Tj
4.2179 0 Td
(mean,)Tj
2.6703 0 Td
(arithmetic)Tj
4.3538 0 Td
(mean)Tj
2.4435 0 Td
(and)Tj
1.7291 0 Td
(harmonic)Tj
4.1216 0 Td
(mean.)Tj
2.6702 0 Td
(Further-)Tj
-33.8227 -1.2983 Td
(more,)Tj
2.5682 0 Td
(the)Tj
1.4683 0 Td
(quadratic)Tj
4.0082 0 Td
(trigonometric)Tj
5.7883 0 Td
(spline)Tj
2.5852 0 Td
(function)Tj
3.6283 0 Td
(\()Tj
0.83 0.64 0.02 0 k
(1)Tj
0 g
(\))Tj
1.3776 0 Td
(has)Tj
1.542 0 Td
(the)Tj
1.4683 0 Td
(following)Tj
/F10 1 Tf
3.9798 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 490.337 543.5716 cm
BT
/F5 1 Tf
1 TL
-0.0053 Tc
0 0 Td
(1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 496.1196 539.1496 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(interpolating)Tj
5.4141 0 Td
(prop-)Tj
-35.0246 -1.3039 Td
(erties:)Tj
/F10 1 Tf
0 Tc
8.9121 -2.0012 Td
[(\n)]TJ
/F13 1 Tf
0.6633 0 Td
()Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 304.3275 503.6031 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 306.4818 506.0976 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
0.6576 0 Td
()Tj
/F10 1 Tf
1.0488 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 328.7622 503.6031 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 335.9055 506.0976 cm
BT
/F15 1 Tf
1 TL
0 0 Td
[()8()4()]TJ
/F10 1 Tf
1.9899 0 Td
[(\n)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 362.4944 510.2362 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
()Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 370.6015 506.0976 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 379.1622 503.6031 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 381.2598 506.0976 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
0.6633 0 Td
()Tj
/F10 1 Tf
1.0488 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 405.3543 503.6031 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 407.5086 506.0976 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F10 1 Tf
0.9354 0 Td
[()]TJ
/F13 1 Tf
0.5499 0 Td
()Tj
/F12 1 Tf
1.0488 0 Td
(0)Tj
/F14 1 Tf
(;)Tj
/F12 1 Tf
0.9354 0 Td
(1)Tj
/F14 1 Tf
(;)Tj
/F12 1 Tf
0.9411 0 Td
(.)Tj
0.4309 0 Td
(.)Tj
0.4365 0 Td
(.)Tj
/F14 1 Tf
0.4366 0 Td
(;)Tj
/F10 1 Tf
0.4365 0 Td
[()]TJ
/F13 1 Tf
9.445 0 Td
()Tj
/F5 1 Tf
(2)Tj
/F13 1 Tf
()Tj
/F5 1 Tf
-0.004 Tc
-36.3455 -2.1996 Td
(where)Tj
/F10 1 Tf
2.6475 0 Td
[(\n)]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 232.3842 488.5795 cm
BT
/F5 1 Tf
1 TL
-0.0053 Tc
0 0 Td
(\(1\))Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 241.0582 484.1007 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(\()Tj
/F10 1 Tf
[()]TJ
/F5 1 Tf
(\))Tj
1.3662 0 Td
(represents)Tj
4.3143 0 Td
(the)Tj
1.4741 0 Td
(derivatives)Tj
4.5013 0 Td
(with)Tj
1.9956 0 Td
(respect)Tj
3.0387 0 Td
(to)Tj
1.0262 0 Td
(the)Tj
1.4683 0 Td
(variable)Tj
/F10 1 Tf
3.3619 0 Td
[()]TJ
/F5 1 Tf
(.)Tj
0.9071 0 Td
(The)Tj
1.7858 0 Td
(functions)Tj
/F10 1 Tf
3.9855 0 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 537.9023 482.4566 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
/F5 1 Tf
(,)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 543.2881 484.1007 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(\()Tj
/F10 1 Tf
[()]TJ
/F5 1 Tf
(,)Tj
/F11 1 Tf
()Tj
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 560.1259 482.4566 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 562.1102 484.1007 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(,)Tj
