Skip to main content
SpringerLink
Log in
Book cover

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging pp 1–28Cite as

Filter Design for Image Decomposition and Applications to Forensics

  • Living reference work entry
  • First Online:

Abstract

Employing image filters in image processing applications, essentially matrix convolution operators, has been an active field of research since a long time, and it is so very much still today. In the first part, we give a brief overview of imaging methods with emphasis on applications in fingerprint recognition and shoeprint forensics. In the second part, we propose a generalized discrete scheme for image decomposition that encompasses many of the existing methods. Due to its generality, it has the potential to learn, for specific use cases, a highly flexible set of imaging filters that are related to one another by rather general conditions.

This is a preview of subscription content, log in via an institution.

References

  • Nist fingerprint quality (NFIQ) (2015) https://www.nist.gov/services-resources/software/nist-biometric-image-software-nbis. Accessed: 2017-12-04

  • Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal. 29(3), 845–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: Minimizing total variation flow. Differ. Integral Equ. 14(3), 321–360 (2001)

    MathSciNet  MATH  Google Scholar 

  • Arridge, S., Maass, P., Öktem, O., Schönlieb, C.-B.: Solving inverse problems using data-driven models. Acta Numerica 28, 1–174 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Aubert, G., Kornprobst, P.: Mathematical problems in image processing, volume 147 of Applied Mathematical Sciences. Springer, New York, 2nd edn. Partial differential equations and the calculus of variations, With a foreword by Olivier Faugeras (2006)

    Google Scholar 

  • Aujol, J.-F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vision 22(1), 71–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)

    Article  MATH  Google Scholar 

  • Aujol, J.-F., Gilboa, G.: Constrained and SNR-based solutions for TV-Hilbert space image denoising. J. Math. Imaging Vision 26(1–2), 217–237 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition—modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006)

    Article  MATH  Google Scholar 

  • Bartůněk, J., Nilsson, M., Sällberg, B., Claesson, I.: Adaptive fingerprint image enhancement with emphasis on preprocessing of data. IEEE Trans. Image Process. 22(2), 644–656 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York. With a foreword by Hédy Attouch (2011)

    Google Scholar 

  • Bazen, A., Gerez, S.: Systematic methods for the computation of the directional fields and singular points of fingerprints. IEEE Trans. Pattern Anal. Mach. Intell. 24(7), 905–919 (2002)

    Article  Google Scholar 

  • Bengio, Y.: Learning deep architectures for AI. Found. Trends Mach. Learn. 2(1), 1–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Computer Science and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York/London (1982)

    Google Scholar 

  • Bigun, J.: Vision with Direction. Springer, Berlin/Germany (2006)

    MATH  Google Scholar 

  • Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  • Bredies, K., Dong, Y., Hintermüller, M.: Spatially dependent regularization parameter selection in total generalized variation models for image restoration. Int. J. Comput. Math. 90(1):109–123 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imag. Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Breiman, L.: Random forests. Mach. Learn. 45, 5–32 (2001)

    Article  MATH  Google Scholar 

  • Buades, A., Le, T., Morel, J.-M., Vese, L.: Fast cartoon + texture image filters. IEEE Trans. Image Process. 19(8), 1978–1986 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Burger, M., Gilboa, G., Moeller, M., Eckardt, L., Cremers, D.: Spectral decompositions using one-homogeneous functionals. SIAM J. Imag. Sci. 9(3), 1374–1408 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, J.-F., Dong, B., Osher, S., Shen, Z.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Calatroni, L., Cao, C., De Los Reyes, J.C., Schönlieb, C.-B., Valkonen, T.: Bilevel approaches for learning of variational imaging models. Variational Meth Imaging Geometric Control 18(252), 2 (2017)

    MathSciNet  MATH  Google Scholar 

  • Candès, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. Multiscale Model. Simul. 5(3), 861–899 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8(6), 679–698 (1986)

