Inversion of strongly scattered data: shape and permittivity recovery U. Shahid, M. A. Fiddy and M. E. Testorf* Center for Optoelectronics and Optical Communications University of North Carolina, Charlotte, NC 28223 *Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755-8000 ABSTRACT Reconstructing an object from scattered field data has always been very challenging, especially when dealing with strong scatterers. Several techniques have been proposed to address this problem but either they fail to provide a good estimate of the object or they are computationally very expensive. We have proposed a straightforward non-linear signal processing method in which we first process the scattered field data to generate a minimum phase function in the object domain. This is accomplished by adding a reference wave whose amplitude and phase satisfy certain conditions. Minimum-phase functions are causal transforms and their phase is continuous in the interval –π and +π, i.e. it is always unwrapped. Following this step, we compute the Fourier transform of the logarithm of this minimum phase function, referred to as its cepstrum. In this domain one can filter cepstral frequencies arising from the object from those of the scattered field. Cepstral data are meaningless for non-minimum phase functions because of phase wraps. We apply low pass filters in the cepstral domain to isolate information about the object and then perform an inverse transform and exponentiation. We have applied this technique to measured data provided by Institut Fresnel (Marseille, France) and investigated in a systematic way the dependence of the approach on the properties of the reference wave and filter. We show that while being a robust method, one can identify optimal parameters for the reference wave that result in a good reconstruction of a penetrable, strongly scattering permittivity distribution. Keywords: inverse scattering, minimum phase condition, homomorphic filtering, imaging 1. INTRODUCTION The inverse problem in scattering and diffraction is an important problem in imaging and remote sensing. For penetrable scattering objects, V = k 2 [ε (r) - 1]/4π, in a homogeneous background, the solution requires knowledge of the total field within the scattering volume. Since this is a function of the complex permittivity, ε (r), of that volume, the problem is inherently nonlinear and severely ill-posed. In weakly scattering approximations (the first Born approximation) or slowly varying permittivity approximations (the first Rytov approximation), one can linearize the inversion problem, making a solution possible at least in principle 1 . In cases for which this cannot be done, the inversion or backpropagation step can be still executed using a Fourier transform, assuming one is working with far field data, which provides information about the so-called secondary source or contrast source function. This is related to the product of the fluctuation of the permittivity distribution about the mean and the total field, Ψ, inside the scattering volume, F(x 1, x 2 ) = ∫∫ V(u,v)F T (u,v)exp(-ik(ux 1 +vx 2 ))dudv (1) where F T = Ψ/Ψ ο and when the total field Ψ =Ψ ο , the incident field, the first Born approximation is satisfied and one can recover an estimate of the scattering object, whose fidelity then primarily depends on the number and location of data samples available. Recovery of the permittivity distribution even from continuous noise-free data is not straightforward and many iterative methods (e.g. distorted wave Born, iterative Born, modified gradient techniques 2 ) have been developed to try to compute a quantitative image. These are typically limited in their application because of computational complexity. To spur on algorithm development, various groups have provides real data from known scattering objects. Oldest amongst these is Image Reconstruction from Incomplete Data V, edited by Philip J. Bones, Michael A. Fiddy, Rick P. Millane, Proc. of SPIE Vol. 7076, 707606, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.796137 Proc. of SPIE Vol. 7076 707606-1 2008 SPIE Digital Library -- Subscriber Archive Copy