Phase retrieval from very few measurements

In many applications, signals are measured according to a linear process, but the phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as phase retrieval. This paper focuses on completely determining signals with as few intensity measurements as possible, and on efficient phase retrieval algorithms from such measurements. For the case of complex M-dimensional signals, we construct a measurement ensemble of size 4M-4 which yields injective intensity measurements; this is conjectured to be the smallest such ensemble. For the case of real signals, we devise a theory of "almost" injective intensity measurements, and we characterize such ensembles. Later, we show that phase retrieval from M+1 almost injective intensity measurements is NP-hard, indicating that computationally efficient phase retrieval must come at the price of measurement redundancy.Comment: 18 pages, 1 figur

Saving phase: Injectivity and stability for phase retrieval

Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles lend stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4M-4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M=2,3. Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer-Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.Comment: 22 page

Climate and Crop Yields Sanborn County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Climate and Crop Yields Union County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Climate and Crop Yields Hand County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Climate and Crop Yields Clay County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Climate and Crop Yields Moody County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Climate and Crop Yields Marshall County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Climate and Crop Yields Tripp County

The County Pamphlet Series in Agricultural Economics is intended to make available to each county economic data concerning its farm history and present agricultural situation. It is hoped that these facts will be of use to county planning groups, individual farmers, research and extension workers and other persons interested in the agriculture of the counties. Each pamphlet will treat one subject for one county, and is to be released when completed. Pamphlets on various other economic subjects for the different counties will be prepared as soon as possible. A few copies of each pamphlet will be placed with the county extension agent and a limited number will be sent to private persons upon request. The project was initiated by the Department of Agricultural Economics and the work is under the direction of its regular staff

Farm Adjustment Possibilities to Increase Income in the Wellton-Mohawk District of Yuma County

This item is part of the Agricultural Experiment Station archive. It was digitized from a physical copy provided by the University Libraries at the University of Arizona. For more information, please email CALS Publications at [email protected]