A Harmonic Analysis View on Neuroscience Imaging

  • Paul Hernandez—Herrera
  • David Jiménez
  • Ioannis A. Kakadiaris
  • Andreas Koutsogiannis
  • Demetrio Labate
  • Fernanda Laezza
  • Manos Papadakis
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


After highlighting some of the current trends in neuroscience imaging, this work studies the approximation errors due to varying directional aliasing, arising when 2D or 3D images are subjected to the action of orthogonal transformations. Such errors are common in 3D images of neurons acquired by confocal microscopes. We also present an algorithm for the construction of synthetic data (computational phantoms) for the validation of algorithms for the morphological reconstruction of neurons. Our approach delivers synthetic data that have a very high degree of fidelity with respect to their ground-truth specifications.


Synthetic tubular data Synthetic dendrites Directional aliasing Approximation error Dendritic arbor segmentation Confocal microscopy 



This work was supported in part by NSF grants DMS 0915242, DMS 1005799, and DMS 1008900 and by NHARP grant 003652-0136-2009.


  1. 1.
    Al-Kofahi, K., Lasek, S., Szarowski, D., Pace, C., Nagy, G.: Rapid automated three-dimensional tracing of neurons from confocal image stacks. IEEE Trans. Information Technology in Biomedicine 6(2), 171–187 (2002)CrossRefGoogle Scholar
  2. 2.
    Ascoli, G.A.: Progress and perspectives in computational neuroanatomy. Anat. Record. 257(6), 195–207 (1999)CrossRefGoogle Scholar
  3. 3.
    Bai, W., Zhou, X., Ji, L., Cheng, J., Wong, S.T.C.: Automatic dendritic spine analysis in two-photon laser scanning microscopy images. Cytometry Part A 71A(10), 818–826 (2007). DOI 10.1002/cyto.a.20431. URL
  4. 4.
    Bodmann, B., Melas, A., Papadakis, M., Stavropoulos, T.: Analog to digital revisited: Controlling the accuracy of reconstruction. Sampl. Theory Signal Image Process. 5(3), 321–340 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Boor, C.D., DeVore, R., Ron, A.: Approximation from shift-invariant subspaces of l 2( d). Trans. Amer. Math. Soc. 341(2), 787–806 (1994)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bouix, S., Siddiqi, K., Tannenbaum, A.: Flux driven automatic centerline extraction. Med. Image Anal. 9(3), 209–221 (2005)CrossRefGoogle Scholar
  7. 7.
    Brown, K., Barrionuevo, G., Canty, A., Paola, V., Hirsch, J., Jefferis, G., Lu, J., Snippe, M., Sugihara, I., Ascoli, G.: The DIADEM data sets: representative light microscopy images of neuronal morphology to advance automation of digital reconstructions. Neuroinform. 9(2–3), 143–157 (2011)CrossRefGoogle Scholar
  8. 8.
    Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise C 2singularities. Comm. Pure Appl. Math. 57(2), 219–266 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Caņero, C., Radeva, P.: Vesselness enhancement diffusion. Pattern Recogn. Lett. 24(16), 3141–3151 (2003)CrossRefGoogle Scholar
  10. 10.
    CBL: ORION: Online Reconstruction and functional Imaging Of Neurons (2008). URL
  11. 11.
    Cheng, J., Zhou, X., Miller, E., Witt, R., Zhu, J., Sabatini, B., Wong, S.: A novel computational approach for automatic dendrite spines detection in two-photon laser scan microscopy. J. Neurosci. Methods 165(1), 122–134 (2007)CrossRefGoogle Scholar
  12. 12.
    Cheng, J., Zhou, X., Miller, E., Alvarez, V., Sabatini, B., Wong, S.: Oriented markov random field based dendritic spine segmentation for fluorescence microscopy images. Neuroinformatics 8, 157–170 (2010). URL 10.1007/s12021-010-9073-yGoogle Scholar
  13. 13.
