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Using the Logarithm of Odds to Define a Vector Space on Probabilistic Atlases

Institution:	Surgical Planning Laboratory, Harvard Medical School and Brigham and Women’s Hospital, Boston, MA, USA.
Publisher:	Med Image Anal
Publication Date:	Oct-2007
Volume Number:	11
Issue Number:	5
Pages:	465-477
Citation:	Med Image Anal. 2007 Oct;11(5):465-77.
PubMed ID:	17698403
PMCID:	PMC2423493
Keywords:	Projects:ShapeBasedSegmentationAndRegistration, Logarithm of odds, Logistic function, Anatomical variations, Shape Representation, Distance map, Bayesian Modeling, Interpolation, Statistical classification
Appears in Collections:	NAC, NA-MIC, NCIGT, PNL, SPL
Sponsors:	Department of Veterans Affairs Merit Awards Brain Science Foundation US Army SBIR W81XWH-04-C0031 NIH R01 MH40799 NIH K05 MH70047 NIH R01 MH50747 NIH U41 RR019703 NIBIB NAMIC U54 EB005149 NCRR mBIRN U24 RR021382 NCRR NAC P41 RR13218 NINDS R01-NS051826 NIAAA R01 AA016748-01 NSF JHU ERC CISST
Generated Citation:	Pohl K.M., Fisher III J.W., Bouix S., Shenton M.E., McCarley R., Grimson W.E.L., Kikinis R., Wells III W.M. Using the Logarithm of Odds to Define a Vector Space on Probabilistic Atlases. Med Image Anal. 2007 Oct;11(5):465-77. PMID: 17698403. PMCID: PMC2423493.
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The logarithm of the odds ratio (LogOdds) is frequently used in areas such as artificial neural networks, economics, and biology, as an alternative representation of probabilities. Here, we use LogOdds to place probabilistic atlases in a linear vector space. This representation has several useful properties for medical imaging. For example, it not only encodes the shape of multiple anatomical structures but also captures some information concerning uncertainty. We demonstrate that the resulting vector space operations of addition and scalar multiplication have natural probabilistic interpretations. We discuss several examples for placing label maps into the space of LogOdds. First, we relate signed distance maps, a widely used implicit shape representation, to LogOdds and compare it to an alternative that is based on smoothing by spatial Gaussians. We find that the LogOdds approach better preserves shapes in a complex multiple object setting. In the second example, we capture the uncertainty of boundary locations by mapping multiple label maps of the same object into the LogOdds space. Third, we define a framework for non-convex interpolations among atlases that capture different time points in the aging process of a population. We evaluate the accuracy of our representation by generating a deformable shape atlas that captures the variations of anatomical shapes across a population. The deformable atlas is the result of a principal component analysis within the LogOdds space. This atlas is integrated into an existing segmentation approach for MR images. We compare the performance of the resulting implementation in segmenting 20 test cases to a similar approach that uses a more standard shape model that is based on signed distance maps. On this data set, the Bayesian classification model with our new representation outperformed the other approaches in segmenting subcortical structures.

Pohl-LogOddsAward-MICCAI2007.jpg (3607.372kB)