Abstract

Whole Slide Imaging (WSI) systems are high-throughput automated microscopes for digital pathology applications. We present a method for testing and monitoring the optical quality of WSI-systems using a measurement of the through-focus Optical Transfer Function (OTF) obtained from the edge response of a custom made resolution target, composed of sagittal and tangential edges. This enables quantitative analysis of a number of primary aberrations. The curvature of the best focus as a function of spatial frequency is indicative for spherical aberration, the argument of the OTF quantifies for coma, and the best focus as a function of field position for sagittal and tangential edges allows assessment of astigmatism and field curvature. The statistical error in the determined aberrations is typically below 20 mλ. We use the method to compare different tube lens designs and to study the effect of objective lens aging. The results are in good agreement with direct measurement of aberrations based on Shack-Hartmann wavefront sensing with a typical error ranging from 10 mλ to 40 mλ.

© 2015 Optical Society of America

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References

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2013 (2)

J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. 249(2), 124–135 (2013).
[Crossref] [PubMed]

K. He, J. Sun, and X. Tang, “Guided image filtering,” IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1397–1409 (2013).
[Crossref] [PubMed]

2012 (2)

S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. 59(5), 1205–1218 (2012).
[Crossref] [PubMed]

S. Al-Janabi, A. Huisman, and P. J. Van Diest, “Digital pathology: current status and future perspectives,” Histopathology 61(1), 1–9 (2012).
[Crossref] [PubMed]

2011 (3)

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

P. L. Schechter and R. S. Levinson, “Generic misalignment aberration patterns in wide-field telescopes,” Publ. Astron. Soc. Pac. 123(905), 812–832 (2011).
[Crossref]

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Effects of sampling on the phase transfer function of incoherent imaging systems,” Opt. Express 19(24), 24609–24626 (2011).
[Crossref] [PubMed]

2009 (2)

2006 (3)

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

2005 (3)

2004 (1)

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

2001 (1)

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

1994 (2)

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

1993 (1)

1988 (1)

C. J. R. Sheppard, “Depth of field in optical microscopy,” J. Microsc. 149(1), 73–75 (1988).
[Crossref]

1987 (1)

I. A. Cunningham and A. Fenster, “A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation,” Med. Phys. 14(4), 533–537 (1987).
[Crossref] [PubMed]

1982 (1)

1980 (1)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 0251, 146–153 (1980).
[Crossref]

Al-Janabi, S.

S. Al-Janabi, A. Huisman, and P. J. Van Diest, “Digital pathology: current status and future perspectives,” Histopathology 61(1), 1–9 (2012).
[Crossref] [PubMed]

Anthony, L.

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

Barker, G.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Bartels, P. H.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Beck, A. H.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Bhakta, V. R.

Bhattacharyya, A. K.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Boddeke, F. R.

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

Boucheron, L. E.

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

Cakmakci, O.

Can, A.

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

Christensen, M. P.

Cunningham, I. A.

I. A. Cunningham and A. Fenster, “A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation,” Med. Phys. 14(4), 533–537 (1987).
[Crossref] [PubMed]

Davis, J. R.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

De Haan, G.

J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. 249(2), 124–135 (2013).
[Crossref] [PubMed]

de Jong, P.

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

Descour, M. R.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Doyle, S.

S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. 59(5), 1205–1218 (2012).
[Crossref] [PubMed]

Dunn, B. E.

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Feldman, M.

S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. 59(5), 1205–1218 (2012).
[Crossref] [PubMed]

Fenster, A.

I. A. Cunningham and A. Fenster, “A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation,” Med. Phys. 14(4), 533–537 (1987).
[Crossref] [PubMed]

Fienup, J. R.

García, G. B.

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

García, J. G.

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

Gilbertson, J. R.

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

Goodall, J. F.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Graham, A. R.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Gurcan, M. N.

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

He, K.

K. He, J. Sun, and X. Tang, “Guided image filtering,” IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1397–1409 (2013).
[Crossref] [PubMed]

Ho, J.

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

Huisman, A.

S. Al-Janabi, A. Huisman, and P. J. Van Diest, “Digital pathology: current status and future perspectives,” Histopathology 61(1), 1–9 (2012).
[Crossref] [PubMed]

Jukic, D. M.

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

Kayser, K.

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Koller, D.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Krupinski, E. A.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Leung, S.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Levinson, R. S.

P. L. Schechter and R. S. Levinson, “Generic misalignment aberration patterns in wide-field telescopes,” Publ. Astron. Soc. Pac. 123(905), 812–832 (2011).
[Crossref]

Liang, C.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Madabhushi, A.

S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. 59(5), 1205–1218 (2012).
[Crossref] [PubMed]

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

Marinelli, R. J.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Mateos, C. P.

