Multibaseline Research Articles

Multi-Baseline Bistatic SAR Three-Dimensional Imaging Method Based on Phase Error Calibration Combining PGA and EB-ISOA

Tomographic synthetic aperture radar (TomoSAR) is an advanced three-dimensional (3D) synthetic aperture radar (SAR) imaging technology that can obtain multiple SAR images through multi-track observations, thereby reconstructing the 3D spatial structure of targets. However, due to system limitations, the multi-baseline (MB) monostatic SAR (MonoSAR) encounters temporal decorrelation issues when observing the scene such as forests, affecting the accuracy of the 3D reconstruction. Additionally, during TomoSAR observations, the platform jitter and inaccurate position measurement will contaminate the MB SAR data, which may result in the multiplicative noise with phase errors, thereby leading to the decrease in the imaging quality. To address the above issues, this paper proposes a MB bistatic SAR (BiSAR) 3D imaging method based on the phase error calibration that combines the phase gradient autofocus (PGA) and energy balance intensity-squared optimization autofocus (EB-ISOA). Firstly, the signal model of the MB one-stationary (OS) BiSAR is established and the 3D imaging principle is presented, and then the phase error caused by platform jitter and inaccurate position measurement is analyzed. Moreover, combining the PGA and EB-ISOA methods, a 3D imaging method based on the phase error calibration is proposed. This method can improve the accuracy of phase error calibration, avoid the vertical displacement, and has the noise robustness, which can obtain the high-precision 3D BiSAR imaging results. The experimental results are shown to verify the effectiveness and practicality of the proposed MB BiSAR 3D imaging method.

Position and Attitude Determination in Urban Canyon with Tightly Coupled Sensor Fusion and a Prediction-Based GNSS Cycle Slip Detection Using Low-Cost Instruments.

We present a position and attitude estimation algorithm of moving platforms based on the tightly coupled sensor fusion of low-cost multi baseline GNSS, inertial, magnetic and barometric observations obtained by low-cost sensors and affordable dual-frequency GNSS receivers. The sensor fusion algorithm is realized by an Extended Kalman Filter and estimates the states including GNSS receiver inter-channel biases, integer ambiguities and non-GNSS receiver biases. Tightly coupled sensor fusion increases the reliability of the position and attitude solution in challenging environments such as urban canyons by utilizing the inertial observations in case of GNSS outage. Moreover, GNSS observations can be efficiently used to mitigate IMU sensor drifts. Standard GNSS cycle slips detection methods, such as the application of triple differences or linear combinations such as Melbourne-Wübbena combination and the phase ionospheric residual extended TurboEdit method. However, these techniques are not well suited for the localization in quickly changing environments such as urban canyons. We present a new method of tightly coupled sensor fusion supported by a prediction based cycle slip detection technique, applied to a GNSS setup using three antennas leading to multiple moving baselines on the platform. Thus, not only the GNSS signal properties but also the dynamics of the moving platform are considered in the cycle slip detection. The developed algorithm is tested in an open-sky validation measurement and two sets of measurement in an urban canyon area. The sensor fusion algorithm processes the data sets using the proposed prediction-based cycle slip method, the loss-of-lock indicator-based, and for comparison, the Melbourne-Wübbena and the TurboEdit cycle slip detection methods are also included. The obtained position and attitude estimation results are compared to the internal solution of raw data source GNSS receivers and to the observations of a high-accuracy GNSS/INS unit including a fiber optic gyro. The validation test confirms the proper cycle slip detection in an ideal environment. The more challenging urban canyon test results show the reliability and the accuracy of the proposed method. In the case of the second urban canyon test, the proposed method improved the integer ambiguity resolution success rate by 19% and these results show the lowest horizontal and vertical coordinate distortion in comparison of the linear combination and the loss-of-lock-based cycle slip methods.

