Dense Motion Analysis in Fluid Imagery

An embedded coding scheme for dense motion (optical flow) fields is proposed. Such a scheme is particularly useful in scalable video compression where one must compensate for inter-frame motion at various visual qualities and resolutions. However, the high cost of coding such fields has often made this option prohibitive. Using our previously developed ‘breakpoint’-adaptive wavelet transform, we show that it is possible to code dense motion fields efficiently while simultaneously endowing the coded motion representation with embedded resolution and quality scalability attributes. Performance comparisons with the traditional non-scalable block-based model are also made and presented with the aid of a modified H.264/AVC JM reference encoder.

SynopsisWhen a rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity field and the pressure field of the fluid motion and the angular velocity of the rigid body.In this paper we obtain existence and uniqueness results for the steady state problem in which a rigid body rotates about an axis of symmetry in a viscous incompressible fluid.

In this paper, a dense motion field estimation technique based on the Bayesian framework is proposed to estimate the true dense motion fields of video sequences. Previous stochastic techniques of dense motion field estimation adopts piecewise smooth motion model and use MAP estimation to find the motion field with joint minimization of motion compensation errors and maximization of motion smoothness. However, such random process does not guarantee to converge to the true motion field. In this paper, the motion of landmark points in the video sequence is introduced into the MAP estimation process to regularize the estimated motion field. Experimental results show that the proposed algorithm produces estimated motion fields which preserve piecewise smooth nature and are visually close to the true motion of the video sequences.

