Logarithm odds maps for shape representation

Abstract

The concept of the Logarithm of the Odds (LogOdds) is frequently used in areas such as artificial neural networks, economics, and biology. Here, we utilize LogOdds for a shape representation that demonstrates desirable properties for medical imaging. For example, the representation encodes the shape of an anatomical structure as well as the variations within that structure. These variations are embedded in a vector space that relates to a probabilistic model. We apply our representation to a voxel based segmentation algorithm. We do so by embedding the manifold of Signed Distance Maps (SDM) into the linear space of LogOdds. The LogOdds variant is superior to the SDM model in an experiment segmenting 20 subjects into subcortical structures. We also use LogOdds in the non-convex interpolation between space conditioned distributions. We apply this model to a longitudinal schizophrenia study using quadratic splines. The resulting time-continuous simulation of the schizophrenic aging process has a higher accuracy then a model based on convex interpolation.

Fig. 1

The binary map of the circle in (a) corresponds to the SDM 𝒟 in (b). In 𝒟, positive values (bright) define the circle, while negative values (dark) represent the background. Panel (c) shows the corresponding SCP. Voxels inside the circle have probabilities greater 0.5.

Fig. 2

(a) the effect of α on α ⊛ p. (b) Resulting SCP SCP_res from combining 2 SCPs of circles (SCP_l and SCP_r) with different centers, radii, and α values.

Fig. 3

Different views of a 3D model of the thalamus (gray), the caudate (black), and the ventricles (light gray). The model is based on a segmentation generated by IMPL-𝒫. The graph to the right summarizes the results of our experiment. For both structures IMPL-𝒫 performs much better then IMPL-𝓗.

Fig. 4

The first row shows a sample slice of the quadratic spline interpolation of a longitudinal schizophrenia study. Each image represents an SCP of the gray matter at a specific point in time of the study. Bright indicates high and dark low probability of the gray matter. The second row shows the SCP of the thalamus (a) with black indicating the voxels that are interpolated over three time point in (b)+(c) and two time points in (d) +(e). Graph (b)+(d) were produce by convex interpolation, while the smoother quadratic spline interpolation is representation (c)+(e).