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Deformable Object Tracking: A Variational Optimization Framework

This paper presents a novel framework for tracking deformable objects through variational optimization. The proposed method combines robust deformation capabilities with effective recognition by leveraging joint probability density estimation of level set functions and image intensities. Utilizing Kullback-Leibler divergence for optimization, it achieves stable convergence even with large time steps through a semi-implicit numerical scheme. This approach is validated by applying it to cine MRI frames for cardiac motion tracking, demonstrating superior performance in challenging sequences.

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Deformable Object Tracking: A Variational Optimization Framework

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  1. Deformable Object Tracking: A Variational Optimization Framework CMPUT 615 Nilanjan Ray

  2. Tracking Deformable Objects • Desirable properties of deformable models: • Adapt with deformations (sometimes drastic deformations, depending on applications) • Ability to learn object and background: • Ability to separate foreground and background • Ability to recognize object from one image frame to the next, in an image sequence

  3. Some Existing Deformable Models • Deformable models: • Highly deformable • Examples: snake or active contour, B-spline snakes, … • Good deformation, but poor recognition (learning) ability • Not-so-deformable • Examples • Active shape and appearance models • G-snake • … • Good recognition (learning) capability, but of course poor deformation ability So, how about good deformation and good recognition capabilities?

  4. Technical Background: Level Set Function • A level set function represents a contour or a front geometrically • Consider a single-valued function (x, y) over the image domain; intersection of the x-y plane and  represents a contour: (X(x, y), Y(x, y)) is the point on the curve that is closest to the (x, y) point • Matlab demo (lev_demo.m)

  5. Technical Background: Non-Parametric Density Estimation Normalized image intensity histogram: I(x, y) is the image intensity at (x, y) i is the standard deviation of the Gaussian kernel C is a normalization factor that forces H(i) to integrate to unity

  6. Technical Background: Similarity and Dissimilarity Measures for PDFs Kullback-Leibler (KL) divergence (a dissimilarity measure): Bhattacharya coefficient (a similarity measure): P(z) and Q(z) are two PDFs being compared

  7. Proposed Method: Tracking Deformable Object • Deformable Object model (due to Leventon [1]): • From the first frame learn the joint pdf of level set function and image intensity (image feature) • Tracking: • From second frame onward search for similar joint pdf [1] M. Leventon, Statistical Models for Medical Image Analysis, Ph.D. Thesis, MIT, 2000.

  8. Deformable Object Model • Joint probability density estimation with Gaussian kernels: Level set function value: l Image intensity: i J(x, y) is the image intensity at (x, y) point on the first image frame (x, y) is the value of level set function at (x, y) on the first image frame C is a normalization factor We learn Q on the first video frame given the object contour (represented by the level set function)

  9. Proposed Object Tracking • On the second (or subsequent) frame compute the density: • Match the densities P and Q by KL-divergence: • Minimize KL-divergence by varying the level set function (x, y) Note that here only P is a function of (x, y) I(x, y) is the image intensity at (x, y) on the second/subsequent frame (x, y) is the level set function at on the second/subsequent frame

  10. Minimizing KL-divergence • In order to minimize KL-divergence we use Calculus of variations • After applying Calculus of variations the rule of update (gradient descent rule) for the level set function becomes: t : iteration number t : timestep size

  11. Minimizing KL-divergence: Implementation • There is a compact way of expressing the update rule: convolution is a function defined simply as: Where g1 is a convolution kernel:

  12. Minimizing KL-divergence: A Stable Implementation • The previous implementation is called explicit scheme and is unstable for large time steps; if small time step is used then the convergence will be extremely slow • One remedy is a semi-implicit scheme of numerical implementation: Where g is a convolution kernel: is a function defined simply as: In this numerical scheme t can be large and still the solution will be convergent; So very quick convergence is achieved in this scheme

  13. Results: Tracking Cardiac Motion A few cine MRI frames and delineated boundaries on them Show videos

  14. Numerical Results and Comparison Sequence with slow heart motion Sequence with rapid heart motion Comparison of mean performance measures

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