Wavefront reconstruction of discontinuous phase objects from optical deflectometry.

One of the challenges of phase measuring deflectometry is to retrieve the wavefront from objects that present discontinuities or non-differentiable gradient fields. Here, we propose the integration of such gradient fields based on an L p-norm minimization problem. The solution of this problem results in a nonlinear partial differential equation, which can be solved with a fast and well-known numerical method and does not depend on external parameters. Numerical reconstructions on both synthetic and experimental data are presented that demonstrate the capability of the proposed method.

Total retardance measurements based on the complex Fourier coefficients for the rotating polarizer analyzer system.

We propose a demodulation algorithm based on the calculus of the complex Fourier coefficients; we used a dual rotating polarizer-analyzer polarimeter to show the feasibility of our proposal. Our demodulation algorithm considers the frequency response obtained by the system, and its possible to calculate the total retardation, fast axis orientation and ellipticity of a sample. Our proposal does not require recovering the full Mueller matrix from getting those parameters. In addition, as the proposal does not use retarders for the measurement, the system presents potential applications for multi-wavelength measurements on phase retardation samples. We show experimental results showing the capabilities of our proposal in characterizing a polarization retardance sample.

Smart Multi-Sensor System for Remote Air Quality Monitoring Using Unmanned Aerial Vehicle and LoRaWAN.

Deaths caused by respiratory and cardiovascular diseases have increased by 10%. Every year, exposure to high levels of air pollution is the cause of 7 million premature deaths and the loss of healthy years of life. Air pollution is generally caused by the presence of CO, NO2, NH3, SO2, particulate matter PM10 and PM2.5, mainly emitted by economic activities in large metropolitan areas. The problem increases considerably in the absence of national regulations and the design, installation, and maintenance of an expensive air quality monitoring network. A smart multi-sensor system to monitor air quality is proposed in this work. The system uses an unmanned aerial vehicle and LoRa communication as an alternative for remote and in-situ atmospheric measurements. The instrumentation was integrated modularly as a node sensor to measure the concentration of carbon monoxide (CO), nitrogen dioxide (NO2), ammonia (NH3), sulfur dioxide (SO2), and suspended particulate mass PM10 and PM2.5. The optimal design of the multi-sensor system has been developed under the following constraints: A low weight, compact design, and low power consumption. The integration of the multi-sensor device, UAV, and LoRa communications as a single system adds aeeded flexibility to currently fixed monitoring stations.

Least-squares gamma estimation in fringe projection profilometry.

This paper introduces a novel, to the best of our knowledge, method to estimate and compensate the nonlinear gamma factor introduced by the optical system in fringe projection profilometry. We propose to determine this factor indirectly by adjusting the least-squares plane to the estimated phase coming from the reference plane. We only require a minimal set of three fringe sinusoidal images to estimate the gamma factor. This value can be used to rectify computational legacy data and also to generate and project the new set of fringe patterns for which we perform the inverse gamma compensation. Experimental results demonstrate the feasibility of the proposed method to estimate and correct the gamma distortion.

Tukey's robust M-estimator for phase demodulation of interferograms with nonuniform shifts.

In this paper, we introduce an iterative scheme for phase demodulation of interferograms with nonuniformly spaced phase shifts. Our proposal consists of two stages: first, the phase map is obtained through a least squares fitting; second, the phase steps are retrieved using a statistical robust estimator. In particular, we use Tukey's biweighted M-estimator because it can cope with both noisy data and outliers in comparison with the ordinary least squares estimator. Furthermore, we provide the frequency description of the algorithm and the phase demodulation allowing us to analyze the procedure and estimation according to the frequency transfer function (FTF) formalism for phase-shifting algorithms. Results show that our method can accurately retrieve the phase map and phase shifts, and it converges by the 10th iteration.

Exhaustive dithering algorithm for 3D shape reconstruction by fringe projection profilometry.

Three-dimensional (3D) shape reconstruction by projection of defocused binary patterns overcomes the nonlinearity introduced by the projector. Current patch-based procedures that generate dithered patterns are time consuming and are affected by the harmonics introduced through the tiling process. To overcome this problem, we propose a novel idea, to the best of our knowledge, to generate dithering patterns using the composition of two-dimensional patches as a stack of one-dimensional arrays obtained through an efficient deterministic approach. This procedure is a one-dimension optimization problem in the intensity domain, employing only a quarter of the fringe pitch. Furthermore, the unwanted distorting harmonics are eliminated using a Hilbert transform method. Both numerical simulations and experimental results verify the effectiveness of the proposal.

