1
D
ata-driven system matrix manipulation enabling fast functional imaging and
intra-image nonrigid motion correction in tomography
Pen
g Hu
1
, X
in Tong
1
, L
i Lin
1,
2
, an
d Lihong V. Wang
1,
*
1
C
altech Optical Imaging Laboratory, Andrew and Peggy Cherng Department of Medical
Engineering, Department of Electrical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
2
P
resent address: College of Biomedical Engineering and Instrument Science, Zhejiang
University, Hangzhou 310027, China
*Correspondence should be addressed to L.V.W. (LVW@caltech.edu).
A
bstract
T
omographic imaging modalities are described by large system matrices. Sparse sampling and
tissue motion degrade system matrix and image quality. Various existing techniques improve the
image quality without correcting the system matrices. Here, we compress the system matrices to
improve computational efficiency (e.g., 42 times) using singular value decomposition and fast
Fourier transform. Enabled by the efficiency, we propose (1) fast sparsely sampling functional
imaging by incorporating a densely sampled prior image into the system matrix, which maintains
the critical linearity while mitigating artifacts and (2) intra-image nonrigid motion correction by
incorporating the motion as subdomain translations into the system matrix and reconstructing the
translations together with the image iteratively. We demonstrate the methods in 3D photoacoustic
computed tomography with significantly improved image qualities and clarify their applicability
to X-ray CT and MRI or other types of imperfections due to the similarities in system matrices.
In
troduction
T
omographic imaging modalities X-ray computed tomography (CT), magnetic resonance imaging
(MRI), and photoacoustic computed tomography (PACT) produce cross-sectional images of tissue
b
y detection of penetrating X-rays
1
, n
uclear-magnetic-resonance-induced radio waves
2,
3
,
and
light-absorption-induced ultrasonic waves
4
,
respectively. Each modality with a certain setup is
described by a system matrix
5–1
0
. A
ccurate image reconstruction poses requirements to the system
matrix, which are often violated. For example, to achieve high temporal resolution for functional
.
CC-BY-NC-ND 4.0 International license
available under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprint
this version posted January 8, 2024.
;
https://doi.org/10.1101/2024.01.07.574504
doi:
bioRxiv preprint
2
i
maging, the spatial sampling density is often sacrificed, which introduces artifacts in the
reconstructed image
11–13
an
d may affect the functional signal extraction. Also, tissue motions such
as heart beating
1
4–16
,
breathing
17–19
, ab
dominal movement
20
,21
,
and fetal movement
2
2
, cau
se
complex geometric errors in each system matrix, which introduce artifacts in the reconstructed
image and compromise valuable image features.
Numerous methods have been proposed to compensate for system-matrix imperfections from
image-domain
12
,23–27
,
signal-domain
28–33
, an
d cross-domain
34–38
p
erspectives. However, due to the
large size of each system matrix, these methods tend not to manipulate or correct the system matrix
directly and have limitations. For sparse sampling functional imaging, traditional
methods
1
2,25,26,32,38
m
itigate artifacts in images but their performances drop sharply as the sampling
density reduces. Deep neural networks (DNNs)
2
7–29,31,34,37,39
sh
ow high performance in mitigating
artifacts but tend to generate false image features when the sampling density is low, and they
require imaging-modality- and device-dependent datasets, which are not always available.
Moreover, most of the methods introduce nonlinearity while mitigating artifacts, which disrupts
the functional signals that are often much weaker than background signals.
For intra-image nonrigid motion correction, gating- and binning-based methods
1
6,30,33,35,36
ar
e
commonly used. However, they require repeated data acquisition, which is time-consuming and
infeasible for unrepeated motions. DNNs have also been used for motion correction
23
,24
,
however,
they need specific training datasets that are not universally available, and it is challenging to reject
falsely generated features in DNNs. Two system-matrix-level methods
4
0,41
h
ave been proposed for
motion correction. In the first method
40
,42
, t
he authors approximate general nonrigid motions with
localized linear translations, identify possible motion paths from multichannel navigator data, and
estimate the motion at each pixel using localized gradient-entropy metric in the image domain.
However, quantifying localized motion from only navigator data is not robust, especially when the
motion amplitude and noise level increase. In the second method
4
1
, t
he authors express breathing-
and heartbeat-induced motions in basis functions by performing singular value decomposition
(SVD) and resolve these motions in imaging. However, for general motions, especially unrepeated
motions, the method’s performance is unknown. The high computation cost of SVD in the method
also restricts its application to 3D imaging.
.
CC-BY-NC-ND 4.0 International license
available under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprint
this version posted January 8, 2024.
;
https://doi.org/10.1101/2024.01.07.574504
doi:
bioRxiv preprint
3
H
ere, we compress the system matrices using SVD and fast Fourier transform (FFT), which
enables efficient system matrix slicing and manipulation. Then, we use two new methods for
functional imaging and motion correction, respectively, by performing data-driven manipulations
of these matrices. Both methods are applicable to CT, MRI, and PACT. For sparse sampling
functional imaging, we incorporate a prior image into the system matrix to reduce unknown
variables in image reconstruction. Special configurations in the method maintain linearity in image
reconstruction while substantially mitigating artifacts, which is critical for weak functional signal
extraction. For intra-image nonrigid motion correction, we approximate the motion with localized
linear translations
4
0
.
Starting from an initially reconstructed motion-blurred image, we first
estimate the translation of each subdomain of the object by minimizing the difference between the
simulated signals from the subdomain and the detected signals. With the estimated translations,
we update the system matrix and reconstruct the image again. We iterate the correction-
reconstruction process to obtain the final image. This method does not require repeated data
acquisition and is effective for unrepeated motions.
In this work, we use 3D PACT for demonstration due to its representatively large size in
tomography: light-absorption-induced ultrasonic wave from every voxel in an image is detected
by every transducer element and the system matrix is intrinsically a 6D tensor
10
. We
apply the
proposed methods to both numerical simulations and
in vivo
experiments: mouse brain functional
imaging with sparse sampling and intra-image motion correction in human breast imaging. We
demonstrate that both methods substantially improve the functional and structural image qualities.
R
esults
Sy
stem matrix compression based on SVD and FFT
In this study, we use the 3D PACT system reported previously by Lin et al.
4
3
,
which consists of
four 256-element arc transducer arrays (central frequency of 2.25 MHz and one-way bandwidth of
98%), and we assume a homogenous medium. The rotation of the four arc-arrays forms a virtual
hemispherical array truncated at the bottom for light delivery (marked as blue arcs in
Fig. 1a
), and
the detected ultrasonic signals are used to form a 3D image (marked as a rectangular cuboid). We
denote the number of voxels in the image and the number of virtual elements in the 2D array as