/F11 1 Tf
-36.2095 -1.2982 Td
()Tj
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 204.8881 469.474 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 206.8724 471.1181 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(\);)Tj
/F10 1 Tf
0.7823 0 Td
[()]TJ
/F5 1 Tf
0.4592 0 Td
(=)Tj
0.7994 0 Td
(0,1,2,3)Tj
2.7949 0 Td
(in)Tj
1.0375 0 Td
(\()Tj
0.83 0.64 0.02 0 k
(1)Tj
0 g
(\))Tj
1.3776 0 Td
(are)Tj
1.4457 0 Td
(the)Tj
1.4683 0 Td
(quadratic)Tj
4.0139 0 Td
(trigonometric)Tj
5.7883 0 Td
(basis)Tj
2.1486 0 Td
(functions)Tj
3.9912 0 Td
(defined)Tj
3.2258 0 Td
(in)Tj
1.0375 0 Td
(each)Tj
2.0239 0 Td
(sub)Tj
1.6157 0 Td
(inter-)Tj
-34.6957 -1.2982 Td
(vals)Tj
1.712 0 Td
([)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 225.1842 456.4346 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 227.2251 458.1354 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(,)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 234.085 456.4346 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
/F5 1 Tf
(+1)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 244.0062 458.1354 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(])Tj
0.5612 0 Td
(as:)Tj
/F10 1 Tf
0 Tc
5.1874 -1.8935 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 306.085 436.7055 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(0)Tj
/F14 1 Tf
(;)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 312.8314 439.2 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F14 1 Tf
(;)Tj
/F17 1 Tf
1.3946 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 331.8236 436.7055 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 333.9212 439.2 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F17 1 Tf
0.4365 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 343.8425 436.7055 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 345.9401 439.2 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
0.6633 0 Td
()Tj
1.0488 0 Td
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
1.1055 0 Td
( )Tj
/F10 1 Tf
0.9921 0 Td
[()10()12()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
3.1974 0 Td
( )Tj
/F17 1 Tf
0.9978 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 430.9795 436.7055 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 433.0771 439.2 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()10()12()]TJ
/F13 1 Tf
()Tj
/F10 1 Tf
-13.1583 -1.6951 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 306.085 419.7543 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(1)Tj
/F14 1 Tf
(;)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 312.8314 422.2488 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F14 1 Tf
(;)Tj
/F17 1 Tf
1.3946 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 331.8236 419.7543 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 333.9212 422.2488 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F17 1 Tf
0.4365 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 343.8425 419.7543 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 345.9401 422.2488 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
0.6633 0 Td
()Tj
1.0488 0 Td
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
1.1055 0 Td
()Tj
/F17 1 Tf
0.9921 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 389.0834 419.7543 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 391.1811 422.2488 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()10()12()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
