12
SUPPLEMENTARY MATERIAL
D
ERIVATION OF
A
LTERNATIVE
M
ODEL
In this section, we explore the parameter estimation for
an alternative model. Specifically, letting
M
i
be the set of
missing voxels of patch
y
i
, we treat
y
M
i
i
as latent variables,
instead of explicitly modeling a low-dimensional representa-
tion
x
. We show the maximum likelihood updates of the model
parameters under the likelihood (5). We employ the Expecta-
tion Conditional Maximization (ECM) [14], [18] variant of
the Generalized Expectation Maximization, where parameter
updates depend on the previous parameter estimates.
The complete data likelihood is
p
p
Y
;
θ
q
¹
i
̧
k
π
k
N
p
y
O
i
i
,y
M
i
i
;
μ
O
i
k
,
Σ
O
i
O
i
k
q
.
(25)
The
expectation step
updates the statistics of the missing
data, computed based on covariates of the known and unknown
voxels:
γ
ik
IE
r
k
i
s
π
k
N
p
y
O
i
i
;
μ
O
i
k
,
Σ
O
i
k
q
°
k
π
k
N
p
y
O
i
i
;
μ
O
i
k
,
Σ
O
i
k
q
(26)
p
y
ij
IE
r
y
ij
s
#
y
ij
if
y
ij
is observed
μ
ij
Σ
j
O
i
i
p
Σ
O
i
O
i
i
q
1
p
y
O
i
i
μ
O
i
q
otherwise
(27)
p
s
ijl
IE
r
y
ij
y
il
s
IE
r
y
ij
s
IE
r
y
il
s
#
0
if
y
ij
or
y
il
is observed
Σ
jl
i
p
Σ
O
i
j
i
q
T
p
Σ
O
i
O
i
i
q
1
Σ
O
i
l
i
otherwise
(28)
where the correction in
p
s
ijl
can be interpreted as the uncer-
tainty in the covariance estimation due to the missing values.
Given estimates for the missing data, the
maximization step
leads to familiar Gaussian Mixture Model parameters updates:
μ
k
1
γ
ik
̧
i
γ
ik
p
y
ik
(29)
Σ
k
1
γ
ik
̧
i
γ
ik
p
p
y
ik
μ
k
qp
p
y
ik
μ
k
q
T
S
T
i
.
(30)
π
k
1
N
̧
i
γ
ik
(31)
where
r
S
i
s
jl
p
s
ijl
.
In additional to the latent missing voxels, we can still model
each patch as coming from a low dimensional representation.
We form
C
k
W
k
W
T
k
σ
2
k
I
as in (3), leading to the complete
data likelihood:
p
p
Y
;
θ
q
¹
i
̧
k
π
k
N
p
y
O
i
i
,y
M
i
i
;
μ
O
i
k
,C
O
i
O
i
k
q
.
(32)
The
expectation steps
are then unchanged from (26)-(28)
with
C
k
replacing
Σ
k
. The
maximization steps
are unchanged
from (29)-(31), with
Σ
k
now the
empirical
covariance in (30).
We let
U
Λ
V
T
SVD
p
Σ
k
q
be the singular value decomposi-
tion of
Σ
k
, leading to the low dimensional updates
σ
2
k
Ð
1
d
q
d
̧
j
d
1
Λ
p
j,j
q
(33)
W
k
Ð
U
p
Λ
σ
2
I
q
1
{
2
.
(34)
Finally, we let
C
k
W
k
W
T
k
σ
2
k
I
.
Unfortunately, both learning procedures involve estimating
all of the missing voxel covariances, leading to a large and
unstable optimization.
13
NLM
Our method
Ground truth
Linear
Subject
3
Subject
4
Subject 5
Subject
6
Subject
7
Fig. 11: Additional restorations in the ADNI dataset
. Reconstruction by NLM, linear interpolation, and our method, and the original high resolution images.
14
Our method
Linear
NLM
Patient 3
Patient 4
Patient 5
Patient 6
Patient 7
Fig. 12: Additional restorations in the clinical dataset
. Reconstruction using NLM, linear interpolation and our method.