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Parameterization of a cylinder patch (400 points, top) using the isomap (middle) and ARAP (bottom) algorithms. The scatterplots show the exact geodesic distance on the true underlying surface between all 79,800 pairs of points plotted against the Euclidean distance between the corresponding estimated (u, v) points provided by each method.
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Reconstructing a free-form surface from 3-dimensional (3D) noisy measurements is a central problem in inspection, statistical quality control, and reverse engineering. We present a new method for the statistical reconstruction of a free-form surface patch based on 3D point cloud data. The surface is represented parametrically, with each of the thre...
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... in our case d D (w 1 , w 2 ) = |w 1 − w 2 |, the Euclidean dis- tance on D ⊂ E 2 . An isometric mapping can be thought of as a transformation that bends the surface S into a different shape without changing the intrinsic distances between points on S. Hence, it can be shown that an isometry also preserves areas on S and angles between curves on S (i.e., it is a conformal mapping). An isometric mapping is also a geodesic mapping, in which geodesic distances between points in one space (d D ) map into geodesic distances d S on the image space ( Kreyszig 1991, Theorem 94.2). But as it is well-known in cartography, finding a perfectly isometric mapping is possible only if the surface is developable, that is, if the surface has a Gaussian curvature of zero everywhere (Kreyszig 1991, p. 181). Some popular parameterization algorithms in the computer graphics literature find a conformal mapping, which has nice mathematical properties (Floater and Hormann 2005) but re- sult in pronounced area deformations. Extensive work on the surface parameterization problem over the past decade has re- sulted in algorithms that instead attempt to preserve areas, or that minimize a weighted sum of distortions due to differences in angles and due to differences in areas, achieving in this way an "as isometric as possible" mapping (e.g., Degener, Meseth, and Klein 2003; Sorkine and Alexa 2007; Liu et al. 2008). This type of parameterization methods are particularly useful for our approach, since we assume correlations are a function of the geodesic distances on the surface, and these are provided by an isometric mapping. Figure 4 shows two instances of surface patches, observed with noise, and their near-isometric parame- terization. Figure 5 shows scatterplots of the exact geodesic distances between points p(u, v) i and p(u, v) j on a cylindrical patch plot- ted against the Euclidean distance between the corresponding (u i , v i ) and (u j , v j ) points (for 400 points there are 79,800 such TECHNOMETRICS, FEBRUARY 2015, VOL. 57, NO. 1 Table 1. Correlation coefficients between Euclidean and geodesic distances obtained with different parameterization algorithms applied to the 79,800 pairs of points from a grid of 400 noise-free observations generated on a half cylindrical patch pairs) obtained with two parameterization algorithms, isomap (Tenenbaum, de Silva, and Langford 2000) and the "as-rigid- as-possible" (ARAP) method ( Liu et al. 2008) that we describe more fully below and in the supplementary materials. As it can be seen, both methods are near isometries, since the scatters are close to a 45 • line (in view of (5), the correlation coef- ficient of the scatters is a measure of near-isometry) with the estimated correlations exceeding 0.995 for each method. Table 1 shows the estimated correlation coefficients of similar scatter- plots (not depicted) obtained with other algorithms used for the parameterization step, applied to 400 noisy observations taken from a half cylinder (here we added noise generated with a GGP with an exponential correlated function with parameters φ • = 1, σ 2 = τ 2 = 0.0001 to the true points on the cylinder, see next section for a description of the covariance model ...
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... an algorithm is unable to "unfold" this particularly simple, developable surface, it will typically be unable to unfold near isometrically more complicated, nondevelopable surfaces. In particular, the first algorithm on the table (least squares con- formal map or LSCM, Levy et al. 2002) shows how conformal parameterization algorithms from the computer graphics field are not useful for our purposes, since they severely distort dis- tances. A complete survey of parameterization methods from the manifold learning literature up to 2009 was given by van der Maaten, Postma, and van der Herik (2009). These authors also provided a very useful library of Matlab programs some of which were used to prepare Table 1. For our purposes, all that is necessary is to find a reliable near-isometric parameter- ization method, perhaps one that is fast to compute for large point clouds, and both isomap and ARAP have these proper- ties. Although we suggest using either method, it is important to point out their weaknesses: as it can be seen in Figure 5, ARAP typically distorts the boundaries of the object (this is also a problem, but of lesser magnitude, for isomap). Likewise, (see Figure 6) Isomap distorts a surface near a "hole" (ISOMAP proof of asymptotic convergence to a near isometry rests on the assumption observations lie on a geodesically convex manifold, see main theorem in Bernstein et al. (2000), an assumption that is false if the surface has holes). ARAP scales better with the number of points than Isomap, which needs to be modified for large datasets (see ...
