Stable recovery of signals from frame coefficients with erasures at unknown locations  

作　　者：Deguang Han Fusheng Lv Wenchang Sun 

机构地区：[1]Department of Mathematics, University of Central Florida [2]School of Mathematical Sciences and the Key Laboratory of Pure Mathematics and Combinatorics,Ministry of Education, Nankai University

出　　处：《Science China Mathematics》2018年第1期151-172,共22页中国科学：数学（英文版）

基　　金：supported by National Science Foundation of USA(Grant Nos.DMS-1403400 and DMS-1712602);National Natural Science Foundation of China(Grant Nos.11171151,11371200,11525104 and 11531013)

摘　　要：In an earlier work, we proposed a frame-based kernel analysis approach to the problem of recovering erasures from unknown locations. The new approach led to the stability question on recovering a signal from noisy partial frame coefficients with erasures occurring at unknown locations. In this continuing work, we settle this problem by obtaining a complete characterization of frames that provide stable reconstructions. We show that an encoding frame provides a stable signal recovery from noisy partial frame coefficients at unknown locations if and only if it is totally robust with respect to erasures. We present several characterizations for either totally robust frames or almost robust frames. Based on these characterizations several explicit construction algorithms for totally robust and almost robust frames are proposed. As a consequence of the construction methods, we obtain that the probability for a randomly generated frame to be totally robust with respect to a fixed number of erasures is one.In an earlier work, we proposed a frame-based kernel analysis approach to the problem of recovering erasures from unknown locations. The new approach led to the stability question on recovering a signal from noisy partial frame coefficients with erasures occurring at unknown locations. In this continuing work, we settle this problem by obtaining a complete characterization of frames that provide stable reconstructions. We show that an encoding frame provides a stable signal recovery from noisy partial frame coefficients at unknown locations if and only if it is totally robust with respect to erasures. We present several characterizations for either totally robust frames or almost robust frames. Based on these characterizations several explicit construction algorithms for totally robust and almost robust frames are proposed. As a consequence of the construction methods, we obtain that the probability for a randomly generated frame to be totally robust with respect to a fixed number of erasures is one.

关 键 词：FRAMES ERASURES signal recovery almost robust frames totally robust frames 

分 类 号：N0[自然科学总论—科学技术哲学]