Leonid Kunyansky
Work Summary
I develop mathematics of biomedical imaging. All modalities of tomography imaging rely heavily on mathematical algorithms for forming an image. I develop the theory and the algorithm enabling this technology.
I develop mathematics of biomedical imaging. All modalities of tomography imaging rely heavily on mathematical algorithms for forming an image. I develop the theory and the algorithm enabling this technology.
Abstract:
We propose and test stable algorithms for the reconstruction of the internal conductivity of a biological object using acousto-electric measurements. Namely, the conventional impedance tomography scheme is supplemented by scanning the object with acoustic waves that slightly perturb the conductivity and cause the change in the electric potential measured on the boundary of the object. These perturbations of the potential are then used as the data for the reconstruction of the conductivity. The present method does not rely on 'perfectly focused' acoustic beams. Instead, more realistic propagating spherical fronts are utilized, and then the measurements that would correspond to perfect focusing are synthesized. In other words, we use synthetic focusing. Numerical experiments with simulated data show that our techniques produce high-quality images, both in 2D and 3D, and that they remain accurate in the presence of high-level noise in the data. Local uniqueness and stability for the problem also hold. © 2011 IOP Publishing Ltd.
Abstract:
A fast, high-order algorithm for the solution of problems of acoustic scattering from smooth surfaces in three dimensions is presented. Numerical experiments indicate that this algorithm performs exceptionally well.
Abstract:
This paper surveys recent analytic and numerical results on asymptotic models of spectra of electromagnetic (EM) waves in two-dimensional (2-D) thin high-contrast photonic bandgap (PBG) materials. These models lead to discovery of interesting phenomena, including extremely narrow bands that can be used for spontaneous emission enhancement, gaps in the long wave regions, and asymptotic periodicity of the spectrum. The asymptotic results provide unexpectedly good qualitative (and sometimes quantitative) description of spectral behavior for materials of finite contrast. In some cases, simple ordinary differential models can be derived that yield a good approximation of the spectra. In such situations, one can obtain approximate analytic formulas for the dispersion relations.