An FPGA Architecture for Extracting Real-Time Zernike Coefficients from Measured Phase Gradients

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Zernike modes are commonly used in adaptive optics systems to represent optical wavefronts. However, real-time calculation of Zernike modes is time consuming due to two factors: the large factorial components in the radial polynomials used to define them and the large inverse matrix calculation needed for the linear fit. This paper presents an efficient parallel method for calculating Zernike coefficients from phase gradients produced by a Shack-Hartman sensor and its real-time implementation using an FPGA by pre-calculation and storage of subsections of the large inverse matrix. The architecture exploits symmetries within the Zernike modes to achieve a significant reduction in memory requirements and a speed-up of 2.9 when compared to published results utilising a 2D-FFT method for a grid size of 8×8. Analysis of processor element internal word length requirements show that 24-bit precision in precalculated values of the Zernike mode partial derivatives ensures less than 0.5% error per Zernike coefficient and an overall error of <1%. The design has been synthesized on a Xilinx Spartan-6 XC6SLX45 FPGA. The resource utilisation on this device is <3% of slice registers, <15% of slice LUTs, and approximately 48% of available DSP blocks independent of the Shack-Hartmann grid size. Block RAM usage is <16% for Shack-Hartmann grid sizes up to 32×32.

[1] Huang, D., Swanson, E. A., Lin, C. P., Schuman, J. S., Stinson, W. G., Chang, W., Hee, M. R., Flotte, T., Gregory, K., Puliafito, C. A., Fujimoto, J. G. (1991). Optical coherence tomography. Science, 254, 1178-1181.

[2] Podoleanu, A. (2005). Optical coherence tomography. British journal of radiology, 78 (935), 976-988.

[3] Li, J., Sarunic, M., Shannon, L. (2011). Scalable, high performance Fourier domain optical coherence tomography: Why FPGAs and not GPGPUs. In IEEE 19th Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM), 1-3 May 2011.IEEE, 49-56.

[4] Cubalchini, R. (1979). Modal wave-front estimation from phase derivative measurements. Journal of the Optical Society of America, 69 (7), 972-977.

[5] Southwell, W. H. (1980). Wave-front estimation from wave-front slope measurements. Journal of the Optical Society of America, 70 (8), 998-1006.

[6] Noll, R. J. (1976). Zernike polynomials and atmospheric turbulence. Journal of the Optical Society of America, 66 (3), 207-211.

[7] Shack, R. V., Platt, B. (1971). Production and use of a lenticular Hartmann screen. Journal of the Optical Society of America, 61 (5), 656.

[8] Navarro, R., Moreno-Barriuso, E. (1999). Laser raytracing method for optical testing. Optics letters, 24 (14), 951-953.

[9] The MathWorks Inc. (2012). MATLAB 7.10.0 (R2012a).

[10] Analog Devices Inc. (1999). A technical tutorial on digital signal synthesis.

[11] Li, D., Hu, L., Mu, Q., Cao, Z., Peng, Z., Liu, Y., Yao, L., Yang, C., Lu, X., Xuan, L. (2014). Wavefront processor for liquid crystal adaptive optics system based on graphics processing unit. Optics Communications, 316, 211-216

[12] Rodríguez-Ramos, J., Castelló, E. M., Conde, C. D., Valido, M. R., Marichal-Hernández, J. (2008). 2D-FFT implementation on FPGA for wavefront phase recovery from the CAFADIS camera. In Adaptive Optics Systems, Proc. SPIE 7015.

[13] Saunter, C., Love, G., Johns, M., Holmes, J. (2005). FPGA technology for high-speed low-cost adaptive optics. In: 5th International Workshop on Adaptive Optics for Industry and Medicine. International Society for Optics and Photonics, 60181G.

Measurement Science Review

The Journal of Institute of Measurement Science of Slovak Academy of Sciences

Journal Information

IMPACT FACTOR 2017: 1.345
5-year IMPACT FACTOR: 1.253

CiteScore 2017: 1.61

SCImago Journal Rank (SJR) 2017: 0.441
Source Normalized Impact per Paper (SNIP) 2017: 0.936


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