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... 1. A cylindrical surface patch. Table 2 shows the performance metrics of a series of simulations taking the cylin- drical patch of Figure 5 as the true underlying surface. Geodesi- cally correlated Gaussian noise was added to a grid of points on the surface, as described before, with correlation ...
Citations
... Use of geodesic distance based kernels in the literature In the field of surface reconstruction, Del Castillo et al. [53] propose to use a Gaussian process to infer the surface that passes through a point cloud. They have shown that using geodesic distances in the kernel yields better results than using the Euclidean distance. ...
Three-dimensional (3d) scans are essential in evaluating and following dermatological and cosmetic treatments. Unfortunately, the scanned body parts may be altered by artifacts from the reconstruction or unsought motions from the patients. For instance, facial expressions makes the 3d face scans improper for measurements. Hence, it is essential to develop methods to standardize the 3d scans to improve the measurement accuracy. Besides, evaluating treatments and labeling 3d scans is time-consuming, prone to errors, and undergoes rater's variability. For example, studies requiring the evaluation of skin aging or tissue sagging are easily distorted by inconsistencies between evaluators. Thus, it would be a significant leap for physicians to have data-driven algorithms that automatically evaluate the treatments. In this thesis, we develop statistical modeling methods for the face to address these problems.The first step is to register the 3d scans, which have different mesh connectivity and a different number of vertices. The Gaussian Process Morphable Models (GPMMs) framework can register 3d face scans with neutral expressions but struggle when facial expressions change. Hence, we propose a new kernel for GPMMs based on geodesic distances that allow more flexible and realistic deformations. Our new kernel allows for fitting the template mesh toward faces with different facial expressions. Also, our registration formulation uses weighted least squares to select areas such as hair that will not be registered. Furthermore, the recent advances in estimating geodesic distances make the extra computation time cost negligible in most applications.Then, we built a statistical shape model to quantify the skin sagging on the face. We use Partial Least Square Regression (PLSR) to find a linear relationship between the skin sagging score and geometric features on the face surface. The interpretability potential of linear models such as PLSR makes these methods ideal candidates for shape analysis on small data sets where generalization is essential. Furthermore, we propose visualization techniques to interpret the parameters of the PLSR model. Our jawline sagging model and the rater agree on 73% of the time, which is in the same range as the raters' variability. The visualization of the PLSR latent variables shows that the model has captured jawline sagging deformations that are coherent with the jawline sagging illustrated on the scales used by the physicians.Finally, we propose to use a morphable face model to neutralize facial expressions from 3d face scans. Unfortunately, in existing models, there is an entanglement between the deformations related to identity changes and those from facial expressions. Consequently, modifying a facial expression also modifies the person's identity, limiting such models' usefulness. We added an orthogonality penalization into the training procedure to address this issue, leading to a quasi-orthogonality between the expression and identity sub-spaces. The quasi orthogonality allows for a better disentanglement of facial expression deformations from face morphology. Averaging on all expressions, the neutralization with quasi-orthogonality produces face meshes that are 20% closer to the ground truth meshes. The effect of quasi-orthogonality is more visible on large amplitude facial expressions, such as opening the mouth. Our visualizations show very convincing facial expression neutralizations.
... Wang and Tsung (2018) developed a Bayesian generative modeling technique to address the calibration problem for a low-resolution scanner, considering both variance and bias for the scanner. However, their study focused only on the deviations and randomness of in-plane, or 2-D, scanning profiles in the x-y plane, whereas variations in the z-direction can be more severe in practice (Del Castillo et al., 2015). Furthermore, their method focused on the calibration of a specific shape profile based on a highresolution system, whereas the application of the calibration model to other shapes of interest may require involved shape approximation methods (Huang et al., 2014). ...
