Abstract

A generalized model is developed to quantitatively describe the smoothing effects from different polishing tools used for optical surfaces. The smoothing effect naturally corrects mid-to-high spatial frequency errors that have features small compared to the size of the polishing lap. The original parametric smoothing model provided a convenient way to compare smoothing efficiency of different polishing tools for the case of sinusoidal surface irregularity, providing the ratio of surface improvement via smoothing to the bulk material removal. A new correlation-based smoothing model expands the capability to quantify smoothing using general surface data with complex irregularity. For this case, we define smoothing as a band-limited correlated component of the change in the surface and original surface. Various concepts and methods, such as correlation screening, have been developed and verified to manipulate the data for the calculation of smoothing factor. Data from two actual polishing runs from the Giant Magellan Telescope off-axis segment and the Large Synoptic Survey Telescope monolithic primary-tertiary mirror were processed, and a quantitative evaluation for the smoothing efficiency of a large pitch lap and a conformal lap with polishing pads is provided.

© 2013 Optical Society of America

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References

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  1. R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra34–39 (1963).
  2. S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. B. Davison, T. J. Trebisky, S. T. Derigne, and B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt.33(34), 8094–8100 (1994).
    [CrossRef] [PubMed]
  3. D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
    [CrossRef]
  4. D. D. Walker, R. Freeman, R. Morton, G. McCavana, and A. Beaucamp, “Use of the ‘Precessions’TM process for prepolishing and correcting 2D & 2(1/2)D form,” Opt. Express14(24), 11787–11795 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. R. Angel, “Very large ground-based telescopes for optical and IR astronomy 4,” Nature295(5851), 651–657 (1982).
    [CrossRef]
  7. R. Angel, “Future optical and infrared telescopes 3,” Nature409(6818), 427–430 (2001).
    [CrossRef] [PubMed]
  8. H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).
  9. J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
    [CrossRef]
  10. D. W. Kim, W. H. Park, H. K. An, and J. H. Burge, “Parametric smoothing model for visco-elastic polishing tools,” Opt. Express18(21), 22515–22526 (2010).
    [CrossRef] [PubMed]
  11. J. S. Taylor, G. E. Sommargren, D. W. Sweeney, and R. M. Hudyma, “The fabrication and Testing of Optics for EUV Projection Lithography,” presented at the 23rd Annual International Symposium on Microlithography, Santa Clara, California, USA, 22–27 Feb. 1998.
    [CrossRef]
  12. D. W. Kim, S. W. Kim, and J. H. Burge, “Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions,” Opt. Express17(24), 21850–21866 (2009).
    [CrossRef] [PubMed]
  13. N. J. Brown, P. C. Baker, and R. E. Parks, “The polishing-to-figuring transition in turned optics,” SPIE’s 25th Annual International Technical Symposium,(SPIE, 1982).
    [CrossRef]
  14. R. A. Jones, “Computer simulation of smoothing during computer-controlled optical polishing,” Appl. Opt.34(7), 1162–1169 (1995).
    [CrossRef] [PubMed]
  15. P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” in Optomechanical Engineering and Vibration Control, E. A. Derby, C. G. Gordon, D. Vukobratovich, P. R. Yoder Jr., and C. H. Zweben, eds., Proc. SPIE 3786, 447 (1999).
  16. M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semiflexible tools,” Opt. Eng.41(7), 1473–1474 (2002).
    [CrossRef]
  17. D. W. Kim, H. M. Martin, and J. H. Burge, “Control of Mid-spatial-frequency Errors for Large Steep Aspheric Surfaces,” in Optical Fabrication and Testing (OF&T) Technical Digest (Optical Society of America, Washington, DC), OM4D.1 (2012).
  18. P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt.49(23), 4404–4412 (2010).
    [CrossRef] [PubMed]

2012 (1)

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

2010 (3)

2009 (1)

2006 (1)

2004 (1)

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

2002 (1)

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semiflexible tools,” Opt. Eng.41(7), 1473–1474 (2002).
[CrossRef]

2001 (1)

R. Angel, “Future optical and infrared telescopes 3,” Nature409(6818), 427–430 (2001).
[CrossRef] [PubMed]

1999 (1)

D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
[CrossRef]

1995 (1)

1994 (1)

1982 (1)

R. Angel, “Very large ground-based telescopes for optical and IR astronomy 4,” Nature295(5851), 651–657 (1982).
[CrossRef]

1963 (1)

R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra34–39 (1963).

