Einführung in die medizinische Bildverarbeitung SS 2013

directions.
In computed tomography, the meaning of the mathematical term inverse problem
is immediately apparent. In contrast to the situation shown in Fig. 1.1,thespatial
distribution of the attenuating objects that produce the projection shadow is
not known a priori. This, actually, is the reason for acquiring the projections along
the rotating detector coordinate ξ over a projection angle interval of at least 180 ∘ .
Figure 1.2 illustrates this situation. It is an inversion of integral transforms. From
‣ Aus asequenceofmeasuredprojectionshadowsp einer geringen Anzahl an Projektionen, lässt γ1 (ξ), sich p γ2
nicht (ξ), pmit γ3 (ξ),...,thespa-
absoluter Gewissheit das
Originalbild rekonstruieren
Grundlagen - Bilderzeugung
Computertomographie
‣ Projektionen
[Quelle: Buzug, T., Computed Tomography - From Photon Statistics to Modern Cone-Beam CT]
Fig. 1.2. Schematic illustration of the inverse problem posed by CT. Attenuation profiles,
p γ (ξ),havebeenmeasuredforasetofprojectionangulations,γ 1 and γ 2 . Theunknown
geometry, or the object with its associated spatial distribution of attenuation coefficients, has
to be calculated from a complete set of attenuation profiles p γ1 (ξ), p γ2 (ξ), p γ3 (ξ),...
Dabei repräsentiert ρ γ i
( ξ ) den Radon Raum des Objektes
© Stephan Gimbel Einführung in die medizinische Bildverarbeitung
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