Four-dimensional imaging of periodically moving objects via post-acquisition synchronization of nongated slice-sequences
When the studied motion is periodic, such as for a beating heart, it is possible to acquire successive sets of two dimensional plus time data slice-sequences at increasing depths over at least one time period which are later rearranged to recover a three dimensional time sequence. Since gating signals are either unavailable or cumbersome to acquire in microscopic organisms, the invention is a method for reconstructing volumes based solely on the information contained in the image sequences. The central part of the algorithm is a least-squares minimization of an objective criterion that depends on the similarity between the data from neighboring depths. Owing to a wavelet-based multiresolution approach, the method is robust to common confocal microscopy artifacts. The method is validated on both simulated data and in-vivo measurements.
1. A method for synchronizing nongated confocal data slice sequences to generate dynamic three dimensional volumes comprising:
sequentially acquiring confocal slice-sequences at different depths; and
reassembling the confocal slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume
where reassembling the slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume comprises registering slice-sequence pairs with respect to time for a given time shift in a manner to maximize similarity in a predetermined metric between two temporally adjacent slices,
where registering slice-sequence pairs with respect to time for a given time shift in a manner to maximize similarity in a predetermined metric between two temporally adjacent slices comprises measuring data intensity, I m , at each point, x, z,
I m ( x,z k ,t )=∫∫∫ I ( x′,z,t−s k ) PSF ( x−x′,z−z k ) dx′dz
where t is time, the s k are the unknown time shifts, defined modulo a predetermined period, at each measured depth z k =kh, k=0, . . . ,N z , with h the axial slice spacing, PSF is a point spread function (PSF) expressed by the Dirac delta function
PSF ( x, z )=δ( x )δ( z )
and where the axial slice spacing h, the vertical distance between two adjacent slides, is smaller than the PSF extent in z, or where the imaged object undergoes sufficiently smooth and homogeneous deformations to provide statistically significant similarity between adjacent slides.
2. A method for synchronizing nongated confocal data slice sequences to generate dynamic three dimensional volumes comprising:
sequentially acquiring confocal slice-sequences at different depths; and
reassembling the confocal slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume,
where reassembling the slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume comprises:
determining a period of the object motion;
cropping data in the acquired slice-sequences;
interpolating data in the cropped slice-sequences;
determining relative shifts between pairs of slice-sequences;
determining absolute shifts between pairs of slice-sequences with respect to
a first slice-sequence;
synchronizing slice-sequences according to the absolute shifts; and post-processing the synchronized slice-sequences
where determining a period of the object motion comprises selecting a period in the range T(z k )∈|T min ,T max |, and determining a period T(z k ) by iteratively minimizing
T ( z k )= arg min T′ D ( z k ,T ′)
where image sequences are acquired at times, T i =ih T , i=0, . . . , N T −1, where h t is the acquisition sampling step and N T is the number of acquired frames, where for a selected period T′, the time-positions of every pixel are phase locked to the first period
τ′ i =τ i −[τ i /T′]T′
where a bijective mapping i=i(j) (respectively j=j(i)) such that T′ j−1 ≦ T′ j ≦ T′ j+1 is defined, and where an estimate of dispersion of a phase locked signal, D, is given by the length of a graph ( 1 ′ j , l m (x, z k , T′ i(j) )) j=0, . . . , N T that joins the newly ordered samples on a normalized time scale cumulated over the whole image, where
D
(
zk
,
T
′
)
=
∑
m
∈
Z
2
∑
j
=
1
N
τ
-
1
[
I
m
(
x
m
,
z
k
,
τ
i
(
j
)
)
-
I
m
(
x
m
,
z
k
,
τ
i
(
j
-
1
)
)
2
+
τ
j
′
-
τ
j
-
1
′
2
/
T
′2
]
1
2
with
x
m
=
m
h
xy
.