    Article  Google Scholar 

  • Caselles, V., Chambolle, A., Novaga, M.: Total variation in imaging. In Handbook of Mathematical Methods in Imaging. Vol. 1, 2, 3. Springer, New York (2015), pp. 1455–1499

    Google Scholar 

  • Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1–2), 89–97 (2004)

    MathSciNet  MATH  Google Scholar 

  • Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numerische Mathematik 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numerica 25, 161–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y., Pock, T.: Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration. IEEE Trans. Pattern Anal. Mach. Intell. 39(6), 1256–1272 (2017)

    Article  Google Scholar 

  • Chikkerur, S., Cartwright, A., Govindaraju, V.: Fingerprint image enhancement using STFT analysis. Pattern Recogn. 40(1), 198–211 (2007)

    Article  MATH  Google Scholar 

  • Chui, C.K.: An introduction to Wavelets, Volume 1 of Wavelet Analysis and its Applications. Academic Press, Inc., Boston (1992)

    Google Scholar 

  • Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation. I. Fast and exact optimization. J. Math. Imaging Vision 26(3), 261–276 (2006a)

    Article  Google Scholar 

  • Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation. II. Levelable functions, convex priors and non-convex cases. J. Math. Imaging Vision 26(3), 277–291 (2006b)

    Google Scholar 

  • Daubechies, I.: Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)

    Google Scholar 

  • De los Reyes, J.C., Schönlieb, C.-B.: Image denoising: learning the noise model via nonsmooth PDE-constrained optimization. Inverse Probl. Imaging 7(4), 1183–1214 (2013)

    Google Scholar 

  • Donoho, D.L., Johnstone, J.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3, Ser. A), 293–318 (1992)

    Google Scholar 

  • Frick, K., Marnitz, P., Munk, A., et al. Statistical multiresolution dantzig estimation in imaging: fundamental concepts and algorithmic framework. Electron. J. Stat. 6, 231–268 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Garnett, J.B., Le, T.M., Meyer, Y., Vese, L.A.: Image decompositions using bounded variation and generalized homogeneous Besov spaces. Appl. Comput. Harmon. Anal. 23(1), 25–56 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Garris, M.D., McCabe, R.M.: Nist special database 27: Fingerprint minutiae from latent and matching tenprint images. Technical Report 6534, National Institute of Standards and Technology, Gaithersburg (2000)

    Google Scholar 

  • Gilboa, G.: A total variation spectral framework for scale and texture analysis. SIAM J. Imag. Sci. 7(4), 1937–1961 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein, T., O’Donoghue, B., Setzer, S., Baraniuk, R.: Fast alternating direction optimization methods. SIAM J. Imag. Sci. 7(3), 1588–1623 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imag. Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Gottschlich, C.: Curved-region-based ridge frequency estimation and curved Gabor filters for fingerprint image enhancement. IEEE Trans. Image Process. 21(4), 2220–2227 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Gottschlich, C., Huckemann, S.: Separating the real from the synthetic: Minutiae histograms as fingerprints of fingerprints. IET Biom. 3(4), 291–301 (2014)

    Article  Google Scholar 

  • Gottschlich, C., Mikaelyan, A., Olsen, M., Bigun, J., Busch, C.: Improving fingerprint alteration detection. In: Proceedings of 9th International Symposium on Image and Signal Processing and Analysis (ISPA 2015), pp. 83–86, Zagreb (2015)

    Google Scholar 

  • Gottschlich, C., Schönlieb, C.-B.: Oriented diffusion filtering for enhancing low-quality fingerprint images. IET Biom. 1(2), 105–113 (2012)

    Article  Google Scholar 

  • Gragnaniello, D., Poggi, G., Sansone, C., Verdoliva, L.: Wavelet-Markov local descriptor for detecting fake fingerprints. Electron. Lett. 50(6), 439–441 (2014)

    Article  Google Scholar 

  • Grossmann, T.G., Korolev, Y., Gilboa, G., Schönlieb, C.-B.: Deeply learned spectral total variation decomposition. arXiv preprint arXiv:2006.10004 (2020)

