{"title":"IFF:一种多测量的超分辨率算法","authors":"Zetao Fei, Hai Zhang","doi":"10.1137/23m1568569","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2175-2201, December 2023. <br/> Abstract. We consider the problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval [math]. This problem is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length [math]. In this paper, we propose a superresolution algorithm, called iterative focusing-localization and iltering, to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm has a distinct feature in that it reconstructs the point sources one by one in an iterative manner and hence requires no prior information about the source numbers. The new feature also allows for a subsampling strategy that can reconstruct sources using small-sized Hankel matrices and thus circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods. In addition, the algorithm can be paralleled. A theoretical analysis of the methods behind the algorithm is also provided. The derived results imply a phase transition phenomenon in the reconstruction of source locations which is confirmed in the numerical experiment. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit. The efficiency and robustness of the algorithm have also been tested. This algorithm can be generalized to higher dimensions.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"25 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"IFF: A Superresolution Algorithm for Multiple Measurements\",\"authors\":\"Zetao Fei, Hai Zhang\",\"doi\":\"10.1137/23m1568569\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2175-2201, December 2023. <br/> Abstract. We consider the problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval [math]. This problem is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length [math]. In this paper, we propose a superresolution algorithm, called iterative focusing-localization and iltering, to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm has a distinct feature in that it reconstructs the point sources one by one in an iterative manner and hence requires no prior information about the source numbers. The new feature also allows for a subsampling strategy that can reconstruct sources using small-sized Hankel matrices and thus circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods. In addition, the algorithm can be paralleled. A theoretical analysis of the methods behind the algorithm is also provided. The derived results imply a phase transition phenomenon in the reconstruction of source locations which is confirmed in the numerical experiment. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit. The efficiency and robustness of the algorithm have also been tested. This algorithm can be generalized to higher dimensions.\",\"PeriodicalId\":49528,\"journal\":{\"name\":\"SIAM Journal on Imaging Sciences\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-11-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Journal on Imaging Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/23m1568569\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Imaging Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1568569","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
IFF: A Superresolution Algorithm for Multiple Measurements
SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2175-2201, December 2023. Abstract. We consider the problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval [math]. This problem is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length [math]. In this paper, we propose a superresolution algorithm, called iterative focusing-localization and iltering, to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm has a distinct feature in that it reconstructs the point sources one by one in an iterative manner and hence requires no prior information about the source numbers. The new feature also allows for a subsampling strategy that can reconstruct sources using small-sized Hankel matrices and thus circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods. In addition, the algorithm can be paralleled. A theoretical analysis of the methods behind the algorithm is also provided. The derived results imply a phase transition phenomenon in the reconstruction of source locations which is confirmed in the numerical experiment. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit. The efficiency and robustness of the algorithm have also been tested. This algorithm can be generalized to higher dimensions.
期刊介绍:
SIAM Journal on Imaging Sciences (SIIMS) covers all areas of imaging sciences, broadly interpreted. It includes image formation, image processing, image analysis, image interpretation and understanding, imaging-related machine learning, and inverse problems in imaging; leading to applications to diverse areas in science, medicine, engineering, and other fields. The journal’s scope is meant to be broad enough to include areas now organized under the terms image processing, image analysis, computer graphics, computer vision, visual machine learning, and visualization. Formal approaches, at the level of mathematics and/or computations, as well as state-of-the-art practical results, are expected from manuscripts published in SIIMS. SIIMS is mathematically and computationally based, and offers a unique forum to highlight the commonality of methodology, models, and algorithms among diverse application areas of imaging sciences. SIIMS provides a broad authoritative source for fundamental results in imaging sciences, with a unique combination of mathematics and applications.
SIIMS covers a broad range of areas, including but not limited to image formation, image processing, image analysis, computer graphics, computer vision, visualization, image understanding, pattern analysis, machine intelligence, remote sensing, geoscience, signal processing, medical and biomedical imaging, and seismic imaging. The fundamental mathematical theories addressing imaging problems covered by SIIMS include, but are not limited to, harmonic analysis, partial differential equations, differential geometry, numerical analysis, information theory, learning, optimization, statistics, and probability. Research papers that innovate both in the fundamentals and in the applications are especially welcome. SIIMS focuses on conceptually new ideas, methods, and fundamentals as applied to all aspects of imaging sciences.