Probabilistic Multi-Shape Representation Using an Isometric Log-Ratio Mapping

Abstract

Abstract. 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 logratio (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.

Convex multi-region probabilistic segmentation with shape prior in the isometric log-ratio transformation space

“…In this paper, we achieve the four goals (convexity, shape prior, multi-region, and probabilistic) by using a new representation of a segmentation based on performing the isometric log-ratio (ILR) [11,6] transformation on the multiregional probabilities, bijectively mapping a probabilistic segmentation to a point in a vector space of real numbers. This mapping removes the need for constraints to guarantee a valid segmentation [9] (with the probabilities at each pixel summing to unity).…”

“…If we can represent probabilistic segmentations by elements of a vector space and then define energies over this vector space, then we will not need explicit constraints on our energy minimization, as any element of the vector space will correspond to a valid segmentation. We will do this by mapping probabilities in the range [0, 1] to real numbers using the ILR transform [11,6]. We note ILR is defined for probability vectors of any length, allowing us to easily handle multiple regions.…”

“…We represent the probabilistic segmentation of an image into D regions as a vector-valued function over Ω using the ILR transform [11,6]. We use the ILR transform because it has several desirable properties: the transformed probabilities form a vector space; it is an isomorphism between the probability simplex and vectors of real numbers; and it is symmetric between regions.…”

The Generalized Log-Ratio Transformation: Learning Shape and Adjacency Priors for Simultaneous Thigh Muscle Segmentation

“…equidistant) (Fig. 3b) [30], [31], [33], [34]. This results in all boundaries being treated equally, so the ILR transformation is an appropriate choice when lacking prior knowledge about the segmentation labels.…”

Mammography segmentation with maximum likelihood active contours

“…However, lesions do not have a predefined shape hence incorporating shape prior is ineffective. Even more signed distance functions are not closed under linear combinations rendering statistical shape modeling difficult (Changizi and Hamarneh, 2010;Andrews et al, 2011).…”