A Cluster-Analysis and Convex Hull-Based Fast Large-Scale Phase Unwrapping Method for Single- and Multibaseline SAR Interferograms

For synthetic aperture radar (SAR) interferometry (InSAR), phase unwrapping (PU) is an important and difficult step. Due to the high computational complexities of the classical and skilled PU methods, the size and number of interferograms to be processed together must be considered in InSAR applications. However, with the rapid development of InSAR technology, the scale of interferometric data has significantly increased, which has brought two difficulties to InSAR processing: 1) excessive memory consumption and 2) unacceptably long computing time. To solve these problems, a fast large-scale PU method based on cluster analysis and convex hull (CCFLS) is proposed in this paper. It was developed for single-baseline (SB) InSAR and refined under the multibaseline (MB) two-stage programming approach InSAR (TSPA-InSAR) framework. The main idea of CCFLS is to reduce the consumption of computing resources by discarding the PU processing of worthless low quality areas. The key to this work is to determine the discarded region at a low computational cost while ensuring the same PU accuracy for the remaining region as global processing. The theoretical analysis demonstrates the advantage of the CCFLS method for large-scale InSAR data processing, and experiments also verify that CCFLS can greatly reduce the consumption of computing resources while ensuring the PU accuracy of the solved region.

A Novel Phase Compensation Method for Urban 3D Reconstruction Using SAR Tomography

Synthetic aperture radar (SAR) tomography (TomoSAR) has been widely used in the three-dimensional (3D) reconstruction of urban areas using the multi-baseline (MB) SAR data. For urban scenarios, the MB SAR data are often acquired by repeat-pass using the spaceborne SAR system. Such a data stack generally has long time baselines, which result in different atmospheric disturbances of the data acquired by different tracks. These factors can lead to the presence of phase errors (PEs). PEs are multiplicative noise for observation data, which can cause diffusion and defocus in TomoSAR imaging and seriously affect the extraction of target 3D information. In this paper, we combine the methods of the block-building network (BBN) and phase gradient autofocus (PGA) to propose a novel phase compensation method called BBN-PGA. The BBN-PGA method can effectively and efficiently compensate for PEs of the MB SAR data over a wide area and improve the accuracy of 3D reconstruction of urban areas. The applicability of this proposed BBN-PGA method is proved by using simulated data and the spaceborne MB SAR data acquired by the TerraSAR-X satellite over an area in Barcelona, Spain.

BM3D Denoising for a Cluster-Analysis-Based Multibaseline InSAR Phase-Unwrapping Method

Multibaseline (MB) phase unwrapping (PU) is a key processing technique in MB interferometric synthetic aperture radar (InSAR). As one of the most popular methods, the cluster analysis (CA)-based MBPU method often suffers from the problem of low noise robustness. Therefore, the block-matching and 3D filtering (BM3D) algorithm, one of the most effective filtering methods for image denoising, is applied to improve the performance of the method. Five different filtering strategies for applying BM3D are proposed in the paper: interferogram filtering (IFF), intercept filtering (ICF), cluster number filtering (CNF), unwrapped phase filtering (UPF), and simultaneous filtering (STF). In particular, while keeping the general structure of BM3D, four different similarity measures are defined for interferograms, intercepts, clusters, and unwrapped phases to accommodate the special characteristics of different filtering objects. Experiments on synthesized and real InSAR datasets prove their feasibility and effectiveness, and the experiment results show that (1) the PU accuracy and robustness of the CA-based MBPU method can be greatly improved by adding BM3D denoising; (2) simultaneous filtering of interferograms, intercepts, cluster numbers, and unwrapped phases works best, but with the worst time complexity; (3) when filtering is performed for only one object of the CA-based MBPU method, the filtering effect of CNF and UPF is better than that of IFF and ICF; and (4), considering the three indicators of PUSR, NRSE, and time consumption, CNF and UPF should be the best choices.