... Furthermore, their method focused on the calibration of a specific shape profile based on a highresolution system, whereas the application of the calibration model to other shapes of interest may require involved shape approximation methods (Huang et al., 2014). Del Castillo et al. (2015) modeled point cloud data of features using a geodesic Gaussian process that can account for pointwise randomness. However, the focus of their model was on surface fitting, not variance modeling. ...
... We observe from Table 2 that the prediction error and its standard deviation for the case of the z-coordinate are significantly higher than those for the cases of the x-and y-coordinates. This is because the magnitude of the z-coordinate variances is greater than the other two coordinates (Del Castillo et al., 2015), which corresponds to the greater importance of predicting z-coordinate variance in RE and metrology process planning (Geng et al., 2022). Furthermore, Freeform 1 and Freeform 2 yield better variance model performances than those obtained from the small number of initiation scans, and those obtained from the BELM models trained by Half Ball (which exhibit comparatively worse performance). ...
Three-dimensional (3-D) point cloud data are increasingly being used to describe a wide range of physical objects in detail, corresponding to customized and flexible shape designs. The advent of a new generation of optical sensors has simplified and reduced the costs of acquiring 3-D data in near-real-time. However, the variation of the acquired point clouds, and methods to describe them, create bottlenecks in manufacturing practices such as Reverse Engineering (RE) and metrology in additive manufacturing. We address this issue by developing an automated variance modeling algorithm that utilizes a physical object’s local geometric descriptors and Bayesian Extreme Learning Machines (BELMs). Our proposed ensemble and residual BELM-variants are trained by a scanning history that is composed of multiple scans of other, distinct objects. The specific scanning history is selected by a new empirical Kullback–Leibler divergence we developed to identify objects that are geometrically similar to an object of interest. A case study of our algorithm on additively manufactured products demonstrates its capability to model the variance of point cloud data for arbitrary freeform shapes based on a scanning history involving simpler, and distinct, shapes. Our algorithm has utility for measuring the process capability of 3-D scanning for RE processes.
... When considering correlation functions on geodesic surfaces, it is essential to note that correlation functions that satisfy Bochner's theorem using Euclidean distances may not satisfy the theorem when using geodesic distances [59,60,61]. In general, the admissibility of a covariance function must be checked before being used on geometry or type of distance. ...
Structures contain inherent deviations from idealized geometry and material properties. Quantifying the effects of such random variations is of interest when determining the reliability and robustness of a structure. Generating fields that follow complex shapes is not trivial. Generating random fields on simple shapes such as a cylinder can be done using series-expansion methods or analytically computed distances as input for a decomposition approach. Generating geodesic random fields on a mesh representing complex geometric shapes using these approaches is very complex or not possible. This paper presents a generalized approach to generating geodesic random fields representing variations in a finite element setting. Geodesic distances represent the shortest path between points within a volume or surface. Computing geodesic distances of structural points is achieved by solving the heat equation using normalized heat gradients originating from every node within the structure. Any element (bar, beam, shell, or solid) can be used as long as it can solve potential flow problems in the finite element program. Variations of the approach are discussed to generate fields with defined similarities or fields that show asymmetric behavior. A numerical example of a gyroid structure demonstrates the effect of using geodesic distances in field generation compared to Euclidean distances. An anisotropic cylinder with varying Young’s modulus and thickness is taken from literature to verify the implementation. Variations of the approach are analyzed using a composite cylinder in which fiber angles are varied. Although the focus of this paper is thin-walled structures, the approach works for all types of finite element structures and elements.
... and it is not straightforward to convert them to functional data in the form of a response variable as a function of some domain variables so that methods for profile/functional data (e.g., kriging models (del Castillo et al. 2015)) can be used. Third, point cloud data are spatially dense (e.g., millions of measured points per each point cloud), which creates computational challenges and calls for efficient methods. ...