    Google Scholar 

  • Hait, E., Gilboa, G.: Spectral total-variation local scale signatures for image manipulation and fusion. IEEE Trans. Image Process. 28(2), 880–895 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  • Hopper, T., Brislawn, C., Bradley, J.: WSQ gray-scale fingerprint image compression specification. Technical report, Federal Bureau of Investigation (1993)

    Google Scholar 

  • Horesh, D., Gilboa, G.: Separation surfaces in the spectral tv domain for texture decomposition. IEEE Trans. Image Process. 25(9), 4260–4270 (2016)

    MathSciNet  MATH  Google Scholar 

  • Kennedy, J.R.E., Shi, Y.: Swarm Intelligence. Academic, San Diego (2001)

    Google Scholar 

  • Le, T.M., Vese, L.A.: Image decomposition using total variation and div(BMO). Multiscale Model. Simul. 4(2), 390–423 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (2015)

    Article  Google Scholar 

  • Ma, J., Plonka, G.: The curvelet transform. IEEE Signal Process. Mag. 27(2), 118–133 (2010)

    Article  Google Scholar 

  • Mallat, S.: A Wavelet Tour of Signal Processing. Academic, San Diego (2008)

    MATH  Google Scholar 

  • Maltoni, D., Maio, D., Jain, A.K., Prabhakar, S.: Handbook of Fingerprint Recognition. Springer, London (2009)

    Book  MATH  Google Scholar 

  • Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. American Mathematical Society, Boston (2001)

    Book  MATH  Google Scholar 

  • Moeller, M., Diebold, J., Gilboa, G., Cremers, D.: Learning nonlinear spectral filters for color image reconstruction. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 289–297 (2015)

    Google Scholar 

  • Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Neal, R.M.: Bayesian Learning for Neural Networks, Vol. 118. Springer Science & Business Media (2012)

    Google Scholar 

  • Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H−1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Papafitsoros, K., Bredies, K.: A study of the one dimensional total generalised variation regularisation problem. Inverse Prob. Imaging 9(2), 511 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Perona, P.: Orientation diffusions. IEEE Trans. Image Process. 7(3), 457–467 (1998)

    Article  Google Scholar 

  • Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  • Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Optimization (Symposium, University of Keele, Keele, 1968), pp. 283–298. Academic, London (1969)

    Google Scholar 

  • Richter, R.: Cartoon-Residual Image Decompositions with Application in Fingerprint Recognition. Ph.D. thesis, Georg-August-University of Goettingen (2019)

    Google Scholar 

  • Richter, R., Gottschlich, C., Mentch, L., Thai, D., Huckemann, S.: Smudge noise for quality estimation of fingerprints and its validation. IEEE Trans. Inf. Forensics Secur. 14(8), 1963–1974 (2019)

    Article  Google Scholar 

  • Richter, R., Thai, D.H., Huckemann, S.: Generalized intersection algorithms with fixpoints for image decomposition learning. arXiv preprint arXiv:2010.08661 (2020)

    Google Scholar 

  • Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer (2009)

    Google Scholar 

  • Schmidt, M.F., Benning, M., Schönlieb, C.-B.: Inverse scale space decomposition. Inverse Prob. 34(4), 1–34 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Schoelkopf, B., Smola, A.: Learning with Kernels. MIT Press, Cambridge (2002)

    Google Scholar 

  • Shen, J.: Piecewise H−1 + H0 + H1 images and the Mumford-Shah-Sobolev model for segmented image decomposition. AMRX Appl. Math. Res. Express (4), 143–167 (2005)

    Article  MATH  Google Scholar 

  • Steidl, G., Weickert, J., Brox, T., Mrázek, P., Welk, M.: On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides. SIAM J. Numer. Anal. 42(2), 686–713 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Prob. 19(6), S165–S187 (2003). Special section on imaging

    Google Scholar 

  • Thai, D., Gottschlich, C.: Directional global three-part image decomposition. EURASIP J. Image Video Process. 2016(12), 1–20 (2016a)

