By Charles L. Epstein
On the center of each scientific imaging expertise is a cosmopolitan mathematical version of the dimension method and an set of rules to reconstruct a picture from the measured information. This booklet offers an organization starting place within the mathematical instruments used to version the measurements and derive the reconstruction algorithms utilized in such a lot imaging modalities in present use. within the strategy, it additionally covers many vital analytic ideas and strategies utilized in Fourier research, critical equations, sampling idea, and noise research.
this article makes use of X-ray computed tomography as a "pedagogical desktop" to demonstrate vital principles and accommodates huge discussions of historical past fabric making the extra complex mathematical subject matters obtainable to readers with a much less formal mathematical schooling. The mathematical suggestions are illuminated with over two hundred illustrations and various workouts.
New to the second one variation are a bankruptcy on magnetic resonance imaging (MRI), a revised part at the dating among the continuum and discrete Fourier transforms, a brand new part on Grangreat s formulation, a more robust description of the gridding process, and a brand new part on noise research in MRI.
viewers The publication is suitable for one- or two-semester classes on the complex undergraduate or starting graduate point at the mathematical foundations of contemporary scientific imaging applied sciences. The textual content assumes an realizing of calculus, linear algebra, and simple mathematical research.
Contents Preface to the second one variation; Preface; the way to Use This publication; Notational Conventions; bankruptcy 1: Measurements and Modeling; bankruptcy 2: Linear versions and Linear Equations; bankruptcy three: A easy version for Tomography; bankruptcy four: advent to the Fourier remodel; bankruptcy five: Convolution; bankruptcy 6: The Radon remodel; bankruptcy 7: advent to Fourier sequence; bankruptcy eight: Sampling; bankruptcy nine: Filters; bankruptcy 10: enforcing Shift Invariant Filters; bankruptcy eleven: Reconstruction in X-Ray Tomography; bankruptcy 12: Imaging Artifacts in X-Ray Tomography; bankruptcy thirteen: Algebraic Reconstruction strategies; bankruptcy 14: Magnetic Resonance Imaging; bankruptcy 15: likelihood and Random Variables; bankruptcy sixteen: purposes of chance; bankruptcy 17: Random tactics; Appendix A: heritage fabric; Appendix B: uncomplicated research; Index.
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Additional info for Introduction to the Mathematics of Medical Imaging
N denote the vector with a 1 in the jth place and (1, 0, 0, ... ,0), e2 = (0, 1, 0, ... ,0), ... , en = (0, ... ,0, 1). 1. Show that if x = (Xl, ... ,xn ), then n X = LXjej. j=l 2. 1; that is, show that there is a vector Y j so that I(x) = (x, Y j). Give a formula for Y j. 3. " Prove this statement. 6. In this exercise we use calculus to prove the Cauchy-Schwarz inequality. Let x, Y E ~n be nonzero vectors. Define the function F(t) = (x + ty, x + ty). Use calculus to find the value of t, where F assumes its minimum value.
For many years clinicians used medical x-ray images without employing a precise quantitative model. X-rays were thought of as high frequency 'light' with three very useful properties: 1. If x-rays are incident on a human body, some fraction of the incident radiation is absorbed or scattered, though a sizable fraction is transmitted. The fraction absorbed or scattered is proportional to the total density of the material encountered. The overall decrease in the intensity of the x-ray beam is called attenuation.
This gives a different way to reconstruct an approximation to D from a finite set of measurements. This method is not as robust as the first technique because it requires the measured data to be differentiated. In order for the finite difference hCB~~=;;:~+d to be a good approximation to h' Cej), it is generally necessary for lej - ej+11 to be small. Moreover, Chapter 1. Measurements and Modeling 28 the errors in the measurements of h(ej) and h(ej+d must also be small compared to lej ej+11. This difficulty arises in solution of the reconstruction problem in x-ray CT; the exact reconstruction formula calls for the measured data to be differentiated.
Introduction to the Mathematics of Medical Imaging by Charles L. Epstein