Edge Points Research Articles

Improved method for positioning crane grab boom corner points using Hough transform and k‐means clustering

AbstractIn the process of automatic grabbing of bridge segment beams, it is crucial to accurately locate and align the corner points of the crane's boom with the beam's lifting holes. This requires the utilization of image processing techniques to precisely detect and locate the corner points of the crane's boom. Existing feature matching methods face challenges such as low detection accuracy and unsuitability for this specific scenario. This article proposes a novel approach for corner point localization by using the intersection points of lines, facilitating the matching of feature points between left and right images. The method consists of three steps: first, a grayscale difference map is constructed by utilizing the R and G channels of the RGB color space. This enhances the bimodal characteristics of the grayscale histograms between the foreground and background, which facilitates the subsequent binarization process. Additionally, opening and closing operations are employed to remove small artifacts from the Canny edge detection results, effectively reducing noise. Second, an adaptive thresholding method based on the mean and variance of Hough transform voting scores is proposed. This method filters out interference lines from the clustering results by selecting appropriate voting scores. Furthermore, an improved centroid calculation method is introduced, which utilizes weighted formulas based on different proportions of voting scores. These weighted formulas replace the original clustering centroids as the basis for line fitting. Finally, the corner coordinates of the crane's boom are computed based on the line fitting results, and the recognition accuracy is compared under different lighting conditions. Experimental results demonstrate that the proposed algorithm exhibits smaller detection errors and higher robustness compared to other corner detection algorithms, particularly when there are numerous interference edge points in the edge detection results. The computed corner coordinates achieve pixel‐level accuracy. The algorithm performs optimally under strong supplementary lighting conditions, with an average detection error percentage of 97.1% within 0–2 pixels and a recognition accuracy of 98.6%. The recognition success rates under different lighting conditions are all above 92.9%. This method is superior to traditional corner detection methods, meets the requirements for automatic grabbing of the boom, and holds practical engineering value. It provides a basis for addressing the accuracy and robustness challenges of crane algorithms influenced by multiple environmental factors.

Rectangular and radial region of interests on the edge of cylindrical phantom for spatial resolution measurement

<span lang="EN-US">The purpose of this study was to evaluate the effect of rectangular region of interest (ROI) size on modulation transfer function (MTF), to develop the radial ROI, and to compare both ROIs performances for MTF measurement using a cylindrical polymethyl methacrylate (PMMA) phantom. The PMMA phantom used in this study was rotated 45°. Four rectangular ROIs and a radial ROI were created to measure the MTF value. The rectangular ROI sizes were 3×41, 21×41, 41×41, and 61×41 pixels; each was placed at upper phantom edge. The radial ROI’s length was 41 pixels and placed at several points in phantom edge. The MTF calculation was automatically conducted using MATLAB. The MTFs from rectangular ROIs and radial ROI were then compared. The comparison of the MTF measurement was also conducted using three different filters. The MTF which used radial ROI was smoother than those of rectangular ROI for all filters. This indicated that radial ROI was more resistant to noise than rectangular ROI. Rectangular ROI with the 41×41 pixels had similar 50% and 10% MTF values with the radial ROI. The MTF value which was obtained using radial ROI is more accurate and robust than those obtained using rectangular ROI.</span>

Path Planning Algorithm Based on Obstacle Clustering Analysis and Graph Search

Path planning is receiving considerable interest in mobile robot research; however, a large number of redundant nodes are typically encountered in the path search process for large-scale maps, resulting in decreased algorithmic efficiency. To address this problem, this paper proposes a graph search path planning algorithm that is based on map preprocessing for creating a weighted graph in the map, thus obtaining a structured search framework. In addition, the reductions in the DBSCAN algorithm were analyzed. Subsequently, the optimal combination of the minPts and Eps required to achieve an efficient and accurate clustering of obstacle communities was determined. The effective edge points were then found by performing obstacle collision detection between special grid nodes. A straight-line connection or A* planning strategy was used between the effective edge points to establish a weighted, undirected graph that contained the start and end points, thereby achieving a structured search framework. This approach reduces the impact of map scale on the time cost of the algorithm and improves the efficiency of path planning. The results of the simulation experiments indicate that the number of nodes to be calculated in the search process of the weighted graph decreases significantly when using the proposed algorithm, thus improving the path planning efficiency. The proposed algorithm offers excellent performance for large-scale maps with few obstacles.

