Figure 1: Magnitude of the magnetic field 0,1 mm above a 4 A direct current carrying track with 1 mm holes and a 0,3 mm crack.

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# Image Processing and Inverse Methods in Magnetic Tomography

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Dipl.-Ing. Dominik Hofer

Can we 'see' current in conductors? Is it possible to make flaws within steel visible?

Magnetic field measurements yield answers for both questions, albeit in an ambiguous form. New magnetic field sensors are capable of simultaneous capture of 3d magnetic field data in space. Their spatial resolution reaches below 0,1 mm and their resolution in time is only limited by the data acquisition hardware. Unfortunately even the best sensors will only measure an integral magnetic field caused by an infinite number of either elementary current or magnetic dipoles. These sensors map only the outer field of any test object and thus multiple current distributions or different flaws in steel yield identical magnetic field readings. Only the special case of 'flat' current distribution within a plane has a unique solution, in all other case special knowledge or limitations are needed to solve the problem.

Above Questions are described by mathematical Models which are of a convolutional type, i.e. their inversions are ill-posed and thus hard to solve. The main part of this dissertation are investigations on boundary conditions, mathematical models of the physical problem and the results of different solving algorithms. Image processing can be applied to the field images and coordinate transformations may yield results in other mathematical domains, for instance in the spectral domain. Finite elements simulations may provide insight into the problem and can produce starting solutions for iterative solvers or they may be used to compensate for boundary effects due to the limited scan area.

Figure 2: Magnitude of a possible current distribution which causes the magnetic field in image 1.

The main tasks of this work are investigations on flat, two dimensional current distributions within conductive traces and planes of a printed circuit board or the localization of the passage of a current carrying conductor through a closed surface. Extensions to three dimensional current distributions with appropriate limitations are also of interest. Magnetic field images taken by a non destructive testing system with externally applied magnetic fields are a subject to image processing techniques to enhance their quality.