The EM Algorithm With Applications To Image Epitome

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Presentation transcript:

The EM Algorithm With Applications To Image Epitome Kiera Henning, Jiwon Kim,

EM Problem Given data U determine the best parameters Q without knowing the values of the hidden variables J. Example: color segmentation (U: color at each pixel, theta: mean color of each cluster, J: cluster assignment for each pixel)

EM Algorithm Maximize the posterior probability of the parameters Q given U, marginalizing over J. To do this, alternate between estimating J (the E-step) and Q (the M-step). More precisely, E step: estimate “distribution over” J. Alternatively, E step: compute tight lower bound to posterior; M step: maximize bound w.r.t. theta.

Using the EM Algorithm Identify U, Q, J and derive update equations

Using the EM Algorithm Identify U, Q, J and derive update equations Convergence guaranteed, but slow

Using the EM Algorithm Identify U, Q, J and derive update equations Convergence guaranteed, but slow: variational EM Instead of directly estimating distribution over J in the E step, use parameterized form for the distribution with a set of variational parameters. This reduces the number of computation steps involved in the update and speeds up the algorithm. However, may lose quality of solution (bound decreased, not maximized). Also up to the designer to choose parameterization.

Using the EM Algorithm Identify U, Q, J and derive update equations Convergence guaranteed, but slow: variational EM Each implementation is application specific No general code that can be plugged in, like graph cuts.

Image Epitome “The epitome of an image is its miniature, condensed version containing the essence of the textural and shape properties of the image.”

Applying EM Data: Parameters to be estimated: Hidden variables: set of patches Parameters to be estimated: epitome Hidden variables: mappings from epitome to image

Examples High initial noise 1000 patches 4000 patches After 10 iterations Low initial noise 1000 patches 4000 patches EM iteration

Examples 4000 Patches

Examples 4000 Patches (high initial noise) 4000 Patches (low initial noise) Reconstructed image

Epitome Extensions Suggestions Courtesy of Nebojsa Jojjic. Video Epitome Motion Epitome Video Epitome: image epitome extended to three dimensions Motion Epitome: articulated object trajectories, as sequences of joint angles

Video Epitome Idea: Extend the concept of an Image Epitome to a Video. The epitome of a video would then be a much smaller video containing the textural, shape, and motion properties of the video.

Motion Epitome Idea: Apply the concept of image epitomes to motion data. The motion epitome would then be significantly smaller than the motion data while still containing most of the motion information. The epitome of motion data is a significantly smaller amount of data which captures the essence of the motion.

Motion Epitome

References Advances in Algorithm for Inference and Learning in Complex Probability Models, Brendan J. Frey and Nebojsa Jojic Epitomic Analysis of Appearance and Shape, Nebojsa Jojic, Brendan J. Frey, and Anitha Kannan Epitome Representation of Images and Notes on Epitome-based Silhouette Generation, Shinko Y. Cheng The Expectation Maximization Algorithm, Frank Dellaert Motion Segmentation using EM – a short Tutorial, Yair Weiss