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  1. ArticulatedBodies Tracking Eran Sela

  2. Articulated Body • Every general 3D motion can be perceived by a moving group of joints and links. • An articulated body has only joints and fixed length limbs.

  3. Motivation • Based on input data such as depth map, • color, silhouette map – We’ll see today • two works about: • How to implement realtime • skeleton tracking on the articulated body. • The tracking can be used to move • computers graphic models & to capture • 3D motion of human’s body.

  4. Tracking Methods • Supervised or semi-supervised learning trackers: Training sorts of decision trees or other statistical models based on labeled & unlabeled data. • Model based skeleton tracking: Modeling the human body with primitives/surfaces and fitting the model to the data using an optimization scheme. • Image processing based tracking: Generate skeleton based on mathematical condition the data conform to.

  5. Presentation timeline • Articulated Soft Objects for Video-based Body Modeling • Modeling the articulated body • Optimization framework to the data (Least squares). • Data constraints • Results • A Multiple Hypothesis Approach to Figure Tracking • Introduction • The 2D Scaled Prismatic Model • Mode-based Multiple-Hypothesis Tracking • Multiple Modes as Piecewise Gaussians • Results

  6. Articulated Soft Objects for Video-based Body Modeling Input: Video sequence containing: Depth map (using stereo cameras or other method). Silhouette map (The points where the line of sight from the camera is tangent to the surface). Output: A set of 3D ellipsoid primitives with translation, orientation and scale corresponding to the articulated body parts.

  7. Modelling with Primitives vs Soft objects Problem: primitive models such as cylinder and spheres are too crude for precise recovery of both shape and motion • Solution: use Soft objects. • Each primitive defines a field function and the skin • is taken to be a level set of the sum of these fields. • Has the following advantages: • Effective use of stereo and silhouette data • Accurate shape description by a small number of parameters. • Explicit modeling of 3–D geometry

  8. Modelling with Primitives vs Soft objects Problem: primitive models such as cylinder and spheres are too crude for precise recovery of both shape and motion • Solution: use Soft objects. • Each primitive defines a field function and the skin • is taken to be a level set of the sum of these fields. • Has the following advantages: • Effective use of stereo and silhouette data • Accurate shape description by a small number of parameters. • Explicit modeling of 3–D geometry

  9. Modelling with Primitives vs Soft objects Problem: primitive models such as cylinder and spheres are too crude for precise recovery of both shape and motion • Solution: use Soft objects. • Each primitive defines a field function and the skin • is taken to be a level set of the sum of these fields. • Has the following advantages: • Effective use of stereo and silhouette data • Accurate shape description by a small number of parameters. • Explicit modeling of 3–D geometry

  10. Modelling the body parts: State Vector: B – number of body parts N – number of consecutive frames J – number of joints The state vector θ changes on each frame.

  11. Generalized algebraic surfaces Example in 2D:

  12. Generalized algebraic surfaces Example in 3D: =1,

  13. Blinn [2] Metaballs The final surface S is found where the density function F equals some threshold amount, in our case: d(x,y,z) is an algebraic ellipsoidal distance function. Will be defined next it approaches zero slowly. Metaballs (Generalized algebraic surfaces), are defined by a summation over n 3-dimensional Gaussian density distributions, each called a source or primitive.

  14. Ellipsoids as sources Next we define the 3D quadratic distance Function d() from the (x,y,z) point to each ellipsoid source. Why choosing ellipsoids as sources for metaballs? They are simple Allow accurate modeling of human limbs with relatively few primitives Their shape is controlled by higher level width and length parameters And thus problems like over-fitting to high-curvature regions do not occur.

  15. 3D Quadratic distance For a specific metaball and a state vector θ we define 4x4 matrix: Is the scaling and translation along the major axis of the ellipsoid is the radii of the ellipsoid (half the axis length along the principal directions. is the primitive’s center. are the coefficients from the state vector.

  16. 3D Quadratic distance • ,) is initial scaling of each ellipsoid to be proportional to the body part dimensions and prevent over-fitting to high curvature regions. This parameters are constant per each part for all the frames. • is the per frame ellipsoid scaling and they are changing at each frame. The scaling is identical for x and y axis.

  17. World frame and joint frame What changes every frame? Joint 1 frame Rotation The translation of each ellipsoid center from the world frame is constant (The vector C). E is per joint rotation matrix to the quadratic frame and is constant per frame. The scaling of each joint doesn’t change per frame. Joint 2 frame Rotation World frame Translation -Rotation

  18. 3D Quadratic distance is the skeleton induced transformation. A 4x4 rotation-translation matrix From the world frame to the frame to which the metaball is attached. Given the rotation of a joint J, we write: Is homogenous 4x4 transformation from the joint frame to the quadric frame. Is transformation from the world frame to joint frame. Is the ellipsoidal quadratic distance field.

