On optimal current patterns for electrical impedance tomography
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Slide 1 :
On optimal current patterns for electrical impedance tomography Eugene Demidenko Based on the paper published in IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 2, FEBRUARY 2005
Slide 2 :
Motivation In Electrical Impedance Tomography (EIT) current is injected at the periphery of object with voltage measured at the same location (say, 16 electrodes). Using the inverse to Laplace equation that governs the potential distribution one can reconstruct electromagnetic properties Inside of the media. Since the current to inject may be arbitrary the question of the optimal current patterns (vector with 16 components) arises. We develop a statistical criterion for optimal patterns in planar circular electrical impedance tomography. These patterns minimize the total variance of the estimation for the resistance or conductance matrix. It is shown that trigonometric patterns (Isaacson, 1986), originally derived from the concept of distinguishability, are a special case of our optimal statistical patterns. New optimal random patterns are introduced.
Slide 3 :
Method Instead of solving a difficult LAPLACE Partial Differential Equation we use the generalized Ohm’s law by expressing the vector of voltages through the vector of currents via a linear statistical model In this model i indicates the electrode (there are n of those, say, n=16); vi is the nx1 vector of voltages, ci is the nx1 vector of currents and R is the nxn resistance matrix to be determined; ei is the error vector. We define optimal current vector ci which minimizes the total variance of matrix R estimation.
Slide 4 :
Solution We prove that the optimal pattern with a fixed norm should be orthogonal to each other. Below you can see several systems of optimal patterns, they all have trigonometric form.
Slide 5 :
Analytical solution: full trigonometric patterns assuming that medium is completely heterogeneous--matrix R is arbitrary.
Slide 6 :
Optimal patterns for ahomogeneous medium(matrix R is a function of one argument)
Slide 7 :
Tank experimentsBelow are reconstructed conductivity with homogeneous/saline medium and metal inclusion using our optimal patternsWe can discriminate a metal pin of 2mm in diameter from a 3 mm pinin a tank of 10 cm diameter
Slide 8 :
Further applications forhardware optimization Use the variance criterion to determine the optimal number of electrodes Determine the optimal number of electrodes. Adaptive optimal patterns.
Slide 9 :
More literature on statistical EITby the same author Eugene Demidenko, Alex Hartov, Keith Paulsen. “Statistical Estimation of Resistance/Conductance by Electrical Impedance Tomography Measurements”, IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 7, JULY 2004 829. Eugene Demidenko. “Separable Laplace equation, magic Toeplitz matrix, and generalized Ohm’s law”, Applied Mathematics and Computation 181 (2006) 1313–1327.
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