3. A method for synchronizing nongated confocal data slice sequences to generate dynamic three dimensional volumes comprising:
sequentially acquiring confocal slice-sequences at different depths; and
reassembling the confocal slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume,
where reassembling the slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume comprises:
determining a period of the object motion;
cropping data in the acquired slice-sequence;
interpolating data in the cropped slice-sequences;
determining relative shifts between pairs of slice-sequences;
determining absolute shifts between pairs of slice-sequences with respect to
a first slice-sequence;
synchronizing slice-sequences according to the absolute shifts; and post-processing the synchronized slice-sequences
where interpolating data in the acquired slice-sequences comprises resampling the cropped data to cover an integer number of periods, where the resampled data are taken at times t i =ih t , i=0, . . . ,N t −1, with sampling step h t =L/N t −1), where N t is the number of considered frames over the total time L=N T T h and N T is the number of considered periods, and using a linear spline interpolation.
4. A method for synchronizing nongated confocal data slice sequences to generate dynamic three dimensional volumes comprising:
sequentially acquiring confocal slice-sequences at different depths; and
reassembling the confocal slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume,
where reassembling the slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume comprises:
determining a period of the object motion;
cropping data in the acquired slice-sequence;
interpolating data in the cropped slice-sequences;
determining relative shifts between pairs of slice-sequences;
determining absolute shifts between pairs of slice-sequences with respect to
a first slice-sequence;
synchronizing slice-sequences according to the absolute shifts; and post-processing the synchronized slice-sequences
where determining relative shifts between pairs of slice-sequences comprises using a least-squares criterion defined by
Q
k
,
k
′
(
s
)
=
∫
∫
R
2
L
∫
0
I
m
(
x
,
z
k
,
t
)
-
I
m
(
x
,
z
k
′
,
t
-
s
)
2
ⅆ
t
ⅆ
x
where the shift s∈R, where
Q
k
,
k
′
(
s
)
=
∫
∫
R
2
∫
0
L
I
m
(
x
,
z
k
,
t
)
2
+
I
m
(
x
,
z
k
′
,
t
-
s
)
2
ⅆ
t
ⅆ
x
-
2
∫
∫
R
2
∫
0
L
I
m
(
x
,
z
k
,
t
)
I
m
(
x
,
z
k
′
,
t
-
s
)
ⅆ
t
ⅆ
x
=
C
-
2
∫
∫
R
2
∫
0
L
I
m
(
x
,
z
k
,
t
)
I
m
(
x
,
z
k
′
,
t
-
s
)
ⅆ
t
ⅆ
x
;
computing Q k,k′ (s) up to the constant C for a number of regularly spaced shifts s=ih t , with h t =L/(N t −1), i=0, . . . ,N t −1, to obtain the relative shifts s k,k′ , between any two pairs of z-slices by finding the shifts s that minimizes Q k,k′ (s), which relative shifts are represented by the antisymmetric matrix S, whose elements are
s k,k′ =arg min s=kt,k=1, . . . ,N t Q k,k′ ( s ).
