Abstract

The benefits of making an effective use of impressive computational power offered by multi-core platforms are investigated for the computation of φ-polynomials used in the description of freeform surfaces. Specifically, we devise parallel algorithms based upon the recurrence relations of both Zernike polynomials and gradient orthogonal Q-polynomials and implement these parallel algorithms on Graphical Processing Units (GPUs) respectively. The results show that more than an order of magnitude improvement is achieved in computational time over a sequential implementation if these recurrence-based parallel algorithms are adopted in the computation of the φ-polynomials.

© 2013 Optical Society of America

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References

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    [CrossRef] [PubMed]
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  3. O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
    [CrossRef]
  4. J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009).
    [CrossRef]
  5. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica1(7–12), 689–704 (1934).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  17. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Chap. 22.
  18. NVIDIA, CUDA C Programming Guide (NVIDIA, 2012).
  19. NVIDIA, CUDA C Best Practices Guide, (NVIDIA, 2012).
  20. Intel, “Intel core 7 processor specifications” (Intel, 2013), http://www.intel.com/content/www/us/en/processors/core/core-i7-processor/Corei7Specifications.html .
  21. C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Tables Other Aids Comput.9, 118–120 (1995).

2013

I. Kaya and J. P. Rolland, “Hybrid RBF and local phi-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2013).

G. W. Forbes, “Fitting freeform shapes with orthogonal bases,” Opt. Express21(16), 19061–19081 (2013).
[CrossRef] [PubMed]

Y. Jian, K. Wong, and M. V. Sarunic, “Graphics processing unit accelerated optical coherence tomography processing at megahertz axial scan rate and high resolution video rate volumetric rendering,” J. Biomed. Opt.18(2), 026002 (2013).
[CrossRef] [PubMed]

2012

2011

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express19(22), 21919–21928 (2011).
[CrossRef] [PubMed]

C. L. Phillips, J. A. Anderson, and S. C. Glotzer, “Pseudo-random number generation for Brownian dynamics and dissipative particle dynamics simulations on GPU devices,” J. Comput. Phys.230(19), 7191–7201 (2011).
[CrossRef]

2010

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010).
[CrossRef] [PubMed]

O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
[CrossRef]

2009

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009).
[CrossRef]

2008

1995

C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Tables Other Aids Comput.9, 118–120 (1995).

1934

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica1(7–12), 689–704 (1934).
[CrossRef]

Anderson, J. A.

C. L. Phillips, J. A. Anderson, and S. C. Glotzer, “Pseudo-random number generation for Brownian dynamics and dissipative particle dynamics simulations on GPU devices,” J. Comput. Phys.230(19), 7191–7201 (2011).
[CrossRef]

Benitez, P.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009).
[CrossRef]

Cakmakci, O.

O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
[CrossRef]

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” SID16, 1089–1098 (2008).

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express16(3), 1583–1589 (2008).
[CrossRef] [PubMed]

Clenshaw, C. W.

C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Tables Other Aids Comput.9, 118–120 (1995).

Cote, J.

O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
[CrossRef]

Dunn, C.

Forbes, G. W.

Foroosh, H.

Fuerschbach, K.

Glotzer, S. C.

C. L. Phillips, J. A. Anderson, and S. C. Glotzer, “Pseudo-random number generation for Brownian dynamics and dissipative particle dynamics simulations on GPU devices,” J. Comput. Phys.230(19), 7191–7201 (2011).
[CrossRef]

Gray, R. W.

Jian, Y.

Y. Jian, K. Wong, and M. V. Sarunic, “Graphics processing unit accelerated optical coherence tomography processing at megahertz axial scan rate and high resolution video rate volumetric rendering,” J. Biomed. Opt.18(2), 026002 (2013).
[CrossRef] [PubMed]

Kaya, I.

I. Kaya and J. P. Rolland, “Hybrid RBF and local phi-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2013).

I. Kaya, K. P. Thompson, and J. P. Rolland, “Comparative assessment of freeform polynomials as optical surface descriptions,” Opt. Express20(20), 22683–22691 (2012).
[CrossRef] [PubMed]

Li, P.

Liu, S.

Luo, Q.

Miñano, J. C.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009).
[CrossRef]

Moore, B.

Phillips, C. L.

C. L. Phillips, J. A. Anderson, and S. C. Glotzer, “Pseudo-random number generation for Brownian dynamics and dissipative particle dynamics simulations on GPU devices,” J. Comput. Phys.230(19), 7191–7201 (2011).
[CrossRef]

Rolland, J. P.

