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1.
Am J Transplant ; 2024 Aug 07.
Artigo em Inglês | MEDLINE | ID: mdl-39097095

RESUMO

Hybrid immunity, resulting from a combination of SARS-CoV-2 infection and vaccination, offers robust protection against COVID-19 in the general population. However, its impact on immunocompromised patients remains unexplored. We investigated the effect of hybrid immunity against the Omicron variant in a population of kidney transplant recipients receiving the fourth dose mRNA monovalent vaccination. By extracting data from the clinical records and performing individual interviews, participants were categorized into the hybrid cohort (previously infected and vaccinated individuals) and the vaccine cohort (vaccinated-only individuals). The study comprised 1114 participants, 442 in the hybrid and 672 in the vaccine cohorts. From April 2022 to August 2023, 286 infections, 38 hospitalizations and 9 deaths were reported. The cumulative incidence of infection was 12.1% (95% confidence interval [CI], 9.03-16.03) for the hybrid cohort and 36.54% (95% CI, 32.81-40.54) for the vaccine cohort after 300 days of follow-up. Hybrid immunity was associated to a 72% lower risk of infection (adjusted hazard ratio, 0.28; 95% CI, 0.21-0.38) and a 96% lower risk of hospitalization (adjusted hazard ratio, 0.04; 95% CI, 0.01-0.32). No deaths occurred in the hybrid cohort. Hybrid immunity was associated with a lower incidence of SARS-CoV-2 infection and severe COVID-19, underscoring its importance for risk stratification in this vulnerable patient population.

2.
Sensors (Basel) ; 24(2)2024 Jan 14.
Artigo em Inglês | MEDLINE | ID: mdl-38257607

RESUMO

The massive deployment of smart meters in most Western countries in recent decades has allowed the creation and development of a significant variety of applications, mainly related to efficient energy management. The information provided about energy consumption has also been dedicated to the areas of social work and health. In this context, smart meters are considered single-point non-intrusive sensors that might be used to monitor the behaviour and activity patterns of people living in a household. This work describes the design of a short-term behavioural alarm generator based on the processing of energy consumption data coming from a commercial smart meter. The device captured data from a household for a period of six months, thus providing the consumption disaggregated per appliance at an interval of one hour. These data were used to train different intelligent systems, capable of estimating the predicted consumption for the next one-hour interval. Four different approaches have been considered and compared when designing the prediction system: a recurrent neural network, a convolutional neural network, a random forest, and a decision tree. By statistically analysing these predictions and the actual final energy consumption measurements, anomalies can be detected in the undertaking of three different daily activities: sleeping, breakfast, and lunch. The recurrent neural network achieves an F1-score of 0.8 in the detection of these anomalies for the household under analysis, outperforming other approaches. The proposal might be applied to the generation of a short-term alarm, which can be involved in future deployments and developments in the field of ambient assisted living.


Assuntos
Inteligência Ambiental , Humanos , Inteligência , Redes Neurais de Computação , Sono
3.
Sensors (Basel) ; 21(19)2021 Sep 30.
Artigo em Inglês | MEDLINE | ID: mdl-34640856

RESUMO

The research interest on location-based services has increased during the last years ever since 3D centimetre accuracy inside intelligent environments could be confronted with. This work proposes an indoor local positioning system based on LED lighting, transmitted from a set of beacons to a receiver. The receiver is based on a quadrant photodiode angular diversity aperture (QADA) plus an aperture placed over it. This configuration can be modelled as a perspective camera, where the image position of the transmitters can be used to recover the receiver's 3D pose. This process is known as the perspective-n-point (PnP) problem, which is well known in computer vision and photogrammetry. This work investigates the use of different state-of-the-art PnP algorithms to localize the receiver in a large space of 2 × 2 m2 based on four co-planar transmitters and with a distance from transmitters to receiver up to 3.4 m. Encoding techniques are used to permit the simultaneous emission of all the transmitted signals and their processing in the receiver. In addition, correlation techniques (match filtering) are used to determine the image points projected from each emitter on the QADA. This work uses Monte Carlo simulations to characterize the absolute errors for a grid of test points under noisy measurements, as well as the robustness of the system when varying the 3D location of one transmitter. The IPPE algorithm obtained the best performance in this configuration. The proposal has also been experimentally evaluated in a real setup. The estimation of the receiver's position at three particular points for roll angles of the receiver of γ={0°, 120°, 210° and 300°} using the IPPE algorithm achieves average absolute errors and standard deviations of 4.33 cm, 3.51 cm and 28.90 cm; and 1.84 cm, 1.17 cm and 19.80 cm in the coordinates x, y and z, respectively. These positioning results are in line with those obtained in previous work using triangulation techniques but with the addition that the complete pose of the receiver (x, y, z, α, ß, γ) is obtained in this proposal.

