Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2016 | OriginalPaper | Buchkapitel

1. Ill-Posed Problems in Imaging and Computer Vision

verfasst von : Mongi A. Abidi, Andrei V. Gribok, Joonki Paik

Erschienen in: Optimization Techniques in Computer Vision

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In many image processing problems, we need to estimate the original complete data sets, generally from incomplete and, most often, from degraded observations. One simplified example is to estimate the original pixel intensity value, which has been attenuated in the imaging system, without any correlation with neighboring pixels. If we know the nonzero attenuation factor for the imaging system, we can easily estimate the original value by multiplying by this attenuation factor. Figure 1.1 shows the corresponding attenuation and the restoration processes.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Selection of the Regularization Parameter
Fußnoten
1
If u is defined on the discrete space, each element of f in (1.3) represents a weighted sum, and we call it the linear or the first order operation. On the other hand, if u is defined on the real space, the corresponding operation is called the first class instead of the first order.
 
Literatur
[courant62]
R. Courant, D. Hilbert, Methods of Mathematical Physics, II, Partial Differential Equations (Interscience, New York, 1962)MATH
[golub96]
G.H. Golub, C.F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1996)MATH
[jain89]
A.K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Upper Saddle River, 1989)MATH
[kats91]
A.K. Katsaggelos (ed.), Digital Image Restoration (Springer, Heidelberg, 1991)
[nashed81]
[tikhonov77]
A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, Washington, DC, 1977)MATH
[bardinet95]
E. Bardinet, L.D. Cohen, N. Ayache, A parametric deformable model to fit unstructured 3D data. Project Report, Programme 4-Robotique, image et vision, Utite de rescherche INRIA Sophia-Antipolis (1995)
[sederberg86]
T.W. Sederberg, S.R. Parry, Free-form deformation of solid geometric models, in Proc. 1986 SIGGRAPH, vol. 20, pp. 151–160, 1986
[solina90]
F. Solina, R. Bajcsy, Recovery of compact volumetric models for shape representation of single-part objects. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-12(2), 131–147 (1990)
[tekalp95]
A.M. Tekalp, Digital Video Processing (Prentice-Hall, Upper Saddle River, 1995)
[ber88]
M. Bertero, T.A. Poggio, V.Torre, Ill-posed problems in early vision. Proc. IEEE 76(8), 869–889 (1988)
[eng00]
W.H. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic Publishers, Dordrecht, 2000)MATH
[gro93]
C.W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Bruanschweig, Wiesbaden, 1993)CrossRefMATH
[han98]
Per Christian Hansen, Rank Deficient and Discrete Ill-Posed Problem (SIAM, Philadelphia, 1998)
[ber98]
M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, Bristol, 1998)CrossRefMATH
[vog02]
C. Vogel, Computational Methods for Inverse Problems (SIAM, Philadelphia, 2002)
[kir96]
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer, New York, 1996)CrossRefMATH
[sab87]
P.C. Sabatier (ed.), Inverse Problems: An Interdisciplinary Study (Academic, London, 1987)
[two96]
S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover Publications, New York, 1996)
[tikh98]
A.N. Tikhonov, A.S. Leonov, A.G. Yagola, Nonlinear Ill-Posed Problems, vols. 1, 2 (Chapman and Hall, London, 1998)
[mor84]
V.A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer, New York, 1984)CrossRef
[lav61]
M.M. Lavrent’ev, Ill-Posed Problems in Mathematical Physics (Nauka, Novosibirsk (in Russian), 1961). English translation in Springer, Berlin-Heidelberg