A Method for Measuring Parameters of Defective Ellipse Based on Vision.

Ellipse detection has a very wide range of applications in the field of object detection, especially in the geometric size detection of inclined microporous parts. However, due to the processing methods applied to the parts, there are certain defects in the features. The existing ellipse detection methods do not meet the needs of rapid detection due to the problems of false detection and time consumption. This article proposes a method of quickly obtaining defective ellipse parameters based on vision. It mainly uses the approximation principle of circles to repair defective circles, then combines this with morphological processing to obtain effective edge points, and finally uses the least squares method to obtain elliptical parameters. By simulating the computer-generated images, the results demonstrate that the center fitting error of the simulated defect ellipses with major and minor axes of 600 and 400 pixels is less than 1 pixel, the major and minor axis fitting error is less than 3 pixels, and the tilt angle fitting error is less than 0.1°. Further, experimental verification was conducted on the engine injection hole. The measurement results show that the surface size deviation was less than 0.01 mm and the angle error was less than 0.15°, which means the parameters of defective ellipses can obtained quickly and effectively. It is thus suitable for engineering applications, and can provide visual guidance for the precise measurement of fiber probes.

Surface texture topography evaluation and classification by considering the tool posture changes in five-axis milling

Surface texture topography affects the performance of industrial parts, such as precision and service life. Different tool postures may lead to various texture topographies in five-axis milling. In this paper, surface texture topography quality and texture trend evaluation indexes are proposed considering the influence of tool posture in five-axis milling. Based on the evaluation, the ranges of tool posture angles corresponding to surface texture topographies of different characteristic are obtained through surface texture topographies classification. The result can provide reference for singularity avoidance in path planning and machining error compensation in five-axis milling. The influence of tool posture change on the surface texture quality could be avoided while adjusting the tool posture angle in the determined range. Firstly, the general tool cutting edge point expression with the typical structure parameters of the end mills is proposed. The general digital model of surface texture topography is established with the improvement of the unit time calculation and computing vertex judgment in simulation. Secondly, several evaluation indexes are proposed to represent the quality and the overall trend of the surface texture topography with digital extraction of indexes realized on the surface digital model. On this basis, the classification of the surface texture topographies and the determination of the range of tool posture angle for each class of textures are proposed. Thirdly, the evaluation and classification of the ball-end milling plane texture topographies are carried out, which divide the texture topographies into zero-inclination angle textures and non-zero inclination angle textures. The surface textures are further divided to detailed five categories with the tool posture angle range determined. Different tool posture angles of the same class are used to generate a texture topography for the comparison to verify the evaluation indexes and the classification result. Finally, the plane cutting orthogonal experiments are carried out. The general surface digital model is verified by the comparison in surface texture characteristic of different types of mills between digital model and machined surface. Based on the ball-end milling plane classification result, the machined surface textures and extracted indexes are compared to verify the effectiveness of the proposed evaluation indexes and the classification result. The fillet-end milling surface textures are further classified and verified based on the method above. In addition, the spherical surface textures of ball-end milling are carried out for further verification.

WireframeNet: A novel method for wireframe generation from point cloud

Generating wireframe from point clouds is a challenging task. To make this process easier, we introduce WireframeNet, a deep neural network that transforms point clouds into wireframes. The network inputs a set of disordered points and outputs a complete wireframe structure. We use the insight of the medial axis transform to filter the original point cloud, then predict a set of edge points by learning the geometric transformation, and finally analyze the connectivity between the edge points to construct the complete wireframe structure. We train and evaluate publicly available wireframe datasets and compare the results quantitatively and qualitatively with traditional and other deep learning-based methods. Extensive experiments have demonstrated the robust and efficient performance of our proposed WireframeNet for the task of wireframe structure extraction from point clouds.

Slope Stability Analysis Based on the Coupling of SA and Limit Equilibrium Mechanics

The limit equilibrium strip method of slope has become mature, but because of the complexity of slope instability with many degrees of freedom and high nonlinear, a more three-dimensional and mature method is needed for slope problems. Based on the overall force balance and moment balance of slope, a unified model of three dimensional limit balance methods is established in this paper. Given different assumptions, the analytical expressions of each traditional model are obtained to avoid the problem of difficult boundary treatment when the original method is divided into bars and columns. The influence of the trailing edge point B and shear outlet A on the central axis of the sliding body, and the control arc radius variable t on the calculated value of the three-dimensional slope stability coefficient is discussed in detail. Then, based on the simulated annealing algorithm, the state generating function, state accepting function and temperature updating function are constructed, and the calculation method of optimizing the sliding surface search of the slope by using the simulated annealing algorithm is proposed, and the stability analysis of the slope of a hydropower reservoir dam area in Guangxi is carried out. The results show that the position of the sliding surface obtained by searching around the design value K = 1.10 is basically consistent with the actual one, which proves that the mechanical analysis method of coupling SA and limit equilibrium is convenient and efficient in the slope stability analysis.

