FRG: Solutions for Inverse Problems

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In the work proposed here, four researchers at Rensselaer Polytechnic Institute work in teams with postdocs and graduate students to solve inverse problems. The goal is to identify material properties or surface features from indirectly related data sets. Each problem is modeled as an elastic, electromagnetic, or acoustic medium and the mathematical model is strongly taken into account to develop solution techniques. Two of the four problems to be considered are: (1) find variations in stiffness in biological tissue so that regions that are abnormal (usually with 7 to 17 times stiffer than normal tissue) can be identified. The model is the equations of elasticity and measurement of low frequency propagating shear waves using Doppler ultrasound provide the data. Noise reduction, use of models with random media, determination of the minimum size of abnormal regions that can be identified, reconstruction algorithms and images are all targets of this investigation; and (2) create radar images from satellite or airborne radar equipment by developing solutions that correct for deviations from ideal flight paths, that correctly identify object positions (break left-right symmetry problems) and that can image objects that are moving. The problems involve establishing mathematical results, utilizing engineering expertise and development and implementation of numerical algorithms.In this work, four principal investigators, together with postdocs and graduate students, solve problems where noninvasive sensing is followed by the creation of images. To solve these problems, the researchers work in teams to combine mathematical analysis, engineering and numerical computation to achieve results. Two of the problems that will be addressed are: (a) elastography with ultrasound measurements of tissue movement created by a second low level propagated signal yields data that can distinguish the stiffer tissue of cancerous tumors from normal tissue. The goal, then, is to determine algorithms for computing the location of the stiff and normal regions, to find the minimum size of regions that can be identified, and to create images in 'real time';(b) airborne and satellite topographic and object sensing where signals reflected from the earth's surface are used to locate positions of objects and topographic features. The goal is to improve resolution and accuracy, to reduce the size of objects and features that can be identified and to correctly represent features that are partially hidden from view.