Page 136 - Artificial Intelligence for Computational Modeling of the Heart

108 Chapter 3 Learning cardiac anatomy

uniform batch-wise sampling of Ξ(k,m):

2
(i) (i)
ˆ θ y − Q(s,a;θ | L d ,m) , (3.6)

k,m = argmin E (s,a,r,s )∼Ξ(k,m) k,m
(i)
θ
k,m
(i)
where θ denotes the parameters of the deep neural network
k,m
search model for landmark k on scale level m at training iteration
i> 0 and Ξ(k,m) is the associated memory array of past state tran-
sitions. The reference value y represents the maximum expected
reward for a trajectory starting at the current state, estimated us-
ing the update-delay technique [261], based on model parameters

(i) (i )

θ ¯ := θ from a past training iteration i <i:
k,m k,m
(i)

y = r + γ maxQ s ,a ;θ ¯ k,m | L d ,m . (3.7)
a
We apply the same convergence criterion as defined in [262].
This is determined as the center of gravity of an oscillation cycle.
3.2.2.4 Robust spatially-coherent landmark detection
To better cope with incomplete data, i.e., partial fields of view,
we propose to model the spatial distribution of the anatomi-
cal landmarks using robust statistical shape modeling. In other
words, we constrain the output of the global search model θ M−1
to ensure a consistent distribution of the agent positions. Consid-
ering a set of N anatomical landmarks in translation and scale-
normalized space, we model the distribution of each individual
landmark i ∈[0,...,N − 1] via a multi-variate normal distribution
p i ∼ N(μ ,Σ i ),where μ and Σ i are estimated using maximum
i
i
likelihood. This defines a mean shape-model for the landmark

set as μ = μ ,...,μ N−1 . Given an unseen configuration of de-
0

tected points at scale M − 1 as P =[ ˜ p 0 , ˜ p 1 ,...] , a robust shape-
˜
model is fitted using M-estimator sample consensus [271]based
on 3-point samples from the set of all triples I 3 (P).Theoptimal
˜
mean-model fit with maximum consensus is obtained by mini-
mizing the following cost function based on the redescending M-
estimator [271]:
| ˜ P|

1 −1
S ← argmin min φ( ˜ p i ) − μ i
Σ i φ( ˜ p i ) − μ , 1 , (3.8)
ˆ
i
Z i
S∈I 3 ( ˜ P) i=0
x−t
where φ(x) = is a projector to normalized shape-space with
s
the estimated fit ˆ w =[t,s] on set S. More details on this model of
the spatial consistency can be found in [264].