Umer Shahid

Compressive Optical MONTAGE Photography Initiative: Noise and Error Analysis

High resolution images are calculated from sub-Nyquist sampled data. The COMP-I program uses focal plane coding to set sub-bandlimited sampling. An analysis of this approach to noise and alignment errors is presented.

Minimum-phase-based inverse scattering algorithm applied to Institut Fresnel data

Laboratory controlled data were recently provided by the Institut Fresnel to assist with the development and validation of inverse scattering algorithms. A nonlinear signal processing method is applied to invert the scattered field data from strongly scattering objects. Successful filtering of this type requires that the data being processed represent a minimum phase function. The properties of minimum phase functions are well understood in 1D problems and the condition can be enforced using an appropriate reference wave. In 2D or higher dimensional problems, we describe the conditions for minimum phase and show how a reference wave can be numerically combined with measured complex scattering data in order to enforce this condition. We present the results using the data provided and comment on the practical implementation of this method.

Minimum-phase-based inverse scattering method applied to IPS008

We discuss an approach to solving the inverse scattering problem using homomorphic filtering and the difficulties that have been experienced in the past in trying to implement it in practice. Solving this problem has important consequences for a number of imaging and remote sensing problems as well as structure-synthesis problems. We show that the problem reduces to one of needing to preprocess the measured data in order that the nonlinear filtering succeeds and gives meaningful recontstructions. We discuss the steps taht have to be taken to achieve this and show that a sufficient condition to obtain a solution is that the data-derived function to be filtered is made close to a minimum phase function. This minimum-phase property is well understood in one dimensional problems but less so in two or higher dimensions. Another significant practical issue is that for inverse scattering problems, in contrast to inverse synthesis problems, only limited noisy data are available from which to compute the structure. These factors are discussed and we note that solving the inverse scattering problem immediately provides a solution to the inverse synthesis problem.

Legacies of the Gerchberg-Saxton algorithm.

To celebrate W. Owen Saxtons 65th birthday, this paper presents some of the impact that the Gerchberg-Saxton algorithm has had over the last 40 years. We explore some of the fundamental concepts underlying the success of the Gerchberg-Saxton algorithm, in the context of how it stimulated many related methods for estimating fields and deepening the understanding of the relationships between complex objects, images and their Fourier transforms.

Sampling rates and image reconstruction from scattered fields

Cepstral filtering is reviewed as a suitable and efficient method to solve the inverse scattering problem in the case of strongly scattering permittivity distributions. The number and distribution of measured scattered field data required is discussed, as is the effective number of degrees of freedom available to describe the scattering structure. The latter is identified as a key parameter determining the performance of the cepstral method. This is of particular importance for strong scattering and nonlinear image processing methods since many data sets are compiled based on the sampling requirements of weakly scattering objects. We find that the domain of the object support and the maximum permittivity contrast are important prior information for determining the minimum number of data samples necessary while maximizing use of the available degrees of freedom; examples are presented.

Inversion of strongly scattered data: shape and permittivity recovery

Reconstructing an object from scattered field data has always been very challenging, especially when dealing with strong scatterers 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 non-linear signal processing method in which we fir first process the scattered field data to generate a minimum phase function in the object st domain. This is accomplished by adding a reference wave whose amplitude and phase satisfy certain conditions. Minimum Minimum-phase functions are causal transforms and their ph 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 frequencie frequencies arising from the object from those of the s scattered field. Cepstral data are meaningless for non 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.

Inversion of strongly scattering data: imaging and structure synthesis

In this paper we explain some of the difficulties encountered in implementing this approach and how they have been successfully addressed. We show examples of reconstructions using real data taken from the US Air Force Research Laboratorys (AFRL) Ipswich data set and also show examples of synthesizing structures similar to these which have prescribed scattering patterns in certain directions

Inverse scattering methods and microstructure design

Recent advances in methods for inverting scattered field data, originally pursued for imaging penetrable objects, can be considered for the design and synthesis of structures that have prescribed scattered field patterns. Unlike imaging, lack of uniqueness can be useful for synthesis problems, The method used for structure synthesis is described.