Poisson-based implicit shape space analysis with application to CT liver segmentation
Keywords: manifold learning
shape representation
PCA
implicit shape
level-set segmentation

A patient-specific model of the liver can supply accurate volume measurements for oncologists and lesion locations and liver visualisation for surgeons. Our work seeks to enable an automatic computational tool for liver quantification. To create this model, the liver shape must be segmented from 3D CT images. In doing so, we can quantify liver volume and restrict the region of interest to ease the task of tumour and vascular segmentation.
The main objective of liver segmentation developed into a mission to fluently describe liver shape a priori in level-set methods. This thesis looks at the utility of an implicit shape representation based on the Poisson equation to describe highly variable shapes, with application to image segmentation.
Our first contribution is analyses on four implicit shape representations based on the heat equation, the signed distance function, Poisson’s equation, and the logarithm of odds. For four separate shape case studies, we summarise the class of shapes through their shape representation using Principal Component Analysis (PCA). Each shape class is highly variable across a population, but have a characteristic structure. We quantitatively compare the implicit shape representations, within each class, by evaluating its compactness, and in the last case, also completeness. To the best of our knowledge, this study is novel in comparing several shape representations through a single dimension reduction method.
Our second contribution is a hybrid region-based level set segmentation that simultaneously infers liver shape given the image data, integrates the Poisson-based shape function prior into the segmentation, and evolves the level set according to the image data. We test our algorithm on exemplary 2D liver axial slices. We compare results for each image to results from (a) level-set segmentation without a shape prior and (b) level-set segmentation with a shape prior based on the Signed Distance Transform (SDT). In both priors, shapes are projected from shape space through the sample population mean and its modes of variation (the minimum number of principal components to comprise at least 95% of the cumulative variance). We compare results on four individual cases using the Dice coefficient and the Hausdorff distance.
This thesis introduces an implicit shape representation based on Poisson’s equation in the field of medical image segmentation, showing its influence on shape space summary and projection. We analyse the shape space for compactness, showing that it is more compact in each of our case studies by at least two-fold and as much as three-fold. For 3D liver shapes, we show that it is more complete than the other three implicit shape representations. We utilise its description efficiency for use in 2D liver image segmentation, implementing the first shape function prior based on the Poisson equation. We show a qualitative and quantitative improvement over segmentation results without any shape prior and comparable results to segmentation with a SDT shape prior.