4.
Sensors (Basel) ; 19(11)2019 Jun 01.
Artigo em Inglês | MEDLINE | ID: mdl-31159431

RESUMO

A method for infrared and cameras sensor fusion, applied to indoor positioning in intelligent spaces, is proposed in this work. The fused position is obtained with a maximum likelihood estimator from infrared and camera independent observations. Specific models are proposed for variance propagation from infrared and camera observations (phase shifts and image respectively) to their respective position estimates and to the final fused estimation. Model simulations are compared with real measurements in a setup designed to validate the system. The difference between theoretical prediction and real measurements is between 0.4 cm (fusion) and 2.5 cm (camera), within a 95% confidence margin. The positioning precision is in the cm level (sub-cm level can be achieved at most tested positions) in a 4 × 3 m locating cell with 5 infrared detectors on the ceiling and one single camera, at distances from target up to 5 m and 7 m respectively. Due to the low cost system design and the results observed, the system is expected to be feasible and scalable to large real spaces.

5.
Sensors (Basel) ; 18(10)2018 Oct 12.
Artigo em Inglês | MEDLINE | ID: mdl-30322007

RESUMO

This paper presents a novel approach for indoor acoustic source localization using microphone arrays, based on a Convolutional Neural Network (CNN). In the proposed solution, the CNN is designed to directly estimate the three-dimensional position of a single acoustic source using the raw audio signal as the input information and avoiding the use of hand-crafted audio features. Given the limited amount of available localization data, we propose, in this paper, a training strategy based on two steps. We first train our network using semi-synthetic data generated from close talk speech recordings. We simulate the time delays and distortion suffered in the signal that propagate from the source to the array of microphones. We then fine tune this network using a small amount of real data. Our experimental results, evaluated on a publicly available dataset recorded in a real room, show that this approach is able to produce networks that significantly improve existing localization methods based on SRP-PHAT strategies and also those presented in very recent proposals based on Convolutional Recurrent Neural Networks (CRNN). In addition, our experiments show that the performance of our CNN method does not show a relevant dependency on the speaker's gender, nor on the size of the signal window being used.

6.
Surg Endosc ; 31(1): 456-461, 2017 01.
Artigo em Inglês | MEDLINE | ID: mdl-27129565

RESUMO

BACKGROUND: Augmented Reality (AR) is a technology that can allow a surgeon to see subsurface structures. This works by overlaying information from another modality, such as MRI and fusing it in real time with the endoscopic images. AR has never been developed for a very mobile organ like the uterus and has never been performed for gynecology. Myomas are not always easy to localize in laparoscopic surgery when they do not significantly change the surface of the uterus, or are at multiple locations. OBJECTIVE: To study the accuracy of myoma localization using a new AR system compared to MRI-only localization. METHODS: Ten residents were asked to localize six myomas (on a uterine model into a laparoscopic box) when either using AR or in conditions that simulate a standard method (only the MRI was available). Myomas were randomly divided in two groups: the control group (MRI only, AR not activated) and the AR group (AR activated). Software was used to automatically measure the distance between the point of contact on the uterine surface and the myoma. We compared these distances to the true shortest distance to obtain accuracy measures. The time taken to perform the task was measured, and an assessment of the complexity was performed. RESULTS: The mean accuracy in the control group was 16.80 mm [0.1-52.2] versus 0.64 mm [0.01-4.71] with AR. In the control group, the mean time to perform the task was 18.68 [6.4-47.1] s compared to 19.6 [3.9-77.5] s with AR. The mean score of difficulty (evaluated for each myoma) was 2.36 [1-4] versus 0.87 [0-4], respectively, for the control and the AR group. DISCUSSION: We developed an AR system for a very mobile organ. This is the first user study to quantitatively evaluate an AR system for improving a surgical task. In our model, AR improves localization accuracy.