Quantitative evaluation of crack based on the sparse decomposition of array Lamb wave propagation

Crack damage is one of the significant factors that may accumulate at the stress concentration area of engineering structures and cause catastrophic accidents. In this paper, we proposed a novel approach to identify the crack location and size by exploiting the reflections and diffractions of Lamb waves. The interaction mechanism between the crack and Lamb wave has been analyzed thoroughly, our analysis of the interaction between the crack and Lamb wave revealed that both reflections and diffractions carry valuable damage information that reflects the size and orientation of the crack. As the interaction coefficients between these two components and the crack are different, there are supposed to exist differences between them in the amplitude value. We implemented a threshold to classify the signals received from all paths into two groups: reflections and diffractions. Then we constructed an overcomplete dictionary of waveforms corresponding to different propagation distances to extract the damage information. Using sparse decomposition, the received signals were mapped to their corresponding propagation distances without the use of baseline signals. The diffractions allow us to determine the crack's tip points, while the reflections provide information about the edge points. The kinked crack's size and orientation were visualized based on the time-domain signals acquired in our experiment. We provided a comprehensive description of our algorithm and verified it through numerical simulation and experimental data. Our results show high agreement with actual cracks, demonstrating the efficacy of our proposed method.

Accuracy Evaluation Trial of Mixed Reality-Guided Spinal Puncture Technology.

To evaluate the accuracy of mixed reality (MR)-guided visualization technology for spinal puncture (MRsp). MRsp involved the following three steps: 1. Lumbar spine computed tomography (CT) data were obtained to reconstruct virtual 3D images, which were imported into a HoloLens (2nd gen). 2. The patented MR system quickly recognized the spatial orientation and superimposed the virtual image over the real spine in the HoloLens. 3. The operator performed the spinal puncture with structural information provided by the virtual image. A posture fixation cushion was used to keep the subjects' lateral decubitus position consistent. 12 subjects were recruited to verify the setup error and the registration error. The setup error was calculated using the first two CT scans and measuring the displacement of two location markers. The projection points of the upper edge of the L3 spinous process (L3↑), the lower edge of the L3 spinous process (L3↓), and the lower edge of the L4 spinous process (L4↓) in the virtual image were positioned and marked on the skin as the registration markers. A third CT scan was performed to determine the registration error by measuring the displacement between the three registration markers and the corresponding real spinous process edges. The setup errors in the position of the cranial location marker between CT scans along the left-right (LR), anterior-posterior (AP), and superior-inferior (SI) axes of the CT bed measured 0.09 ± 0.06 cm, 0.30 ± 0.28 cm, and 0.22 ± 0.12 cm, respectively, while those of the position of the caudal location marker measured 0.08 ± 0.06 cm, 0.29 ± 0.18 cm, and 0.18 ± 0.10 cm, respectively. The registration errors between the three registration markers and the subject's real L3↑, L3↓, and L4↓ were 0.11 ± 0.09 cm, 0.15 ± 0.13 cm, and 0.13 ± 0.10 cm, respectively, in the SI direction. This MR-guided visualization technology for spinal puncture can accurately and quickly superimpose the reconstructed 3D CT images over a real human spine.

LiDAR Point Cloud Data Combined Structural Analysis Based on Strong Form Meshless Method Using Essential Boundary Condition Capturing.