[win91]
G. Milton Wing, A Primer on Integeral Equations of the First Kind (SIAM, Philadelphia, 1991)CrossRefMATH
[bau87]
[hh93]
Martin Hanke, Per Christian Hansen, Regularization methods for large scale problems. Surv. Math. Ind. 3, 253–315 (1993)
[eng93]
W.H. Engl, Regularization methods for the stable solution of inverse problems. Surv. Math. Ind. 3, 71–143 (1993)MathSciNetMATH
[col00]
D. Colton, H.W. Engl, A.K. Louis, J. McLaughlin, W. Rundell, Surveys on Solution Methods for Inverse Problems (Springer, Wien/New York Herausgeber, 2000)CrossRefMATH
[tar87]
A. Tarantola, Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation (Elsevier, New York, 1987)MATH
[hen91]
E. Hensel, Inverse Theory and Applications for Engineers (Prentice-Hall, Englewood Cliffs, 1991)
[tru97]
D.M. Trujillo, H.R. Busby, Practical Inverse Analysis in Engineering (CRC Press, Boca Raton, 1997)MATH
[tikh95]
A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed-Problems (Kluwer Academic Press, Dordrecht, 1995)CrossRefMATH
[ber85]
M. Bertero, C. De Mol, E.R. Pikes, Linear inverse problems with discrete data. I: General formulation and singular system analysis. Inverse Prob. 1, 301–330 (1985)
[ber88a]
M. Bertero, C. De Mol, E.R. Pikes, Linear inverse problems with discrete data. II: Stability and Regularization. Inverse Prob. 4, 573–594 (1988)
[par77]
R.L. Parker, Understanding inverse theory. Annu. Rev. Earth Planet. Sci. 5, 35–64 (1977)CrossRef
[iva62a]
V.K. Ivanov, Integral equations of the first kind and the approximate solution of an inverse potential problem. Dokl. Acad. Nauk. SSSR 142, 997–1000 (in Russian). English translation in Soviet. Math. Dokl. (1962)
[iva62b]
V.K. Ivanov, On linear ill-posed problems. Dokl. Acad. Nauk. SSSR 145, 270–272 (in Russian). English translation in Soviet. Math. Dokl. (1962)
[ph62]
D.L. Phillips, A technique for the numerical solution of certain integral equations of the first kind. JACM 9, 84–97 (1962)MathSciNetCrossRefMATH
[two63]
S. Twomey, On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. JACM 10, 97–101 (1963)CrossRefMATH
[tikh63]
A.N. Tikhonov, Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl. 4, 1035–1038 (1963)MATH
[mil70]
[ber88b]
M. Bertero, P. Brianzi, E.R. Pike, L. Rebolia, Linear regularizing algorithms for positive solutions of linear inverse problems. Proc. R. Soc. Lond. A 415, 257–275 (1988)MathSciNetCrossRefMATH
[ber81a]
M. Bertero, V. Dovi, Regularized and positive-constrained inverse methods in the problem of object restoration. Opt. Acta 28, 1635–1649 (1981)
[ber81b]
M. Bertero, C. De Mol, Ill-posedness, regularization and number of degrees of freedom. Atti della Fondazione Giorgio Ronchi 36, 619–632 (1981)
[ber92]
M. Bertero, E.R. Pike (eds.), Inverse Problems in Scattering and Imaging (Adam Hilger, Bristol, 1992)
[ber89]
M. Bertero, Linear inverse and ill-posed problems, in Advances in Electronics and Electron Physics, ed. by P.W. Hawkes, vol 75 (Academic Press, New York, 1989), pp. 1–120
[ber86]
M. Bertero, Regularization methods for linear inverse problems, in Inverse Problems, ed. by G. Talenti. Lecture Notes in Mathematics, vol 1225 (Springer, Berlin, 1986), pp. 52–112CrossRef
[ber80]
M. Bertero, C. De Mol, G.A. Viano, The stability of inverse problems, in Inverse Scattering Problems in Optics, ed. by H.P. Baltes. Topics in Current Physics, vol 20 (Springer, Berlin, 1980), pp. 161–214CrossRef
[rog96]
M.C. Roggeman, B.M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, 1996)
[good96]
J.W. Goodman, Introduction to Fourier Optics, 2nd edn. (McGraw-Hill, New York, 1996)
[har98]
