CONSECUENCIAS JURÍDICAS DE LA TUTELA DE DERECHOS

In the nonlinear manifold on which m-reps are situated, the addition operation is not commu- tative. The separation of ∆Minto ∆Ms_⊕∆Tn_{is thus not equivalent to ∆T}n_⊕∆Ms_{. In our}

iterative algorithm that estimates the self variations ∆Ms and the neighbor effects ∆Tn, this non-commutativity is ignored and ∆Ms, ∆Tn are treated as if they are interchangeable. It should be tested whether the effect of our method’s assumption of commutativity is significant. It would also be useful to test the four assumptions stated in section 4.4.1 to show our interpretation on joint probability holds. The interpretation and the simplification of joint probability of multiple objects rely on these assumptions about correlations among the self variations and the neighbor effects of objects in a multi-object complex. The validity of these assumptions in turn provides ground of using the estimated self variations and the neighbor effects as a shape prior for segmentation of a multi-object complex since the simplified joint probability (Eq. (4.12)) is not sensitive to the order of objects.

The applicability of the method described in this chapter depends on the type of statistical variation in the population. Problems to which statistical shape analysis can apply can be

classified according to the target data, namely within patient and across patient. Within- patient data are relatively easier to deal with than cross-patient data since the amount of variations in the configuration and shapes of multiple anatomic objects within a patient are small compared to those of anatomic objects across patients. Objects in a region of a patient are bound together by muscles and tissues among the objects, so the positioning and the distribution of objects in a patient are rather stationary, and the shape changes of the objects in a patient are limited within the configuration. As a result, the configuration and the shape changes of multiple objects within a patient are diffeomorphic. However, this diffeomorphic property in the configuration and the shape changes of objects within a patient cannot be guaranteed when the same multiple objects are considered across patients. The configuration of the objects changes. For example, sliding of one object along the other object can be observed between two adjacent objects across different patients. But also the shape of each object in the same multi-object complexes is much more variable in cross-patient data than in within-patient data.

Thus, the configuration and the shape variations of objects of a patient can be predicted reasonably well from some mean or reference configuration and shapes of the objects even with a limited number of training models of the patient. Cross-patient data have much more non-diffeormorphic and geometric variations than within-patient data in the configuration of anatomic objects in a region, and in the shape variation of each object, which makes it challenging to capture the inter-object relation in different patients.

The experiments reported in this chapter are limited to data within a patient. They suggest that the new approach to separate out inherent variation of an object itself and effects from its neighboring objects is promising for within-patient cases. It is interesting to consider whether the new approach also applies to cross-patient cases.

For both within-patient and cross-patient data a further analysis is required to verify that the estimates truly reflect the self and neighbor effects of multi-objects. One possible approach is to simulate the obvious neighbor effects and self variations independent from neighbor effects on multiple ellipsoids m-reps and apply this approach on the simulated multiple ellipsoids m- reps to determine what are self variations and neighbor effects.

of the self variation and the neighbor effects. For within-patient data the obvious choice is the most highly correlated neighboring atoms with those in the target object. Separate research is now being carried out to measure this correlation [Jung and Marron, 2008], and the results of that research will provide a firmer basis for this choice than the distance criterion presently being used. For cross-patient data the choice is not so simple because of the issue of correspondence.

The methods of augmentation and prediction assume the correspondence of selected atoms in neighboring objects to augment the target object across training models since this correspon- dence captures the relation of the neighboring objects to the target object. This assumption is the limiting factor of the proposed shape models for cross-patient data. As mentioned be- fore, it is fair to assume this correspondence for training models within a patient since the changes in the configuration and shapes of organs are diffeomorphic and the amount of ge- ometric variations are relatively limited. On the other hand, it is not reasonable to assume that the relation that the selected atoms in neighboring objects have relative to the target object in one patient carries over to the other patient. For example, positions of dents that can happen on the bladder by the nearby prostate of one patient can be explained by some atoms in the prostate. However, it is not feasible to expect that the same set of atoms (atoms of the same indices) in the prostate of the other patient can predict positions of dents on the bladder, considering that the positions of dents on the bladder vary from patients to patients. For the proposed shape model to work for cross-patient data, the methods of augmentation and prediction need an improvement that solves this issue of the correspondence of augmented objects across patients.

Chapter 5

Conditional Shape Statistics

This chapter presents an indirect approach to the challenging problem of automatic segmen- tation of an object of ambiguous boundary when its nearby objects are easier to segment. As mentioned throughout this dissertation, segmentation of human organs is one of the cen- tral problems in medical image analysis. Anatomical structure is complex: some organs are clumped together and touch each other, and low physical contrast in the image between such structures can make significant parts of the boundary of the target organ ambiguous. Seg- menting such an object slice-by-slice manually can be a particularly hard and laborious task even to clinicians, and certainly also to automatic segmentation methods. However, if image intensities at the boundaries of some objects near the target object are easier to identify than the target object, then positions, orientations or material properties of these objects relative to those of the target object can be used as guides to pick out the target object.

The goal of the approach is to make use of this observation and to incorporate into shape priors the relations of the target object to its neighboring objects that can be easily identified so that the shape variation of the target object is constrained not only by its own shape variations but also by its positional, orientational information, and even shape deformations relative to neighbor objects over a population. A natural and intuitive way to include this supplementary information into the shape prior is by means of conditional probability distribution. Thus, the conditional shape probability distribution (CSPD) of the target shape model given neighboring shape models is considered for segmentation of the target object. The CSPD provides a shape

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prior for finding a most probable shape of the target object given the configuration of its neighboring objects in relation to the target object.

In chapter 4 a novel approach to estimate the shape probability distribution of multi-objects was introduced. The basic objective of the conditional model is similar to that described in the previous chapter in the sense that both approaches aim to capture the inter-relations of objects in the shape probability distribution of each object. The main difference is that this conditional approach is geared towards finding the target object using its neighboring objects. That is, the conditional approach relies heavily on the segmentation quality of the neighboring objects and is likely to succeed when the segmentation quality of the neighboring objects is as good as that of the training models. The multi-object statistical shape model in the previous chapter is more general than the conditional shape model since the inter-dependency among objects is reflected on the self variation and the neighbor effect per object that is estimated iteratively.

Section 5.1 reviews the Gaussian conditional probability distribution. Section 5.2 presents a method to estimate the CSPD using the principal components regression method that was described in section 2.1.3.1 of chapter 2. Finally, section 5.3 demonstrates the application of the conditional shape model to a set of synthetic multi-object complex data and to real multi-object complex data.