    Choy, S., Chen, K., Zhang, Y., Baron, M., Teylan, M., Kim, Y., Tong, C.S., Song, Z., Wong, S.: Multi scale and slice-based approach for automatic spine detection. In: Engineering in Medicine and Biology Society (EMBC), pp. 4765–4768 (2010)Google Scholar
  14. 14.
    Curtis, H.J., Cole, K.S.: Transverse electric impedance of the squid giant axon. J. General Physiol. 21, 757–765 (1938)CrossRefGoogle Scholar
  15. 15.
    DeVore, R.: Non-linear approximation. Acta Numer. 7, 51–150 (1998)MathSciNetGoogle Scholar
  16. 16.
    DeVore, R.A., Jawerth, B., Popov, V.: Compression of wavelet decomposition. Am. J. Math. 114(4), 737–785 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dumitriu, D., Rodriguez, A., Morrison, J.H.: High-throughput, detailed, cell-specific neuroanatomy of dendritic spines using microinjection and confocal microscopy. Nat. Protocols 6(9), 1391–1411 (2011). DOI 10.1038/nprot.2011.389. URL
  18. 18.
    Fan, J., Zhou, X., Dy, J., Zhang, Y., Wong, S.: An automated pipeline for dendrite spine detection and tracking of 3d optical microscopy neuron images of in vivo mouse models. Neuroinformatics 7, 113–130 (2009). URL 10.1007/s12021-009-9047-0Google Scholar
  19. 19.
    Glaser, J., Glaser, E.: Neuron imaging with neurolucida-a pc-based system for image combining microscopy. Comput. Med. Imaging Graph. 14(5), 307–317 (1990)CrossRefGoogle Scholar
  20. 20.
    Gonzalez, G., Fleuret, F., Fua, P.: Automated delineation of dendritic networks in noisy image stacks. In: Proceedings of European Conference on Computer Vision, pp. 214–227. Marseille, France (2008)Google Scholar
  21. 21.
    Govindarajan, A., Kelleher, R.J., Tonegawa, S.: A clustered plasticity model of long-term memory engrams. Nat. Rev. Neurosci. 7(7), 575–583 (2006). DOI 10.1038/nrn1937. URL Google Scholar
  22. 22.
    Govindarajan, A., Israely, I., Huang, S.Y., Tonegawa, S.: The dendritic branch is the preferred integrative unit for protein synthesis-dependent ltp. Neuron 69(1), 132–146 (2011). DOI 10.1016/j.neuron.2010.12.008. URL Google Scholar
  23. 23.
    Guo, K., Labate, D.: Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal. 39(2007)Google Scholar
  24. 24.
    Hines, M., Carnevale, N.: NEURON: a tool for neuroscientists. Neuroscientist 7, 123–135 (2001)CrossRefGoogle Scholar
  25. 25.
    Hodgkin, A.L., Huxley, A.F., Katz, B.: Measurement of current-voltage relations in the membrane of the giant axon of loligo. J. Physiol. 116, 424448 (1952)Google Scholar
  26. 26.
    Jingfan, L., Gensun, F.: On truncation error bound for multidimensional sampling expansion Laplace transform. Anal. Theory Appl. 1, 52–57 (2004)CrossRefGoogle Scholar
  27. 27.
  28. 28.
    Jetter, K., Plonka, G.: A survey on L 2-approximation order from shift-invariant spaces. In: Dyn, N., Leviatan, D., Levin, D., Pinkus, A. (eds.) Multivariate Approximation and Applications, pp. 73–111. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  29. 29.
    Kakadiaris, I., Santamaría-Pang, A., Colbert, C., Saggau, P.: Morphological reconstruction of living neurons. In: Rittscher, J., Machiraju, R., Wong, S. (eds.) Microscopic Image Analysis for Life Science Applications. Artech House Publishers, Norwood (2007)Google Scholar
  30. 30.
    Kakadiaris, I., Santamaría-Pang, A., Colbert, C., Saggau, P.: Automatic 3-D morphological reconstruction of neuron cells from multiphoton images. In: Rittscher, J., Machiraju, R., Wong, S. (eds.) Microscopic Image Analysis for Life Science Applications, pp. 389–399. Artech House, Norwood (2008). DOI 978-1-59693-236-4Google Scholar
  31. 31.
    Katz, B.: Nerve, Muscle and Synapse. McGraw-Hill, New York (1966)Google Scholar
  32. 32.
