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### Tracking Using A Highly Deformable Object Model

Nilanjan Ray

Department of Computing Science

University of Alberta

Overview of Presentation

- 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

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

Show cine MRI video

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)

Applications of Level Set: Image Segmentation

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

Show videos

Level Set Applications: Computer Graphics

- Morphing
- Simulation
- Animation
- ….

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

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

for amazing videos

More Applications of Level Set Methods

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

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

Numerical Results and Comparison

Sequence with slow heart motion

Sequence with rapid heart motion

Comparison of mean performance measures

Extensions: Tracking Objects in Color Video

- 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

Extensions: Adding Object Shape Constraint

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

Minimize:

where

Application in Computer Graphics: Morphing

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)

Morphing: Formulation

- 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

Morphing: Preliminary Results

Show videos

Summary

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

Acknowledgements

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

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