Comparative Study of DEM Reconstruction Accuracy Between Single- and Multibaseline InSAR Phase Unwrapping

Phase unwrapping (PU) is a key processing step in interferometric synthetic aperture radar (InSAR). To date, a number of skillful single-baseline (SB) and multibaseline (MB) PU methods exhibiting different advantages have been proposed. However, as the basic principles of SB and MB PUs are essentially different, it is difficult to effectively and systematically compare the performance of SB and MB PUs, despite the knowledge that this type of analysis is important for allowing the ever-increasing number of InSAR practitioners to choose a suitable approach for practical applications and to optimally plan future InSAR satellite missions. Recently, the framework of the two-stage programming approach (TSPA) was proposed, and this allows for the majority of the existing SB PU methods to be transplanted into the MB domain, to allow practitioners to feasibly and comprehensively compare SB and MB PU methodologies. In this study, the digital elevation model (DEM) reconstruction accuracy is compared between the classical SB PU methods and their corresponding TSPA-framework-based MB PU methods using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L^{p}$ </tex-math></inline-formula> -norm model. Interestingly, we observed that although the number of PU residues in the MB case is larger than that in the SB case, the MB PU performance is better. The reason for this is that the type of MB residue is typically dipole, so the average length of the required MB branch-cut is shorter than that of SB. It is also demonstrated that the TSPA framework can effectively improve the PU accuracy of many existing SB PU methods when the number of input interferograms is sufficient.

CANet: An Unsupervised Deep Convolutional Neural Network for Efficient Cluster-Analysis-Based Multibaseline InSAR Phase Unwrapping

Multibaseline (MB) phase unwrapping (PU) is a vital processing procedure for MB synthetic aperture radar interferometry (InSAR) signal processing and can improve the traditional InSAR by changing the ill-posed problem to the well-posed problem. The existing research has shown that the MB PU problem can be successfully converted into an unsupervised cluster analysis problem. Using the high feature descriptiveness of the deep learning technique, an unsupervised deep convolutional neural network, referred to as CANet, is proposed to cluster all the pixels into different groups according to the input’s recognizable pattern of the ambiguity number of the MB interferometric phase. Subsequently, we extend our previous two-stage programming-based MB processing approach (TSPA) to processing MB PU on a sparse irregular network, which is established from the clustering result of CANet. Both theoretical analysis and experimental results show that the proposed method is an effective MB PU method, and its execution time is drastically lower than those of many classical MB PU methods.

Artificial Intelligence In Interferometric Synthetic Aperture Radar Phase Unwrapping: A Review

Interferometric synthetic aperture radar (InSAR) is a radar technique widely used in geodesy and remote sensing applications, e.g., topography reconstruction and subsidence estimation. Phase unwrapping (PU) is one of the key procedures of InSAR signal processing. Artificial intelligence (AI) techniques have proven to be potentially powerful in many fields and have been introduced into the PU domain, achieving superior performance. In this article, we provide a comprehensive overview of AI-based PU techniques in InSAR. We survey the AI-based single-baseline (SB) PU methods and then review the AI techniques related to multibaseline (MB) PU. In addition, we show several experimental examples of these methods, from both simulated and real InSAR data sets, which gives readers an overview of AI-based PU processing's potential and limitations. It is our hope that this article will provide researchers with guidelines and inspiration to further enhance the development of AI-based PU.

Improved DEM Reconstruction Method Based on Multibaseline InSAR

Digital elevation models (DEMs) are vital in the geosciences and many other fields. Interferometric synthetic aperture radar (InSAR), an advanced earth observation technology, has shown its potential in DEM reconstruction. Multi baseline InSAR (MB-InSAR) is currently improving the precision of DEM reconstruction by combining multiple interferograms. However, MB-InSAR for DEM generation can result in severe decorrelation, which may cause significant gaps in the final DEM product. To solve this problem, an improved MB-InSAR DEM reconstruction method is proposed in this study, which we term as the dynamic DEM calculation algorithm. The proposed method can estimate the DEM pixel-by-pixel, which allowed us to select the interferograms dynamically and therefore minimize the void values. For the performance test of the proposed method, 25 ascending and 20 descending TerraSAR-X images over Heifangtai (China) were collected to form repeat-pass interferograms and produce the DEM using the proposed method. Results showed that the number of valid pixels increased by approximately 20% compared with the traditional MB-InSAR DEM reconstruction method without loss of precision, thereby illustrating the feasibility of the proposed method.

