The Annals of Applied Statistics

Improving PSF calibration in confocal microscopic imaging—estimating and exploiting bilateral symmetry

Nicolai Bissantz, Hajo Holzmann, and Mirosław Pawlak

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A method for estimating the axis of reflectional symmetry of an image f(x, y) on the unit disc D={(x, y) : x2+y2≤1} is proposed, given that noisy data of f(x, y) are observed on a discrete grid of edge width Δ. Our estimation procedure is based on minimizing over β∈[0, π) the L2 distance between empirical versions of f and τβf, the image of f after reflection at the axis along (cos β, sin β). Here, f and τβf are estimated using truncated radial series of the Zernike type. The inherent symmetry properties of the Zernike functions result in a particularly simple estimation procedure for β. It is shown that the estimate β̂ converges at the parametric rate Δ−1 for images f of bounded variation. Further, we establish asymptotic normality of β̂ if f is Lipschitz continuous. The method is applied to calibrating the point spread function (PSF) for the deconvolution of images from confocal microscopy. For various reasons the PSF characterizing the problem may not be rotationally invariant but rather only reflection symmetric with respect to two orthogonal axes. For an image of a bead acquired by a confocal laser scanning microscope (Leica TCS), these axes are estimated and corresponding confidence intervals are constructed. They turn out to be close to the coordinate axes of the imaging device. As cause for deviation from rotational invariance, this indicates some slight misalignment of the optical system or anisotropy of the immersion medium rather than some irregular shape of the bead. In an extensive simulation study, we show that using a symmetrized version of the observed PSF significantly improves the subsequent reconstruction process of the target image.

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Ann. Appl. Stat., Volume 4, Number 4 (2010), 1871-1891.

First available in Project Euclid: 4 January 2011

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Image analysis semiparametric estimation reflection symmetry two-dimensional functions Zernike polynomials confocal microscopy


Bissantz, Nicolai; Holzmann, Hajo; Pawlak, Mirosław. Improving PSF calibration in confocal microscopic imaging—estimating and exploiting bilateral symmetry. Ann. Appl. Stat. 4 (2010), no. 4, 1871--1891. doi:10.1214/10-AOAS343.

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  • Amayeh, G., Erol, A., Bebis, G. and Nicolescu, M. (2005). Accurate and efficient computation of high order Zernike moments. In Advances in Visual Computing 462–469. Springer, Berlin.
  • Anscombe, F. J. (1948). The transformation of Poisson, binomial and negative-binomial data. Biometrika 35 246–254.
  • Atallah, M. J. (1985). On symmetry detection. IEEE Trans. Comput. 34 663–666.
  • Bailey, R. R. and Srinath, M. (1996). Orthogonal moment feature for use with parametric and non-parametric classifiers. IEEE Trans. Pattern Anal. Mach. Intell. 18 389–396.
  • Bewersdorf, J., Schmidt, R. and Hell, S. W. (2006). Comparison of I5M and 4Pi-microscopy. J. Microscopy 222 105–117.
  • Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD.
  • Bissantz, N., Holzmann, H. and Pawlak, M. (2009). Testing for image symmetries—with application to confocal miscroscopy. IEEE Trans. Inform. Theory 55 1841–1855.
  • Bissantz, N., Holzmann, H. and Pawlak, M. (2010). Estimating bilateral symmetry: Technical details. Supplement to “Improving PSF calibration in confocal microscopic imaging—estimating and exploiting bilateral symmetry.” DOI: 10.1214/10-AOAS343SUPP.
  • Bhatia, A. B. and Wolf, E. (1954). On the circle polynomials of Zernike and related orthogonal sets. Proc. Cambridge Philos. Soc. 50 40–48.
  • Conway, J. H., Burgiel, H. and Goodman-Strauss, C. (2008). The Symmetry of Things. A. K. Peters, Wellesley, MA.
  • Dieterlen, A., Debailleul, M., De Meyer, A., Simon, B., Georges, V., Colicchio, B. and Haeberlé, O. (2008). Recent advances in 3-D fluorescence microscopy: Tomography as a source of information. In Eighth International Conference on Correlation Optics (M. Kujawinska and O. V. Angelsky, eds.). Proc. SPIE 7008 70080S–70080S-8. SPIE.
  • Dieterlen, A., Xu, C., Haeberle, O., Hueber, N., Malfara, R., Colicchio, B. and Jacquey, S. (2004). Identification and restoration in 3D fluorescence microscopy. In Sixth International Conference on Correlation Optics (O. V. Angelsky, ed.). Proc. SPIE 5477 105–113. SPIE.
  • Friedberg, S. A. (1986). Finding axes of skewed symmetry. Comput. Vision Graphics Image Process 32 138–155.
  • Johnstone, I. M. and Silverman, B. W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 251–280.
  • Jones, M. C. and Silverman, B. W. (1989). An orthogonal series density estimation approach to reconstructing positron emission tomography images. J. Appl. Statist. 16 177–191.
  • Kim, W.-Y. and Kim, Y.-S. (1999). Robust rotation angle estimator. IEEE Trans. Pattern Anal. Mach. Intell. 21 768–773.
  • Khotanzad, A. and Hong, Y. H. (1990). Invariant image recognition by Zernike moments. IEEE Trans. Pattern Anal. Mach. Intell. 12 489–498.
  • Lehr, J., Sibarita, J.-B. and Chassery, J.-M. (1998). Image restoration in X-ray microscopy: PSF determination and biological applications. IEEE Trans. Image Processing 7 258–263.
  • Liu, Y., Collins, R. T. and Tsin, Y. (2004). A computational model for periodic pattern perception based on frieze and wallpaper groups. IEEE Trans. Pattern Anal. Mach. Intell. 26 354–371.
  • Lucy, L. B. (1974). An iterative technique for the rectification of observed distributions. Astron. J. 79 745–754.
  • Mukundan, R. and Ramakrishnan, K. (1998). Moment Functions in Image Analysis: Theory and Applications. World Scientific, River Edge, NJ.
  • Munk, A., Bissantz, N., Wagner, T. and Freitag, G. (2005). On difference-based variance estimation in nonparametric regression when the covariate is high dimensional. J. Roy Statist. Soc. B 67 19–41.
  • Pankajakshan, P., Zhang, B., Blanc-Féraud, L., Kam, Z., Olivo-Marin, J.-C. and Zerubia, J. (2008). Blind deconvolution for diffraction-limited fluorescence microscopy. IEEE International Symposium on Biomedical Imaging.
  • Pawlak, M. and Liao, S. X. (2002). On the recovery of a function on a circular domain. IEEE Trans. Inform. Theory 48 2736–2753.
  • Revaud, J., Lavoue, G. and Baskurt, A. (2008). Improving Zernike moments comparison for optimal similarity and rotation angle retrieval. IEEE Trans. Pattern Anal. Mach. Intell. 30 954–971.
  • Richardson, W. H. (1972). Bayesian-based iterative method of image restoration. J. Opt. Soc. Am. 62 55–59.
  • Shepp, I. A. and Vardi, Y. (1982). Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imaging 1 113–122.
  • Viana, M. A. G. (2008). Symmetry Studies: An Introduction to the Analysis of Structured Data in Applications. Cambridge Univ. Press, Cambridge.
  • Zernike, F. (1934). Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. Physica 1 689–701.

Supplemental materials

  • Supplementary material: Estimating bilateral symmetry: Technical details. Here we provide the technical proofs for our results in the paper “Improving PSF calibration in confocal microscopic imaging—estimating and exploiting bilateral symmetry.”.