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Deformable Object Tracking: A Variational Optimization Framework. CMPUT 615 Nilanjan Ray. Tracking Deformable Objects. Desirable properties of deformable models: Adapt with deformations (sometimes drastic deformations, depending on applications) Ability to learn object and background:
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Deformable Object Tracking: A Variational Optimization Framework CMPUT 615 Nilanjan Ray
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
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?
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)
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
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
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.
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)
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
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
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:
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
Results: Tracking Cardiac Motion A few cine MRI frames and delineated boundaries on them Show videos
Numerical Results and Comparison Sequence with slow heart motion Sequence with rapid heart motion Comparison of mean performance measures