Inverse Problems In Medical Imaging

Transkript

1 Inverse Problems In Medical Imaging Simon Krug physics654: Seminar Medical Imaging University of Bonn

2 Inverse Problems Problem solving Inverse problems Noise Object and Image Radon transform Silent Sources Models for electrophysical activites Silent Sources Summary 2/17

3 Problem solving Forward problem (e.g.): is known; calculation of gives unique solution! Inverse problem is the opposite: The inverse calculation is NOT unique! Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Sarvas, J.: Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol., 1987, Vol. 32, No 1, Ch. 5 3/17

4 Inverse Problems 1) The problem is under-determined: Measurement of values Reconstruction of variables interpolation, discretization and quantization 2) The problem is ill-posed: Little errors (= noise) have huge effect on solution solutions are unstable under small perturbations a) Measured (e.g.) allows many solutions (huge errors) b) Silent sources: Outside measurement is unaware of inside activities Flower, M. A.: Webb s Physics of Medical Imaging, 2nd Edition, CRC Press, 2012, Ch Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch /17

5 Object and Image Object System Image exists! Linear and translation-invariant! Fundamental Theorem of System Theory Fourier transform Convolution Theorem Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch. 2.6 Flower, M. A.: Webb s Physics of Medical Imaging, 2nd Edition, CRC Press, 2012, Ch Lehmann, T. et al.: Bildverarbeitung für die Medizin, Springer-Verlag, 1997, Ch /17

6 Noise terms Realistic: Noise Uncorrelated noise Estimation of Example of a system in a CT Flower, M. A.: Webb s Physics of Medical Imaging, 2nd Edition, CRC Press, 2012, Ch /17

7 Radon transform Idea: Express through infinitely many line integrals But: Infinitely many integrals are identical! Ordered procedure Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Radon, J.: Über die Bestimmung von Funktionen durch ihre Integeralwerte längs gewisser Mannigfaltigkeiten, Akad. Wiss., Vol 69, 1917, p. 1 Lehmann, T. et al.: Bildverarbeitung für die Medizin, Springer-Verlag, 1997, Ch /17

8 Radon transform New procedure: 1) Choose fixed 2) Take line integral of all Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Radon, J.: Über die Bestimmung [...] Mannigfaltigkeiten, Akad. Wiss., Vol 69, 1917, p. 1 Lehmann, T. et al.: Bildverarbeitung für die Medizin, Springer-Verlag, 1997, Ch /17

9 Radon transform Unit vector in direction Direction of line integration Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Radon, J.: Über die Bestimmung [...] Mannigfaltigkeiten, Akad. Wiss., Vol 69, 1917, p. 1 Lehmann, T. et al.: Bildverarbeitung für die Medizin, Springer-Verlag, 1997, Ch /17

10 Radon transform Now choose: is fixed. Now do this projection for every Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Radon, J.: Über die Bestimmung [...] Mannigfaltigkeiten, Akad. Wiss., Vol 69, 1917, p. 1 Lehmann, T. et al.: Bildverarbeitung für die Medizin, Springer-Verlag, 1997, Ch /17

11 Radon transform 11/17

12 Radon transform 1) Example without noise 2) Four limited objects with sharp edges (hard spheres) 3) Red areas are distinct greater than 0 is possible (little detour, is ill-posed and non-unique) BUT: Assumptions are unrealistic! realistic is much more problematic! Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Radon, J.: Über die Bestimmung [...] Mannigfaltigkeiten, Akad. Wiss., Vol 69, 1917, p. 1 Lehmann, T. et al.: Bildverarbeitung für die Medizin, Springer-Verlag, 1997, Ch /17

13 Models for electrophysical activities Assume static : Introduce primary current : with conductivity (unknown) (Poisson s equation) Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Sarvas, J.: Basic mathematical [...] problem, Phys. Med. Biol., 1987, Vol. 32, No 1, Ch Hämäläinen, M. et al.: Magnetoencephalography, Reviews of Modern Physics, Vol. 65, No. 2, 1993, Ch. III.A+B+C 13/17

14 Models for electrophysical activities Solution is already known: Discretization of volume: Compare with pot. for dipoles: is measurable! ( s, respectively) is a superposion of dipoles Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Sarvas, J.: Basic mathematical [...] problem, Phys. Med. Biol., 1987, Vol. 32, No 1, Ch Hämäläinen, M. et al.: Magnetoencephalography, Reviews of Modern Physics, Vol. 65, No. 2, 1993, Ch. III.A+B+C 14/17

15 Silent Sources = dipoles inside the body which have no or little contribution to outside the body Silent Sources can be added to accessibly measurable results s Unfortunately only results available! without changing the Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch Sarvas, J.: Basic mathematical [...] problem, Phys. Med. Biol., 1987, Vol. 32, No 1, Ch Hämäläinen, M. et al.: Magnetoencephalography, Reviews of Modern Physics, Vol. 65, No. 2, 1993, Ch. III.A+B+C 15/17

16 Silent Sources Be the causes explained by solutions and silent sources with solutions Then the sum of these causes are also explained by Dössel, O.: Bildgebende Verfahren in der Medizin, Springer-Verlag, 2000, Ch /17

17 Summary 1) Inverse problems have non-unique solutions problem is under-determined and ill-posed 2) Uncorrelated noise perturbs the measurement small perturbation yields huge deviation 3) Silent Sources have no contribution to Correspondent dipoles cannot be sensed 17/17