A Deterministic Theory for Exact Non-Convex Phase Retrieval

In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we show that the convergence of the WF iterates to a true solution is attained geometrically under a single condition on the lifted forward model. As a result, a deterministic relationship between the accuracy of spectral initialization and the validity of {the regularity condition} is derived. In particular, we determine that a certain concentration property on the spectral matrix must hold uniformly with a sufficiently tight constant. This culminates into a sufficient condition that is equivalent to a restricted isometry-type property over rank-1, positive semi-definite matrices, and amounts to a less stringent requirement on the lifted forward model than those of prominent low-rank-matrix-recovery methods in the literature. We characterize the performance limits of our framework in terms of the tightness of the concentration property via novel bounds on the convergence rate and on the signal-to-noise ratio such that the theoretical guarantees are valid using the spectral initialization at the proper sample complexity.Comment: In Revision for IEEE Transactions on Signal Processin

Second-order stationary statistical models for inverse frequency processes

In this thesis, we have proposed two new classes of nonstationary stochastic processes for 1/f phenomena, developed efficient and convenient spectral representations, and explore some potential applications. The first class is a family of self-similar processes with the correlation structure of the form E\lbrack X(t)X(\lambda t)\rbrack=t\sp{2H}\lambda\sp HR(\lambda) where $t,\lambda\u3e0, {-\infty}0, {-\infty

ESTIMATION OF RADAR TARGET REFLECTIVITY IN ULTRAWIDEBAND REGIME – A GROUP THEORETIC APPROACH

This paper presents a fundamentally new mathematical framework based on the group representation theory for the modeling and processing of signals in ultrawideband (UWB) regime. A group theoretical approach is motivated by the fact that in UWB regime, the underlying mathematical structure of the inverse scattering is governed by the affine group. In particular, received echo can be viewed as the affine Fourier transform of the range-Doppler image evaluated at the transmitted waveform. Based on group representation theory and a novel Wiener filtering method over the affine group, we derived a regularized closed form analytical estimate of the range-Doppler target reflectivity density in the presence of nonstationary noise and clutter in UWB regime. This estimate also leads to a method of data fusion to form synthetic UWB high resolution images from multiple radars with narrowband transmission. 1

Direct Reconstruction of Pharmacokinetic-Rate Images of Optical Fluorophores From NIR Measurements

In this paper, we present a new method to form pharmacokinetic-rate images of optical fluorophores directly from near infra-red (NIR) boundary measurements. We first derive a mapping from spatially resolved pharmacokinetic rates to NIR boundary measurements by combining compartmental modeling with a diffusion based NIR photon propagation model. We express this mapping as a state-space equation. Next, we introduce a spatio-temporal prior model for the pharmacokinetic-rate images and combine it with the state-space equation. We address the image formation problem using the extended Kalman filtering framework. We analyze the computational complexity of the resulting algorithms and evaluate their performance in numerical simulations. An important feature of our approach is that the reconstruction of fluorescence concentrations and compartmental modeling are combined into a single step 1) to take advantage of the inherent temporal correlations in dynamic NIR measurements, and 2) to incorporate spatio-temporal a priori information on pharmacokinetic-rate images, Simulation results show that the resulting algorithms are more robust and lead to higher signal-to-noise ratio as compared to existing approaches where the reconstruction of concentrations and compartmental modeling are treated separately. Additionally, we reconstructed pharmacokinetic-rate images using in vivo data obtained from three patients with breast tumors. The reconstruction results show that the pharmacokinetic rates of indocyanine green are higher inside the tumor region as compared to the surrounding tissue

Synthetic aperture radar imaging of moving targets using ultra-narrowband continuous waveforms

Abstract. We consider a synthetic aperture radar (SAR) system that uses ultra-narrowband continuous waveforms (CWs) as an illumination source. Such a system has many practical advantages, such as the use of relatively simple low-cost and low-power transmitters, and in some cases, using the transmitters of opportunity, such as television and radio stations. Additionally, ultra-narrowband CW signals are suitable for motion estimation due to their ability to acquire high resolution Doppler information. In this paper, we present a novel synthetic aperture imaging method for moving targets using a bistatic SAR system transmitting ultra-narrowband CWs. Our method exploits the high Doppler resolution provided by ultra-narrowband CW signals both to image the scene reflectivity and to determine the velocity of multiple moving targets. Starting from the first principle, we develop a novel forward model based on the temporal Doppler induced by the movement of antennas and moving targets. We form the reflectivity image of the scene and estimate the motion parameters using a filtered-backprojection technique combined with a contrast optimization method. Analysis of the point spread function of our image formation method shows that reflectivity images are focused when the motion parameters are estimated correctly. We present analysis of the velocity resolution and the resolution of reconstructed reflectivity images. We analyze the error between the correct and reconstructed positions of targets due to errors in velocity estimation. Extensive numerical simulations demonstrate the performance of our method and validate the theoretical results

BREAST CANCER DETECTION BASED ON ULTRASOUND B-SCAN TEXTURE ANALYSIS- AND PATIENT AGE INFORMATION

Abslracl-In this paper we apply the Fractional Differencing Autoregressive Moving Average (FARMA) model for ultrasonic breast tissue characterization using RF echo signals. We present estimation techniques to extract the model parameters, namely features, for classification purposes and tissue characterization. Along with the model parameters, we use patient age information as an additional feature to improve ROC results. We evaluate the performance of the proposed method using in vivo ultrasound breast images with benign and malignant tumors. I