A Modified Capon Method for SAR Tomography Over Forest

The 3-D structure of forests is an important indicator for evaluating forest health and can provide data support for ecological monitoring and protection. Synthetic aperture radar (SAR) tomography (TomoSAR) is an important technology for forest structure estimation using the multibaseline (MB) SAR data stacks. The spectral estimators, such as the Capon method, are usually used to estimate the 3-D reflectivity along elevation direction. In this letter, a modified Capon (M-Capon) method is proposed to improve the estimated profiles of ground and canopy scatterers in the elevation direction, including the estimation accuracy and resolution. This method combines the ideas of the CLEAN algorithm with the Capon method and uses iteration to improve the resolution and accuracy of the estimation. The effectiveness of the M-Capon method is demonstrated using the simulated data and the MB SAR data acquired by the P-band airborne SAR system over the Saihanba Forest Farm in Hebei, China.

The Performance of Relative Height Metrics for Estimation of Forest Above-Ground Biomass Using L- and X-Bands TomoSAR Data

Both synthetic aperture radar tomography (TomoSAR) profiles and light detection and ranging (LiDAR) waveforms are the responses of a 3-D canopy structure. Relative height (RH) metrics are extensively applied for forest biophysical parameters [i.e., the forest height and above-ground biomass (AGB)] estimation in full-waveform LiDAR studies. However, the use of RH metrics to forest biophysical estimation with TomoSAR profiles is limited due to the estimation error in the ground peak. To overcome this problem, RH metrics were redefined to avoid directly estimating the ground peak in this article. The redefined RH metrics were examined based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> - and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> -bands multibaseline (MB) SAR data simulated by LandSAR, which was a coherent backscattering model of 3-D forest canopies with the capability of MB data simulation at a landscape scale. First, the performance of LandSAR in modeling the tomographic features was verified over mountainous areas using the reference DSM and LiDAR digital terrain model acquired in 2012 in the frame of the Daxinganling campaign. Subsequently, the tomographic profiles were retrieved from the simulated MB data by using the Capon method. Additionally, redefined RH metrics were derived and then applied for retrieving the forest height. The highest performance corresponded to the combination of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> -band redefined RH metrics, with a correlation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0.863. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> -band redefined RH metrics performed worst due to limited penetration. Finally, the estimated forest height was used for the AGB retrieval with a height-to-biomass allometry. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and root-mean-square error of the forest AGB estimated using the combined model were 0.814 and 28.566 t/ha, respectively, compared with the reference forest AGB. All findings demonstrated that the redefined RH metrics had the potential of forest height and AGB retrieval using low frequencies TomoSAR data.

Baseline Design for Multibaseline InSAR System: A Review

Multibaseline (MB) synthetic aperture radar (SAR) interferometry (InSAR) is an exciting and growing remote sensing technique, which can improve the traditional InSAR from the ill-posed problem to the well-posed problem. Baseline lengths play an important role in the MB InSAR system design. It is fair to say that the baseline lengths of the MB InSAR system directly determine the quality of the MB InSAR products. In this article, we present an overview of the established MB baseline design criteria and requirements for the MB InSAR digital elevation model (DEM) generation, which includes the effect of the baseline lengths and their estimation errors on ambiguity height, DEM accuracy, decorrelation, and MB phase unwrapping robustness. In addition, we prospectively apply the geometric programming technique to combine those effects together for obtaining the global optimal baseline lengths of the MB InSAR system. It is our hope that this review will provide guidelines to future researchers to enhance further the new airborne and spaceborne MB InSAR system developments.

Rapid Surface Large-Change Monitoring by Repeat-Pass GEO SAR Multibaseline Interferometry

Fast observations of rapid surface large-changes are demanded in disaster evaluations and scientific studies. Digital elevation model (DEM) differencing before and after the events is an effective way to retrieve the changes. Owing to a short repeat cycle, geosynchronous synthetic aperture radar (GEO SAR) systems can quickly obtain repeat-pass data and generate postevent DEMs by interferometry. However, interferometric baselines under its quick revisit cases are short, resulting in generating low-accuracy postevent DEMs. Moreover, surface large-changes can bring height ambiguity problems under the single-baseline interferometric processing. In this letter, we address the problem through a multibaseline (MB) processing. Since GEO SAR MB data can derive from the repeat-pass interferometric data of different subapertures and revisits, a subaperture-decomposition-based temporal and spatial MB method is proposed. The simulation results verify the effectiveness of the proposed method, where the quickly generated postevent DEM can help to realize the rapid large-elevation change observations.

