Machine Theory

# The Mathematics of Medical Imaging by Charles L. Epstein

By Charles L. Epstein

This publication presents an built-in presentation of arithmetic and its software to difficulties in clinical imaging. Key subject matters contain info assortment, sign processing and noise research. The ebook will be compatible for self learn by way of a stimulated individual with a high-quality mathematical heritage attracted to clinical imaging.

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Roughly speaking the support of a function is the set of points where the function does not vanish. To get a useful concept we need to add points that are very near to points where the function is non-zero. As the definition is the same in all dimensions we give it for functions defined on Ê n . 1. Let f (x) be a function defined on Ê n . A point x belong to the support of f if there is a sequence of points < xn > such that (1). f (xn ) = 0, (2). limn→∞ xn = x. This set is denoted by supp(f ). CHAPTER 2.

Beer’s law state that di d(log i) = −m(s)i(s) or = −m(s). ds ds Integrating this equation from s = a to s = b gives b i(b) log =− i(a) m(s)ds. 4. We consider a two dimensional example. 2(a). The X-ray source is isotropic which means the outgoing flux is the same in all directions. CHAPTER 2. A BASIC MODEL FOR TOMOGRAPHY 38 r o S source (x,y) (a) A point source of X-rays (b) The flux through a curve not enclosing the source is zero. 2: Analysis of an isotropic point source. Because the source is isotropic, the intensity of the beam is only a function of the distance to the source.

Let e(θ) ≡ 0 be a function that satisfies e(θ) + e(θ + π) = 0. 2 we know that if hD has two derivatives such that hD + hD > 0 then hD (θ) is the shadow function of a strictly convex region. 3. LINEARITY 21 as well, then hD + e is also the shadow function for a different strictly convex region. Observe that hD (θ) + e(θ) is the shadow function for a different region, D which has the same width of shadow for each direction as D. That is wD (θ) = (hD (θ) + e(θ)) + (hD (θ + π) + e(θ + π)) = wD (θ).