In this paper, the potential flow equations are converted to ordinary differential equations through a Galerkin approach in which velocity and pressure potential functions are expanded in terms of closedform solutions to Laplace's equation. Because the method gives differential equations for the flow in terms of a relatively few generalized coordinates (that represent modes of the flow field) the resultant equations can be used effectively in preliminary design, real-time simulations, and dynamic eigenvalue analysis for aeroelasticity. This new theory is more general than the Peters-He dynamic wake model since it has a more rigorous derivation and includes inflow modes previously neglected in the Peters-He model. Results are presented in the frequency domain for simple harmonic motion. The axial velocity components are obtained by this new methodology on and off the disk, for axial and skewed flows, and for different pressure distributions and are compared with the Peters-He model and with an exact solution obtained by a convolution integral. INTRODUCTION Dynamic inflow models go back over 50 years in terms of their development and application. Sissingh first formulated a mathematical model that described how gradients in inflow could result from cyclic pitch changes or from fuselage pitch and roll rates. He further showed how these inflow gradients could create cyclic changes in blade angle of attack that significantly changed the actual rotor response. Curtiss and Shupe refined this idea into a lift deficiency function for rotor cyclic variations. Ormiston and Peters showed how the formulation could be generalized into an inflow model * AIAA Fellow Copyright © 2002 The American Institute of Aeronautics and Astronautics Inc. All rights reserved. for both uniform and side-to-side variations in inflow. Peters generalized these concepts to the unsteady case by including apparent mass terms to account for the time delay involved in the development of the rotor flow field. Comparisons with static data in hover and forward flight showed that the previous methodologies were inadequate in forward flight. Pitt and Peters used principles of potential flow theory to show that these concepts could be developed from first principles without ad hoc assumptions on time delay or on the effects of forward flight. They obtained an unsteady flow model from potential functions that gave spectacular correlation with wind tunnel response data throughout the frequency and advance ratio range. However, despite the unquestioned success of this inflow model, it still remained a theory with only three inflow degrees of freedom: 1) uniform, 2) fore-to-aft, and 3) side-to-side. Later, Peters and He showed how the Pitt-Peters ideas could be truly generalized to a theory with an arbitrary number of inflow harmonics and an arbitrary number of radial shape functions per harmonic. They showed how the general theory could reduce in special cases to the old Pitt-Peters model as well as to Loewy theory and Prandtl tip-loss theory. This new theory was used to correlate unsteady wind tunnel data from the NASA Langley wind tunnel for various planforms, thrust coefficients, and advance ratios. The match in both steady and unsteady distributions was excellent The Peters-He model represents a mature dynamic inflow model that is now used in many production codes, including 2GCHAS (Government code), FLIGHTLAB (Advanced Rotorcraft Technology), COPTER (Bell helicopter), ONERA-DFVLR (European Community) and many university applications. More recent extensions of the theorv include applications to the off-axis coupling problem, and to ground effect in flight simulation. 1 American Institute of Aeronautics and Astronautics (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. The inflow models described in the above paragraphs are formulated such that the states of the model represent induced flow distributions on the rotor disk (expanded radially as polynomials and azimuthally as Fourier coefficients). The first-order differential equations for these states depend on the free-stream impingement angle, flight speed, and thrust level. However, they are all in closed form in terms of a mass matrix and a damping matrix, such that they are easily assembled either into a comprehensive code, an eigenvalue analysis, or a flight simulator. This makes the model extremely efficient. However, in many of the applications, such as the ground effect work, it is necessary to find not just the normal flow at the disk, but all three components of flow off of the disk as well. Peters and Morillo presented a consistent methodology for computing all three components of the flow in axial flow, both on and off of the rotor disc within the context of a finite-state model. The model is formulated in a manner fairly similar to previous work (in that the potential functions in ellipsoidal coordinates are used). However, in contrast with the previous work, the states represent velocity potentials rather than individual flow components. In addition, all potential functions are considered (not just the ones that have a pressure discontinuity across the disk); and the derivation of the equation coefficients is done in a more consistent and rigorous manner than in the earlier derivations. The result is a simpler derivation and a more complete inflow theory for the velocity both off and on the disk. Previous dynamic inflow models in axial flow are shown to be special cases of the new model when offdisk coupling is neglected. This new methodology based on a Galerkin approach provides the exact solution on as well as off the disk for the axial velocity component in axial flow. In this paper, the concepts introduced by Peters and Morillo are extended to analyze skewed flow cases. FLUID DYNAMICS EQUATIONS The three-dimensional potential flow equations (momentum and continuity equations) for the pressure and velocity fields P and v , with a free-stream velocity V^are f;-Jp-^ (1) V • v = 0 (2) These equations have been non-dimensionalized by defining P as pressure divided by pVj, v as induced velocity divided by V^ and time as a reduced time T, (i.e., time multiplied by VJR.) The variable £ is the non-dimensional coordinate along the free-stream line, positive upstream. All lengths are divided by the rotor radius R. Figure 1 shows the coordinate system. Fig.l Coordinate system. From continuity, Eq. (2), it is observed that v can be expressed by a velocity potential that satisfies Laplace's equation. It can also be shown that P satisfies Laplace's equation. Therefore, P can be expressed as a summation of pressure potentials, 0; and v can be expressed as a summation of the gradient of velocity potentials, V. NEW FORMULATION Pressure Potentials* and Velocity Potentials To transform Eqs. (1) and (2) by a Galerkin method, it is required to expand the pressure potential, 4>, and the velocity potentials, *P, in terms of complete set of functions that satisfy Laplace's equation. In addition, they have to fulfill the boundary conditions for pressure in the case of <&, and for velocity in the case of V. The boundary conditions for pressure are given by a discontinuity across the rotor disk. The use of an ellipsoidal coordinate system, has the advantage that any odd function in v, will allow a representation of a discontinuity across the rotor disk. An additional advantage of using an ellipsoidal coordinate system is that an analytical solution of Laplace's equation is known and can be expressed as,

A superresolution imaging method suitable for imaging objects moving in a dynamic scene is described. The main operations are performed over three threads: the computation of a dense interframe 2D motion-field induced by the moving objects at subpixel resolution in the first thread. Concurrently, each video image frame is enlarged by cascade of an ideal low-pass filter and a higher rate sampler, essentially stretching each image onto a larger grid. Then, the main task is to synthesize a higher resolution image, from the stretched image of the first frame, and that of the subsequent frames subjected to suitable motion compensation. A simple averaging process and/or a simplified Kalman filter may be used to minimize the spatio-temporal noise, in the aggregation process. The method takes advantage of widely used MPEG-4 encoding hardware/software tools. A few experimental cases are presented with a basic description of the key operations performed in the overall process.

This study conducts a thorough numerical analysis of a complex problem involving magnetohydrodynamic flow, radiation, activation energy, Soret, the Dufour effect, and heat transfer in a permeable channel for a Maxwell fluid. Governing equations for momentum, concentration, and heat are transformed into ordinary differential equations through a similarity transformation, enabling focused analysis. Utilizing a fourth-order Runge-Kutta scheme, we compute dimensionless velocity, temperature, and concentration fields for the steady motion of a Maxwell fluid in a channel. To ensure accuracy, our results are validated against existing research, showing excellent agreement. The study investigates the impact of various physical parameters on fluid motion, revealing insights such as decreased velocity and concentration with increased chemical reaction, and rising fluid temperature. Additionally, concentration distribution decreases with a chemical reaction and a higher Schmidt number. This research enhances understanding of complex fluid dynamics, providing valuable insights for engineering applications, particularly in industries requiring precise fluid behavior control.

It is shown that the solutions for the non-linear equations governing the motion of a stratified self-gravitating isothermal fluid are generated by a sine-Gordon equation (SG) which has well-known soliton solutions. Starting with two solutions of the SG-equation which are related via an auto-Baecklund-transformation, the physical quantities, density, velocity and gravitational potential are expressed in a parametric form in terms of characteristic coordinates. The zero-solution and the one-soliton solution of SG generate a single hump in the density moving with constant velocity. The typical non-linear feature is the relation between amplitude A and width (y = square of the speed of sound). The 1-soliton together with the 2-sollton solution of SG produces two moving humps in the density which come to rest asymptotically. As a result of the interaction the ratio amplitude/width is different from the corresponding ratio of the single soliton. Due to a certain exchange symmetry in the transformation procedure relating the SG to the gravity system it is possible to generate another solution out of the 2-soliton. Which represents a single hump moving on a homogeneous background. The remarkable property is that the density becomes infinite in a finite time depending on the amplitude. The “breather”-solutlon of SG produces a density distribution that represents two pulses colliding onto a central hump at rest which are separated from each other by a discontinuity in the density and the velocity field. These solutions may have applications to the non-linear evolution of flat structures arising in the process of galaxy formation.

Electroosmotic flow is fluid motion driven by an electric field acting on the net fluid charge produced by charge separation at a fluid-solid interface. Under many conditions of practical interest, the resulting fluid velocity is proportional to the local electric field, and the constant of proportionality is everywhere the same. Here we show that the main conditions necessary for this similitude are a steady electric field, uniform fluid and electric properties, an electric Debye layer that is thin compared to any physical dimension, and fluid velocities on all inlet and outlet boundaries that satisfy the Helmholtz-Smoluchowski relation normally applicable to fluid-solid boundaries. Under these conditions, the velocity field can be determined directly from the Laplace equation governing the electric potential, without solving either the continuity or momentum equations. Three important consequences of these conditions are that the fluid motion is everywhere irrotational, that fluid velocities in two-dimensional channels bounded by parallel planes are independent of the channel depth, and that such flows exhibit no dependence on the Reynolds number. Similitude is demonstrated by comparing measured and computed fluid streamlines with computed electric flux lines.