Assuntos
Laparoscopia/métodos , Leiomioma/cirurgia , Modelos Anatômicos , Cirurgia Assistida por Computador/métodos , Miomectomia Uterina/métodos , Neoplasias Uterinas/cirurgia , Feminino , Procedimentos Cirúrgicos em Ginecologia/métodos , Ginecologia/educação , Humanos , Internato e Residência , Leiomioma/diagnóstico por imagem , Imageamento por Ressonância Magnética , Software , Interface Usuário-Computador , Neoplasias Uterinas/diagnóstico por imagem
7.
Sensors (Basel) ; 17(10)2017 Oct 14.
Artigo em Inglês | MEDLINE | ID: mdl-29036886

RESUMO

This paper presents a sequential non-rigid reconstruction method that recovers the 3D shape and the camera pose of a deforming object from a video sequence and a previous shape model of the object. We take PTAM (Parallel Mapping and Tracking), a state-of-the-art sequential real-time SfM (Structure-from-Motion) engine, and we upgrade it to solve non-rigid reconstruction. Our method provides a good trade-off between processing time and reconstruction error without the need for specific processing hardware, such as GPUs. We improve the original PTAM matching by using descriptor-based features, as well as smoothness priors to better constrain the 3D error. This paper works with perspective projection and deals with outliers and missing data. We evaluate the tracking algorithm performance through different tests over several datasets of non-rigid deforming objects. Our method achieves state-of-the-art accuracy and can be used as a real-time method suitable for being embedded in portable devices.

8.
Artigo em Inglês | MEDLINE | ID: mdl-39014177

RESUMO

PURPOSE: Augmented reality guidance in laparoscopic liver resection requires the registration of a preoperative 3D model to the intraoperative 2D image. However, 3D-2D liver registration poses challenges owing to the liver's flexibility, particularly in the limited visibility conditions of laparoscopy. Although promising, the current registration methods are computationally expensive and often necessitate manual initialisation. METHODS: The first neural model predicting the registration (NM) is proposed, represented as 3D model deformation coefficients, from image landmarks. The strategy consists in training a patient-specific model based on synthetic data generated automatically from the patient's preoperative model. A liver shape modelling technique, which further reduces time complexity, is also proposed. RESULTS: The NM method was evaluated using the target registration error measure, showing an accuracy on par with existing methods, all based on numerical optimisation. Notably, NM runs much faster, offering the possibility of achieving real-time inference, a significant step ahead in this field. CONCLUSION: The proposed method represents the first neural method for 3D-2D liver registration. Preliminary experimental findings show comparable performance to existing methods, with superior computational efficiency. These results suggest a potential to deeply impact liver registration techniques.

9.
Sensors (Basel) ; 12(10): 13781-812, 2012 Oct 15.
Artigo em Inglês | MEDLINE | ID: mdl-23202021

RESUMO

This paper presents a novel approach for indoor acoustic source localization using sensor arrays. The proposed solution starts by defining a generative model, designed to explain the acoustic power maps obtained by Steered Response Power (SRP) strategies. An optimization approach is then proposed to fit the model to real input SRP data and estimate the position of the acoustic source. Adequately fitting the model to real SRP data, where noise and other unmodelled effects distort the ideal signal, is the core contribution of the paper. Two basic strategies in the optimization are proposed. First, sparse constraints in the parameters of the model are included, enforcing the number of simultaneous active sources to be limited. Second, subspace analysis is used to filter out portions of the input signal that cannot be explained by the model. Experimental results on a realistic speech database show statistically significant localization error reductions of up to 30% when compared with the SRP-PHAT strategies.

10.
IEEE Trans Pattern Anal Mach Intell ; 44(10): 6409-6423, 2022 Oct.
Artigo em Inglês | MEDLINE | ID: mdl-34133273

RESUMO

Non-Rigid Structure-from-Motion (NRSfM) reconstructs a deformable 3D object from keypoint correspondences established between monocular 2D images. Current NRSfM methods lack statistical robustness, which is the ability to cope with correspondence errors. This prevents one to use automatically established correspondences, which are prone to errors, thereby strongly limiting the scope of NRSfM. We propose a three-step automatic pipeline to solve NRSfM robustly by exploiting isometry. Step (i) computes the optical flow from correspondences, step (ii) reconstructs each 3D point's normal vector using multiple reference images and integrates them to form surfaces with the best reference and step (iii) rejects the 3D points that break isometry in their local neighborhood. Importantly, each step is designed to discard or flag erroneous correspondences. Our contributions include the robustification of optical flow by warp estimation, new fast analytic solutions to local normal reconstruction and their robustification, and a new scale-independent measure of 3D local isometric coherence. Experimental results show that our robust NRSfM method consistently outperforms existing methods on both synthetic and real datasets.

