lecture13.ppt

CSci 6971: Image Registration Lecture 13: Robust EstimationFebruary 27, 2004
Prof. Chuck Stewart, RPI
Dr. Luis Ibanez, Kitware
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Lecture 13
Today’s Lecture
Motivating problem:
Mismatches and missing features
Robust estimation:
Reweighted least-squares
Scale estimation
Implementation in rgrl
Solving our motivating problem
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Lecture 13
Motivation
What happens to our registration algorithm when there are missing or extra features?
Mismatches tend to occur, causing errors in the alignment
Our focus today is how to handle this situation
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Lecture 13
Robust Estimation
Down-grade or eliminate the “influence” of mismatches (more generally, gross errors or “outliers”)
Base the estimate of the transformation on correct matches (“inliers”)
Major question is how to distinguish inliers from outliers
Studying ways to answer this question has occupied statisticians puter vision researchers for many years.
We’ll look at just a few of the answers.
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Lecture 13
A Simplified Problem: Location Estimation
Given:
A set of n scalar values {xi}
Problem:
Find the mean and variance (or standard deviation) of these values
Standard solutions:
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Lecture 13
Maximum Likelihood Estimation
The calculation of the average from the previous slide can be derived from standard least-squares estimation, which in turn derives from Maximum-Likelihood Estimation
The (normal) probability of a particular random xi value:
The probability of n independent random values:

The negative log likelihood:
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Lecture 13
Estimating the Mean
Focusing only on m, we can ignore the first two terms, and focus on the least-squares problem of minimizing:
Computing the derivative with respect to m and setting it equal to 0 yields:
Solving leads to the original estimate of the mean:
Remember this
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Lecture 13
Estimating the Variance
Substituting the esti