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

Mullikin, J. C.

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

Netten, H.

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

Nielsen, T. O.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Olszak, A. G.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Parwani, A. V.

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

Rajpoot, N. M.

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

Rennels, M.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Richter, L.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Rojo, M. G.

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

Rolland, J. P.

Russum, W. C.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Sangoi, A. R.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Schechter, P. L.

P. L. Schechter and R. S. Levinson, “Generic misalignment aberration patterns in wide-field telescopes,” Publ. Astron. Soc. Pac. 123(905), 812–832 (2011).
[Crossref]

Schmid, T.

Scott, K. M.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 0251, 146–153 (1980).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, “Depth of field in optical microscopy,” J. Microsc. 149(1), 73–75 (1988).
[Crossref]

Somayaji, M.

Stallinga, S.

Sun, J.

K. He, J. Sun, and X. Tang, “Guided image filtering,” IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1397–1409 (2013).
[Crossref] [PubMed]

Szymus, J.

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

Tang, X.

K. He, J. Sun, and X. Tang, “Guided image filtering,” IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1397–1409 (2013).
[Crossref] [PubMed]

Thompson, K.

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
[Crossref] [PubMed]

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 0251, 146–153 (1980).
[Crossref]

Thompson, K. P.

Tomaszewski, J.

S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. 59(5), 1205–1218 (2012).
[Crossref] [PubMed]

van de Rijn, M.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

van de Vijver, M. J.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Van Deurzen, C. H.

J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. 249(2), 124–135 (2013).
[Crossref] [PubMed]

Van Diest, P. J.

S. Al-Janabi, A. Huisman, and P. J. Van Diest, “Digital pathology: current status and future perspectives,” Histopathology 61(1), 1–9 (2012).
[Crossref] [PubMed]

Van Leeuwen, M. B.

J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. 249(2), 124–135 (2013).
[Crossref] [PubMed]

van Vliet, L. J.

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

Vanderfeltz, G.

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

Vanvliet, L. J.

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

Vicente, M. C.

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

Vink, J. P.

J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. 249(2), 124–135 (2013).
[Crossref] [PubMed]

Weinstein, R. S.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

West, R. B.

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Williams, B. H.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Wyant, J. C.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Yagi, Y.

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

Yener, B.

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

Young, I. T.

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

Zhou, P.

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

Appl. Opt. (4)

Bioimaging (1)

H. Netten, L. J. van Vliet, F. R. Boddeke, P. de Jong, and I. T. Young, “A fast scanner for fluorescence microscopy using a 2‐D CCD and time delayed integration,” Bioimaging 2(4), 184–192 (1994).
[Crossref]

BMC Clin. Pathol. (1)

J. R. Gilbertson, J. Ho, L. Anthony, D. M. Jukic, Y. Yagi, and A. V. Parwani, “Primary histologic diagnosis using automated whole slide imaging: a validation study,” BMC Clin. Pathol. 6(1), 4 (2006).
[Crossref] [PubMed]

Histopathology (1)

S. Al-Janabi, A. Huisman, and P. J. Van Diest, “Digital pathology: current status and future perspectives,” Histopathology 61(1), 1–9 (2012).
[Crossref] [PubMed]

Hum. Pathol. (3)

R. S. Weinstein, M. R. Descour, C. Liang, A. K. Bhattacharyya, A. R. Graham, J. R. Davis, K. M. Scott, L. Richter, E. A. Krupinski, J. Szymus, K. Kayser, and B. E. Dunn, “Telepathology overview: from concept to implementation,” Hum. Pathol. 32(12), 1283–1299 (2001).
[Crossref] [PubMed]

J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. 37(3), 322–331 (2006).
[Crossref] [PubMed]

R. S. Weinstein, M. R. Descour, C. Liang, G. Barker, K. M. Scott, L. Richter, E. A. Krupinski, A. K. Bhattacharyya, J. R. Davis, A. R. Graham, M. Rennels, W. C. Russum, J. F. Goodall, P. Zhou, A. G. Olszak, B. H. Williams, J. C. Wyant, and P. H. Bartels, “An array microscope for ultrarapid virtual slide processing and telepathology. Design, fabrication, and validation study,” Hum. Pathol. 35(11), 1303–1314 (2004).
[Crossref] [PubMed]

IEEE Rev Biomed Eng. (1)

M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: a review,” IEEE Rev Biomed Eng. 2, 147–171 (2009).
[Crossref] [PubMed]

IEEE Trans. Biomed. Eng. (1)

S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “A boosted Bayesian multiresolution classifier for prostate cancer detection from digitized needle biopsies,” IEEE Trans. Biomed. Eng. 59(5), 1205–1218 (2012).
[Crossref] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