    Labate, D., Lim, W., Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. In: Unser, M. (ed.) Proceedings of Wavelets XI, SPIE Proceedings, vol. 5914, pp. 247–255 (2005)Google Scholar
  33. 33.
    Leviatan, D., Temlyakov, V.N.: Simultaneous approximation by greedy algorithms. Advances in Computational Mathematics 25(1), 73–90 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Li, Q., Zhou, X., Deng, Z., Baron, M., Teylan, M., Kim, Y., Wong, S.: A novel surface-based geometric approach for 3d dendritic spine detection from multi-photon excitation microscopy images. In: Proceedings of Biomedical Imaging: From Nano to Macro, 2009. ISBI ’09. IEEE International Symposium on, pp. 1255–1258 (2009). DOI 10.1109/ISBI.2009.5193290Google Scholar
  35. 35.
    Losavio, B., Reddy, G., Colbert, C., Kakadiaris, I., Saggau, P.: Combining optical imaging and computational modeling to analyze structure and function of living neurons. In: Proceedings of 28th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, vol. 1, pp. 668–670. New York, NY (2006). DOI 10.1109/IEMBS.2006.259552Google Scholar
  36. 36.
    Losavio, B., Liang, Y., Santamaria-Pang, A., Kakadiaris, I., Colbert, C., Saggau, P.: Live neuron morphology automatically reconstructed from multiphoton and confocal imaging data. J. Neurophysiol. 100, 2422–2429 (2008). DOI 10.1152/jn.90627.2008CrossRefGoogle Scholar
  37. 37.
    Lu, J.: Neuronal tracing for connectomic studies. Neuroinformatics 9(2–3), 159–166 (2011)CrossRefGoogle Scholar
  38. 38.
    Luisi, J., Narayanaswamy, A., Galbreath, Z., Roysam, B.: The farsight trace editor: an open source tool for 3-D inspection and efficient pattern analysis aided editing of automated neuronal reconstructions. Neuroinformatics 9(2–3), 305–315 (2011). DOI 10.1007/s12021-011-9115-0CrossRefGoogle Scholar
  39. 39.
    Meijering, E.: Neuron tracing in perspective. Cytometry Part A 77A(7), 693–704 (2010). DOI 10.1002/cyto.a.20895. URL
  40. 40.
    Narayanaswamy, A., Wang, Y., Roysam, B.: 3-D image pre-processing algorithms for improved automated tracing of neuronal arbors. Neuroinformatics 9(2–3), 219–231 (2011)CrossRefGoogle Scholar
  41. 41.
    Olenko, A., Pogány, T.: A precise upper bound for the error of interpolation of stochastic processes. Theory Probab. Math. Stat. 71, 151–163 (2005)CrossRefGoogle Scholar
  42. 42.
    Pelt, J.v., Schierwagen, A.: Morphological analysis and modeling of neuronal dendrites. Math. Biosci. 188(1–2), 147–155 (2004)Google Scholar
  43. 43.
    Peng, H., Ruan, Z., Atasoy, D., Sternson, S.: Automatic reconstruction of 3D neuron structures using a graph-augmented deformable model. Bioinformatics 26(12) (2010). URL
  44. 44.
    Plonka, G.: Approximation order provided by refinable function vectors. Constr. Approx. 13(2), 221–244 (1997)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Q., L., Z., D.: A surface-based 3d dendritic spine detection approach from confocal microscopy images. IEEE Trans. Image Process. (2011). To appearGoogle Scholar
  46. 46.
    Reddy, G.D., Saggau, P.: Development of a random-access multi-photon microscope for fast three-dimensional functional recording of neuronal activity (2007).Google Scholar
  47. 47.
    Rodriguez, A., Ehlenberger, D., Kelliher, K., Einstein, M., Henderson, S., Morrison, J., Hof, P., Wearne, S.: Automated reconstruction of three-dimensional neuronal morphology from laser scanning microscopy images. Methods 30(1), 94–105 (2003)CrossRefGoogle Scholar
  48. 48.
    Rodriguez, A., Ehlenberger, D., Hof, P., Wearne, S.: Rayburst sampling, an algorithm for automated three-dimensional shape analysis from laser-scanning microscopy images. Nat. Protoc. 1, 2156–2161 (2006)CrossRefGoogle Scholar