11.
Sensors (Basel) ; 11(9): 8339-57, 2011.
Artigo em Inglês | MEDLINE | ID: mdl-22164079

RESUMO

This paper describes a relative localization system used to achieve the navigation of a convoy of robotic units in indoor environments. This positioning system is carried out fusing two sensorial sources: (a) an odometric system and (b) a laser scanner together with artificial landmarks located on top of the units. The laser source allows one to compensate the cumulative error inherent to dead-reckoning; whereas the odometry source provides less pose uncertainty in short trajectories. A discrete Extended Kalman Filter, customized for this application, is used in order to accomplish this aim under real time constraints. Different experimental results with a convoy of Pioneer P3-DX units tracking non-linear trajectories are shown. The paper shows that a simple setup based on low cost laser range systems and robot built-in odometry sensors is able to give a high degree of robustness and accuracy to the relative localization problem of convoy units for indoor applications.


Assuntos
Lasers , Robótica
12.
Sensors (Basel) ; 10(4): 3261-79, 2010.
Artigo em Inglês | MEDLINE | ID: mdl-22319297

RESUMO

This paper presents a method for obtaining the motion segmentation and 3D localization of multiple mobile robots in an intelligent space using a multi-camera sensor system. The set of calibrated and synchronized cameras are placed in fixed positions within the environment (intelligent space). The proposed algorithm for motion segmentation and 3D localization is based on the minimization of an objective function. This function includes information from all the cameras, and it does not rely on previous knowledge or invasive landmarks on board the robots. The proposed objective function depends on three groups of variables: the segmentation boundaries, the motion parameters and the depth. For the objective function minimization, we use a greedy iterative algorithm with three steps that, after initialization of segmentation boundaries and depth, are repeated until convergence.

13.
Sensors (Basel) ; 10(4): 3655-80, 2010.
Artigo em Inglês | MEDLINE | ID: mdl-22319318

RESUMO

This paper presents a sensor system for robot localization based on the information obtained from a single camera attached in a fixed place external to the robot. Our approach firstly obtains the 3D geometrical model of the robot based on the projection of its natural appearance in the camera while the robot performs an initialization trajectory. This paper proposes a structure-from-motion solution that uses the odometry sensors inside the robot as a metric reference. Secondly, an online localization method based on a sequential Bayesian inference is proposed, which uses the geometrical model of the robot as a link between image measurements and pose estimation. The online approach is resistant to hard occlusions and the experimental setup proposed in this paper shows its effectiveness in real situations. The proposed approach has many applications in both the industrial and service robot fields.

14.
Sensors (Basel) ; 10(5): 4825-37, 2010.
Artigo em Inglês | MEDLINE | ID: mdl-22363203

RESUMO

In this paper, a new method of classification and localization of reflectors, using the time-of-flight (TOF) data obtained from ultrasonic transducers, is presented. The method of classification and localization is based on Generalized Principal Component Analysis (GPCA) applied to the TOF values obtained from a sensor that contains four ultrasound emitters and 16 receivers. Since PCA works with vectorized representations of TOF, it does not take into account the spatial locality of receivers. The GPCA works with two-dimensional representations of TOF, taking into account information on the spatial position of the receivers. This report includes a detailed description of the method of classification and localization and the results of achieved tests with three types of reflectors in 3-D environments: planes, edges, and corners. The results in terms of processing time, classification and localization were very satisfactory for the reflectors located in the range of 50-350 cm.

15.
Sensors (Basel) ; 10(10): 9232-51, 2010.
Artigo em Inglês | MEDLINE | ID: mdl-22163406

RESUMO

This paper presents a complete implementation of the Principal Component Analysis (PCA) algorithm in Field Programmable Gate Array (FPGA) devices applied to high rate background segmentation of images. The classical sequential execution of different parts of the PCA algorithm has been parallelized. This parallelization has led to the specific development and implementation in hardware of the different stages of PCA, such as computation of the correlation matrix, matrix diagonalization using the Jacobi method and subspace projections of images. On the application side, the paper presents a motion detection algorithm, also entirely implemented on the FPGA, and based on the developed PCA core. This consists of dynamically thresholding the differences between the input image and the one obtained by expressing the input image using the PCA linear subspace previously obtained as a background model. The proposal achieves a high ratio of processed images (up to 120 frames per second) and high quality segmentation results, with a completely embedded and reliable hardware architecture based on commercial CMOS sensors and FPGA devices.