  19. Least Square Framework • In each frame we’re given 3D data • Based on the state vector θ, we define an observation to be the difference between the total field function and the threshold value. • This constrains the point to lie on the surface parameterized by the state vector θ:

  20. Least Square Framework Least squares optimization framework is used to estimate the state vector parameters: is the observation equation for the least squares framework. Each weight can be determined by the object space coordinates, silhouette rays, or temporal constraints.

  21. Least Square Framework Solution to the optimization problem is based on Levenberg-Marquardt algorithm For solving the least squares problem, and find the new state vector θ. The Jacobian matrix is calculated for any point x:

  22. Silhouettes Observations The silhouette points defined as the points where the line of sight from the camera Is perpendicular to the normal of the surface. Why silhouette data is important?

  23. Integrate silhouette constraint

  24. Integrate silhouette constraint • We integrate silhouette observations into our framework by performing an initial search (using Brent’s line minimization) along the line of sight to find the point that is closest to the model at its current configuration. • Then when we find the closest silhouette point to the model we give it a higher weight in the P weight matrix, so the silhouette points are more significant for the fitting.

  25. Fitting Result • Sensor configuration: • Depth is acquired by 3 cameras in an L configuration taking non-interlaced images at 30 frames/sec, with an effective resolution of 640 x 400. • stereo algorithm produced very dense point clouds which are then filtered yielding about 4000 evenly distributed 3–D points on the surface of the subject • In the top row are the original sequences of upper body motions of different persons. Results of the tracking and fitting are shown in the bottom row. Although the two persons have very different body sizes the system adjusts the generic model accordingly.

  26. Fitting Result First person: Second person:

  27. End of topic 1

  28. Presentation timeline • Articulated Soft Objects for Video-based Body Modeling • Modeling the articulated body • Optimization framework to the data (Least squares). • Data constraints • Results • A Multiple Hypothesis Approach to Figure Tracking • Introduction • The 2D Scaled Prismatic Model • Mode-based Multiple-Hypothesis Tracking • Multiple Modes as Piecewise Gaussians • Results

  29. A Multiple Hypothesis Approach to Figure Tracking A 2D human figure tracking. Probability approach to estimate the 2D human figure model. Maintaining a set of possible tracking solutions. Every possible track can be potentially updated with every new update. Over time, the track branches into many possible directions.

  30. A Multiple Hypothesis Approach to Figure Tracking A 2D human figure tracking. Probability approach to estimate the 2D human figure model. Maintaining a set of possible tracking solutions. Every possible track can be potentially updated with every new update. Over time, the track branches into many possible directions.

  31. A Multiple Hypothesis Approach to Figure Tracking A 2D human figure tracking. Probability approach to estimate the 2D human figure model. Maintaining a set of possible tracking solutions. Every possible track can be potentially updated with every new update. Over time, the track branches into many possible directions.

  32. A Multiple Hypothesis Approach to Figure Tracking A 2D human figure tracking. Probability approach to estimate the 2D human figure model. Maintaining a set of possible tracking solutions. Every possible track can be potentially updated with every new update. Over time, the track branches into many possible directions.

  33. A Multiple Hypothesis Approach to Figure Tracking A 2D human figure tracking. Probability approach to estimate the 2D human figure model. Maintaining a set of possible tracking solutions. Every possible track can be potentially updated with every new update. Over time, the track branches into many possible directions.

  34. Used in radars • The MHT is designed for situations in which the target motion model is very unpredictable, as all potential track updates are considered. •  As each radar update is received every possible track can be potentially updated with every new update. Over time, the track branches into many possible directions.

  35. The 2D Scaled Prismatic Model • Scaled Prismatic Models (SPM): • Each link in a scaled prismatic model describes the image plane projection • of an associated rigid link in an underlying 3D kinematic chain. • Each link has 2 DOF: the distance between the joint centers of adjacent links, and the rotation angle at its joint center around an axis which is perpendicular to the image plane. • It captures the foreshortening that occurs when 3D links rotate into and out of the image plane. How we can enforce 3D kinematic constraints of the model that conform to the 2D monocular image data?

  36. The 2D Scaled Prismatic Model • Scaled Prismatic Models (SPM): • Each link in a scaled prismatic model describes the image plane projection • of an associated rigid link in an underlying 3D kinematic chain. • Each link has 2 DOF: the distance between the joint centers of adjacent links, and the rotation angle at its joint center around an axis which is perpendicular to the image plane. • It captures the foreshortening that occurs when 3D links rotate into and out of the image plane. How we can enforce 3D kinematic constraints of the model that conform to the 2D monocular image data?

  37. The 2D Scaled Prismatic Model • Scaled Prismatic Models (SPM): • Each link in a scaled prismatic model describes the image plane projection • of an associated rigid link in an underlying 3D kinematic chain. • Each link has 2 DOF: the distance between the joint centers of adjacent links, and the rotation angle at its joint center around an axis which is perpendicular to the image plane. • It captures the foreshortening that occurs when 3D links rotate into and out of the image plane. How we can enforce 3D kinematic constraints of the model that conform to the 2D monocular image data?