5. The method of claim 4 determining relative shifts between pairs of slice-sequences comprises using a least-squares criterion comprises using multiresolution and noise decorrelation properties of wavelet decomposition using a separable orthogonal wavelet basis of L 2 (R 2 ),
{ψ j,m 1 (x),ψ j,m 2 (x),ψ j,m 3 (x)} j∈Z,m∈Z 2 (10)
where the two-dimensional wavelets
ψ
k
,
m
p
(
x
)
=
1
2
j
ψ
p
(
x
2
j
-
m
)
are constructed with separable the one dimensional scaling function φ(x) and wavelet ψ(x)
ψ 1 ( x )=φ( x )ψ( y ),
ψ 2 ( x )=φ( y )ψ( x ),
ψ 3 ( x )=ψ( x )ψ( y )
where the basis functions are indexed with a single index k that includes the scale j∈Z, translation m∈Z 2 , and wavelet type p∈{1, 2, 3}:
ψ k ( x )=ψ j,m p ( x ), {right arrow over (k)}=( p, j, m )
to expand the slice-sequence at a fixed depth z k and time-point t in the wavelet basis as
I
m
(
x
,
z
k
,
t
)
=
∑
k
c
k
,
k
->
(
t
)
ψ
k
->
(
x
)
where the coefficients are given by the inner products (wavelet transform)
c
k
,
k
->
(
t
)
=
〈
I
m
(
•
,
z
k
,
t
)
,
ψ
k
_
〉
=
∫
∫
I
m
(
x
->
,
z
k
,
t
)
ψ
k
_
(
x
->
)
ⅆ
x
and since the basis functions are orthogonal, i.e. <ψ k , ψ k′ >=δ k,k′ <ψ k , ψ k′ >=δ k,k′ ,
Q
k
,
k
′
(
jh
t
)
=
C
-
2
∫
0
L
∑
k
->
c
k
,
k
→
(
t
)
c
k
′
,
k
→
(
t
-
jh
t
)
≈
C
-
2
h
t
∑
k
->
∑
i
=
0
N
t
-
1
c
k
,
k
→
(
ih
t
)
c
k
′
,
k
→
[
(
i
-
j
)
h
t
]
where only a finite number of scales and translations for k are considered because of the finite resolution and support of the image, and appropriate boundary conditions, where the fine resolution coefficients are discarded to downsize the data complexity, where the low-pass coefficients are discarded, where a soft threshold is applied to the remaining coefficients to limit the influence of noise sources, where D(z k , T′) is derived from a reduced data set of wavelet coefficients instead of sampled image pixels, I m , using
D
~
(
z
k
,
T
′
)
=
∑
k
->
∑
j
=
1
N
t
-
1
{
c
k
,
k
->
[
τ
i
(
j
)
]
-
c
k
,
k
->
[
τ
i
(
j
-
1
)
]
2
+
τ
j
′
-
τ
j
-
1
′
2
/
T
′2
}
1
2
thus gaining robustness, reducing required memory and decreasing computation time.
6. The method of claim 5 where using a separable orthogonal wavelet basis of L 2 (R 2 ) comprises using Daubechies 9/7 wavelets.
7. A method for synchronizing nongated confocal data slice sequences to generate dynamic three dimensional volumes comprising:
sequentially acquiring confocal slice-sequences at different depths; and
reassembling the confocal slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume,
where reassembling the slice-sequences at different depths in synchronization with each other without the use of a gating signal to recover a dynamic three dimensional volume comprises:
determining a period of the object motion;
cropping data in the acquired slice-sequences;
interpolating data in the cropped slice-sequences;
determining relative shifts between pairs of slice-sequences;
determining absolute shifts between pairs of slice-sequences with respect to
a first slice-sequence;
synchronizing slice-sequences according to the absolute shifts; and post-processing the synchronized slice-sequences
where determining absolute shifts between pairs of slice-sequences with respect to a first slice-sequence comprises determining the absolute shifts in relation to the relative shifts s k,k′ , by setting
s 1 =0
s k −s k′ =s k,k′ with k,k= 1 , . . . ,N z and k<k′;
assigning different weights w |k′−k| w |k′−k| to equations that involve the estimated shifts s k,k′ depending on the distance j=|k′−k, lower weights w j being assigned to equations for slice-sequence pairs less likely to exhibit similarities, namely when the distance |k′−k| between the slice-sequence pairs increases as summarized in matrix algebraic form as A t=s along with the diagonal weighting matrix, where A is an assignment matrix, t an absolute shift matrix and s a relative shift matrix along with a diagonal weighting matrix, W=diag (1, w 1 , w 1 , w 1 , w 1 , w 2 , w 2 , w 2 ); and
determining a weighted least-squares solution of A t=s by solving A † W † WAt=A † W † Ws, where (▪) † denotes transposition.
8. The method of claim 7 further comprising setting appropriate weights in W=diag (1, w 1 , w 1 , w 1 , w 1 , w 2 , w 2 , w 2 ) to include supplementary information which is not image-intrinsic, such as electrocardiograms (ECG).