Allen, R. G.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

An, H. K.

Anderson, B.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semiflexible tools,” Opt. Eng.41(7), 1473–1474 (2002).
[CrossRef]

Angel, R.

R. Angel, “Future optical and infrared telescopes 3,” Nature409(6818), 427–430 (2001).
[CrossRef] [PubMed]

R. Angel, “Very large ground-based telescopes for optical and IR astronomy 4,” Nature295(5851), 651–657 (1982).
[CrossRef]

Angel, R. P.

Beaucamp, A.

Borden, M. R.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Burge, J. H.

Campbell, J. H.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Davison, W. B.

Derigne, S. T.

Dumas, P.

D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
[CrossRef]

Feit, M. D.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Freeman, R.

Golini, D.

D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
[CrossRef]

Hackel, R. P.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Hawley-Fedder, R. A.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Hille, B. B.

Hogan, S.

D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
[CrossRef]

Jones, R. A.

Kim, D. W.

Kim, S. W.

Kingsley, J. S.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

Kordonski, W. I.

D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
[CrossRef]

Law, K.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

Lutz, R. D.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

Martin, H. M.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. B. Davison, T. J. Trebisky, S. T. Derigne, and B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt.33(34), 8094–8100 (1994).
[CrossRef] [PubMed]

McCavana, G.

Menapace, J. A.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Morton, R.

Nagel, R. H.

Park, W. H.

Parks, R. E.

Riley, M. O.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Runkel, M.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Stolz, C. J.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Strittmatter, P. A.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

Su, P.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt.49(23), 4404–4412 (2010).
[CrossRef] [PubMed]

Trebisky, T. J.

Tuell, M. T.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semiflexible tools,” Opt. Eng.41(7), 1473–1474 (2002).
[CrossRef]

Walker, D. D.

Wang, L.

West, S. C.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. B. Davison, T. J. Trebisky, S. T. Derigne, and B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt.33(34), 8094–8100 (1994).
[CrossRef] [PubMed]

Whitman, P. K.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Young, R. S.

Yu, J.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Zhou, P.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

Appl. Opt. (3)

in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE (1)

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, K. Law, R. D. Lutz, P. A. Strittmatter, P. Su, M. T. Tuell, S. C. West, and P. Zhou, “Production of 8.4 m segments for the Giant Magellan Telescope,” in Modern Technologies in Space-and-Ground-based Telescopes and Instrumentation II, Proc. SPIE8450, 84502D (2012).

Nature (2)

R. Angel, “Very large ground-based telescopes for optical and IR astronomy 4,” Nature295(5851), 651–657 (1982).
[CrossRef]

R. Angel, “Future optical and infrared telescopes 3,” Nature409(6818), 427–430 (2001).
[CrossRef] [PubMed]

Opt. Eng. (1)

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semiflexible tools,” Opt. Eng.41(7), 1473–1474 (2002).
[CrossRef]

Opt. Express (4)

Photon. Spectra (1)

R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra34–39 (1963).

Proc. SPIE (2)

D. Golini, W. I. Kordonski, P. Dumas, and S. Hogan, “Magnetorheological finishing (MRF) in commercial precision optics manufacturing,” Proc. SPIE3782, 80–91 (1999).
[CrossRef]

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: An overview,” Proc. SPIE5341, 84–101 (2004).
[CrossRef]

Other (4)

J. S. Taylor, G. E. Sommargren, D. W. Sweeney, and R. M. Hudyma, “The fabrication and Testing of Optics for EUV Projection Lithography,” presented at the 23rd Annual International Symposium on Microlithography, Santa Clara, California, USA, 22–27 Feb. 1998.
[CrossRef]

N. J. Brown, P. C. Baker, and R. E. Parks, “The polishing-to-figuring transition in turned optics,” SPIE’s 25th Annual International Technical Symposium,(SPIE, 1982).
[CrossRef]

P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” in Optomechanical Engineering and Vibration Control, E. A. Derby, C. G. Gordon, D. Vukobratovich, P. R. Yoder Jr., and C. H. Zweben, eds., Proc. SPIE 3786, 447 (1999).

