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)

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Tracking Using A Highly Deformable Object Model

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Tracking using a highly deformable object model

Tracking Using A Highly Deformable Object Model

Nilanjan Ray

Department of Computing Science

University of Alberta


Overview of presentation

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

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

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

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

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 image denoising

Level Set Applications: Image Denoising

  • Two example videos

Show video


Level set applications robotics

Level Set Applications: Robotics

  • Finding shortest path

Show video


Level set applications computer graphics

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

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

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

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

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

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

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

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

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

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

Results: Tracking Cardiac Motion

A few cine MRI frames and delineated boundaries on them

Show videos


Numerical results and comparison

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

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

Extensions: Adding Object Shape Constraint

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

Minimize:

where


Application in computer graphics morphing

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

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

Morphing: Preliminary Results

Show videos


Summary

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

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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