  49. 49.
    Rodriguez, A., Ehlenberger, D., Hof, P., Wearne, S.: Three-dimensional neuron tracing by voxel scooping. J. Neurosci. Methods 184(1), 169–175 (2009). DOI 10.1016/j. jneumeth.2009.07.021CrossRefGoogle Scholar
  50. 50.
    Romero, J., Alexander, S., Baid, S., Jain, S., Papadakis, M.: The geometry and the analytic properties of isotropic multiresolution analysis. Adv. Computat. Math. 31, 283–328 (2009). DOI 10.1007/s10444-008-9111-6MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Rusakov, D., Stewart, M.: Quantification of dendritic spine populations using image analysis and a tilting dissector. J. Neurosci. Methods 60, 11–21 (1995)CrossRefGoogle Scholar
  52. 52.
    Santamaria-Pang, A., Bildea, T., Colbert, C., Saggau, P., Kakadiaris, I.: Towards segmentation of irregular tubular structures in 3D confocal microscope images. In: Proceedings of MICCAI Workshop in Microscopic Image Analysis and Applications in Biology, pp. 78–85. Denmark, Copenhagen (2006). DOI Scholar
  53. 53.
    Santamaria-Pang, A., Colbert, C., Losavio, B., Saggau, P., Kakadiaris, I.: Automatic morphological reconstruction of neurons from optical images. In: Proceedings of International Workshop in Microscopic Image Analysis and Applications in Biology. Piscataway, NJ (2007)Google Scholar
  54. 54.
    Santamaria-Pang, A., Colbert, C., Saggau, P., Kakadiaris, I.: Automatic centerline extraction of irregular tubular structures using probability volumes from multiphoton imaging. In: Proceedings of Medical Image Computing and Computer-Assisted Intervention, pp. 486–494. Brisbane, Australia (2007). DOI 10. 1007/978-3-540-75759-759Google Scholar
  55. 55.
    Santamaria-Pang, A., Herrera, P.H., Papadakis, M., Prott, A., Shah, S., Kakadiaris, I.: Automatic morphological reconstruction of neurons from multiphoton and confocal microscopy images (2011). SubmittedGoogle Scholar
  56. 56.
    Scorcioni, R., Polavaram, S., Ascoli, G.A.: L-measure: a web-accessible tool for the analysis, comparison and search of digital reconstructions of neuronal morphologies. Nat. Protoc. 3(5), 866–876 (2008). DOI doi:10.1038/nprot.2008.51CrossRefGoogle Scholar
  57. 57.
    Senft, S.: A brief history of neuronal reconstruction. Neuroinformatics 9(2–3), 119–128 (2011)CrossRefGoogle Scholar
  58. 58.
    Shen, H., Sesack, S., Toda, S., Kalivas, P.: Automated quantification of dendritic spine density and spine head diameter in medium spiny neurons of the nucleus accumbens. Brain Struct. Funct. 213, 149–157 (2008). URL 10.1007/s00429-008-0184-2
  59. 59.
    Shen, H.W., Toda, S., Moussawi, K., Bouknight, A., Zahm, D.S., Kalivas, P.W.: Altered dendritic spine plasticity in cocaine-withdrawn rats. J. Neurosci. 29(9), 2876–2884 (2009). DOI 10.1523/JNEUROSCI.5638-08.2009. URL Google Scholar
  60. 60.
    Strohmer, T., Tanner, J.: Implementations of Shannon’s sampling theorem, a time-frequency approach. Sampl. Theory Signal Image Process. 4(1), 1–17 (2005)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Swanger, S., Yao, X., Gross, C., Bassell, G.: Automated 4D analysis of dendritic spine morphology: applications to stimulus-induced spine remodeling and pharmacological rescue in a disease model. Mol. Brain 4, 1–14 (2011). URL 10.1186/1756-6606-4-38
  62. 62.
