Material for Chapter 6: Basic Principles of Tomography M I A Integral Equations in Visual Computing Material

Transcription

1 Material for Chapter : Integral Equations in Visual Computing Material Basic Principles of Tomography c 00 Bernhard Burgeth 0

2 Source: Images Figure : Radon Transform: ttenuation 0

3 Images Parallel scanning geometry I Figure : Dual motion scanner. Top: One source, one detector. Bottom: One source, several detectors. Source: Natterer & Wübbeling, 00. 0

4 Images Parallel scanning geometry II Figure : Fan beam scannng. Left: Rotating detector-source system. Right: Stationary detector ring, rotating source. Source: Natterer & Wübbeling, 00. 0

5 Projections I Images Figure : Object, f(x, y), and its projection, P θ (t ), shown for angle θ. Source: Kak,. 0

6 Images Parallel Projections Figure : Parallel projections are taken by measuring a set of parallel rays for a number of different angles. Source: Rosenfeld & Kak,. 0

7 Images Fan Beam Projections Figure : Kak,. fan beam projection is colleted if all the rays meet in one location. Source: Rosenfeld & 0

8 For tests: head phantom Images Figure : Left: The Shepp and Logan head phantom. Right: The phantom is a superposition of 0 ellipses. Source: Rosenfeld & Kak,. 0

9 Images Example: Projected Ellipses I Figure : nalytic expression is shown for the projection of an ellipse. For computer simulations a projection can be generated by simply summing the projection of each individual ellipse. Source: Rosenfeld & Kak,. 0

10 Images Example: Projected Ellipses II Figure : nalytical calculation for an ellipse with its center located at (x, y ) and its major axis rotated by α. Source: Rosenfeld & Kak,. 0

11 Images 0 Fourier Slice Theorem Figure 0: The Fourier Slice Theorem relates the Fourier transform of a projection to the Fourier transform of the object along a radial line. Source: Pan & Kak,. 0

12 Images Interpolation in frequency domain??? Figure : Collecting projections of the object at a number of angles gives estimates of the Fourier transform of the object along radial lines. The dots represent the actual location of estimates of the object s Fourier transform. Source: Pan & Kak,. 0

13 Images Backprojection I Figure : Reconstructions are often done using a procedure known as backprojection. Here a filtered projection is smeared back over the reconstruction plane along lines of constant t. The filtered projection at a point t makes the same contribution to all pixels along the line LM in the x y plane. Source: Rosenfeld & Kak,. 0

14 Images Backprojection II Figure : The result of backprojecting the projection of an ellipse. (a) shows the result of backprojecting for a single angle, (b) shows the effect of backprojecting over angles, (c) shows angles, and (d) shows angles. 0

15 Images Example: Reconstruction Figure : Reconstruction of the head phantom. The dark regions at the top and at the bottom are visible artefacts. This reconstruction was made from 0 projections with rays in each direction. Source: Rosenfeld & Kak, 0

16 Literature Literature F. Natterer, F. Wübbeling: Mathematical Methods in Image Processing. SIM monographs on mathematical modeling and computation, 00. (textbook with mathematical in-depth treatment of tomography).c. Kak:Computerized tomography with X-ray emission and ultrasound sources. Proc. IEEE, vol., pp. -,.. Rosenfeld and.c. Kak: Digital Picture Processing. nd edition, cademic Press, New York,. S.X. Pan and.c. Kak: computational study of reconstruction algorithms for diffraction tomography: Interpolation vs. filtered- backpropagation. IEEE Trans. coustic Speech Signal Processing, vol. SSP-, pp. -,. 0

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