This study proposes a novel hybrid simulation technique for analyzing structural deformation and stress using light detection and ranging (LiDAR)-scanned point cloud data (PCD) and polynomial regression processing. The method estimates the edge and corner points of the deformed structure from the PCD. It transforms into a Dirichlet boundary condition for the numerical simulation using the particle difference method (PDM), which utilizes nodes only based on the strong formulation, and it is advantageous for handling essential boundaries and nodal rearrangement, including node generation and deletion between analysis steps. Unlike previous studies, which relied on digital images with attached targets, this research uses PCD acquired through LiDAR scanning during the loading process without any target. Essential boundary condition implementation naturally builds a boundary value problem for the PDM simulation. The developed hybrid simulation technique was validated through an elastic beam problem and a three-point bending test on a rubber beam. The results were compared with those of ANSYS analysis, showing that the technique accurately approximates the deformed edge shape leading to accurate stress calculations. The accuracy improved when using a linear strain model and increasing the number of PDM model nodes. Additionally, the error that occurred during PCD processing and edge point extraction was affected by the order of polynomial regression equation. The simulation technique offers advantages in cases where linking numerical analysis with digital images is challenging and when direct mechanical gauge measurement is difficult. In addition, it has potential applications in structural health monitoring and smart construction involving machine leading techniques.

3D Cartographic Generalization of LiDAR Point Clouds Based on the Principle of Self-Similarity of a Deterministic Fractal Structure

The rendering of virtual three-dimensional (3D) structures represented by Point Cloud (PC) allows the representation of internal and/or external environments to buildings. However, the compilation of 3D geometric models is influenced by the intrinsic characteristics of PCs, which can be mitigated by the application of an PC simplification operator. According to the mathematical norms of fractal geometry, it was assumed that a PC is characterized by self-similarity. Two experimental datasets acquired with an SLT in static mode indoors were used. Four tasks were accomplished: sampling and structuring of a PC to solve the problem of random distribution, from an octree structure; estimation of the curvature of the points and the roughness of a neighbourhood for the extraction of edge points by the analysis of self-similarity and application of the Statistical Outliers Remove (SOR) algorithm, for the elimination of outliers points; uniform voxelization, to simplify the intermediate points; application of the Iterative Closest Point (ICP) algorithm to register the sets generated in the same local coordinate system. The use of voxelization was satisfactory, but once the voxel size is manually defined, the PC can be oversimplified and lose essential characteristics. This can be minimized by the primary analysis of the edge points, generating a set that is uniform, less noisy, and self- similar to the original set. To achieve a minimum density of points to model an environment three-dimensionally, one must analyse the geometric self- similarity characteristics of the PC to produce a simplified set self-similar to the original, considering the premises of fractal geometry. It is recommended to create an automatic simplification process to minimize the subjectivity coming from the analyst.

Color UAV Image Edge Detection Based on Improved Fireworks Algorithm

Image edge detection plays a crucial role in image analysis and recognition. However, when dealing with color images captured by unmanned aerial vehicles (UAVs), there are certain limitations, such as large operations, multiple noise sources, easy distortion, and missing information in edge detection. To address these shortcomings, this study proposes a UAV color image edge detection method based on an enhanced fireworks algorithm. In this method, the color image pixels of the UAV are represented using quaternions. The explosion amplitude formula of the fireworks is divided into two categories based on the mean value of the number of fireworks explosions. For each category, an explosion formula is proposed, and the explosion mutation operator of the fireworks algorithm is improved accordingly. By applying the proposed algorithm, the preliminary edges of a UAV color image are obtained. Additionally, a novel approach for color image edge refinement is introduced. This approach involves classifying the edge points based on their degree of attachment, which leads to the formation of the edges in a UAV color image. Experimental results demonstrate that the algorithm proposed in this study offers several advantages, including fast calculation, strong denoising capability, and high-quality edge detection.

Research on Visual Measurement of Aperture and Hole Group Center Distance Based on Geometric Constraint

Hole is the most common symmetrical geometric structure of mechanical parts, and aperture and hole center distance are important measured dimensions in machining. However, existing visual measurement methods often require high equipment installation accuracy and low measurement accuracy. To solve the problem of projection deformation of the measured hole: Firstly, a local coordinate system is created, and the coordinate plane of the coordinate is parallel to the end face of the measured hole. Secondly, a cone is established, the end face of the measured hole is alike to the bottom face of the cone, and the optical center of the camera is the vertex. The space equation of the cone is represented by the coordinates of the center point of the hole. Finally, a cutting plane is established near the imaging plane of the camera, and the intersection line equation between the plane and the cone is obtained, and the diameter of the measured hole and the coordinates of the center of the circle are obtained through the parameters of the intersection line equation. In order to solve the problem of errors in edge points of the hole, geometric constraints based on the center of the circle are used to improve measurement accuracy. This experiment takes the automobile insert mold as the measured part and uses the measurement model of this paper to measure the aperture and hole center distance of the mold. The measurement results show that the measurement accuracy of the aperture is 0.018 mm, the measurement accuracy of the center distance of the hole is 0.05 mm, and the maximum relative error is 2.2%.