    Temlyakov, V.N.: Greedy algorithms with regard to multivariate systems with special structure. Constr. Approx. 16(3), 399–425 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Temlyakov, V.N.: Weak greedy algorithms. Adv. Comput. Math. 12(2–3), 213–227 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Temlyakov, V.N.: Greedy algorithms in banach spaces. Adv. Comput. Math. 14(3), 277–292 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Tropp, J.: Algorithms for simultaneous sparse approximation. part ii: convex relaxation. Signal Processing, special issue “Sparse approximations in signal and image processing” 86, 589–602 (2006)Google Scholar
  66. 66.
    Tropp, J., Gilbert, A., Strauss, M.: Algorithms for simultaneous sparse approximation. part I: greedy pursuit. Signal Processing, special issue “Sparse approximations in signal and image processing” 86, 572–588 (2006)Google Scholar
  67. 67.
    Uehara, C., Colbert, C.M., Saggau, P., Kakadiaris, I.: Towards automatic reconstruction of dendrite morphology from live neurons. In: Proceedings of 26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 1798–1801. San Francisco, CA (2004). DOI 10.1109/IEMBS.2004.1403537Google Scholar
  68. 68.
    Vasilkoski, Z., Stepanyants, A.: Detection of the optimal neuron traces in confocal microscopy images. J. Neurosci. Methods 178(1), 197–204 (2009)CrossRefGoogle Scholar
  69. 69.
    Wang, Y., Narayanaswamy, A., Tsai, C.L., Roysam, B.: A broadly applicable 3-D neuron tracing method based on open-curve snake. Neuroinformatics 9(2–3), 193–217 (2011). DOI 10.1007/s12021-011-9110-5CrossRefGoogle Scholar
  70. 70.
    Wearne, S., Rodriguez, A., Ehlenberger, D., Rocher, A., Henderson, S., Hof, P.: New techniques for imaging, digitization and analysis of three-dimensional neural morphology on multiple scales. Neuroscience 136(3), 661–680 (2005). DOI 10.1016/ j.neuroscience.2005.05.053. Quantitative Neuroanatomy: from molecules to system. A special issue in honor of the late Professor Theodor W. BlackstadCrossRefGoogle Scholar
  71. 71.
    Weaver, C.M., Hof, P.R., Wearne, S.L., Lindquist, W.B.: Automated algorithms for multiscale morphometry of neuronal dendrites. Neural Comput. 16(7), 1353–1383 (2004). DOI 10.1162/089976604323057425. URL Google Scholar
  72. 72.
    Yuste, R.: Dendritic Spines. MIT, Cambridge (2009)Google Scholar
  73. 73.
    Yuste, R., Denk, W.: Dendritic spines as basic functional units of neuronal integration. Nature 375(6533), 682–684 (1995)CrossRefGoogle Scholar
  74. 74.
    Yuste, R., Bonhoeffer, T.: Genesis of dendritic spines: insights from ultrastructural and imaging studies. Nat. Rev. Neurosci. 5(1), 24–34 (2004). DOI 10.1038/nrn1300.CrossRefGoogle Scholar
  75. 75.
    Zhang, Y., Zhou, X., Witt, R., Sabatini, B., Adjeroh, D., Wong, S.: Dendritic spine detection using curvilinear structure detector and LDA classifier. Neuroimage 36(2), 346–360 (2007) URL Google Scholar
  76. 76.
    Zhang, Y., Chen, K., Baron, M., Teylan, M., Kim, Y., Song, Z., Greengard, P., Wong, S.: A neurocomputational method for fully automated 3d dendritic spine detection and segmentation of medium-sized spiny neurons. NeuroImage 50(4), 1472–1484 (2010). DOI DOI:10.1016/j.neuroimage.2010.01.048. URL Google Scholar

Copyright information

© Birkhäuser Boston 2013

Authors and Affiliations

  • Paul Hernandez—Herrera
    • 1
  • David Jiménez
    • 2
  • Ioannis A. Kakadiaris
    • 1
  • Andreas Koutsogiannis
    • 3
  • Demetrio Labate
    • 2
  • Fernanda Laezza
    • 4
  • Manos Papadakis
    • 2
  1. 1.Computational Biomedicine Lab, Department of Computer ScienceUniversity of HoustonHoustonUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA
  3. 3.Department of MathematicsUniversity of Athens, GreeceZografouGreece
  4. 4.Department of Pharmacology and ToxicologyUniversity of Texas Medical BranchGalvestonUSA

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