108
F. Rathke et al.
include all possibilities for k, i, j: For example b •, j ∈ R
K are all boundary positions
in column j. When possible, we will drop • and write b j to denote b •, j . Finally, b \ j
denotes all entries of b except those for column j.
Appearance p( y|c). Given a segmentation c, we can assign class labels x i, j ∈ X to
each pixel from
X = {X l , X t },
X l = {l 1 , . . . , l K +1 }, X t = {t 1 , . . . , t K },
where labels in X l indicate affiliation of pixel y i, j to retina layers 1 to K + 1 (K = 10
in this work) and labels in X t to boundaries between them, c.f. Fig. 5.1. To obtain
a mapping c → x = x(c) consistent with physiology, we require c to satisfy the
ordering constraint
1 ≤ c 1, j < c 2, j < · · · < c K , j ≤ N , ∀ j = 1, . . . , M.
(5.2)
We will use a patch-based model. Since OCT scans display a large variability
in brightness and contrast, each patch is normalized by subtracting its mean and
projected onto a low-dimensional subspace using PCA. We define the probability of
pixel y i, j belonging to the class x i, j as
p(y i, j |x i, j (c)) = N ( ˜
y i, j ; μ x i, j , , x i, j ),
(5.3)
where ˜
y i, j is the low-dimensional projection of the patch around pixel y i, j . Note
that in this definition, p(y i, j |x i, j (c)) does not integrate to 1 over the range of y i, j .
This is taken care of, when we introduce discriminative appearance terms in the next
section. The class-conditional moments μ x , , x for all x ∈ X are learned offline.
Regularized estimates for x are obtained by utilizing the graphical lasso approach
[19], imposing sparsity on
−1
x .
We define pixels y i, j to be conditionally independent given c. Furthermore, in
Rathke et al. [20] it was shown that the model performs best when restricted to terms
belonging to transition classes t k . Thus p(y|c) factorizes into
p(y|c) =
M
j=1
i:x i, j ∈X l
p(y i, j |x i, j (c))
0
i:x i, j ∈X t
p(y i, j |x i, j (c)),
(5.4)
where we do not take into account pixel with labels X
l
= {l 1 , . . . , l 10 }.
Shape Prior p(b). We use a shape model to represent typical shape variations, due
to both biological variability as well as to the image formation process. We model b
as random vector with Gaussian distribution
p(b) = N (b; μ, ,),
(5.5)
where parameters μ and again are learned offline. We regularize the estimation of
by probabilistic PCA [21], which assumes a linear Gaussian model for b:
F. Rathke et al.
include all possibilities for k, i, j: For example b •, j ∈ R
K are all boundary positions
in column j. When possible, we will drop • and write b j to denote b •, j . Finally, b \ j
denotes all entries of b except those for column j.
Appearance p( y|c). Given a segmentation c, we can assign class labels x i, j ∈ X to
each pixel from
X = {X l , X t },
X l = {l 1 , . . . , l K +1 }, X t = {t 1 , . . . , t K },
where labels in X l indicate affiliation of pixel y i, j to retina layers 1 to K + 1 (K = 10
in this work) and labels in X t to boundaries between them, c.f. Fig. 5.1. To obtain
a mapping c → x = x(c) consistent with physiology, we require c to satisfy the
ordering constraint
1 ≤ c 1, j < c 2, j < · · · < c K , j ≤ N , ∀ j = 1, . . . , M.
(5.2)
We will use a patch-based model. Since OCT scans display a large variability
in brightness and contrast, each patch is normalized by subtracting its mean and
projected onto a low-dimensional subspace using PCA. We define the probability of
pixel y i, j belonging to the class x i, j as
p(y i, j |x i, j (c)) = N ( ˜
y i, j ; μ x i, j , , x i, j ),
(5.3)
where ˜
y i, j is the low-dimensional projection of the patch around pixel y i, j . Note
that in this definition, p(y i, j |x i, j (c)) does not integrate to 1 over the range of y i, j .
This is taken care of, when we introduce discriminative appearance terms in the next
section. The class-conditional moments μ x , , x for all x ∈ X are learned offline.
Regularized estimates for x are obtained by utilizing the graphical lasso approach
[19], imposing sparsity on
−1
x .
We define pixels y i, j to be conditionally independent given c. Furthermore, in
Rathke et al. [20] it was shown that the model performs best when restricted to terms
belonging to transition classes t k . Thus p(y|c) factorizes into
p(y|c) =
M
j=1
i:x i, j ∈X l
p(y i, j |x i, j (c))
0
i:x i, j ∈X t
p(y i, j |x i, j (c)),
(5.4)
where we do not take into account pixel with labels X
l
= {l 1 , . . . , l 10 }.
Shape Prior p(b). We use a shape model to represent typical shape variations, due
to both biological variability as well as to the image formation process. We model b
as random vector with Gaussian distribution
p(b) = N (b; μ, ,),
(5.5)
where parameters μ and again are learned offline. We regularize the estimation of
by probabilistic PCA [21], which assumes a linear Gaussian model for b: