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Image-Based Photonic Techniques for Microfluidics
Published in Sushanta K. Mitra, Suman Chakraborty, Fabrication, Implementation, and Applications, 2016
David S. Nobes, Mona Abdolrazaghi, Sushanta K. Mitra
A general PTV algorithm consists of a number of steps, including camera calibration, particle identification, three-dimensional particle reconstruction, and tracking of particles in the three-dimensional space (Maas et al., 1993). Camera calibration determines each camera’s unique projection matrix, which specifies the viewing direction and camera position. Images from the multicamera system are then collected and processed in a general scheme that is outlined in Figure 1.8. The scheme consists of anywhere from one to nc cameras that collect synchronized images of the field of view at a known Δt apart. Figure 1.8 shows an arbitrary series of acquired images, with particles visible in each image. Particles are identified within the two-dimensional images, and a list of their centroid locations is developed. Particles are then reconstructed from these positions and each camera’s unique projection matrix. This process takes the particle locations from images acquired from all cameras at a single time step and determines the three-dimensional positions of particles using epipolar lines to match the particles and their positions from different views (Kurada et al., 1997; Ouellette et al., 2006). Particle reconstruction is completed for each individual time step.
Visual Guidance for Autonomous Vehicles: Capability and Challenges
Published in Shuzhi Sam Ge, Frank L. Lewis, Autonomous Mobile Robots, 2018
Andrew Shacklock, Jian Xu, Han Wang
This equation is linear because we use homogeneous coordinates by augmenting the position vectors with a scalar (X=[X˜T1]T∈ℝ4) and likewise the image point (x=[xyw]T∈ℝ3:x˜=x/w). A powerful and more natural way of treating image formation is to consider the ray model as an example of projective space. P is the projection matrix and encodes the position of the camera and its intrinsic parameters. We can rewrite (1.1) as: () x=K[RT]X˜:K∈ℝ3×3,R∈SO(3),T∈ℝ3
Vision-based methodology for characterizing the flow of a high-density crowd
Published in Nigel Powers, Dan M. Frangopol, Riadh Al-Mahaidi, Colin Caprani, Maintenance, Safety, Risk, Management and Life-Cycle Performance of Bridges, 2018
J. Van Hauwermeiren, P. Van den Broeck, K. Van Nimmen, M. Vergauwen
R and t are a 3 × 3 matrix and 3 × 1 vector describing the orientation and position of the camera in world coordinates respectively. The matrix P is a 3×4 matrix and is called the projection matrix.
Hidden dimensions of the data: PCA vs autoencoders
Published in Quality Engineering, 2023
Davide Cacciarelli, Murat Kulahci
By setting and using once again Eckart and Young (1936) theorem, the solution that minimizes (7) is given by the projection matrix that projects onto -dimensional subspace spanned by eigenvectors of the covariance matrix of corresponding to the first largest eigenvalues. Even though LAE finds the subspace spanned by these eigenvectors, it cannot necessarily find the exact eigenvectors as and in Equation (7) can simply be replaced with and for any invertible without affecting the minimization in (7). Kunin et al. (2019) show that applying an L2 regularization to the loss function allows for the retrieval of the exact eigenvectors from the left singular vectors of , or the right singular vectors of . In that case, the loss function becomes
A new key performance indicator oriented industrial process monitoring and operating performance assessment method based on improved Hessian locally linear embedding
Published in International Journal of Systems Science, 2022
Hongjun Zhang, Chi Zhang, Jie Dong, Kaixiang Peng
Furthermore, to monitor the whole production process and detect the fault in real-time, the process monitoring statistics are constructed based on the features extracted by IHLLE. In the KPI-oriented subspace, the Hotelling statistics can be calculated as follows: where is the covariance matrix of the projection matrix of the projection vectors in the training dataset. The corresponding control limit can be determined by the kernel density estimation (KDE) based on the training dataset collected from normal conditions. Therefore, in the online monitoring stage, for a sample , the fault detection logic is
Rational (Padé) approximation for estimating the components of the partially-linear regression model
Published in Inverse Problems in Science and Engineering, 2021
Dursun Aydın, Ersin Yılmaz, Nur Chamidah
Notice that linear smoother defined in (2.10) is a projection matrix and known as hat matrix in statistics theory. In this sense, it transforms the vector of response observations into the fitted values in a simple nonparametric model, obtained from partially linear model (1.1) with . From (2.12) we can say that fitted nonparametric function vector in the partially- linear model (1.1) is given by as expressed in the equation (2.10). It should be noted that the matrix plays the same role as the matrix (A5.9) in the smoothing spline method. But there are some important similarities and differences between in (2.13) and in (A5.9), as indicated in section 4.