K. He, J. Sun, and X. Tang, “Guided image filtering,” IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1397–1409 (2013).
[Crossref] [PubMed]

Int. J. Surg. Pathol. (1)

M. G. Rojo, G. B. García, C. P. Mateos, J. G. García, and M. C. Vicente, “Critical comparison of 31 commercially available digital slide systems in pathology,” Int. J. Surg. Pathol. 14(4), 285–305 (2006).
[Crossref] [PubMed]

J. Microsc. (2)

J. P. Vink, M. B. Van Leeuwen, C. H. Van Deurzen, and G. De Haan, “Efficient nucleus detector in histopathology images,” J. Microsc. 249(2), 124–135 (2013).
[Crossref] [PubMed]

C. J. R. Sheppard, “Depth of field in optical microscopy,” J. Microsc. 149(1), 73–75 (1988).
[Crossref]

J. Opt. Soc. Am. A (2)

Med. Phys. (1)

I. A. Cunningham and A. Fenster, “A method for modulation transfer function determination from edge profiles with correction for finite-element differentiation,” Med. Phys. 14(4), 533–537 (1987).
[Crossref] [PubMed]

Opt. Express (1)

Proc. SPIE (2)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 0251, 146–153 (1980).
[Crossref]

J. C. Mullikin, L. J. Vanvliet, H. Netten, F. R. Boddeke, G. Vanderfeltz, and I. T. Young, “Methods for CCD camera characterization,” Proc. SPIE 2173, 73–84 (1994).
[Crossref]

Publ. Astron. Soc. Pac. (1)

P. L. Schechter and R. S. Levinson, “Generic misalignment aberration patterns in wide-field telescopes,” Publ. Astron. Soc. Pac. 123(905), 812–832 (2011).
[Crossref]

Sci. Transl. Med. (1)

A. H. Beck, A. R. Sangoi, S. Leung, R. J. Marinelli, T. O. Nielsen, M. J. van de Vijver, R. B. West, M. van de Rijn, and D. Koller, “Systematic analysis of breast cancer morphology uncovers stromal features associated with survival,” Sci. Transl. Med. 3(108), 108ra113 (2011).
[Crossref] [PubMed]

Other (7)

J. Gu and R. W. Ogilvie, Virtual microscopy and virtual slides in teaching, diagnosis, and research, (Taylor & Francis, 2005).

Photography–Electronic Still Picture Cameras–Resolution Measurements, ISO Standard 12233:2000.

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

G. D. Boerman, Modulation transfer function in optical and electro-optical systems, (SPIE, 2001).

T. Williams, The optical transfer function of imaging systems, (CRC, 1998).

R. Tessieres, “Analysis for alignment of optical systems”, MSc Dissertation (University of Arizona, Tucson, 2003).

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Image-based measurement of phase transfer function,” in Digital Image Processing and Analysis, (2010).

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Figures (12)

Fig. 1
Fig. 1

Schematic layout of the home-built line sensor based WSI system (left) and an image acquired with the setup (right)

Fig. 2
Fig. 2

(a) Custom made resolution target and assigned ROIs. (b) Processing steps for calculating the OTF. Each ROI is split in blocks, the Edge Spread Function (ESF) is found by averaging along the edge followed by edge preserving denoising, the Line Spread Function (LSF) is found by numerical differentiation of the ESF, the OTF then follows by Fourier Transformation (FT). Edge alignment of the different blocks in an ROI is needed to calibrate the linear part of the PTF from each block. The tangential edges give a cross-section of the OTF along the fx-axis, the sagittal edges give a cross-section of the OTF along the fy-axis.

Fig. 3
Fig. 3

Effect of primary aberrations on the through focus OTF (NA = 0.75, λ = 565nm). (a) Through-focus MTF in an ideal optical system. (b) Through-focus MTF with spherical aberration of 36mλ. (c) Best focus line with spherical aberration of 0, 36 and 72mλ (d) Through-focus MTF with coma of 72mλ in the x direction (e) PTF at the best focus with coma of 0, 36 and 72mλ (f) Through-focus MTF with astigmatism of 72mλ at 0 deg.

Fig. 4
Fig. 4

Schematic layout of Shack-Hartmann setup for validation of the proposed method.

Fig. 5
Fig. 5

Result of Zemax simulation for field curvature and astigmatism. T and S stand for the tangential and sagittal best focus, respectively. Blue, green and red pairs are for 435, 565 and 618nm wavelength. (a) Single achromat. (b) Double back to back achromat design. Notice the different scales in (a) and (b).

