By Luis L. Bonilla
Nowadays we face a number of and critical imaging difficulties: nondestructive trying out of fabrics, tracking of business techniques, enhancement of oil construction by means of effective reservoir characterization, rising advancements in noninvasive imaging concepts for clinical reasons - automated tomography (CT), magnetic resonance imaging (MRI), positron emission tomography (PET), X-ray and ultrasound tomography, and so on. within the CIME summer season college on Imaging (Martina Franca, Italy 2002), major specialists in mathematical ideas and purposes offered huge and necessary introductions for non-experts and practitioners alike to many elements of this fascinating box. the quantity includes a part of the above lectures accomplished and up to date by means of extra contributions on different similar topics.
Read or Download Inverse Problems and Imaging: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 15-21, 2002 (Lecture Notes in Mathematics) PDF
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Additional info for Inverse Problems and Imaging: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 15-21, 2002 (Lecture Notes in Mathematics)
Imag. 13, 186-195. 5. E. Danielsson, P. Edholm, J. Eriksson, M. Magnusson Seger, and H. Turbell (1999): The original π-method for helical cone-beam CT, Proc. Int. Meeting on Fully 3D-reconstruction, Egmond aan Zee, June 3-6. 6. A. C. W. Kress (1984): ‘Practical cone–beam algorithm’, J. Opt. Soc. Amer. A 6, 612-619. 7. K. Fourmont (1999): ‘Schnelle Fourier–Transformation bei nicht–¨ aquidistanten Gittern und tomographische Anwendungen’, Dissertation Fachbereich Mathematik und Informatik der Universit¨ at M¨ unster, M¨ unster, Germany.
This is true since f g has bandwidth 2Ω. We put vˆ(σ) = 2(2π)(n−1)/2 |σ|n−1 φ(σ/Ω) . v is called the reconstruction ﬁlter. e. ⎧ ⎨ 1 , |σ| ≤ 1 , φ(σ) = ⎩ 0 , |σ| > 1 . Then, Ω2 1 s 2 sinc u(Ωs) , u(s) = sinc(s) − . 2 4π 2 2 Now we describe the ﬁltered backprojection algorithm for the reconstruction of the function f from g = Rf . e. (14) fˆ(ξ) is negligible in some sense for |ξ| > Ω . 1 that the same is true for g. Hence, by a loose application of the sampling theorem, v(θ, s − sk )g(θ, sk ) (v ∗ g)(θ, s ) = ∆s (15) k where s = ∆s and ∆s ≤ π/Ω.
The simplest source curve one can think of, a circle around the object, does not satisfy the Kirillov–Tuy condition. For a circular source curve an approximate inversion formula, the FDK formula, exists; see Feldkamp et al. (1984). In medical applications the source curve is a helix. 4 The Attenuated Radon Transform Let n = 2. The attenuated Radon transform of f is deﬁned to be e−(Cµ )(x,θ⊥ ) f (x)dx (Rµ f )(θ, s) = x·θ=s 24 F. Natterer where µ is another function in R2 and θ⊥ is the unit vector perpendicular to θ such that det (θ, θ⊥ ) = 1.