Surface point cloud data from three-dimensional optical scanners provide rich information about the surface geometry of scanned parts and potential variation in the surfaces from part-to-part. It is challenging, however, to make full use of these data for statistical process control purposes to identify sources of variation that manifest in a more complex nonparametric manner than variation in some pre-specified set of geometric features of each part. We develop a framework for identifying nonparametric variation patterns that uses dissimilarity representation of the data and dissimilarity-based manifold learning, which helps discover a low-dimensional implicit manifold parameterization of the variation. Visualizing how the parts change as the manifold parameters are varied helps build an understanding of the physical characteristic of the variation. We also discuss utilizing the nominal surface of parts when it is accessible to improve the computational expense and visualization aspects of the framework. Our approaches clearly reveal the nature of the variation patterns in a real cylindrical-part machining example and a simulated square head bolt example.
... In the field of surface reconstruction, Del Castillo et al. [7] propose to use a Gaussian process to infer the surface that passes through a point cloud. They have shown that using geodesic distances into the kernel yields to better results than using the Euclidean distance. ...
This work addresses the problem of non-rigid registration of 3D scans, which is at the core of shape modeling techniques. Firstly, we propose a new kernel based on geodesic distances for the Gaussian Process Morphable Models (GPMMs) framework. The use of geodesic distances into the kernel makes it more adapted to the topological and geometric characteristics of the surface and leads to more realistic deformations around holes and curved areas. Since the kernel possesses hyperparameters we have optimized them for the task of face registration on the FaceWarehouse dataset. We show that the Geodesic squared exponential kernel performs significantly better than state of the art kernels for the task of face registration on all the 20 expressions of the FaceWarehouse dataset. Secondly, we propose a modification of the loss function used in the non-rigid ICP registration algorithm, that allows to weight the correspondences according to the confidence given to them. As a use case, we show that we can make the registration more robust to outliers in the 3D scans, such as non-skin parts.
... GPs are demonstrated to successfully model and predict complex profiles such as turbulent flows [29], liquefaction triggering procedures [30], heterogeneous land data in remote sensing [31]. In addition, GP was used to model complex shape and free-form surfaces of 3D point cloud data from a laser scanner [32]. Therefore, several GP models and their variants are being used to model profiles from manufacturing processes. ...
Additive Manufacturing (AM) enables the direct production of complex geometries from computer-aided designs (CAD). The AM fabrication process is often executed in a layer-by-layer manner, whereby small printing errors in one layer can manifest significant defects in the final part. Real-time quality control is currently limited for AM processes due to low repeatability in key quality measurements. Advanced imaging is increasingly invested in AM and leads to the proliferation of real-time process data, which has the potential to transform quality control of AM from post-build inspection to in-situ quality monitoring. However, existing methodologies for in-situ inspection primarily focus on key characteristics of image profiles that tend to be limited in the ability to analyze the variance components, as well as root causes and failure patterns that are critical to process improvement. This paper presents an Additive Gaussian Process with dependent layerwise correlation (AGP-D) to model the spatio-temporal correlation of layerwise imaging data for AM quality monitoring. The AGP-D consists of three independent GP modules. The first GP approximates the base profile, whereas the second and third GP capture the correlation within the same layer and among layers, respectively. Based on posterior predictions of new layers, Hotelling T^2 and generalized likelihood ratio (GLR) control tests are formulated to detect process shifts in the newly fabricated layer and analyze root causes. The proposed methodology is evaluated and validated using both simulation data and real-world case study of a cylinder build fabricated by a laser powder bed fusion (LPBF) machine. Experimental results show the proposed AGP-D is effective for real-time modeling and monitoring of layerwise-correlated
... temperature distribution over a surface). Recently, manifold Gaussian processes (mGPs) [29] have been developed to map information fields to complex domains and surfaces with heat kernels [30], generalized Matérn kernels [31], and geodesic Gaussian kernels [32], [33]. In this paper we use manifold Gaussian processes with geodesic kernel functions to map the surface information fields and plan informative paths based on the map. ...
... We use an mGP to map the information field f ∼ GP(µ, k) with mean function µ(x) and covariance function (kernel) k(x, x ), which encodes the correlation of field values at the two locations x and x . In this paper, we use the geodesic Matérn 3/2 kernel function to model such spatial correlations on a surface manifold [32], [33]: ...
