The Mathematics of Medical Imaging: A Beginner’s Guide by Timothy G. Feeman

By Timothy G. Feeman

A Beginner's consultant to the maths of scientific Imaging provides the fundamental arithmetic of automated tomography – the CT test – for an viewers of undergraduates in arithmetic and engineering. Assuming no earlier historical past in complicated mathematical research, subject matters similar to the Fourier rework, sampling, and discrete approximation algorithms are brought from scratch and are built in the context of scientific imaging. A bankruptcy on magnetic resonance imaging specializes in manipulation of the Bloch equation, the process of differential equations that's the beginning of this significant technology.

The textual content is self-contained with a number sensible routines, themes for additional examine, and an abundant bibliography, making it excellent to be used in an undergraduate direction in utilized or engineering arithmetic, or through practitioners in radiology who need to know extra in regards to the mathematical foundations in their field.

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DeMoivre’s law. When the equation (eiθ )n = einθ is translated into standard complex number form, we get [cos(θ ) + i sin(θ )]n = cos(nθ ) + i sin(nθ ). 5) This is called DeMoivre’s law. It is just a short step now to define the exponential function for every complex number. Namely, for any complex number z = a + bi, ez = ea+bi = ea · ebi = ea · (cos(b) + i sin(b)). 6) The complex exponential function has many interesting and important properties, not least of which is that it is a conformal mapping.

So, |r| is the modulus of reiθ . The number θ , viewed as an angle now, is called the argument of the complex number reiθ . A simple computation shows that (reiθ ) · (Reiφ ) = r · R · ei(θ +φ ) . Thus, when we multiply two complex numbers, expressed here in their polar forms, the modulus of the product is equal to the product of the individual moduli and the argument of the product is the sum of the individual arguments. DeMoivre’s law. When the equation (eiθ )n = einθ is translated into standard complex number form, we get [cos(θ ) + i sin(θ )]n = cos(nθ ) + i sin(nθ ).

Verify that, for every pair of real numbers a and b, the set of points in the plane that satisfy the polar-coordinate equation 24 3 Back Projection Fig. 3. The back projection of the Radon transform of the Shepp–Logan phantom. r = (a cos(θ ) + b sin(θ )) for 0 ≤ θ ≤ π forms a circle that passes through the origin as well as through the point with Cartesian coordinates (a, b). Find the radius of the circle and the location of its center. 3. 2, for a function F = F(r, θ ) whose inputs are polar coordinates, the value of BF(x, y) is the average value of the function F on the circle determined by the polar-coordinate equation r = (x cos(θ ) + y sin(θ )) for 0 ≤ θ ≤ π .

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