Spectral analysis for the generalized least squares phase-shifting algorithms with harmonic robustness.

We introduce the frequency transfer function (FTF) formalism for generalized least squares phase-shifting algorithms (GLS-PSAs), whose phase shifts are nonuniformly spaced. The GLS-PSA's impulsive response is found by computing the Moore-Penrose pseudoinverse. FTF theory allows analyzing these GLS-PSAs spectrally, as well as easily finding figures of merit such as signal-to-noise ratio (SNR) and harmonic rejection capabilities. We show simulations depicting that the SNR slightly decreases as the harmonic rejection robustness improves.

Dynamic 3D shape measurement by iterative phase shifting algorithms and colored fringe patterns.

In this work, we propose a novel technique to retrieve 3D shape of dynamic objects by the simultaneous projection of a fringe pattern and a homogeneous white light pattern, both coded in an RGB image. The first one is used to retrieve the phase map by an iterative least-squares method. The second one is used to match object pixels in consecutive images, acquired at various object positions. The proposed method successfully accomplishes the requirement of projecting simultaneously two different patterns. One extracts the object's information while the other retrieves the phase map. Experimental results demonstrate the feasibility of the proposed scheme.

Robust interferometer with external phase-shift control.

We describe a robust interferometer with external phase-shift control that does not require moving parts. The optical architecture resembles a common-path device in which the interfering waves propagate together in one collimated beam passing through the test sample. The collimated beam is incident on a calcite plate, which produces a polarization selective lateral translation and superposition of the reference and test waves. The characteristic features of the proposed interferometer, i.e. one-beam single-element scheme combined with external phase-shift control without moving parts, make a highly vibration insensitive device. Validation experiments are presented.

Three-dimensional shape profiling by out-of-focus projection of colored pulse width modulation fringe patterns.

Three-dimensional (3D) shape profiling by sinusoidal phase-shifting methods is affected by the non-linearity of the projector. To overcome this problem, the defocusing technique has become an important alternative to generate sinusoidal fringe patterns. The precision of this method depends on the binary pattern used and on the defocusing applied. To improve the defocusing technique, we propose the implementation of a color-based binary fringe patterns. The proposed technique involves the generation of colored pulse width modulation (PWM) fringe patterns, which are generated with different frequencies at the carrier signal. From an adequate selection of these frequencies, the colored PWM fringe patterns will lead to amplitude harmonics lower than the conventional PWM fringe patterns. Hence, the defocusing can decrease, and the 3D shape profiling can be more accurate. Numerical simulations and experimental results are presented as validation.

Error compensation in a pointing system based on Risley prisms.

Risley prisms are widely used for beam pointing in several optical systems. The exact solution for the inverse problem does not exist, except using numerical methods. However, the errors introduced by misalignment are usually greater than the approximation errors. We present a novel method to compensate alignment errors in pointing systems based on Risley prisms. The prism model that we used is based on paraxial approximation with an additional vector to compensate typical alignment errors. Simulation and experimental results show that the improvement in pointing accuracy is achievable even in comparison with exact ray tracing methods.

Generalized phase-shifting algorithms: error analysis and minimization of noise propagation: erratum.

There are a couple of typos in our recent paper Appl. Opt.55(6), 1461-1469 (2016)APOPAI0003-693510.1364/AO.55.001461.

Image segmentation by nonlinear filtering of optical Hough transform.

The identification and extraction (i.e., segmentation) of geometrical features is crucial in many tasks requiring image analysis. We present a method for the optical segmentation of features of interest from an edge enhanced image. The proposed method is based on the nonlinear filtering (implemented by the use of a spatial light modulator) of the generalized optical Hough transform and is capable of discriminating features by shape and by size. The robustness of the method against noise in the input, low contrast, or overlapping of geometrical features is assessed, and experimental validation of the working principle is presented.

Generalized phase-shifting algorithms: error analysis and minimization of noise propagation.