Assuntos
Processamento de Imagem Assistida por Computador/instrumentação , Processamento de Imagem Assistida por Computador/métodos , Análise de Componente Principal/métodos , Processamento de Sinais Assistido por Computador/instrumentação , Algoritmos , Computadores , Desenho de Equipamento , Modelos Teóricos
16.
Sensors (Basel) ; 10(10): 8865-87, 2010.
Artigo em Inglês | MEDLINE | ID: mdl-22163385

RESUMO

This paper presents a novel system capable of solving the problem of tracking multiple targets in a crowded, complex and dynamic indoor environment, like those typical of mobile robot applications. The proposed solution is based on a stereo vision set in the acquisition step and a probabilistic algorithm in the obstacles position estimation process. The system obtains 3D position and speed information related to each object in the robot's environment; then it achieves a classification between building elements (ceiling, walls, columns and so on) and the rest of items in robot surroundings. All objects in robot surroundings, both dynamic and static, are considered to be obstacles but the structure of the environment itself. A combination of a Bayesian algorithm and a deterministic clustering process is used in order to obtain a multimodal representation of speed and position of detected obstacles. Performance of the final system has been tested against state of the art proposals; test results validate the authors' proposal. The designed algorithms and procedures provide a solution to those applications where similar multimodal data structures are found.


Assuntos
Reconhecimento Automatizado de Padrão/métodos , Robótica/métodos , Algoritmos , Inteligência Artificial , Teorema de Bayes , Análise por Conglomerados , Meio Ambiente
17.
IEEE Trans Pattern Anal Mach Intell ; 42(12): 3011-3026, 2020 Dec.
Artigo em Inglês | MEDLINE | ID: mdl-31180886

RESUMO

3D reconstruction of deformable objects using inter-image visual motion from monocular images has been studied under Shape-from-Template (SfT) and Non-Rigid Structure-from-Motion (NRSfM). Most methods have been developed for simple deformation models, primarily isometry. They may treat a surface as a discrete set of points and draw constraints from the points only or they may use a non-parametric representation and use both points and differentials to express constraints. We propose a differential framework based on Cartan's theory of connections and moving frames. It is applicable to SfT and NRSfM, and to deformation models other than isometry. It utilises infinitesimal-level assumptions on the surface's geometry and mappings. It has the following properties. 1) It allows one to derive existing solutions in a simpler way. 2) It models SfT and NRSfM in a unified way. 3) It allows us to introduce a new skewless deformation model and solve SfT and NRSfM for it. 4) It facilitates a generic solution to SfT which does not require deformation modeling. Our framework is complete: it solves deformable 3D reconstruction for a whole class of algebraic deformation models including isometry. We compared our solutions with the state-of-the-art methods and show that ours outperform in terms of both accuracy and computation time.

18.
Arthroscopy ; 25(1): 106-8, 2009 Jan.
Artigo em Inglês | MEDLINE | ID: mdl-19111226

RESUMO

Varicella-zoster virus-associated arthritis has not been well documented in adults. We present the case of a 27-year-old female patient with monoarthritis of the knee associated with clinical symptoms typical of varicella. Arthroscopic examination showed unusual oval and circular lesions in cartilage, some of which measured 5 +/- 3 mm in diameter in weight-supporting zones. Such lesions have not been described previously and were type III-A lesions on the Noyes scale or grade IV on the Outerbridge scale. On microscopic observation, synovial fluid cultures and hemocultures were negative for the presence of bacteria. A biopsy sample and synovial liquid from the affected knee produced a positive polymerase chain reaction for varicella-zoster virus, genotype E. These findings suggest a strong relation between clinical varicella infection and important lesion invasion in the knee articulation of such a young adult, probably related to the virus. However, it remains necessary to corroborate this relation between cartilage destruction and clinical symptoms of varicella associated with monoarthritis of an adult knee. Nevertheless, it is advisable to initiate the appropriate antiviral treatment in adults with varicella-related gonalgia because the lesions produce the most severe effects on exposure to the knee bone.


