There is evidence that atoms in one object that are near another object are most highly correlated with that other object. Ray et al. [2006], Jung and Marron [2008] On the basis of that evidence the inter-relation of a multi-object complex is described using these nearby atoms through the methods of augmentation and prediction. Augmentation pools medial atoms across objects to reflect the mixed effects local to an object (e.g., bladder filling with urine) with neighbor effects (e.g., bladder push).
4.2.1 Augmentation
Given a multi-object m-rep of l objects, i.e.,{Mk}lk=1 where Mk is an ordered set of medial
atoms per object, an object’s inter-relation with other atoms is dealt with by augmenting highly correlated atoms of its neighboring objects. LetTi be a set of atoms in a target object
Mi that are located near its neighboring objects, and let Ni be a set of atoms in neighboring
objects that are close to Ti. Ti ⊆Mi, andNi ⊆ ∪k6=iMk. The “augmented” representation
of the i-th object is then
Ui=Ti∪Ni = Ti Ni .
Niindicates atoms used to predict the neighbors effect on the target object (predicting atoms),
andTi indicates atoms in the target object affected by the variation of its neighboring objects
(predicted atoms). Fig. 4.2 shows an example of these two sets of atoms selected for bladder and prostate in their multi-object complexes.
As will be described in the next section, this augmentation allows us to predict the changes brought on a target object by the change of the neighboring objects. After this deterministic effect of its neighbor on the target object is taken off from the total variation of the object, what remains in the total variation is considered as the combination of the variable part in the neighbor effect and the self variation of the object. A method called prediction to estimate this
Figure 4.2: Prostate atoms combined with its predicting medial atoms (left). Bladder atoms combined with its predicting medial atoms (right).
deterministic effect is proposed in the next section. The prediction method helps to extract the deterministic effect from its neighbor on the object and to concentrate on the variable part in the neighbor effects. Augmentation is also used in the estimation of the conditional probability of the target object in a multi-object complex described in the next chapter 5.
4.2.2 Prediction
In estimating the probability distribution of Mk, prediction reflects changes in Mi6=k by pre-
dicting how Mk bends, twists or warps from the change of Mi6=k through its augmenting
atoms Nk ⊆Mi6=k and the predicted atoms Tk ⊆Mk . In doing so, the shape space of the
augmented atoms is taken into account as suggested in [Rajamani et al., 2004], but by using PGA in a nonlinear symmetric space rather than PCA as used in [Rajamani et al., 2004].
Now consider an augmented m-rep figure Uk = (Tk∪Nk). Let µ and H be the mean
and the shape space generated by h principal geodesics in the symmetric space of Uk. The
deformation ofUk, i.e., how Tk and Nk change together, can be used to predict the effect of
the other objects Mi6=k on the target object Mk.
LetU∗ = µ|Tk Nk
, whereu|Aindicates taking the elements in the vectoruthat correspond
to the subsetAof all the features of the vectoru. U∗is a vector that concatenates the mean of
inH closest toU∗, that is,
P red(Tk) = argmin
b U|Tk∈H
d(U∗,Ub) (4.5)
Prediction is an attempt to find an element of the highest probability in the shape space H
given the known variation in Nk. The issues in this definition of prediction will be discussed
later in section 4.5.
This prediction can be easily computed by the projection operation [Fletcher, 2004] on the shape space H. The projection of U∗ onH is
P rojH(U∗) = expµ h X l=1 hlogµ(U∗), vlivl ! , (4.6)
where{vl}hl=1 are principal directions in the tangent space atµcorresponding to the principal
geodesics in H. P rojH(U∗) produces also an m-rep. Then the prediction for Mi6=k is defined
as
P red(Tk) :=P rojH(U∗)|Tk. (4.7)
The next section describes a preliminary approach to build multi-object shape models on the basis of these methods. In this preliminary approach the objects are assumed to be in an order of decreasing stability, i.e., whose posterior probability, based on both geometric and intensity variability and edge sharpness, are in decreasing levels of tightness. The probability distribution of each object is estimated in this order. Thus, the neighbor relation among objects is not mutual but directional from the most stable object to the least stable object. The direction of prediction is modified accordingly. The details are explained in the next section.