By Peter Kuchment
This ebook surveys the most mathematical rules and strategies in the back of a few well-established imaging modalities comparable to X-ray CT and emission tomography, in addition to quite a few newly constructing coupled-physics or hybrid innovations, together with thermoacoustic tomography.
The Radon remodel and clinical Imaging emphasizes mathematical innovations and concepts bobbing up around the spectrum of clinical imaging modalities and explains very important recommendations touching on inversion, balance, incomplete facts results, the function of inside info, and different matters severe to all clinical imaging methods.
For nonexperts, the writer presents appendices that hide historical past details on notation, Fourier research, geometric rays, and linear operators. The big bibliography, with over 825 entries, directs readers to a big selection of extra details resources on scientific imaging for additional study.
Audience: Graduate scholars and researchers in arithmetic, physics, and engineering who're drawn to the math in the back of scientific imaging will locate this booklet useful.
Contents: half I: creation; bankruptcy 1: Why Use arithmetic in scientific Imaging?; bankruptcy 2: a short and Incomplete background of CT; bankruptcy three: a few significant CT Modalities and Their good points to monitor for; bankruptcy four: association of the e-book; half II: conventional Computed Tomography strategies and fundamental Geometry; bankruptcy five: commonplace CT and X-ray and Radon Transforms; bankruptcy 6: Emission Tomography; bankruptcy 7: Artifacts, Incomplete facts, Miocrolocal research, and Such; bankruptcy eight: extra approximately 3D Radon and X-ray Transforms; bankruptcy nine: a short evaluate of Numerical tools; bankruptcy 10: MRI, EIT, OT, Elastography, UT; half III: Hybrid (Coupled Physics) Imaging thoughts; bankruptcy eleven: Thermo-, Photo-, and Optoacoustic Tomography (TAT/PAT/OAT); bankruptcy 12: Ultrasound Modulation in EIT and OT; bankruptcy thirteen: Inverse issues of inside info; half IV: Appendices; Appendix A: Notation; Appendix B: short Notes at the Fourier remodel and Harmonic research; Appendix C: Geometric Rays, Trapping, power Decay; Appendix D: a few sessions of Linear Operators and Operator features
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Extra resources for The Radon transform and medical imaging
Sample text
24. 30) commutes with the dilation f (r ) → f (a r ). Dilation invariance shows that one should expect this transform to be a convolution operator on the multiplicative group of positive reals. Thus, Mellin transform might be able to invert it explicitly. Fortunately, this is indeed possible, and was done by A. , [559]). 2, knowledge that the range of R is the whole space (whatever this means) is needed to know that the left inversion is indeed the true inversion and thus is unique. Alas, we will see that the range of R in appropriate function spaces is in fact very small, of infinite co-dimension, and so there exists a huge variety of nonequivalent (for imperfect data) inversion procedures.
5 Sinograms A sinogram is the density plot of the X-ray transform of a 2D function; see Fig. 5. 5. Sinogram (density plot of the X-ray transform) of two squares with parallel sides touching at a vertex. 5. Why is it called a sinogram? Hint: (a) Look at the picture. (b) What is the X-ray transform of a small single dot? 4 Some notation and conventions We will adhere to the following notation (see also Appendix B for more discussion of Fourier transforms): • S1 ⊂ • T := 2 —the circle of unit vectors ω ∈ 2, |ω| = 1.
9. The following commutation relation holds: 1 D R. 9) 1 D (R f )(t , ω). QED. r r 32 Chapter 5. 16) must be useful when studying the Radon transform. 4 Relations with the Fourier transform: Projection-slice theorem The shift invariance shows that the Fourier transform should be a good friend (maybe even a close relative) of the Radon transform. The next statement (called the projection-slice, Fourier-slice, or central slice formula/theorem) is indeed central for studying the X -ray and Radon transforms.