3.1974 0 Td
( )Tj
/F10 1 Tf
0.9921 0 Td
[()10()12()]TJ
/F13 1 Tf
()Tj
/F10 1 Tf
-13.2262 -1.7007 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 305.4047 402.7464 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(2)Tj
/F14 1 Tf
(;)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 312.1511 405.2409 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F14 1 Tf
(;)Tj
/F17 1 Tf
1.3946 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 331.1433 402.7464 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 333.2409 405.2409 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F17 1 Tf
0.4365 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 343.1622 402.7464 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 345.2598 405.2409 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
0.6633 0 Td
()Tj
1.0488 0 Td
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
1.1055 0 Td
()Tj
/F17 1 Tf
0.9978 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 388.9133 402.7464 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 391.011 405.2409 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()7()7()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
3.2428 0 Td
( )Tj
/F10 1 Tf
0.9921 0 Td
[()7()7()]TJ
/F13 1 Tf
()Tj
/F10 1 Tf
-13.2546 -1.6951 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 305.4047 385.7952 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(3)Tj
/F14 1 Tf
(;)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 312.1511 388.2897 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F14 1 Tf
(;)Tj
/F17 1 Tf
1.3946 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 331.1433 385.7952 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 333.2409 388.2897 cm
BT
/F14 1 Tf
1 TL
0 0 Td
(;)Tj
/F17 1 Tf
0.4365 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 343.1622 385.7952 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 345.2598 388.2897 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
0.6633 0 Td
()Tj
1.0488 0 Td
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
1.1055 0 Td
( )Tj
/F10 1 Tf
0.9978 0 Td
[()7()7()]TJ
/F13 1 Tf
()Tj
/F12 1 Tf
(1)Tj
/F13 1 Tf
3.2371 0 Td
( )Tj
/F17 1 Tf
0.9921 0 Td
(b)Tj
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 431.2062 385.7952 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 433.3606 388.2897 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()7()7()]TJ
/F13 1 Tf
()Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 20.0001 456.3212 445.4362 cm
BT
/F16 1 Tf
1 TL
0 0 Td
(\))Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 563.4708 411.1937 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F5 1 Tf
(3)Tj
/F13 1 Tf
()Tj
/F5 1 Tf
-0.004 Tc
-36.3455 -4.9832 Td
(with)Tj
/F10 1 Tf
0 Tc
1.9955 0 Td
[()]TJ
/F13 1 Tf
0.8504 0 Td
()Tj
ET
Q
q
1 j
1 J
0 w
238.9606 364.0252 m
242.3055 364.0252 l
242.3055 363.6283 l
238.9606 363.6283 l
f*
5.9998 0 0 5.9998 238.9606 364.9889 cm
BT
/F17 1 Tf
1 TL
0 0 Td
(p)Tj
/F12 1 Tf
0.0283 -1.1811 Td
(2)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 243.9496 361.3606 cm
BT
/F17 1 Tf
1 TL
0 0 Td
(d)Tj
/F5 1 Tf
-0.004 Tc
0.7199 0 Td
(and)Tj
/F17 1 Tf
0 Tc
1.7292 0 Td
(d)Tj
/F13 1 Tf
0.7823 0 Td
()Tj
ET
Q
q
1 j
1 J
0 w
292.7055 364.0252 m
304.8945 364.0252 l
304.8945 363.6283 l
292.7055 363.6283 l
f*
