Population-based fitting of medial shape models with correspondence optimization

pre-printA crucial problem in statistical shape analysis is establishing the correspondence of shape features across a population. While many solutions are easy to express using boundary representations, this has been a considerable challenge for medial representations. This paper uses a new 3-D medial model that allows continuous interpolation of the medial manifold and provides a map back and forth between it and the boundary. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system.We use these integrals to optimize correspondence during model construction, reducing variability due to the model parameterization that could potentially mask true shape change effects. Discrimination and hypothesis testing of populations of shapes are expected to benefit, potentially resulting in improved significance of shape differences between populations even with a smaller sample size

Swept regions and surfaces: Modeling and volumetric properties

We consider “swept regions” and “swept hypersurfaces” in (and especially ) which are a disjoint union of subspaces or obtained from a varying family of affine subspaces . We concentrate on the case where and are obtained from a skeletal structure . This generalizes the Blum medial axis of a region , which consists of the centers of interior spheres tangent to the boundary at two or more points, with denoting the vectors from the centers of the spheres to the points of tangency. We extend methods developed for skeletal structures so that they can be deduced from the properties of the individual intersections or and a relative shape operator , which we introduce to capture changes relative to the varying family . We use these results to deduce modeling properties of the global in terms of the individual , and determine volumetric properties of regions expressed as global integrals of functions on in terms of iterated integrals over the skeletal structure of which is then integrated over the parameter space

Shape Deformation Statistics and Regional Texture-Based Appearance Models for Segmentation

Transferring identified regions of interest (ROIs) from planning-time MRI images to the trans-rectal ultrasound (TRUS) images used to guide prostate biopsy is difficult because of the large difference in appearance between the two modalities as well as the deformation of the prostate's shape caused by the TRUS transducer. This dissertation describes methods for addressing these difficulties by both estimating a patient's prostate shape after the transducer is applied and then locating it in the TRUS image using skeletal models (s-reps) of prostate shapes. First, I introduce a geometrically-based method for interpolating discretely sampled s-reps into continuous objects. This interpolation is important for many tasks involving s-reps, including fitting them to new objects as well as the later applications described in this dissertation. This method is shown to be accurate for ellipsoids where an analytical solution is known. Next, I create a method for estimating a probability distribution on the difference between two shapes. Because s-reps live in a high-dimensional curved space, I use Principal Nested Spheres (PNS) to transform these representations to instead live in a flat space where standard techniques can be applied. This method is shown effective both on synthetic data as well as for modeling the deformation caused by the TRUS transducer to the prostate. In cases where appearance is described via a large number of parameters, such as intensity combined with multiple texture features, it is computationally beneficial to be able to turn these large tuples of descriptors into a scalar value. Using the inherent localization properties of s-reps, I develop a method for using regionally-trained classifiers to turn appearance tuples into the probability that the appearance tuple in question came from inside the prostate boundary. This method is shown to be able to accurately discern inside appearances from outside appearances over a large majority of the prostate boundary. Finally, I combine these techniques into a deformable model-based segmentation framework to segment the prostate in TRUS. By applying the learned mean deformation to a patient's prostate and then deforming it so that voxels with high probability of coming from the prostate's interior are also in the model's interior, I am able to generate prostate segmentations which are comparable to state of the art methods.Doctor of Philosoph

Atlas Diffeomorphisms Via Object Models

To tackle the problem of segmenting several closely-spaced objects from 3D medical images, I propose a hybrid of two segmentation approaches: one image-based and one model-based. A major contribution takes the image-based approach by diffeomorphically mapping a fully segmented atlas image to a partially segmented target patient image preserving any `correspondence' inferred from the partial segmentation of the target. The mapping is produced by solving the steady-state heat flow equation where the temperature is a coordinate vector and corresponding points have the same temperature. Objects carried over from the atlas into the target serve as reasonable initial segmentations and can be further refined by a model-based segmentation method. Good quality segmentations are added to the list of the initial partial segmentations, and the process is repeated. Another contribution takes the model-based approach in developing shape models of quasi-tubular objects and statistics on those models. Whereas medial models were previously only developed for slab-shaped objects, this contribution provides an approximately medial method to stably represent nearly tubular objects. I test my method on segmenting objects from 3D Computed Tomography (CT) scans of the head and neck obtained for radiotherapy treatment planning

