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Tracking Using A Highly Deformable Object Model

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Tracking Using A Highly Deformable Object Model

Nilanjan Ray

Department of Computing Science

University of Alberta

- Tracking deformable objects
- Motivations: desirable properties of a deformable object model
- An example application (mouse heart tracking)

- Some technical background
- Level set function and its application in image processing
- Non-parametric probability density function (pdf) estimation
- Similarity/dissimilarity measures for pdfs

- Proposed tracking technique
- Results, comparisons and demos
- Ongoing investigations
- Incorporating color cues, and other features
- Adding constraints on object shape
- Application in morphing (?)
- Incorporating object motion information (??)

- Summary
- Acknowledgements

- 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

Show cine MRI video

- 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

- Examples

- Highly deformable

So, how about good deformation and good recognition capabilities?

- 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)

- Matlab segmentation demo (yezzi.m)
- Vessel segmentation
- Brain reconstruction
- Virtual endoscopy
- Trachea fly through
- …tons out there

Show videos

- Two example videos

Show video

- Finding shortest path

Show video

- Morphing
- Simulation
- Animation
- ….

http://www.sci.utah.edu/stories/2004/fall_levelset.html

Go to http://graphics.stanford.edu/~fedkiw/

for amazing videos

- Go to http://math.berkeley.edu/~sethian/2006/Applications/Menu_Expanded_Applications.html

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

Kullback-Leibler (KL) divergence (a dissimilarity measure):

Bhattacharya coefficient (a similarity measure):

P(z) and Q(z) are two PDFs being compared

- 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.

- 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)

- 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

- 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

- There is a compact way of expressing the update rule:

convolution

is a function defined simply as:

Where g1 is a convolution kernel:

- 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

A few cine MRI frames and delineated boundaries on them

Show videos

Sequence with slow heart motion

Sequence with rapid heart motion

Comparison of mean performance measures

- If we want to learn joint distribution of level set function and color channels (say, r, g, b), then non-parametric density estimation suffers from:
- Slowness
- Curse of dimensionality

- Another important theme is combine edge information and region information of objects
- One remedy sometimes is to take a linear combination of r, g, and b channels
- Fisher’s linear discriminant can be used to learn the coefficients of linear combination

- A demo

- Can we constrain the object shape in this computational framework?

Minimize:

where

Initial object Shape and

intensity/texture

Final object Shape and

intensity/texture

(J1, 1)

(I2, 2)

(I1, 1),

(I2, 2),

…..

Morphing: generate realistic intermediate tuples (It, t)

- Generate intermediate shapes, i.e., level set function t (say, via interpolation):
- Next, generate intermediate intensity It by maximizing:
- Once again we get a similar PDE for It

Show videos

- Highly deformable object tracking: Variational minimization of KL-divergence leading to fast and stable partial differential equations
- Several exciting extensions
- Application in morphing

- Baidyanath Saha
- CIMS lab and Prof. Hong Zhang
- Prof. Dipti P. Mukherjee, Indian Statistical Institute
- Department of Computing Science, UofA