5.9998 0 0 5.9998 292.7055 366.0094 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
/F13 1 Tf
( )Tj
/F10 1 Tf
1.2567 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5 0 0 5 303.0803 364.4787 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5.9998 0 0 5.9998 296.3338 357.9023 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
5 0 0 5 299.3952 356.4283 cm
BT
/F10 1 Tf
1 TL
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 286.7527 372.4157 cm
BT
/F16 1 Tf
1 TL
0 0 Td
()Tj
1.8141 0 Td
()Tj
/F5 1 Tf
-0.004 Tc
0.5953 -1.1055 Td
(.)Tj
0.4479 0 Td
(From)Tj
2.4434 0 Td
(Eq)Tj
1.2983 0 Td
(\()Tj
0.83 0.64 0.02 0 k
(3)Tj
0 g
(\))Tj
1.3776 0 Td
(it)Tj
0.7823 0 Td
(can)Tj
1.6271 0 Td
(be)Tj
1.1509 0 Td
(observed)Tj
3.7927 0 Td
(that)Tj
1.7802 0 Td
(the)Tj
1.4683 0 Td
(quadratic)Tj
4.0138 0 Td
(trigonometric)Tj
-31.2658 -1.6894 Td
(functions)Tj
3.9911 0 Td
(are)Tj
1.4457 0 Td
(Bernstein)Tj
4.0421 0 Td
(Be)Tj
0 Tc
0.6009 0.051 Td
()Tj
-0.004 Tc
0.4025 -0.051 Td
(zier)Tj
1.6951 0 Td
(like)Tj
1.6555 0 Td
(basis)Tj
2.1486 0 Td
(functions)Tj
3.9912 0 Td
(which)Tj
2.6475 0 Td
(are)Tj
1.4513 0 Td
(demonstrated)Tj
5.76 0 Td
(in)Tj
0.83 0.64 0.02 0 k
1.0318 0 Td
(Fig)Tj
1.4797 0 Td
(1)Tj
0 g
(.)Tj
-31.1468 -1.3039 Td
(Moreover)Tj
4.1726 0 Td
(these)Tj
2.2563 0 Td
(quadratic)Tj
4.0081 0 Td
(functions)Tj
3.9912 0 Td
(satisfied)Tj
3.4469 0 Td
(the)Tj
1.4683 0 Td
(following)Tj
3.9798 0 Td
(properties:)Tj
-24.5194 -1.8992 Td
(1.)Tj
1.2019 0 Td
(Nonnegative:)Tj
/F10 1 Tf
5.5502 0 Td
[(\r)]TJ
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 272.0692 310.7905 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
/F5 1 Tf
(,)Tj
/F10 1 Tf
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 277.5118 312.4346 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(\()Tj
/F10 1 Tf
[()]TJ
/F5 1 Tf
(,)Tj
/F11 1 Tf
()Tj
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 294.3496 310.7905 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 296.3338 312.4346 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(,)Tj
/F11 1 Tf
()Tj
ET
Q
q
1 j
1 J
0 w
7.4999 0 0 6.6659 303.4771 310.7905 cm
BT
/F10 1 Tf
1 TL
-0.0053 Tc
0 0 Td
[()]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 305.4614 312.4346 cm
BT
/F5 1 Tf
1 TL
-0.004 Tc
0 0 Td
(\))Tj
/F13 1 Tf
0 Tc
0.5612 0 Td
()Tj
/F5 1 Tf
-0.004 Tc
0.9865 0 Td
(0;)Tj
/F10 1 Tf
0.924 0 Td
[()]TJ
/F5 1 Tf
0.4536 0 Td
(=)Tj
0.7994 0 Td
(0,1,2,3,)Tj
/F10 1 Tf
3.0273 0 Td
[()]TJ
/F5 1 Tf
0.4876 0 Td
(=)Tj
0.7994 0 Td
(1,2,)Tj
/F12 1 Tf
0 Tc
(.)Tj
1.8368 0 Td
(.)Tj
0.4309 0 Td
(.)Tj
/F5 1 Tf
-0.004 Tc
(,)Tj
/F10 1 Tf
[()]TJ
/F15 1 Tf
[()]TJ
/F5 1 Tf
(1.)Tj
-20.8515 -2.0919 Td
(2.)Tj
1.2019 0 Td
(Partition)Tj
3.7701 0 Td
(of)Tj
1.0148 0 Td
(unity:)Tj
/F16 1 Tf
0 Tc
2.5398 0.7483 Td
(P)Tj
ET
Q
q
1 j
1 J
0 w
6.9998 0 0 6.9998 295.7669 296.2771 cm
BT
/F12 1 Tf
1 TL
0 0 Td
(3)Tj
/F18 1 Tf
0 -1.0933 Td
[( O)]TJ
/F13 1 Tf
0.3077 0 Td
()Tj
/F12 1 Tf
(0)Tj
ET
Q
q
1 j
1 J
0 w
10 0 0 10 307.3322 291.5149 cm
BT
/F18 1 Tf
1 TL
0 0 Td
[( 6)]TJ
ET
Q
q
1 j
1 J
0 w
6.9998 0 0 6.9998 312.3212 289.9842 cm
BT
/F18 1 Tf
1 TL
0 0 Td
[( O)]TJ
/F14 1 Tf
0.2996 0 Td
(;)Tj
/F18 1 Tf
[( L)]TJ
ET
Q
q
1 j
1 J
0 w
10 0 0 10 318.7275 291.5149 cm
BT
/F13 1 Tf
1 TL
0 0 Td
()Tj
/F10 1 Tf
[()]TJ
/F14 1 Tf
(;)Tj
/F17 1 Tf
1.3946 0 Td
(a)Tj
ET
Q
q
1 j
1 J
0 w
6.9998 0 0 6.9998 337.6629 289.9842 cm
BT
/F18 1 Tf
1 TL
0 0 Td
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