Matching based on variance minimization of component distances using edges of free-form surfaces

The basis for guidance in the field of automated robot processing is modeling by visual scanning. Matching algorithms are the real link that can be made between the ideal model and the subsequent robot processing. The matching algorithm that meets the machining requirements plays a pivotal role in the entire process, which provides the exact location of design models and measurement data. In order to meet the requirement of making the machining allowance uniform, a fine registration method which considers the variance minimization of the normal and tangential distances between the edge neighbors of the scattered point cloud is proposed. The edge neighbors are achieved by local growing of edge seeds based on the minimization of energy of supervoxel, which can represent the model more accurate than edge points extracted directly. The edge neighbor points are used to participate in the calculation, and the objective function of minimizing the variance of the two-way distance with the introduction of weight coefficients is proposed to constrain the iterative process. The effect of the distance in two directions on the result is analyzed, to determine the appropriate weight coefficients so that the matching calculation converges quickly and accurately. In comparison with other classical and state-of-the-art matching methods, the method in this paper performs well in terms of solution efficiency and accuracy of results. Moreover, the ability of this paper’s method to resist Gaussian noise is investigated, and it is found that this paper’s method has good robustness when σ is less than 1 for Gaussian noise. Ultimately, a uniformly distributed residual model is obtained to provide a visually guided basis for subsequent machining.

3D-shape formation of blood vessels based on computer aided design system

This paper proposes and tests a computerized approach for constructing a 3D model of blood vessels from angiogram images. The approach is divided into two steps, image features extraction and solid model formation. In the first step, image morphological operations and post-processing techniques are used for extracting geometrical entities from the angiogram image. These entities are the middle curve and outer edges of the blood vessel, which are then passed to a computer-aided graphical system for the second phase of processing. The system has embedded programming capabilities and pre-programmed libraries for automating a sequence of events that are exploited to create a solid model of the blood vessel. The gradient of the middle curve is adopted to steer the vessel’s direction, while the cross-sections of the blood vessel are formed as a sequence of circles lying in planes that are orthogonal to the gradients of the middle curves. The radii for the circles are estimated as a distance between the intersection points of the blood vessel edges with the orthogonal plane to the middle curve gradient. The system then uses these circles and the middle curve gradients to produce a solid volume that represents the 3D shape of the blood vessel. The method was tested and evaluated using different cases of angiogram images, and showed a reasonable agreement between the generated shapes and the tested images.

Indoor Visible Light Positioning System Based on Point Classification Using Artificial Intelligence Algorithms.

In RSSI-based indoor visible light positioning systems, when only RSSI is used for trilateral positioning, the receiver height needs to be known to calculate distance. Meanwhile, the positioning accuracy is greatly affected by multi-path effect interference, with the influence of the multi-path effect varying across different areas of the room. If only one single processing is used for positioning, the positioning error in the edge area will increase sharply. In order to address these problems, this paper proposes a new positioning scheme, which uses artificial intelligence algorithms for point classification. Firstly, height estimation is performed according to the received power data structure from different LEDs, which effectively extends the traditional RSSI trilateral positioning from 2D to 3D. The location points in the room are then divided into three categories: ordinary points, edge points and blind points, and corresponding models are used to process different types of points, respectively, to reduce the influence of the multi-path effect. Next, processed received power data are used in the trilateral positioning method for calculating the location point coordinates, and to reduce the room edge corner positioning error, so as to reduce the indoor average positioning error. Finally, a complete system is built in an experimental simulation to verify the effectiveness of the proposed schemes, which are shown to achieve centimeter-level positioning accuracy.

Dimensional Measurement Method for PVC Plates Based on Improved Zernike Moment

There are many pixel-level and sub-pixel-level edge localization methods, but some methods perform poorly in the presence of noise and require image pre-processing or iterative methods to improve the localization accuracy, which increases the computational cost. In this paper, we propose an improved Zernike moment subpixel edge localization algorithm based on the ramp model, and combine the idea of bilinear interpolation method to optimize the selection of parameters in the edge model. Through the validation of several examples, it is found that the algorithm outperforms the compared methods for edge localization of noisy images. After the edge detection, in order to improve the measurement accuracy of polyvinyl chloride (PVC) plates, a parallel line fitting method is proposed to fit the edge points, thus avoiding the interference of extraneous noise points and achieving accurate measurement of PVC plate size. The experiments were carried out for several measurements of the sheet length, and the method was verified to have high measurement accuracy.

