کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4605088 1337544 2014 20 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Saving phase: Injectivity and stability for phase retrieval
ترجمه فارسی عنوان
فاز صرفه جویی: جبری بودن و ثبات برای بازیابی فاز
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
چکیده انگلیسی

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 yield 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−44M−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=2,3M=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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Applied and Computational Harmonic Analysis - Volume 37, Issue 1, July 2014, Pages 106–125
نویسندگان
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