Particle image velocimetry (PIV) has matured as a flow measurement technique. It enables the description of the instantaneous velocity field of the flow by analyzing the particle motion obtained from digitally recorded images. Correlation based PIV evaluation technique is widely used because of its good accuracy and robustness. Although very successful, correlation PIV technique has some weakness which can be avoided by optical flow based PIV algorithms. At present, most of the optical flow methods applied to PIV are based on brightness constancy assumption. However, some factors of flow imaging technology and the nature property of the fluids make the brightness constancy assumption less appropriate in real PIV cases. In this paper, an implementation of a 2D optical flow algorithm (GCOF) based on gradient constancy assumption is introduced. The proposed GCOF assumes the edges of the illuminated PIV particles are constant during motion. It comprises two terms: a combined local-global gradient data term and a first-order divergence and vorticity smooth term. The approach can provide accurate dense motion fields. The approach are tested on synthetic images and on two experimental flows. The comparison of GCOF with other optical flow algorithms indicates the proposed method is more accurate especially in conditions of illumination variation. The comparison of GCOF with correlation PIV technique shows that the proposed GCOF has advantages on preserving small divergence and vorticity structures of the motion field and getting less outliers. As a consequence, the GCOF acquire a more accurate and better topological description of the turbulent flow.

In this paper a new iterative motion-based segmentation algorithm is presented for object-based video coding. At first, dense motion fields between image pairs are obtained by a multiresolution motion estimation algorithm. Based on these motion fields a set of global motion vectors are determined by an iterative vector quantisation procedure. Using the global motion vectors, the images are segmented into homogenous regions. Covered and uncovered areas of the background and objects are properly recognized by examining two consecutive motion fields. Experimental results for artificial as well for real-life video have shown that the proposed algorithm results in good quality segmentation label fields while it requires only moderate computations.

Deep learning for fluid velocity field estimation: A review

In this paper the development and a partially verification of an anisotropic modification of linear and nonlineare models of turbulence is presented. This modification departs from the observation that there are some regions of fluid domains in which the Reynolds stresses do not depend on the magnitude of the mean flow gradients. The model is based on the hypothesis that actions of fluctuating motion on mean motion are almost exclusively due to large turbulent eddies, and that the latter contain the same kinetic energy of the small eddies of mean flow. The contribution of large turbulent eddies to the Reynolds stresses is then modelled by a linear combination of an isotropic term and an anisotropic term. The latter is obtained by multiplying a quote of the turbulent kinetic energy by a normalised redistribution tensor. The performances of the model are tested calculating the Reynolds stresses in a fully developed turbulent channel flow, and simulating the separated turbulent flow over a backward-facing step, and a fully developed turbulent flow in a squared duct. The results show that the modified models improve the prediction of the normal Reynolds stresses in a bidimensional channel and of the reattachment point in a backward-facing step. On the other hand, these models do not supply any improvement of the secondary flow in a squared duct. INTRODUCTION Algebraic two-equation models of turbulence for Reynolds averaged Navier-Stokes equations are still, today, among the most widely used in the field of engineering design. The two major alternative to such models are the second-order closure models and large-eddy simulations (LES). The weak points of the first approach are high computational cost, due to the five more equations to be solved, and wall-bounded turbulent flows. On the other hand, LES requires much more computational time than second-order closure models, and improved models for the smallscale and the wall region are needed. For these reasons these two approaches have not become yet a useful engineering tool. The aim of this work is to develop an anisotropic modification for linear and nonlinear turbulence model which improves the prediction of normal Reynolds stresses, without increasing the number of differential equations to be solved. More precisely, the objective is to simulate the contribution to the anisotropy of the turbulent stresses observable in areas characterised by low mean velocity gradients. This phenomenon, which can be noted, for example, in experimental data of the turbulent channel flow in figure 5, is not actually described by standard, explicit algebraic models. The starting point of this model consists in the substitution of turbulent kinetic energy by an appropriate vectorial magnitude which provides information about the anistropy of the turbulent stresses. The quantity is identified by the field of velocity of the small eddies of mean motion. The kinetic energy of these small eddies is obtained by assuming that these t Dr, Center of Studies and Activities for Space (CISAS). Dr, Department of Mechanical Engineering. ' Professor, Department of Mechanical Engineering. Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. have the same kinetic energy content of the large turbulent eddies: the two fields of motion are said to be similar. It has been demonstrated that the field of motion of large turbulent eddies, or coherent structures^ can be obtained directly from measurements of instantaneous velocity once the quota of turbulent kinetic energy has been defined. The models obtained provides improvements in the predictions of normal Reynolds stresses present in the turbulent flow of a bidimensional channel; as for the separated turbulent flow in a backward-facing step appreciable improved results of mean flow and of normal Reynolds stresses have been attained compared to standard algebraic models. On the other side, as it will be shown in the following paragraphs, the model of the turbulent kinetic energy redistribution tensor does not supply any improvement in the case of the fully developed turbulent flow in a squared duct. ANISOTROPIC MODIFICATION OF THE REYNOLDS STRESSES Let us consider a field of turbulent motion of a viscous, homogenous and incompressible fluid; the instantaneous velocity v and the instantaneous pressure p can be set out as follows in a mean and fluctuating parts: v = v -f v' p-p + p (1) The evolution of the velocities and mean pressures can be described completely by the well-known Reynolds equations:

In two-view correspondence learning, prevalent multi-layer perceptron (MLP)-based methods struggle with context capturing. To remedy this issue, recent advances innovatively stack convolutional neural network (CNN)-based Resblocks sequentially, showing an inherent proficiency in local context extraction. Yet, such non-issue-specific designs inherit the drawback of CNN’s difficulty in aggregating global context, leading to performance bottlenecks. To address this problem, this prospective study further explores the potential of the CNN-based framework and proposes MC-Net, a top-performing network that integrates both local and global context elegantly and seamlessly. Specifically, considering that sparse motion vectors and a dense motion field can be converted into each other through interpolation and sampling, we first transform unordered matches into image-structured data by estimating the dense motion field implicitly. Then, we design a hierarchical rectifying module to rectify the error of each ordered motion vector with CNN at multiple levels, enabling MC-Net to perceive global context from coarse-level features and local context from fine-level features simultaneously, which facilitates to tackle the discontinuities of the motion field in case of large scene disparity. Finally, we reconstruct comprehensive context-embedded features from rectified motion fields at all levels. Also, instead of using the residuals between rectified and pre-rectified motion vectors at the same layer to reject outliers as in previous studies, which seriously affects the inlier prediction accuracy, we rethink this operation meticulously and modify it to the difference between motion vectors obtained from each layer’s reconstruction and ones from the first layer before transformation, ensuring purer residuals and enhancing the matching performance without extra computational burden. Extensive experiments show that MC-Net outperforms state-of-the-arts on multiple domains and datasets.

A method to obtain motion artifact-free reconstructed images of the coronary arteries is proposed and evaluated. The method relies on the integration of coronary motion estimation in an iterative computed tomography reconstruction technique. Coronary motion fields are derived from a set of coronary centerlines extracted at multiple cardiac phases within the R-R interval. Start and end points are provided by the user in one time-frame only. Corresponding centerline positions are used to determine the motion fields from phase to phase. Finally, dense motion fields are achieved by thin-plate-spline interpolation and are used to perform a motion-corrected iterative reconstruction of a selected region of interest, which results in an effective improvement of the reconstructed image quality.

Sloshing may induce adverse loads to cause structural instability and damage. A vertical elastic baffle mounted at the inside bottom of a rectangular container is used as an anti-slosh device to attenuate the liquid oscillation. A semi-analytical model is presented to analyze the hydroelastic problem. The liquid is partitioned into four simple sub-domains with three hypothetical interfaces. The velocity potential of each sub-domain is analytically deduced by the separation of variables. The baffle deflection is expanded into the Fourier series by its dry modals. The eigenvalue equation is formulated by plugging the velocity potentials into the sloshing conditions, interface continuity conditions, as well as the dominant equation and compatibility conditions of the baffle. Then, the velocity potential is expressed by a complete basis of the coupled mode shapes for the system considered under lateral excitation. The system response equation is constituted by inserting the velocity potential into wave equations and baffle equation. The proposed method is verified by comparing the present results with the available data. In addition, numerical analyses are performed to examine the effects of baffle parameters on the natural frequencies, mode shapes and dynamic responses of the container. The sloshing frequency may be altered to a higher value due to the installation of the elastic baffle.