J.W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, Oxford, UK, 1998)
[louis92]
[arr97]
S.R. Arridge, J.C. Hebden, Optical imaging in medicine: II. Modelling and reconstruction. Phys. Med. Biol. 42, 841–853 (1997)CrossRef
[arr99]
[bar81]
H.H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981)
[her79]
[her80]
G.T. Herman, The Fundamentals of Computerized Tomography. Image Reconstruction from Projections (Academic, New York, 1980)MATH
[kak87]
A.C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987)MATH
[mac83]
A. Macovski, Medical Imaging Systems. Prentice-Hall Information and System Sciences Series (Prentice-Hall, Upper Saddle River, 1983)
[nat96]
National Research Council Institute of Medicine, National Academy Press, Mathematics and Physics of Emerging Biomedical Imaging (National Research Council Institute of Medicine, National Academy Press, Washington, DC, 1996)
[nat86]
F. Natterer, The Mathematics of Computerized Tomography (B.G. Teubner and Wiley, Chichester, 1986)MATH
[cra86]
I.J.D. Craig, J.C. Brown, Inverse Problems in Astronomy: A Guide to Inversion Strategies for Remotely Sensed Data (Adam Hilger, Boston, 1986)MATH
[cast96]
K.R. Castleman, Digital Image Processing (Prentice Hall, Toronto, 1996)
[myr92]
J. Myrheim, H. Rue, New algorithms for maximum entropy image restoration. CVGIP: Graph. Models Image Process. 54(3), 223–238 (1992)
[bur83]
S.F. Burch, S.F. Gull, J. Skilling, Image restoration by a powerful maximum entropy method. Comput. Vis. Graph. Image Process. 23(2), 113–128 (1983)CrossRef
[gon92]
R.C. Gonzalez, R.E. Woods, Digital Image Processing (Addison-Wesley, New York, 1992)
[arc95]
G. Archer, D.M. Titterington, On some bayesian/regularization methods for image restoration. IEEE Trans. Image Process. 4(7), 989–995 (1995). AV94
[acar94]
[buck91]
B. Buck, V.A. Macaulay (eds.), Maximum Entropy in Action (Clarendon Press, Oxford, 1991)
[craig86]
I.J.D. Craig, J.C. Brown, Inverse Problems in Astronomy (Adam Hilger, Bristol, 1986). Chapter 6.2, CB90
[chou90]
P.B. Chou, C.M. Brown, The theory and practice of Bayesian image modeling. Int. J. Comput. Vis. 4, 185–210 (1990)CrossRef
[dem89]
G. Demoment, Image reconstruction and restoration: overview of common estimation structures and problems. IEEE Trans. Acoust. Speech Signal Process. 37(12), 2024–2036 (1989)CrossRef
[hum87]
R.A. Hummel, B. Kimia, S. Zucker, Deblurring gaussian blur. Comput. Vis. Graph. Image Process. 38, 66–80 (1987)CrossRefMATH
[mar87]
J.L. Marroquin, S. Mitter, T. Poggio, Probabilistic solution of ill-posed problems in computer vision. J. Am. Stat. Assoc. 82, 76–89 (1987)CrossRefMATH
[pog85a]
T. Poggio, V. Torre, C. Koch, Computational vision and regularization theory. Nature 317, 314–319 (1985)CrossRef
[pog85b]
T. Poggio, C. Koch, Ill-posed problems in early vision: from computational theory to analogue networks. Proc. R. Soc. Lond. B. Biol. Sci. 226, 303–323 (1985)CrossRefMATH
[leary94]
D.P. O’Leary, Regularization of ill-posed problems in image restoration, in Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, ed. by J.G. Lewis. (SIAM Press, Philadelphia, 1994), pp. 102–105
[terzopoulos86]
D. Terzopoulos, Regularization of inverse visual problems involving discontinuities. IEEE Trans Pattern Anal. Mach. Intell. 8(4), 413–424 (1986)CrossRef
[blo97]
Peter Blomgren, Tony F. Chan, Pep Mulet, C.K. Wong, Total variation image restoration: numerical methods and extensions, in IEEE Image Proc. Proceedings of the 1997 I.E. International Conference on Image Processing, October 1997
[dob96]
D. Dobson, O. Scherzer, Analysis of regularized total variation penalty methods for denoising. Inverse Prob. 12, 601–617 (1996)MathSciNetCrossRefMATH
