School Reputation/Values Introduction

algorithm, we treat the results of the deformable model as a second patient-specific atlas. We then combine the initial atlas and the results of the deformable model, and give the combined atlas back to the EM-MRF algorithm.

intensities. To generate the atlas, up to 80 manual segmentations were warped onto a randomly chosen segmentation using a non-rigid registration procedure (WRH⁺01).

The MRI corresponding to the randomly chosen segmentation is the template case.

From these aligned segmentations, the probability of each tissue class at each voxel is estimated. These spatial tissue distributions define the atlas.

The atlas is aligned onto the image to be segmented in order to generate an atlas specific to that image. To achieve this alignment, the template MRI is non-rigidly registered to the image to be segmented (GRAM01; Thi98). The same warping is then applied to the atlas. The result is an atlas in the coordinate system of the subject of interest. We call the probability determined by the atlas of tissueγ at voxelx P(γ|x).

As an example of an atlas for a single tissue class, a slice through the probability of white matter is shown in Figure 5-2. A similar slice through the probability for the left-thalamus is shown in Figure 5-5. Note that in both examples, the atlas does not have sharp edges. The atlas is useful in roughly localizing a structure, not in accurately describing its shape.

The atlas has a secondary purpose besides providing localization information. In order to label a particular structure, it is necessary to know the expected intensity distribution of that structure. This intensity distribution will change depending on the particular imager used and imaging modality chosen. One way of obtaining the intensity distribution automatically is to examine the intensities of voxels at locations where the atlas is very confident that a particular tissue class is found. Based on this idea, Pohl (Poh03) estimates the probability p(Y|γ) that intensity Y comes from tissue class γ by a Gaussian distribution fit to the log intensity of the voxels where the atlas is at least 95% certain that tissue class γ will be found.

5.2.2 Inhomogeneity and Tissue Estimation

The core of the segmentation algorithm consists of an expectation maximization (EM) technique used in conjunction with a Markov random field (EM-MRF). The algorithm iterates between an expectation step (E-step) and a maximization step (M-step). For each tissue class, γ, the E-step computes the probability of the tissue class, W(γ|x),

at each voxel x. These probabilities are generally called weights in the EM-MRF framework. The estimate for the next iterationn+ 1 is based on the image intensities Y, the current estimate of additive image inhomogeneities, or bias field, Bⁿ, the probabilities from the atlas p(γ|x), and the estimated intensity distribution of each tissue class p(Y|γ). The estimate is also affected by a neighbor energy function EN(γ, x), which reflects the likelihood that two tissue classes occur in neighboring voxels

Wⁿ⁺¹(γ|x) = 1

Zp(Y(x)|Bⁿ(x), γ)p(γ|x)EN(γ, x), (5.1) where Z is a normalization factor. Equation 5.1 effectively calculates the posterior probability of a tissue class at a particular voxel, given the intensity distribution of the tissue class, the probability that a tissue class is assigned to a particular voxel according to the atlas, and the neighbor energies. Because of the neighbor energy function, Equation 5.1 can be viewed as a description of a Markov random field, and solved approximately using the mean-field approximation. Use of this approximation avoids having to calculate the normalization factor Z.

The M-step calculates the maximum a posteriori (MAP) estimate of the additive bias field, B, which models intensity inhomogeneity artifacts. That is, one finds the value of B which maximally predicts the measured intensities at each voxel given a prior probability on B:

Bⁿ⁺¹ = arg max

B p(B|Y,Wⁿ⁺¹) = arg max

B p(B)p(Y|B,Wⁿ⁺¹). (5.2) The probability distribution of the additive bias field p(B) is assumed to be a sta-tionary, zero mean Gaussian with a given variance, and covariance across neighboring voxels. The covariance gives the additive bias field smoothness properties. The sec-ond term in Equation 5.2 asks how well a given bias fieldBand weights Wpredict the measured intensities. The solution to this equation can be shown to be (WGKJ96)

Bⁿ⁺¹=H·R (5.3)

where H is a smoothing filter determined by the covariance of B, and R is the mean residual field of the predicted image in relation to the real image. In words, the addi-tive bias field is roughly a smoothed version of the difference between the predicted image using the weights and the measured image (see (WGKJ96; PWG⁺02) for more details).

The E- and M- Steps are repeated until the algorithm converges to a local maxi-mum.

5.2.3 Deformable Models

We introduce shape-based information into the EM-MRF framework using a de-formable model very similar to the ones presented by Cootes and Taylor (CHTH92).

Deformable model based segmentation methods use prior information on the shape of an object. They typically begin by describing a set of training shapes using a com-mon description. Depending on the choice of shape descriptor, the representation can be very large, often using thousands of parameters to describe the object. However, researchers argue that the majority of the variations in the data can be captured by a few important modes of variation. Principal component analysis is commonly used to find these modes of variation (CHTH92; DHS01). Given the representations of the examples, PCA finds an mean representation, and a set of orthonormal vectors

~ei, that maximally describe the variation of the training examples about their mean.

PCA additionally finds the variances σ²_i of the examples in each direction ~ei.

To use the results of PCA as the basis of a segmentation process, the mean model is deformed to agree with an image. The mean model is only allowed to deform in the subspace defined by the vectors~ei with largest variance. Choosing enough vectors to account for 98% of the variation in a space is typical. When the model is deforming, the target image voxels inside the final model are identified as the voxels of interest.

The matching process is carried out within a maximum a posteriori (MAP) frame-work. An MAP estimator finds the representation that maximizes a combination of an agreement term between the model and the image, the prior probability on the representation and pose. The prior probability on the representation comes directly

from PCA, which is equivalent to forming a Gaussian prior on the changes in the model with covariance matrix P

i~ei~e^T_i σ_i². The image agreement term can be in the form of the expected intensity that the deformable model will overlap, or a measure of “edge-ness” that the boundaries of the model will overlie. One typically further trains the models to include covariances between image intensity variations and shape, though we will not use these ideas here. Instead, we will fit the deformable model to the image weights W, as we describe in the next section.