  38. Tracking problem representation We model the human 2D figure as a branched SPM chain. Each link in the arms, legs, and head is modeled as an SPM link. Each link 2 DOF, leading to a total body model with 18 DOF’s. The tracking problem consists of estimating a vector of SPM parameters for the figure in each frame of a video sequence, given some initial state.

  39. Probability Density Representation The choice of representation for the probability density of a tracker state is largely dominated by two concerns: The unimodality constraint imposed when using a Gaussian-based parametric representation such as the KalmanFilter is inaccurate when tracking in a cluttered environment. Sample-based representation (such as used in the CONDENSATION algorithm) requires a prohibitive number of samples for encoding the probability distribution of a high-DOF SPM model.

  40. Condensation Algorithm • Condensation algorithm is an application of particle filtering in which: • Observations and hidden states are represented by hand contours. • Contours can be represented as splines, list of angles between phalanxes, etc. • There is a model for P(next state|previousstate). • Can be set manually by studying the anatomy of a hand. • Can be learned by gathering lots of examples of sequences of hand movement. • Learning can be done using special gloves which report exact hand location and shape. • P(state|observation) is estimated using visual features (SIFT,Harris, etc.)

  41. Probability Density Representation A hybrid approach: Supports a multimodal description but requires fewer samples for modeling. The representation is based on retaining only the modes (or peaks) of the probability density and modeling the local neighborhood surrounding each mode with a Gaussian.

  42. MHT Algorithm • Input: • Video sequence containing 1 or more humans • Output: • A state vector per each frame of values for all the DOF of the SPM chains assembling the model.

  43. Mode-based Multiple-Hypothesis Tracking (Bayes rule) is the observed data at time t. the past image observations (i.e. for

  44. The algorithm The stages of the algorithm at each time-frame are: • 1. Generating the new prior density ) by passing the modes of through the Kalman filter prediction step. • 2. Likelihood computation, involving: • (a) Creating initial hypothesis seeds by sampling the distribution of . • (b) Refining the hypotheses through differential state-space search to obtain the modes of the likelihood . • (c) Measure the local statistics associated with each likelihood mode (saving the modes selected in the likelihood to be updated later). • 3. Computing the posterior density via Baye’s Rule, then updating and selecting the set of modes.

  45. The algorithm The stages of the algorithm at each time-frame are: • 1. Generating the new prior density ) by passing the modes of through the Kalman filter prediction step. • 2. Likelihood computation, involving: • (a) Creating initial hypothesis seeds by sampling the distribution of . • (b) Refining the hypotheses through differential state-space search to obtain the modes of the likelihood . • (c) Measure the local statistics associated with each likelihood mode (saving the modes selected in the likelihood to be updated later). • 3. Computing the posterior density via Baye’s Rule, then updating and selecting the set of modes.

  46. The algorithm The stages of the algorithm at each time-frame are: • 1. Generating the new prior density ) by passing the modes of through the Kalman filter prediction step. • 2. Likelihood computation, involving: • (a) Creating initial hypothesis seeds by sampling the distribution of . • (b) Refining the hypotheses through differential state-space search to obtain the modes of the likelihood . • (c) Measure the local statistics associated with each likelihood mode (saving the modes selected in the likelihood to be updated later). • 3. Computing the posterior density via Baye’s Rule, then updating and selecting the set of modes.

  47. The algorithm The stages of the algorithm at each time-frame are: • 1. Generating the new prior density ) by passing the modes of through the Kalman filter prediction step. • 2. Likelihood computation, involving: • (a) Creating initial hypothesis seeds by sampling the distribution of . • (b) Refining the hypotheses through differential state-space search to obtain the modes of the likelihood . • (c) Measure the local statistics associated with each likelihood mode (saving the modes selected in the likelihood to be updated later). • 3. Computing the posterior density via Baye’s Rule, then updating and selecting the set of modes.

  48. The algorithm The stages of the algorithm at each time-frame are: • 1. Generating the new prior density ) by passing the modes of through the Kalman filter prediction step. • 2. Likelihood computation, involving: • (a) Creating initial hypothesis seeds by sampling the distribution of . • (b) Refining the hypotheses through differential state-space search to obtain the modes of the likelihood . • (c) Measure the local statistics associated with each likelihood mode (saving the modes selected in the likelihood to be updated later). • 3. Computing the posterior density via Baye’s Rule, then updating and selecting the set of modes.

  49. Generating Prior Distributions Kalman Filter Obtaining the prior density in the next time frame is similar to the Kalman filter prediction step. is acquired by a naive constant velocity predictor, (e.g. ).

  50. Kalman Filter State Prediction: Measurement Prediction: - state prediction - control signal (Most of the time there is no control signal) - process noise A,B,H - define the physics of interest ( acceleration, position, speed… ) - measurement prediction - measurement noise