Download Mathematical Modeling in Biomedical Imaging I: Electrical by Jin Keun Seo, Eung Je Woo (auth.), Habib Ammari (eds.) PDF

By Jin Keun Seo, Eung Je Woo (auth.), Habib Ammari (eds.)

This quantity offers an advent to a desirable learn sector to utilized mathematicians. it's dedicated to supplying the exposition of promising analytical and numerical strategies for fixing difficult biomedical imaging difficulties, which set off the research of fascinating concerns in quite a few branches of mathematics.

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Extra resources for Mathematical Modeling in Biomedical Imaging I: Electrical and Ultrasound Tomographies, Anomaly Detection, and Brain Imaging

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Je Woo domain Ω ⊂ R3 with a smooth connected boundary ∂Ω and each Ωz0 := Ω ∩ {z = z0 } ⊂ R2 , the slice of Ω cut by the plane {z = z0 }, has a smooth connected boundary. We assume that the conductivity distribution σ in Ω is isotropic. We attach a pair of copper electrodes E + and E − on ∂Ω in order to inject current, and let E + ∪ E − be the portion of the surface ∂Ω where electrodes are attached. 34) where n is the outward unit normal vector on ∂Ω and ds the surface area element. The condition of J×n = 0 on E + ∪E − comes from the fact that copper electrodes are almost perfect conductors.

Let us briefly discuss the boundary conditions that are essentially related with the size of electrodes. The condition ∇u × n|E ± = 0 ensures that each of u|E + and u|E − is a ∂u constant, since ∇u is normal to its level surface. The term ±I = E ± σ ∂n ds means that the total amount of injection current through the electrodes is ∂u |∂Ω . In practice, it is difficult to specify the I mA. Let us denote g := −σ ∂n Neumann data g in a point-wise sense because only the total amount of injection current I is known.

For the image reconstruction, we try to use a weighted frequency-difference voltage ujωk − αb ujωl at each electrode. K. Seo and E. Je Woo and τj = (σb (ωj ) − σa (ωj )) + iωj ( b (ωj ) − σb (ω2 ) + iω2 b (ω2 ) a (ωj )) , j = 1, 2. Hence, the real and imaginary parts of fω2 − αb fω1 correspond to those of τ2 ∇uω2 − τ1 ∇uω1 , respectively. 1. Average directions of ∇v1 , ∇v2 , ∇h1 , ∇h2 on D are approximately parallel or anti-parallel: ∇v2 (r)dr ≈ 0 and ∇v1 (r)dr × D ∇hj (r)dr ≈ 0 ∇vj (r)dr × D D D for j = 1, 2.

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