    Google Scholar 

  • Thai, D., Gottschlich, C.: Global variational method for fingerprint segmentation by three-part decomposition. IET Biom. 5(2), 120–130 (2016b)

    Article  Google Scholar 

  • Thai, D., Huckemann, S., Gottschlich, C.: Filter design and performance evaluation for fingerprint image segmentation. PLoS ONE 11(5), e0154160 (2016)

    Article  Google Scholar 

  • Turroni, F., Maltoni, D., Cappelli, R., Maio, D.: Improving fingerprint orientation extraction. IEEE Trans. Inf. Forensics Secur. 6(3), 1002–1013 (2011)

    Article  Google Scholar 

  • Unser, M., Ville, D.V.D.: Wavelet steerability and the higher-order Riesz transform. IEEE Trans. Image Process. 19(3), 636–652 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Van De Ville, D., Blu, T., Unser, M.: Isotropic polyharmonic B-splines: scaling functions and wavelets. IEEE Trans. Image Process. 14(11), 1798–1813 (2005)

    Article  MathSciNet  Google Scholar 

  • Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillatory patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  • Weickert, J.: Coherence-enhancing diffusion filtering. Int. J. Comput. Vis. 31(2/3), 111–127 (1999)

    Article  Google Scholar 

  • Wiesner, S., Kaplan-Damary, N., Eltzner, B., Huckemann, S.F.: Shoe prints: The path from practice to science. In: Banks, D., Kafadar, K., Kaye, D. (eds.) Handbook of Forensic Statistics, pp. 391–410. Springer (2020a)

    Google Scholar 

  • Wiesner, S., Shor, Y., Tsach, T., Kaplan-Damary, N., Yekutieli, Y.: Dataset of digitized racs and their rarity score analysis for strengthening shoeprint evidence. J. Forensic Sci. 65(3), 762–774 (2020b)

    Article  Google Scholar 

  • Wu, C., Tai, X.-C.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imag. Sci. 3(3), 300–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, Y., Sun, J., Li, H., Xu, Z.: Deep ADMM-net for compressive sensing MRI. In: 30th Conference on Neutral Information Processing Systems (NIPS 2016), pp. 10–18 (2016)

    Google Scholar 

  • Yao, Z., Le Bars, J.-M., Charrier, C., Rosenberger, C.: A literature review of fingerprint quality assessment and its evaluation. IET J. Biom. 5(3), 243–251 (2016)

    Article  Google Scholar 

  • Zeune, L., van Dalum, G., Terstappen, L.W., van Gils, S.A., Brune, C.: Multiscale segmentation via bregman distances and nonlinear spectral analysis. SIAM J. Imaging Sci. 10(1), 111–146 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonimous referee for the valuable comments and the first and last author gratefully acknowledge funding by the DFG within the RTG 2088.

Author information

Authors and Affiliations

  1. Felix-Bernstein-Institute for Mathematical Statistics in the Biosciences, University of Göttingen, Göttingen, Germany

    Robin Richter & Stephan F. Huckemann

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, USA

    Duy Hoang Thai

  3. Institute for Mathematical Stochastics, University of Göttingen, Göttingen, Germany

    Carsten Gottschlich

Authors
  1. Robin Richter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duy Hoang Thai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carsten Gottschlich
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stephan F. Huckemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Robin Richter , Duy Hoang Thai , Carsten Gottschlich or Stephan F. Huckemann .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Dept Appl Math & Theoretical Phys, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics & Stat, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younces

Section Editor information

  1. No affiliation provided

    Ke Chen Ph.D

  2. No affiliation provided

    Xue-Cheng Tai Ph.D

  3. No affiliation provided

    Carola-Bibiane Schönlieb Ph.D

  4. No affiliation provided

    Laurent Younes

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Richter, R., Thai, D.H., Gottschlich, C., Huckemann, S.F. (2021). Filter Design for Image Decomposition and Applications to Forensics. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_92-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-03009-4_92-1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-03009-4

  • Online ISBN: 978-3-030-03009-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation