Micro-phase measuring profilometry: Its sensitivity analysis and phase unwrapping

Tang, Suming; Zhang, Xu; Tu, Dawei

doi:10.1016/j.optlaseng.2015.03.018

Cited by 32 publications

(8 citation statements)

References 36 publications

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“…By the definition of dynamic range, we know that {Q 1 , Q 2 } can be uniquely determined by their residue sets R k (Q 1 , Q 2 ). Consequently, {N 1 , N 2 } can be uniquely reconstructed by using formula (17).…”

Section: B a Generalized Crt For Two Integers With Modulus Set Mmentioning

confidence: 99%

A Robust Generalized Chinese Remainder Theorem for Two Integers

Xia

Wang

et al. 2016

IEEE Trans. Inform. Theory

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A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A robust CRT has also been proposed lately for robustly reconstruct a single integer from its erroneous remainders. In this paper, we consider the reconstruction problem of two integers from their residue sets, where the remainders are not only out of order but also may have errors. We prove that two integers can be robustly reconstructed if their remainder errors are less than M/8, where M is the greatest common divisor (gcd) of all the moduli. We also propose an efficient reconstruction algorithm. Finally, we present some simulations to verify the efficiency of the proposed algorithm. The study is motivated and has applications in the determination of multiple frequencies from multiple undersampled waveforms. Index TermsChinese remainder theorem (CRT), robust CRT, dynamic range, residue sets, remainder errors, frequency determination from undersampled waveforms

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Section: B a Generalized Crt For Two Integers With Modulus Set Mmentioning

confidence: 99%

A Robust Generalized Chinese Remainder Theorem for Two Integers

Xia

Wang

et al. 2016

IEEE Trans. Inform. Theory

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“…The contacting mode of atomic force microscopes is actually in this mechanical category [3]. In addition to the mechanical tools, optical devices exist in diverse modifications [2], light section microscopy [4,5], coherence scanning interferometry [6], speckle metrology [7], stereo projection [8], photogrammetry [9], and various types of light measurement of profiles [10], to mention some of them.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Principles of Object 3D Reconstruction by Shape-from-Focus Methods

Martišek

Mikulášek

2021

Mathematics

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Shape-from-Focus (SFF) methods have been developed for about twenty years. They able to obtain the shape of 3D objects from a series of partially focused images. The plane to which the microscope or camera is focused intersects the 3D object in a contour line. Due to wave properties of light and due to finite resolution of the output device, the image can be considered as sharp not only on this contour line, but also in a certain interval of height—the zone of sharpness. SSFs are able to identify these focused parts to compose a fully focused 2D image and to reconstruct a 3D profile of the surface to be observed.

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“…Note that all the remainders in the CRT reconstruction formula have to be error-free, because a small error in a remainder may cause a large reconstruction error. In this work, we consider a problem of robustly reconstructing a large nonnegative integer when the remainders have errors, called the robust remaindering problem, and its applications can be found in phase unwrapping in radar signal processing [4]- [15], multiwavelength optical measurement [16]- [18], wireless sensor networks [19]- [24], and computational neuroscience [25]- [28]. In this robust remaindering problem, two fundamental questions are of interested: 1) What is the dynamic range of the large integer and how large can the remainder errors be for the robustness to hold?…”

Section: Introductionmentioning

confidence: 99%

Towards Robustness in Residue Number Systems

Xiao

Xia

Huo

2017

IEEE Trans. Signal Process.

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The problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli, namely the robust remaindering problem, may occur in many applications including phase unwrapping, frequency detection from several undersampled waveforms, wireless sensor networks, etc.Assuming that the dynamic range of the large number is the maximal possible one, i.e., the least common multiple (lcm) of all the moduli, a method called robust Chinese remainder theorem (CRT) for solving the robust remaindering problem has been recently proposed. In this paper, by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli, a trade-off between the dynamic range and the robustness bound for two-modular systems is studied. It basically says that a decrease in the dynamic range may lead to an increase of the robustness bound. We first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed-form formula for the robustness to hold. We then propose simple closed-form reconstruction algorithms. Furthermore, the newly obtained two-modular results are applied to the robust reconstruction for multi-modular systems and generalized to real numbers. Finally, some simulations are carried out to verify our proposed theoretical results. Index TermsChinese remainder theorem, dynamic range, frequency estimation from undersamplings, residue number systems, robust reconstruction.

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Micro-phase measuring profilometry: Its sensitivity analysis and phase unwrapping

Cited by 32 publications

References 36 publications

A Robust Generalized Chinese Remainder Theorem for Two Integers

A Robust Generalized Chinese Remainder Theorem for Two Integers

Mathematical Principles of Object 3D Reconstruction by Shape-from-Focus Methods

Towards Robustness in Residue Number Systems

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