The isometric log-ratio transform for probabilistic multi-label anatomical shape representation.

Sources of uncertainty in the boundaries of structures in medical images have motivated the use of probabilistic labels in segmentation applications. An important component in many medical image segmentation tasks is the use of a shape model, often generated by applying statistical techniques to training data. Standard statistical techniques (e.g., principal component analysis) often assume data lies in an unconstrained vector space, but probabilistic labels are constrained to the unit simplex. If these statistical techniques are used directly on probabilistic labels, relative uncertainty information can be sacrificed. A standard method for facilitating analysis of probabilistic labels is to map them to a vector space using the LogOdds transform. However, the LogOdds transform is asymmetric in one of the labels, which skews results in some applications. The isometric log-ratio (ILR) transform is a symmetrized version of the LogOdds transform, and is so named as it is an isometry between the Aitchison geometry, the inherent geometry of the simplex, and standard Euclidean geometry. We explore how to interpret the Aitchison geometry when applied to probabilistic labels in medical image segmentation applications. We demonstrate the differences when applying the LogOdds transform or the ILR transform to probabilistic labels prior to statistical analysis. Specifically, we show that statistical analysis of ILR transformed data better captures the variability of anatomical shapes in cases where multiple different foreground regions share boundaries (as opposed to foreground-background boundaries).

MRI-based 3D shape analysis of thigh muscles patients with chronic obstructive pulmonary disease versus healthy adults.

RATIONALE AND OBJECTIVE: Because lower limb muscles differ in architecture and function, the systemic effects of chronic obstructive pulmonary disease (COPD) and related disuse may result in regional abnormalities. The purpose of this study was to investigate the differences between patients with COPD and healthy controls in three-dimensional shape and size measurements of individual thigh muscles. MATERIALS AND METHODS: Twenty patients with COPD and 20 healthy adults (aged 55-79 years) underwent magnetic resonance imaging of the thighs. After manual segmentation of individual knee extensor and flexor muscles, the three-dimensional shape of each muscle was obtained using specialized software. Eight shape descriptors were computed both globally (for the whole muscle) and regionally (for portions of the muscle). A two-tailed t test with a modified Bonferroni correction was used to compare group differences. RESULTS: Compared to the thigh muscles of healthy subjects, vastus intermedius and semimembranosus showed the most shape abnormalities in the COPD group (P < .01). Greater regional shape anomalies in the COPD group were found in the middle to proximal regions of all knee extensor muscles and the middle region of the semimembranosus muscle, compared to those of the control group (P < .01). In the COPD group, more shape abnormalities were found in the knee extensors than in the knee flexors (P < .01). CONCLUSIONS: A non-uniform distribution of atrophy and size changes was found across knee extensors and flexors in patients with COPD. Further research is required to investigate the underlying mechanisms of regional morphologic abnormalities of the thigh muscles and the increased susceptibility of the knee extensors to atrophy-related anatomic anomalies in COPD.

Extraction of the plane of minimal cross-sectional area of the corpus callosum using template-driven segmentation.

Changes in corpus callosum (CC) size are typically quantified in clinical studies by measuring the CC cross-sectional area on a midsagittal plane. We propose an alternative measurement plane based on the role of the CC as a bottleneck structure in determining the rate of interhemispheric neural transmission. We designate this plane as the Minimum Corpus Callosum Area Plane (MCCAP), which captures the cross section of the CC that best represents an upper bound on interhemispheric transmission. Our MCCAP extraction method uses a nested optimization framework, segmenting the CC as it appears on each candidate plane, using registration-based segmentation. We demonstrate the robust convergence and high accuracy of our method for magnetic resonance images and present preliminary clinical results showing higher sensitivity to disease-induced atrophy.

Probabilistic multi-shape representation using an isometric log-ratio mapping.

Several sources of uncertainties in shape boundaries in medical images have motivated the use of probabilistic labeling approaches. Although it is well-known that the sample space for the probabilistic representation of a pixel is the unit simplex, standard techniques of statistical shape analysis (e.g., principal component analysis) have been applied to probabilistic data as if they lie in the unconstrained real Euclidean space. Since these techniques are not constrained to the geometry of the simplex, the statistically feasible data produced end up representing invalid (out of the simplex) shapes. By making use of methods for dealing with what is known as compositional or closed data, we propose a new framework intrinsic to the unit simplex for statistical analysis of probabilistic multi-shape anatomy. In this framework, the isometric log-ratio (ILR) transformation is used to isometrically and bijectively map the simplex to the Euclidean real space, where data are analyzed in the same way as unconstrained data and then back-transformed to the simplex. We demonstrate favorable properties of ILR over existing mappings (e.g., LogOdds). Our results on synthetic and brain data exhibit a more accurate statistical analysis of probabilistic shapes.