122
F. Rathke et al.
L
∗
a,b for regions of width w = b − a, we need quantities L
∗
a,x , L
∗
x,b for all regions of
smaller width. Given these, the complexity is dominated by O(I ) evaluations of the
base model to determine the rightmost maximum in (5.19).
This leads to an iterative algorithm: for increasing w ∈ {w min , 2w min , . . . , M},
compute L
∗
a,b for all regions of that width. Finally, L
∗
1,M is the optimal selection
of θ, X, K over the whole image. The total number of base model evaluations is
O(I (M/w min )
2
), which is tractable for sufficiently large w min .
5.5 Discussion
5.5.1 Conclusion
We presented a probabilistic approach for the segmentation of retina layers in OCT
scans. The approach entails to infer a full posterior distribution p(b, c|y) that we
evaluate using a variational method. This turned out to be beneficial in several ways:
Since the inference scheme comprises efficient subproblems, segmenting one B-Scan
only requires about 2 s. Furthermore, the quality of the computed segmentation can
be assessed and further used for pathology detection. The segmentation performance
was very good for five datasets, even in the presence of mild pathologies.
5.5.2 Prospective Work
While our major focus has been on healthy data and mild pathologies so far, we
outlined in Sect. 5.4 ongoing work about an adaption of our model to more severe
pathological deformations. We conclude this section by briefly discussing two major
aspects of corresponding ongoing and future work.
Interpretation as Sum-Product Network. The algorithm of Sect. 5.4 has an interpretation as maximum a posteriori (MAP) inference in a probabilistic architecture
called Sum-Product Network (SPN), introduced in [30]. This connection may support
future extensions and applications of our model.
Let X be a set of continuous variables. A SPN S(X ) is a directed acyclic graph
with sums and products as internal nodes and probability distributions {L k (X k )} as
leaves, where X k ⊆ X . In SPNs, product nodes represent factorizations and sum
nodes represent mixtures of the distributions rooted in children nodes, thus creating
a hierarchical mixture model. Inference has linear costs in the number of edges,
and it is performed by first evaluating the leaf distributions and then evaluating the
internal nodes from the leaves to the root. MAP inference is similarly performed
after replacing the sum operator by the max mapping.
Our algorithm can be directly interpreted as MAP inference on a SPN performed in logarithmic space, by substituting the sums with products in Eq. (5.18)
F. Rathke et al.
L
∗
a,b for regions of width w = b − a, we need quantities L
∗
a,x , L
∗
x,b for all regions of
smaller width. Given these, the complexity is dominated by O(I ) evaluations of the
base model to determine the rightmost maximum in (5.19).
This leads to an iterative algorithm: for increasing w ∈ {w min , 2w min , . . . , M},
compute L
∗
a,b for all regions of that width. Finally, L
∗
1,M is the optimal selection
of θ, X, K over the whole image. The total number of base model evaluations is
O(I (M/w min )
2
), which is tractable for sufficiently large w min .
5.5 Discussion
5.5.1 Conclusion
We presented a probabilistic approach for the segmentation of retina layers in OCT
scans. The approach entails to infer a full posterior distribution p(b, c|y) that we
evaluate using a variational method. This turned out to be beneficial in several ways:
Since the inference scheme comprises efficient subproblems, segmenting one B-Scan
only requires about 2 s. Furthermore, the quality of the computed segmentation can
be assessed and further used for pathology detection. The segmentation performance
was very good for five datasets, even in the presence of mild pathologies.
5.5.2 Prospective Work
While our major focus has been on healthy data and mild pathologies so far, we
outlined in Sect. 5.4 ongoing work about an adaption of our model to more severe
pathological deformations. We conclude this section by briefly discussing two major
aspects of corresponding ongoing and future work.
Interpretation as Sum-Product Network. The algorithm of Sect. 5.4 has an interpretation as maximum a posteriori (MAP) inference in a probabilistic architecture
called Sum-Product Network (SPN), introduced in [30]. This connection may support
future extensions and applications of our model.
Let X be a set of continuous variables. A SPN S(X ) is a directed acyclic graph
with sums and products as internal nodes and probability distributions {L k (X k )} as
leaves, where X k ⊆ X . In SPNs, product nodes represent factorizations and sum
nodes represent mixtures of the distributions rooted in children nodes, thus creating
a hierarchical mixture model. Inference has linear costs in the number of edges,
and it is performed by first evaluating the leaf distributions and then evaluating the
internal nodes from the leaves to the root. MAP inference is similarly performed
after replacing the sum operator by the max mapping.
Our algorithm can be directly interpreted as MAP inference on a SPN performed in logarithmic space, by substituting the sums with products in Eq. (5.18)