Assuntos
Artroscopia/métodos , Herpes Zoster/complicações , Osteoartrite do Joelho/etiologia , Adulto , Biópsia , Cartilagem/diagnóstico por imagem , Cartilagem/patologia , Cartilagem/virologia , DNA Viral/análise , Diagnóstico Diferencial , Feminino , Herpes Zoster/diagnóstico , Herpes Zoster/virologia , Herpesvirus Humano 3/genética , Humanos , Osteoartrite do Joelho/diagnóstico , Osteoartrite do Joelho/cirurgia , Tomografia Computadorizada por Raios X
19.
Eur J Ophthalmol ; 29(4): 417-425, 2019 Jul.
Artigo em Inglês | MEDLINE | ID: mdl-30198329

RESUMO

PURPOSE: To compare rotational stability, centration and visual outcomes provided by three trifocal lens models that have the same optical zone design but different material, composition, and/or toricity. METHODS: The study included 78 patients with symmetric bilateral intraocular lens implantation. The lenses under evaluation were trifocal intraocular lenses made of hydrophilic acrylic material: a spherical lens 26% hydrophilic acrylic (POD FineVision), a similar lens but having a toric design (POD Toric FineVision), and a trifocal lens 25% hydrophilic acrylic material (FineVision/MicroF). Moreover, the lenses share the same optical zone design. The lenses' rotational stability and centration were measured by means of the PIOLET software, which relies on recording and image processing techniques to determine lens rotation and centration based on slit-lamp images. We also assessed patients' visual quality by means of 25, 40, and 80 cm VA tests. RESULTS: The best centration results were achieved with the POD Toric FineVision model, although the differences were not statistically significant. As for lens rotation, it was below 5° in all cases under study. Regarding VA, all subjects attained at least 0.3 logMAR for far distance uncorrected VA, at 80 cm VA was about 0.2 logMAR, at 40 cm it was above 0.15 logMAR, and at 25 cm it was about 0.3 logMAR for both lens types. CONCLUSION: All three intraocular lens models yield excellent visual results at far, near as well as intermediate distances. The POD FineVision and POD Toric FineVision models, with double C-loop design, yielded the best results centration-wise and rotation-wise. Differences had no clinical relevance.


Assuntos
Implante de Lente Intraocular , Lentes Intraoculares Multifocais , Facoemulsificação , Acuidade Visual/fisiologia , Idoso , Catarata/complicações , Óculos , Feminino , Humanos , Masculino , Pessoa de Meia-Idade , Estudos Prospectivos , Desenho de Prótese , Pseudofacia/fisiopatologia , Rotação
20.
IEEE Trans Pattern Anal Mach Intell ; 40(10): 2442-2454, 2018 10.
Artigo em Inglês | MEDLINE | ID: mdl-28991733

RESUMO

We study Isometric Non-Rigid Shape-from-Motion (Iso-NRSfM): given multiple intrinsically calibrated monocular images, we want to reconstruct the time-varying 3D shape of a thin-shell object undergoing isometric deformations. We show that Iso-NRSfM is solvable from local warps, the inter-image geometric transformations. We propose a new theoretical framework based on the Riemmanian manifold to represent the unknown 3D surfaces as embeddings of the camera's retinal plane. This allows us to use the manifold's metric tensor and Christoffel Symbol (CS) fields. These are expressed in terms of the first and second order derivatives of the inverse-depth of the 3D surfaces, which are the unknowns for Iso-NRSfM. We prove that the metric tensor and the CS are related across images by simple rules depending only on the warps. This forms a set of important theoretical results. We show that current solvers cannot solve for the first and second order derivatives of the inverse-depth simultaneously. We thus propose an iterative solution in two steps. 1) We solve for the first order derivatives assuming that the second order derivatives are known. We initialise the second order derivatives to zero, which is an infinitesimal planarity assumption. We derive a system of two cubics in two variables for each image pair. The sum-of-squares of these polynomials is independent of the number of images and can be solved globally, forming a well-posed problem for $N\geq 3$ images. 2) We solve for the second order derivatives by initialising the first order derivatives from the previous step. We solve a linear system of $4N-4$ equations in three variables. We iterate until the first order derivatives converge. The solution for the first order derivatives gives the surfaces' normal fields which we integrate to recover the 3D surfaces. The proposed method outperforms existing work in terms of accuracy and computation cost on synthetic and real datasets.

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