Phase shifting is a technique for phase retrieval that requires a series of intensity measurements with certain phase steps. The purpose of the present work is threefold: first we present a new method for generating general phase-shifting algorithms with arbitrarily spaced phase steps. Second, we study the conditions for which the phase-retrieval error due to phase-shift miscalibration can be minimized. Third, we study the phase extraction from interferograms with additive random noise, and deduce the conditions to be satisfied for minimizing the phase-retrieval error. Algorithms with unevenly spaced phase steps are discussed under linear phase-shift errors and additive Gaussian noise, and simulations are presented.

Color-fringe pattern profilometry using a generalized phase-shifting algorithm.

In order to overcome the limitations of the sequential phase-shifting fringe pattern profilometry for dynamic measurements, a color-channel-based approach is presented. The proposed technique consists of projecting and acquiring a colored image formed by three sinusoidal phase-shifted patterns. Therefore, by using the conventional three-step phase-shifting algorithm, only one color image is required for phase retrieval each time. However, the use of colored fringe patterns leads to a major problem, the color crosstalk, which introduces phase errors when conventional phase-shifting algorithms with fixed phase-shift values are utilized to retrieve the phase. To overcome the crosstalk issue, we propose the use of a generalized phase-shifting algorithm with arbitrary phase-shift values. The simulations and experimental results show that the proposed algorithm can significantly reduce the influence of the color crosstalk.

Wrapping-free phase retrieval with applications to interferometry, 3D-shape profiling, and deflectometry.

Phase unwrapping is probably the most challenging step in the phase retrieval process in phase-shifting and spatial-carrier interferometry. Likewise, phase unwrapping is required in 3D-shape profiling and deflectometry. In this paper, we present a novel phase retrieval method that completely sidesteps the phase unwrapping process, significantly eliminating the guessing in phase reconstruction and thus decreasing the time data processing. The proposed wrapping-free method is based on the direct integration of the spatial derivatives of the interference patterns under the single assumption that the phase is continuous. This assumption is valid in most physical applications. Validation experiments are presented confirming the robustness of the proposed method.

Real-time pattern recognition using an optical generalized Hough transform.

We present some pattern recognition applications of a generalized optical Hough transform and the temporal multiplexing strategies for dynamic scale and orientation-variant detection. Unlike computer-based implementations of the Hough transform, in principle its optical implementation does not impose restrictions on the execution time or on the resolution of the images or frame rate of the videos to be processed, which is potentially useful for real-time applications. Validation experiments are presented.

Generation of phase-shifting algorithms with N arbitrarily spaced phase-steps.

Phase-shifting (PS) is an important technique for phase retrieval in interferometry (and three-dimensional profiling by fringe projection) that requires a series of intensity measurements with known phase-steps. Usual PS algorithms are based on the assumption that the phase-steps are evenly spaced. In practice, however, this assumption is often not satisfied exactly, which leads to errors in the recovered phase. In this work we present a systematic algebraic approach for generating general PS algorithms with N arbitrarily spaced phase-steps, which present advantages (e.g., the PS error can be avoided) over known algorithms that assume equally spaced phase-steps. Simulations are presented.

Phase retrieval from one partial derivative.

Phase objects can be characterized using well-known methods such as shear interferometry and deflectometry, which provide information on the partial derivatives of the phase. It is often believed that for phase retrieval it is strictly necessary to have knowledge of two partial derivatives in orthogonal directions. In the praxis, this implies that the measurements have to be performed along two dimensions, which often requires a rotation of the object or rotation of the shear direction. This is time consuming and errors can be easily generated from the process of rotation, especially for image registration in the axial direction. In the present Letter, we will demonstrate that only one partial derivative often suffices to recover the phase, and we will discuss under which conditions that is possible. Simulations and validation experiments are presented.

One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns.

Deflectometry is a well-known method to characterize pure phase objects by measuring the deformation of fringes. In principle, the retrieved magnitude is the partial derivative of the phase along the coordinate orthogonal to the fringes. In order to recover the phase it is necessary to know the derivatives in two orthogonal directions, which is usually achieved by rotating 90° the original fringes and acquiring a new deformed pattern. This "time-multiplexed" two-dimensional deflectometry is a time-consuming operation if the goal is to characterize phase objects in real time. In the present paper we propose a kind of two-dimensional deflectometry that allows acquisition of fringe patterns in two orthogonal directions in a single frame. The proposed procedure utilizes a two-dimensional ("additive") fringe pattern that allows the application of Takeda's method to each coordinate independently. The advantage of the method (with respect to the traditional one) is that it simplifies the setup and reduces the acquisition time. Validation experiments are presented.