| Περίληψη: | The reconstruction of geometric shapes plays an important role in many biomedical
applications. A characteristic example is the patient-specific, computer-aided
intervention and treatment, which requires the generation of explicitly represented
geometric models of anatomical structures depicted in medical images. In the past
few decades, there has been a lot of effort to automate the procedure of geometric
model construction, since the manual segmentation of 3D images can be timeconsuming
and irreproducible. However, most automatic segmentation algorithms
are inherently sensitive to image artifacts and noise, resulting in invalid geometric
representations.
The fundamental hypothesis, pursued in this thesis, is that the aforementioned
problems can be resolved by incorporating the a-priori knowledge of the shape of biological
structures. The basic idea is to capture the essential variations contained in a
given population of a certain class of geometric objects through Statistical Shape Models
(SSMs), and restrict the result of a reconstruction algorithm to the space spanned
by them. We extend the discussion to the construction of multiple - shape models,
in which the relationship between the neighboring anatomical components is
described through Canonical Correlation Analysis.
A fundamental prerequisite for performing statistical shape analysis on a set of
objects is the identification of corresponding points on their associated surfaces. In
this thesis, we address the correspondence problem using the recently proposed
Functional Maps framework, which is a generalization of the notion of point-to-point
shape correspondence, describing how mappings act on real-valued functions defined
on shapes. Additionally, we show that, by incorporating techniques from the
deep learning theory into the Functional Maps framework, we can further enhance
the ability of SSMs to better capture the shape variations within the given dataset.
We present the applicability of our methods on real 3D medical data of the human
knee. In particular, we construct 3D models of the knee complex that can be
transferred across any image modality, and demonstrate their application for improving
the quality of automated segmentation methods and conducting inference
about missing shapes.
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