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Tracking Using A Highly Deformable Object ModelPowerPoint Presentation

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

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

- Examples

- Highly deformable

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 for PDFs

- 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 for PDFs

- 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 for PDFs

- 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 for PDFs

- 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 for PDFs

- 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 for PDFs

- 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 for PDFs

A few cine MRI frames and delineated boundaries on them

Show videos

Numerical Results and Comparison for PDFs

Sequence with slow heart motion

Sequence with rapid heart motion

Comparison of mean performance measures

Extensions: Tracking Objects in Color Video for PDFs

- 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 for PDFs

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

Minimize:

where

Application in Computer Graphics: Morphing for PDFs

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 for PDFs

- 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 for PDFs

Show videos

Summary for PDFs

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

Acknowledgements for PDFs

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

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