Fig. 6
Fig. 6

Results for a Nikon 20 × /0.75 objective lens with single achromat tube lens (case A) (a) Through-focus MTFs for 6 ROIs (red rectangles in custom resolution target) with the best focus line (dashed black line). (b) MTFs (dashed) and PTFs (solid) at the overall best focus. (c) Measured field curvatures. (d) Best focus lines in the scan direction at five position in the FOV from the sagittal to the tangential area showing the absence of tilt in the scan direction and sagittal to tangential jumps due to astigmatism.

Fig. 7
Fig. 7

Results for a Nikon 20 × /0.75 objective lens with double back-to-back achromat tube lens (case B) (a) Through-focus MTFs for 6 ROIs (red rectangles in custom resolution target) with the best focus line (dashed black line). (b) MTFs (dashed) and PTFs (solid) at the overall best focus. (c) Measured field curvatures. (d) Best focus lines in the scan direction at five position in the FOV from the sagittal to the tangential area showing the absence of tilt in the scan direction and sagittal to tangential jumps due to astigmatism.

Fig. 8
Fig. 8

Example of statistical errors for case B in the middle sagittal ROI (a) Average MTF and PTF at the best focus, (b) Measured and fitted best focus as a function of spatial frequency. The small decrease of the error at the highest spatial frequency is due to a coincidence, not all the ROIs show this behavior.

Fig. 9
Fig. 9

Results for an aged Olympus 20 × /0.75 objective lens with single achromat tube lens (case C) (a) Through-focus MTFs for 6 ROIs (red rectangles in custom resolution target) with the best focus line (dashed black line). (b) MTFs (dashed) and PTFs (solid) at the overall best focus. (c) Measured field curvatures. (d) Best focus lines in the scan direction at five position in the FOV from the sagittal to the tangential area showing the absence of tilt in the scan direction and sagittal to tangential jumps due to astigmatism.

Fig. 10
Fig. 10

Comparison of the aberration values for astigmatism, coma and spherical aberration using standard Zernike coefficients representation [23] for case A (Nikon 20 × /0.75 objective lens with single achromat tube lens), case B (Nikon 20 × /0.75 objective lens with double back-to-back achromat tube lens), and case C (aged Olympus 20 × /0.75 objective lens with single achromat tube lens). L, M and R stands for left, middle and right of the FOV, and Ast, cTan and cSag denote astigmatism, tangential coma and sagittal coma respectively. (a) Results obtained with the proposed method. (b) Results obtained with the Shack-Hartmann measurement. It should be noted that the left and right of the FOV are here at ± 0.38 mm in the FOV due to the mechanical constraints in the Shack-Hartmann setup. To make a fair comparison to the aberration values of (b) the values reported in (a) are also at ± 0.38 mm in the FOV and can therefore not be compared directly to the values in section 3.1 to 3.3 which were acquired at ± 0.5 mm field position.

Fig. 11
Fig. 11

Representation of coma and astigmatism measured by the Shack-Hartmann sensor over the FOV for three cases: (a) single achromat tube lens, (b) double achromat tube lens design, and (c) aged objective lens with single achromat tube lens. Note the different scale bar in (c) compared to (a) and (b).

Fig. 12
Fig. 12

Sagittal and tangential best focus lines for the three color channels (red: 618 nm, green: 565 nm, blue: 435 nm) for the (a) single achromat tube lens (case A). (b) double back-to-back achromat tube lens design (case B).

Equations (9)

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Δ x est =arg min Δx { f W(f) (PTF(f)+rem(2πfΔx,2π)) 2 }
z opt ( f x )=[ p 1 ( λ NA f x ) 2 + p 2 λ NA f x + p 3 ] λ 1 1N A 2
W(x,y)= 6 A 22 ( x 2 y 2 )+ z ob [1 1N A 2 ( x 2 + y 2 ) ],
W( x,0 )= 6 A 22 f 1 ( x )+ z ob f 2 ( x ), f 1 ( x )= x 2 f 2 ( x )=1 1N A 2 x 2
W RMS 2 = W 2 W 2
W RMS 2 =6( f 1 2 f 1 2 ) A 22 2 +2  6 ( f 1 f 2 f 1 f 2 ) A 22 z ob +( f 2 2 f 2 2 ) z ob 2
z ob = f 1 f 2 f 1 f 2 f 2 2 f 2 2 6 A 22
S( NA )= f 2 2 f 2 2 2 6 ( f 1 f 2 f 1 f 2 )
f 1 = 1 3 f 2 =1 NA 1 NA 2 +ArcSin[NA] 2NA f 1 2 = 1 5 f 1 f 2 = 1 3 NA 1 NA 2 (1+2 NA 2 )+ArcSin[NA] 8 NA 3   f 2 2 =2 NA 2 3 1 NA 2 ArcSin[NA] NA

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