Research on a Horizon Line Detection Method for Unmanned Surface Vehicles in Complex Environments

A critical step in the visual navigation of unmanned surface vehicles (USVs) is horizon line detection, which can be used to adjust the altitude as well as for obstacle avoidance in complex environments. In this paper, a real-time and accurate detection method for the horizon line is proposed. Our approach first differentiates the complexity of navigational scenes using the angular second moment (ASM) parameters in the grey level co-occurrence matrix (GLCM). Then, the region of interest (ROI) is initially extracted using minimal human interaction for the complex navigation scenes, while subsequent frames are dynamically acquired using automatic feature point matching. The matched ROI can be maximally removed from the complex background, and the Zernike-moment-based edges are extracted from the obtained ROI. Finally, complete sea horizon information is obtained through a linear fitting of the lower edge points to the edge information. Through various experiments carried out on a classical dataset, our own datasets, and that of another previously published paper, we illustrate the significance and accuracy of this technique for various complex environments. The results show that the performance has potential applications for the autonomous navigation and control of USVs.

Machine learning-based segmentation of aerial LiDAR point cloud data on building roof

ABSTRACT Three-dimensional (3D) reconstruction of a building can be facilitated by correctly segmenting different feature points (e.g. in the form of boundary, fold edge, and planar points) over the building roof, and then, establishing relationships among the constructed feature lines and planar patches using the segmented points. Present machine learning-based segmentation approaches of Light Detection and Ranging (LiDAR) point cloud data are confined only to different object classes or semantic labelling. In the context of fine-grained feature point classification over the extracted building roof, machine learning approaches have not yet been explored. In this paper, after generating the ground truth data for the extracted building roofs from three different datasets, we apply machine learning methods to segment the roof point cloud based on seven different effective geometric features. The goal is not to semantically enhance the point cloud, but rather to facilitate the application of 3D building reconstruction algorithms, making them easier to use. The calculated F1-scores for each class confirm the competitive performances over the state-of-the-art techniques, which are more than 95% almost in each area of the used datasets.

Irreducible representation of surface distributions and Piola transformation of external loads sustainable by third gradient continua

In this paper Piola transformations are found that relate the Eulerian and Lagrangian external loads which third gradient continua can sustain. As shown by Gabrio Piola and Paul Germain, the most effective postulation scheme in mechanics is based on the principle of virtual work and therefore continuum mechanics must be mathematically founded based on the theory of distributions and on differential geometry. Using the principle of virtual work, the set of admissible external loads sustainable by third gradient continua is seen to include: i) volume force density, ii) surface density of contact force, iii) surface density of contact double force, iv) surface density of contact triple force, v) line density of edge contact forces, vi) line density of contact edge double forces and vii) contact forces concentrated on wedge points. Following the nomenclature introduced by Paul Germain, forces are dual in virtual work of virtual displacements, surface and line double forces are dual of the derivatives of virtual displacements in the normal direction(s) of the surfaces and edges constituting the boundary of the continuum, and surface triple forces are dual of the second normal derivatives of virtual displacements. Volume and surface forces transform as in first gradient Cauchy continua. Moreover we find that: a) the virtual work expended by Eulerian surface triple force, when transformed into the Lagrangian description, must be represented as the work expended by all the kinds of external Lagrangian loads listed in i)-vii); b) Eulerian surface double force transforms into Lagrangian surface double force, surface contact force and edge contact line force; c) Eulerian edge contact line double force transforms into Lagrangian edge contact line double forces, edge line forces and point concentrated wedge forces; d) Eulerian edge and wedge contact line forces transforms into their Lagrangian counterpart only. The Piola transformation formulas deduced in this paper depend on the first, second and third gradients of placement. The presented results allow for the formulation of well-posed boundary condition problems for third gradient continua in the Lagrangian description, and are relevant in computational mechanics. In view of the obtained Piola transformation formulas, the concept of dead loads needs to be modified. We believe to have given an example of how the Mechanics in the French Style, as developed on the ideas by D’Alembert and Lagrange, is still a fertile tool of invention.