... Thus, this merely captures potentially important work (limited by the pros and cons of using citations as a proxy to a work's quality/importance), where the author(s) continued pursuing this research stream. (e) Among these papers, CPS-related contributions were limited to: Colosimo et al (2014) and Del Castillo et al (2015), who investigated how profile surfaces can be monitored in advanced manufacturing scenarios. ...
With the continued technological advancements in mobile computing, sensors, and artificial intelligence methodologies, computer acquisition of human and physical data, often called cyber-physical convergence, is becoming more pervasive. Consequently, personal device data can be used as a proxy for human operators, creating a digital signature of their typical usage. Examples of such data sources include: wearable sensors, motion capture devices, and sensors embedded in work stations. Our motivation behind this paper is to encourage the quality community to investigate relevant research problems that pertain to human operators. To frame our discussion, we examine three application areas (with distinct data sources and characteristics) for human performance modeling: (a) identification of physical human fatigue using wearable sensors/accelerometers; (b) capturing changes in a driver’s safety performance based on fusing on-board sensor data with online API data; and (c) human authentication for cybersecurity applications. Through three case studies, we identify opportunities for applying industrial statistics methodologies and present directions for future work. To encourage future examination by the quality community, we host our data, Code, and analysis on an online repository.
... temperature distribution over a surface). Recently, manifold Gaussian processes (mGPs) [29] have been developed to map information fields to complex domains and surfaces with heat kernels [30], generalized Matérn kernels [31], and geodesic Gaussian kernels [32], [33]. In this paper we use manifold Gaussian processes with geodesic kernel functions to map the surface information fields and plan informative paths based on the map. ...
... We use an mGP to map the information field f ∼ GP(µ, k) with mean function µ(x) and covariance function (kernel) k(x, x ), which encodes the correlation of field values at the two locations x and x . In this paper, we use the geodesic Matérn 3/2 kernel function to model such spatial correlations on a surface manifold [32], [33]: ...
This paper presents an online informative path planning approach for active information gathering on three-dimensional surfaces using aerial robots. Most existing works on surface inspection focus on planning a path offline that can provide full coverage of the surface, which inherently assumes the surface information is uniformly distributed hence ignoring potential spatial correlations of the information field. In this paper, we utilize manifold Gaussian processes (mGPs) with geodesic kernel functions for mapping surface information fields and plan informative paths online in a receding horizon manner. Our approach actively plans information-gathering paths based on recent observations that respect dynamic constraints of the vehicle and a total flight time budget. We provide planning results for simulated temperature modeling for simple and complex 3D surface geometries (a cylinder and an aircraft model). We demonstrate that our informative planning method outperforms traditional approaches such as 3D coverage planning and random exploration, both in reconstruction error and information-theoretic metrics. We also show that by taking spatial correlations of the information field into planning using mGPs, the information gathering efficiency is significantly improved.
... Although geometric accuracy control in AM involves 3-D shapes [24]- [28], there is a critical lack of major progress on learning 3-D shape data for improving 3-D printing accuracy. Describing the 3-D shape formation through the layer-by-layer fabrication process has been a daunting task. ...
One major impediment to wider adoption of additive manufacturing (AM) is the presence of larger-than-expected shape deviations between an actual print and the intended design. Since large shape deviations/deformations lead to costly scrap and rework, effective learning from previous prints is critical to improve build accuracy of new products for cost reduction. However, products to be built often differ from the past, posing a significant challenge to achieving learning efficacy. The fundamental issue is how to learn a predictive model from a small set of training shapes to predict the accuracy of a new object. Recently an emerging body of work has attempted to generate parametric models through statistical learning to predict and compensate for shape deviations in AM. However, generating such models for 3D freeform shapes currently requires extensive human intervention. This work takes a completely different path by establishing a nonparametric, random forest model through learning from a small training set. One novelty of this approach is to extract features from training shapes/products represented by triangular meshes, as opposed to point-cloud forms. This facilitates fast generation of predictive models for 3D freeform shapes with little human intervention in model specification. A real case study for a fused deposition modeling (FDM) process is conducted to validate model predictions. A practical compensation procedure based on the learned random forest model is also tested for a new part. The overall shape deviation is reduced by 44%, which shows a promising prospect for improving AM print accuracy.