Continuous Medial Models in Two-Sample Statistics of Shape

In questions of statistical shape analysis, the foremost is how such shapes should be represented. The number of parameters required for a given accuracy and the types of deformation they can express directly influence the quality and type of statistical inferences one can make. One example is a medial model, which represents a solid object using a skeleton of a lower dimension and naturally expresses intuitive changes such as "bending", "twisting", and "thickening". In this dissertation I develop a new three-dimensional medial model that allows continuous interpolation of the medial surface and provides a map back and forth between the boundary and its medial axis. It is the first such model to support branching, allowing the representation of a much wider class of objects than previously possible using continuous medial methods. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system. I show how these properties can be used to optimize correspondence during model construction. This improved correspondence reduces variability due to how the model is parameterized which could potentially mask a true shape change effect. Finally, I develop a method for performing global and local hypothesis testing between two groups of shapes. This method is capable of handling the nonlinear spaces the shapes live in and is well defined even in the high-dimension, low-sample size case. It naturally reduces to several well-known statistical tests in the linear and univariate cases

Geometric and statistical models for multi-object shape analysis

Shape analysis of multi-object complexes is important in many applications because it reveals additional information of interest over single-object shape analysis. For example, in medical applications where multiple structures in the human body often deform together, joint shape analysis of those interrelated structures facilitates robust and efficient algorithms. Specifically, shape correlation of functionally related structures allows us to understand the common underlying biological factors (e.g., disease). Also, beyond the within-object shape relations, between-object shape relations provide additional understanding of multi-object complexes. Despite the need of multi-object shape analysis, this field has been challenged by many issues. For instance, shape variation is often coupled with pose and size variation between objects. Moreover, within-object shape variation is often coupled with between-object shape variation. These issues have prevented us from sufficiently understanding multi-object complexes. To address the issues, this dissertation proposes geometric and statistical methods for joint analysis of multi-object complexes. In particular, I base my research on skeletal representations (i.e., s-reps) that are designed to provide intrinsic shape features with good correspondences. This dissertation improves the previous method fitting an s-rep to an object such that the fitted s-reps have desirable geometric and statistical properties. This improvement allows me to analyze intrinsic shape correlation between objects. To this end, this dissertation extends the existing statistical method to effectively extract joint shape variation, leading to a method called Non-EUclidean Joint and Individual Variation Explained (NEUJIVE). NEUJIVE shows notable robustness in analyzing multi-block non-Euclidean data with different variability. Last, to decouple within- and between-object shape variation, I develop non-branching linking structures for statistical analysis of between-object shape features. To capture geometric features that are insensitive to pose variation of multi-object complexes, this dissertation extends fitted local frames on s-reps to affine frames. The fitted local affine frames show special advantage because they free multi-object shape analysis from pre-alignment. The driving problem of the proposed methods involves classifying and testing hypotheses on the shape of the hippocampus-caudate pairs between an autism group and a non-autism group. Also, this dissertation discusses other potential applications that can benefit from the proposed methods.Doctor of Philosoph

Refinement of object-based segmentation

Automated object-based segmentation methods calculate the shape and pose of anatomical structures of interest. These methods require modeling both the geometry and object-relative image intensity patterns of target structures. Many object-based segmentation methods minimize a non-convex function and risk failure due to convergence to a local minimum. This dissertation presents three refinements to existing object-based segmentation methods. The first refinement mitigates the risk of local minima by initializing the segmentation closely to the correct answer. The initialization searches pose- and shape-spaces for the object that best matches user specified points on three designated image slices. Thus-initialized m-rep based segmentations of the bladder from CT are frequently better than segmentations reported elsewhere. The second refinement is a statistical test on object-relative intensity patterns that allows estimation of the local credibility of a segmentation. This test effectively identifies regions with local segmentation errors in m-rep based segmentations of the bladder and prostate from CT. The third refinement is a method for shape interpolation that is based on changes in the position and orientation of samples and that tends to be more shape-preserving than a competing linear method. This interpolation can be used with dynamic structures and to understand changes between segmentations of an object in atlas and target images. Together, these refinements aid in the segmentation of a dense collection of targets via a hybrid of object-based and atlas-based methods. The first refinement increases the probability of successful object-based segmentations of the subset of targets for which such methods are appropriate, the second increases the user's confidence that those object-based segmentations are correct, and the third is used to transfer the object-based segmentations to an atlas-based method that will be used to segment the remainder of the targets