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This paper presents a novel reconstruction method based on multi-refraction to remedy the non-linear refraction, which recovers 3D geometry from 2D images. To simplify calibration, this paper prove that air calibration can replace underwater calibration. Finally, a multi-line rotating structured is designed to improve the measurement efficiency without additional equipment. The results of the clean water experiment show that the measurement accuracy is 1.91 mm within the 1000–2000 mm depth of field. The qualitative analysis demonstrates our method can reconstruct the surface topography of underwater U and N letter boards. In different Nephelometric Turbidity Unit (NTU) (as NTU = 0.156 mg/ml) environment, a circular plate with a diameter of 220 mm is measured, the shape reconstruction error (whether the reconstructed shape is a circular, square, or other) is 1.74 mm; the diameter measurement result is 218 mm, so the size error of the circular plate is 2 mm. [ABSTRACT FROM AUTHOR]

• The under-sampling problem with folded frequencies in multi-surface WPI is analyzed. • Using active frequency folding undersampling, the correct frequency can be obtained as a solid basis for phase demodulation. • The spectral lines near the frequency peaks are redivided, and the frequencies can be accurately solved. • Based on the solved frequency of each harmonic, extended measurement distances can be reached, and efficient phase demodulation is achieved. The difficulty in multi-surface measurements comes from that each measured surface contributes harmonics to the fringe patterns, and the wavelength-shifting technique is an efficient technique that allows the harmonics to have different frequencies. When under-sampling in multi-surface measurements occurs, harmonic frequencies can not be detected by the spectrum directly. As a typical case in under-sampling, the problem of harmonic frequencies folded in the obtained spectrum as false frequencies are addressed in this paper. This contribution demonstrates an efficient algorithm enabling the detection of real and accurate harmonic frequencies, whereby the wavefronts can also be correctly reconstructed. Then the frequency detection accuracy is improved by redividing a smaller spectral interval of the selected frequency bands. Comparative studies based on the Zernike polynomials verify the effectiveness of the proposed method. Experiments for a transparent flat were performed, and the maximum PV Q , PV, and maximum RMS errors for repeatable measurements verified the effectiveness of the proposed algorithm in practical application. [ABSTRACT FROM AUTHOR]

[Display omitted] • The front surface, rear surface, thickness variation, and refractive index inhomogeneity information of the transparent multi-surface optics to be tested can be reconstructed simultaneously from a set of interferograms; • Two convolutional fitted windows have been redesigned which have explicit function expressions and better window performance, they can be stretched to free lengths; • The factors which affect the performance of multi-surface reconstruction are fully analyzed, which provides a solid foundation for the analysis of sampling validity during multi-surface interferometry; • A phase-shift matching method without strict limitation of cavity lengths is developed, and four-surface detection in flexible positions is realized. Consisting of the front and rear surfaces of the measured optics and two reference surfaces, the four-surface interferometry system can obtain parameters of the measured surface profiles and refractive index inhomogeneity by acquiring and decoding the fringe patterns of the measured plate and the rear reference plate respectively. However, the difficulty of such measurement comes from the complex superposition of at least six interference sub-signals, and the setting of the phase-shift interval determines whether the harmonics can be reconstructed or not. To improve the measurement accuracy, two kinds of convolutional fitting window functions are redesigned based on the Hanning window to enhance the function performance. Then, by analyzing the reconstruction errors of each wavefront under different combinations of sampling intervals and cavity lengths during four-surface measurements, an error quantization matrix obtained by triple iteration is constructed. By setting a threshold condition, the optimal phase-shifts under different cavity lengths can be matched adaptively. The multi-surface measurement accuracy and reliability of adaptive phase-shift matching are verified in the simulation. The interferogram acquisition and phase demodulation of two transparent flats were performed using a Fizeau wavelength-tuning interferometer, and the reconstruction errors of the surfaces in the repetitive measurements further verified the reliability of the proposed method. [ABSTRACT FROM AUTHOR]

• Automatic sampling matching and simultaneous full-aperture multi-surface measurement at flexible cavities can be realized. • Semi-blind multi-surface wavelength-shifting algorithm without prior and whole cavity length determinations is devised. • A DTFT-based global spectrum modulation algorithm is applied to obtain accurate harmonic frequencies. • A thorough sampling performance analysis and experiment are performed. The core issue of multi-surface wavelength-shifting interferometry at free cavity lengths is mainly centered on adaptively matching the tuning step. To avoid harmonic aliasing and undersampling, the discrete-time Fourier transform method with global spectrum densification is used to comprehensively analyze the spectral characteristics under different sampling cases and cavity lengths, then a composite multi-surface wavelength-shifting interferometry algorithm is proposed accordingly. The proposed method includes a preprocessing algorithm and three spectral criteria to identify and avoid undesirable sampling problems for in-situ measurements. The algorithm performance is quantitatively evaluated from several aspects of the simulation analysis. A flat plate with an average thickness of 20 mm was repeatedly measured using a Fizeau wavelength-shifting interferometer in the experiments, and the PV Q and RMS errors at different measurement distances were less than 5 mm and 1 mm, respectively. [ABSTRACT FROM AUTHOR]