Bayesian Filtering Multi-Baseline Phase Unwrapping Method Based on a Two-Stage Programming Approach

Phase unwrapping (PU) has been a key step in the processing of interferometric synthetic aperture radar (InSAR) data, and its processing accuracy will directly affect the reconstruction results of digital elevation models (DEMs). The traditional single-baseline (SB) PU must be calculated under continuity assumptions. However, multi-baseline (MB) PU can get rid of the limitation of continuity assumption, so reasonable results can be obtained in regions with large gradient changes. However, the poor noise robustness of MBPU has always been a key problem. To address this issue, we transplant three Bayesian filtering methods with a two-stage programming approach (TSPA), and propose corresponding MBPU models. First, we propose a gradient-estimation method based on the first step of TSPA, and then the corresponding PU model is determined according to different Bayesian filtering. Finally, the wrapped phase can be obtained by unwrapping, one by one, using an effective quality map based on heapsort. These methods can improve the robustness of the MBPU methods. More significantly, this paper establishes a novel TSPA-based Bayesian filtering MBPU framework for the first time. This is of great significance for broadening the research of MBPU. The proposed methods experiments on simulated and real MB InSAR datasets. From the results, we can see that the TSPA-based Bayesian filtering MBPU framework can significantly improve the robustness of the MBPU method.

Extended Phase Unwrapping Max-Flow/Min-Cut Algorithm for Multibaseline SAR Interferograms Using a Two-Stage Programming Approach.

Multi-baseline (MB) phase unwrapping (PU) is a key step of MB synthetic aperture radar (SAR) interferometry (InSAR). Compared with the traditional single-baseline (SB) PU, MB PU is applicable to the area where topography varies violently without obeying the phase continuity assumption. A two-stage programming MB PU approach (TSPA) proposed by H. Yu. builds the link between SB and MB PUs, so many existing classical SB PU methods can be transplanted into the MB domain. In this paper, an extended PU max-flow/min-cut (PUMA) algorithm for MB InSAR using the TSPA, referred to as TSPA-PUMA, is proposed, consisting of a two-stage programming procedure. In stage 1, phase gradients are estimated based on Chinese remainder theorem (CRT). In stage 2, a Markov random field (MRF) model of PUMA is designed for modeling local contextual dependence based on the phase gradients obtained by stage 1. Subsequently, the energy of the MRF model is minimized by graph cuts techniques. The experiment results illustrate that the TSPA-PUMA method can drastically enhance the accuracy of the original PUMA method in the rugged area, and is more efficient than the original TSPA method. In addition, the noise robustness of TSPA-PUMA can be improved through adding more interferograms with different baseline lengths.

Large-Scale Multibaseline Phase Unwrapping: Interferogram Segmentation Based on Multibaseline Envelope-Sparsity Theorem

Multibaseline (MB) phase unwrapping (PU) is a critical processing step for the MB synthetic aperture radar interferometry (InSAR). Compared with the traditional single-baseline (SB) PU, MB PU has wider application scope on the study area with strong phase variation, because it can overcome the limitation of the Itoh condition. Since most of the MB PU methods need to process multiple interferograms simultaneously, the size of the input interferograms will pose unique challenges when it exceeds the limit of computational capabilities. Until now, the research achievements related to large-scale (LS) MB PU have been quite limited. To deal with such case, we propose a technique for applying the two-stage programming-based MB PU method (TSPA) proposed by Yu and Lan to the LS MB InSAR data set in this paper. To be specific, the MB $L^\kappa $ -norm envelope-sparsity theorem is proposed and proved first, which gives a sufficient condition to exactly guarantee the consistency between local and global TSPA solutions. Afterward, based on the MB $L^\kappa $ -norm envelope-sparsity theorem, we put forward an interferogram tiling strategy, whereby each LS interferogram in the input MB InSAR data set is partitioned into a set of several smaller sub-interferograms that can be unwrapped individually by TSPA in parallel or in series. Both theoretical analysis and experimental results show that the proposed tiling strategy is effective for the LS MB PU problem.