D. W. Kim, H. M. Martin, and J. H. Burge, “Control of Mid-spatial-frequency Errors for Large Steep Aspheric Surfaces,” in Optical Fabrication and Testing (OF&T) Technical Digest (Optical Society of America, Washington, DC), OM4D.1 (2012).

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Figures (9)

Fig. 1
Fig. 1

Smoothing simulation for a 60cm infinitely rigid tool [10].

Fig. 2
Fig. 2

The sinusoidal ripple profiles: initial, final and removal (i.e. initial – final).

Fig. 3
Fig. 3

Computer controlled polishing machines equipped with the 0.8m (contact area diameter) stressed lap on the 8.4m diameter Large Synoptic Survey Telescope monolithic primary-tertiary workpiece (left) and the 0.25m diameter RC lap on the 8.4m diameter Giant Magellan Telescope off-axis segment (right) at the Steward Observatory Mirror Lab, University of Arizona [8].

Fig. 4
Fig. 4

Data processing flow of the correlation-based smoothing model.

Fig. 5
Fig. 5

Comparison between well correalated profiles (left) and badly correlated profiles (right).

Fig. 6
Fig. 6

Six exemplary sub-region surface maps showing the local smoothing event for different correlation coefficients. The sub-regions are 0.3m in diameter on the 8.4m GMT segement. (Note: Red means high in the high-pass filtered initial map and more removal in the difference map.)

Fig. 7
Fig. 7

Smoothing factor SF vs. initial sinusoidal error RMSini for a RC lap with LP-66 polyurethane pad on a Pyrex workpiece.

Fig. 8
Fig. 8

Smoothing factor (left) and correlation coefficient ρ histogram (right) for Case 1: 8.4m GMT off-axis primary workpiece, SF vs. RMSini for 1.2m stressed lap with pitch (black solid circle) and 0.25m RC lap with polyurethane pad (red open circle). (The bar represents the spread (+/− standard deviation) of the local SF values. The values in the histogram are the mean+/−standard deviation of the distribution. The ρthreshold value is indicated as a vertical thin line in the histogram.)

Fig. 9
Fig. 9

Smoothing factor (left) and correlation coefficient ρ histogram (right) for Case 2: 5.066m LSST tertiary mirror, SF vs. RMSini for 0.8m stressed lap with pitch and LP-66 polyurethane pad (black solid circle) and 0.35m RC lap with polyurethane pad (red open circle). (The bar represents the spread (+/− standard deviation) of the local SF values. The values in the histogram are the mean+/−standard deviation of the distribution. The ρthreshold value is indicated as a vertical thin line in the histogram.)

Tables (1)

Tables Icon

Table 1 CCOS parameters for the GMT and LSST runs

Equations (3)

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

SF=k( ε ini ε 0 )
ρ M ini , M diff = subregion ( M ini (x,y) M ini ¯ )( M diff (x,y) M diff ¯ )dxdy subregion ( M ini (x,y) M ini ¯ ) 2 dxdy subregion ( M diff (x,y) M diff ¯ ) 2 dxdy = subregion ( M ini (x,y) subregion M ini (x,y)dxdy subregion dxdy )( M diff (x,y) subregion M diff (x,y)dxdy subregion dxdy )dxdy subregion ( M ini (x,y) subregion M ini (x,y)dxdy subregion dxdy ) 2 dxdy subregion ( M diff (x,y) subregion M diff (x,y)dxdy subregion dxdy ) 2 dxdy .
SF= RM S highpass_filtered_initial_map RM S highpass_filtered_final_map Averag e nominal_removal_map = subregion M ini 2 (x,y)dxdy / subregion dxdy subregion M fin 2 (x,y)dxdy / subregion dxdy subregion M nominal (x,y)dxdy / subregion dxdy ,

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