[rud92]
[ban97]
M.R. Banham, A.K. Katsaggelos, Digital image restoration. IEEE Signal Process. Mag. 14, 24–41 (1997)
[kun96]
D. Kundur, D. Hatzinakos, Blind image deconvolution. IEEE Signal Process. Mag. 13(3), 43–64 (1996)CrossRef
[kat91a]
A.K. Katsaggelos, Digital Image Restoration (Springer, New York, 1991)CrossRef
[han00]
P.C. Hansen, Numerical Aspects of Deconvolutions, Lectures Notes (Technical University of Denmark, Department of Mathematical Modelling, Denmark, 2000)
[bie84a]
J. Biemond, A.K. Katsaggelos, A new iterative restoration scheme for noisy blurred images, in Proc. Conf. on Math. Methods in Signal Processing, pp. 74–76, Aachen, W. Germany, September 1984
[bie84b]
J. Biemond, A.K. Katsaggelos, Iterative restoration of noisy blurred images, in Proc. 5th Inf. Theory Symposium in the Benelux, pp. 11–20, Aalten, The Netherlands, May 1984
[kat86]
A.K. Katsaggelos, A general formulation of adaptive iterative image restoration algorithms, in Proc. 1986 Conf. Inf. Sciences and Systems, pp. 42–47, Princeton, NJ, March 1986
[kat84]
A.K. Katsaggelos, J. Biemond, R.M. Mersereau, R.W. Schafer, An iterative method for restoring noisy blurred images, in Proc. 1984 Int. Conf. Acoust., Speech, Signal Processing, pp. 37.2.1–37.2.4, San Diego, CA, March 1984
[and77]
H.C. Andrews, B.R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, 1977)
[gal92]
N.P. Galatsanos, A.K. Katsaggelos, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process. 1, 322–336 (1992)CrossRef
[and76]
G.L. Anderlson, A.N. Natravali, Image restoration based on subjective criterion. IEEE Trans. Syst., Man, Cyber. SMC­6, 845–853 (1976)
[lim90]
J.S. Lim, Two­Dimensional Signal and Image Processing (Prentice­Hall, Englewood Cliffs, 1990)
[kar90]
N.B. Karayiannis, A.N. Venetsanopoulos, Regularization theory in image restoration—the stabilizing functional approach. IEEE Trans. Acoust. Speech Signal Process. 38(7), 1155 (1990)
[kang92]
M.G. Kang, A.K. Katsaggelos, Simultaneous iterative image restoration and evaluation of the regularization parameter. IEEE Trans. Signal Process. 40(9), 2329–2334 (1992)CrossRef
[kat91b]
A.K. Katsaggelos, J. Biemond, R.W. Schafer, R.M. Mersereau, A regularized iterative image restoration algorithm. IEEE Trans. Signal Process. 39(4), 914–929 (1991)CrossRef
[reeves90]
S.J. Reeves, R.M. Mersereau, Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration. Opt. Eng. 29, 446–454 (1990)CrossRef
[reeves92]
S.J. Reeves, R.M. Mersereau, Blur identification by the method of generalized cross-validation. IEEE Trans. Image Process. 1, 301–311 (1992)CrossRef
[ber79]
M. Bertero, C. De Mol, G.A. Viano, On the problems of object restoration and image extrapolation in optics. J. Math. Phys. 20, 509–521 (1979)MathSciNetCrossRef
[ber78a]
M. Bertero, G.A. Viano, On probabilistic methods for the solution of improperly posed problems. Bollettino U.M.I., 15-B, 453–508 (1978)
[ber78b]
M. Bertero, C. De Mol, G.A. Viano, Restoration of optical objects using regularization. Opt. Lett. 3, 51–53 (1978)CrossRef
[ber78c]
M. Bertero, C. De Mol, G.A. Viano, On the regularization of linear inverse problems in Fourier optics, in Applied Inverse Problems, ed. by P.C. Sabatier. Lecture Notes in Physics, vol 85 (Sprinter, Berlin, 1978), pp. 180–199CrossRef
Metadaten
Titel
Ill-Posed Problems in Imaging and Computer Vision
verfasst von
Mongi A. Abidi
Andrei V. Gribok
Joonki Paik
Copyright-Jahr
2016
Buch
Optimization Techniques in Computer Vision

Print ISBN: 978-3-319-46363-6

Electronic ISBN: 978-3-319-46364-3

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-46364-3_1