Santamaria, A.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009).
[CrossRef]

Sarunic, M. V.

Y. Jian, K. Wong, and M. V. Sarunic, “Graphics processing unit accelerated optical coherence tomography processing at megahertz axial scan rate and high resolution video rate volumetric rendering,” J. Biomed. Opt.18(2), 026002 (2013).
[CrossRef] [PubMed]

Thompson, K.

O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
[CrossRef]

Thompson, K. P.

Vallee, P.

O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
[CrossRef]

Vo, S.

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” SID16, 1089–1098 (2008).

Wong, K.

Y. Jian, K. Wong, and M. V. Sarunic, “Graphics processing unit accelerated optical coherence tomography processing at megahertz axial scan rate and high resolution video rate volumetric rendering,” J. Biomed. Opt.18(2), 026002 (2013).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica1(7–12), 689–704 (1934).
[CrossRef]

Adv. Opt. Technol.

I. Kaya and J. P. Rolland, “Hybrid RBF and local phi-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2013).

J. Biomed. Opt.

Y. Jian, K. Wong, and M. V. Sarunic, “Graphics processing unit accelerated optical coherence tomography processing at megahertz axial scan rate and high resolution video rate volumetric rendering,” J. Biomed. Opt.18(2), 026002 (2013).
[CrossRef] [PubMed]

J. Comput. Phys.

C. L. Phillips, J. A. Anderson, and S. C. Glotzer, “Pseudo-random number generation for Brownian dynamics and dissipative particle dynamics simulations on GPU devices,” J. Comput. Phys.230(19), 7191–7201 (2011).
[CrossRef]

Math. Tables Other Aids Comput.

C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Tables Other Aids Comput.9, 118–120 (1995).

Opt. Express

Opt. Rev.

J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009).
[CrossRef]

Physica

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica1(7–12), 689–704 (1934).
[CrossRef]

Proc. SPIE

O. Cakmakci, K. Thompson, P. Vallee, J. Cote, and J. P. Rolland, “Design of a freeform single-element head-worn display,” Proc. SPIE7618, 761803 (2010).
[CrossRef]

SID

O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” SID16, 1089–1098 (2008).

Other

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Chap. 22.

NVIDIA, CUDA C Programming Guide (NVIDIA, 2012).

NVIDIA, CUDA C Best Practices Guide, (NVIDIA, 2012).

Intel, “Intel core 7 processor specifications” (Intel, 2013), http://www.intel.com/content/www/us/en/processors/core/core-i7-processor/Corei7Specifications.html .

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Figures (3)

Fig. 1
Fig. 1

GPU computed low-order φ-polynomials (a) Zernike, Z 9 3 , (c) gradient orthogonal Q-poly, Q 3 3 ; the difference between the parallel and sequential implementations within 14 significant digits (b) Zernike (d) Q-poly.

Fig. 2
Fig. 2

(a) Total execution time of the sequential and parallel algorithms of φ-polynomials on both CPU and GPU as a function of the grid size (b) speedups of φ-polynomials with grid size.

Fig. 3
Fig. 3

Effect of the polynomial order on the computation of φ-polynomials (a) computation times on CPU and GPU (b) speedups through parallelization.

Tables (2)

Tables Icon

Table 1 Effect of the size of the ray grids on the speedup of the computation of φ-polynomials.

Tables Icon

Table 2 Effect of the order of the φ-polynomial over the computation time and speedup.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Z n m ( ρ,φ )= R n ±m ( ρ ){ cosmφ sinmφ },
R n ±m ( ρ )= q=0 n| m | 2 ( 1 ) q ( nq )! q!( n+| m | 2 q )!( n| m | 2 q )! ρ n2q .
R n ±m ( ρ )= ρ m P (nm)/2 ( 0,m ) ( 2 ρ 2 1 ),
P k+1 m (x)=( a k + b k x ) P k m (x) c k P k1 m (x),
a k = (s+1)[ (sk) 2 + k 2 +s] (k+1)(sk+1)s , b k = (s+2)(s+1) (k+1)(sk+1) , c k = (s+2)(sk)k (k+1)(sk+1)s ,
Q n m ( ρ 2 )= A n m ( ρ 2 ) g n1 m Q n1 m ( ρ 2 ) f n m ,

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