Forest SAR Tomography: Principles and Applications

Synthetic-aperture radar (SAR) systems are widely used for monitoring vegetation and forested environments. Those that offer large-scale coverage, short revisiting times, and an under-foliage wave-penetration capability have been broadly employed to extract information about the forest structure. Similarly, 3D forest structures that serve as important indicators of productivity and biomass levels can be efficiently estimated using the SAR tomography (TomoSAR) processing technique through multibaseline (MB) image acquisition. This article provides an overview of the main developments in forest TomoSAR during the past two decades.

Optimal Baseline Design for Multibaseline InSAR Phase Unwrapping

Multibaseline (MB) synthetic aperture radar interferometry (InSAR) has the potential to improve the traditional single-baseline InSAR from the ill-posed problem to the well-posed problem. It is because MB InSAR phase unwrapping (PU) can take advantage of the baseline diversity to significantly increase the ambiguity intervals of interferometric phases, so it completely overcomes the limitation of the phase continuity assumption. However, due to the mathematical foundation of most of the MB PU methods, i.e., the Chinese remainder theorem (CRT), has poor measurement bias robustness (measurement bias could be caused by surface deformation, atmospheric artifact, and phase noise), CRT is sensitive to the normal baseline length. In other words, even if we choose the same CRT-based MB PU method, different system baseline lengths could result in different PU performances. Therefore, how to choose the baseline lengths to optimize CRT performance is crucial to all the CRT-based MB PU methods. In this paper, a nonlinear mixed-integer programming-based baseline design criterion (referred to NIP criterion) is proposed to maximize the measurement bias tolerance of the CRT-based MB PU. The optimality condition of the NIP criterion quantitatively provides important instructions for the CRT-based MB PU applications and offers significant guidance in the development of the practical MB InSAR system. The experiment results are shown to verify the effectiveness of the NIP criterion by using three representative CRT-based MB PU methods.

Impact of Backscatter in Pol-InSAR Forest Height Retrieval Based on the Multimodel Random Forest Algorithm

For forest with complex structure, the vertical structure backscatter is influenced by a combination of factors, including the frequencies of the radar waves and the forest biophysical parameters (i.e., density, species). The backscatter-induced error is thus a critical element in limiting the accuracy of polarimetric synthetic aperture radar interferometry (Pol-InSAR) forest height inversion based on a single model. It is, therefore, necessary to select the optimal backscatter profile from the multiple possible solutions in each pixel of the test site. In this letter, the impacts of backscatter in forest height estimation based on the models of random volume over ground (RVoG) ( $\sigma >0$ ), RVoG ( $\sigma ), and Gaussian vertical backscatter (GVB) were investigated in the complex plane, and then with the combined use of Pol-InSAR and light detection and ranging (LiDAR), a random forest (RF) classifier is trained to obtain the optimal backscatter function and Pol-InSAR forest height from the results based on the different models in each pixel. The proposed method was tested with single- and multi baseline Pol-InSAR data in the P-band, and the root-mean-square errors (RMSEs) of the proposed approach were 2.85 and 2.69 m, respectively, which represented average improvements of 20.6% and 17.7% over the optimal single-model inversion.

Phase Unwrapping in InSAR : A Review

Synthetic aperture radar (SAR) interferometry (InSAR) is primarily used in remote-sensing applications and has created a new class of radar data that has significantly evolved during the last couple of decades. Most of the InSAR applications (e.g., topographic mapping and deformation monitoring) typically use a technique called phase unwrapping (PU). In this article, we present an overview of PU techniques in InSAR signal processing. First, we review the established single-baseline (SB) PU methods and then describe innovative PU techniques and concepts related to multibaseline (MB) PU and large-scale (LS) PU. In addition, we discuss several numerical processing examples of these PU techniques. It is our hope that this review will provide guidelines to future researchers to enhance further PU algorithmic developments.