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
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
Where g1 is a convolution kernel:
is a function defined simply as:

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
Pratt’s FOM
Segmentation Score
Slow Seq.
Rapid Seq.
GVF Snake Method
0.51
0.62
0.44
Proposed Method
0.88
0.77
0.74
0.79