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Image warping using empirical Bayes
Neurohnage
13, Number
6, 2001,
Part 2 of 2 Parts ID
E blw
METHODS
- ANALYSIS
Image Warping using Empirical John Ashburner, Wellcome
Department
of Cognitive
Bayes
K.J. Friston Neurology,
London,
UK.
Image warping procedures are often thought of as a regularised maximum likelihood problem, and involve finding a set of parameters for a warping model that maximise the likelihood of the observed data. What is actually needed is a set of parameters that are maximally likely given the data. In order to achieve this, the problem should be considered within a Bayesian context, and expressed something like: “The posterior probability of the parameters given the data is proportional to the likelihood of observing the data given the parameters times the prior probability of tbe parameters.” Taking logs of this expression converts the Bayesian problem into one of simultaneously minimising two cost function terms. The first of these is usually a measure of intensity correlation between the registered images (likelihood potential), whereas the second is a measure of the roughness of the warps (prior potential). An ongoing argument in the field of image registration is something to the effect of “if there are too many parameters in a registration model, then the model will be overfitted”, versus “if not enough parameters are used, then the images can not be registered properly”. A better way of expressing this problem is in terms of obtaining an optimal balance between the cost function terms. This involves assigning the most appropriate weighting to each. These weights (hyperparameters) can be estimated from the data using an empirical Bayes approach. An algorithm similar to expectation maximisation (EM) can be used to do this. Pure EM is an iterative approach that alternates between two steps: the expectation (E) step and the maximisation (M) step. The algorithm is initialised by assigning some starting estimates to the hyperparameters. The first E step involves computing the expectation of the warp parameters while holding the hyperparameters constant. This is where this approach deviates from a pure EM algorithm, as a MAP estimate is used instead of the expectation of the warping parameters. The M step involves re-estimating the hyperparameters such that the likelihood of observing the data is maximised, while holding the parameters fixed. These hyperparameters are derived from variance estimates of both the residual difference between the images and the roughness of the deformations. Different models can be used to parameterise the variability. For example, it could be assumed to be spatially stationary, whict would allow hyperparameters to be estimated from registering a single pair of images. More complex models could be derived by simultaneously registering several images. This would give a more complete picture of the structural variability expected among a population. This is information that could be used by subsequent image warping procedures in order to make them more accurate
S64
13, Number
6, 2001,
Part 2 of 2 Parts ID
E blw
METHODS
- ANALYSIS
Image Warping using Empirical John Ashburner, Wellcome
Department
of Cognitive
Bayes
K.J. Friston Neurology,
London,
UK.
Image warping procedures are often thought of as a regularised maximum likelihood problem, and involve finding a set of parameters for a warping model that maximise the likelihood of the observed data. What is actually needed is a set of parameters that are maximally likely given the data. In order to achieve this, the problem should be considered within a Bayesian context, and expressed something like: “The posterior probability of the parameters given the data is proportional to the likelihood of observing the data given the parameters times the prior probability of tbe parameters.” Taking logs of this expression converts the Bayesian problem into one of simultaneously minimising two cost function terms. The first of these is usually a measure of intensity correlation between the registered images (likelihood potential), whereas the second is a measure of the roughness of the warps (prior potential). An ongoing argument in the field of image registration is something to the effect of “if there are too many parameters in a registration model, then the model will be overfitted”, versus “if not enough parameters are used, then the images can not be registered properly”. A better way of expressing this problem is in terms of obtaining an optimal balance between the cost function terms. This involves assigning the most appropriate weighting to each. These weights (hyperparameters) can be estimated from the data using an empirical Bayes approach. An algorithm similar to expectation maximisation (EM) can be used to do this. Pure EM is an iterative approach that alternates between two steps: the expectation (E) step and the maximisation (M) step. The algorithm is initialised by assigning some starting estimates to the hyperparameters. The first E step involves computing the expectation of the warp parameters while holding the hyperparameters constant. This is where this approach deviates from a pure EM algorithm, as a MAP estimate is used instead of the expectation of the warping parameters. The M step involves re-estimating the hyperparameters such that the likelihood of observing the data is maximised, while holding the parameters fixed. These hyperparameters are derived from variance estimates of both the residual difference between the images and the roughness of the deformations. Different models can be used to parameterise the variability. For example, it could be assumed to be spatially stationary, whict would allow hyperparameters to be estimated from registering a single pair of images. More complex models could be derived by simultaneously registering several images. This would give a more complete picture of the structural variability expected among a population. This is information that could be used by subsequent image warping procedures in order to make them more accurate
S64