# NeRF Solves Undersampled MRI Reconstruction

Tae Jun Jang1,2 and Chang Min Hyun3,∗

1School of Mathematics and Computing (Computational Science and Engineering),
Yonsei University, Seoul, KR.  2AI Vision Group, Samsung Medison, Seoul, KR.
3Department of Radiology, Perelman School of Medicine, University of
Pennsylvania, Philadelphia, USA. *Corresponding author (e-mail:
chang.minhyun@pennmedicine.upenn.edu).The authors declare that they have no
known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

###### Abstract

This article presents a novel undersampled magnetic resonance imaging (MRI)
technique that leverages the concept of Neural Radiance Field (NeRF). With
radial undersampling, the corresponding imaging problem can be reformulated
into an image modeling task from sparse-view rendered data; therefore, a high
dimensional MR image is obtainable from undersampled $k$-space data by taking
advantage of implicit neural representation. A multi-layer perceptron, which
is designed to output an image intensity from a spatial coordinate, learns the
MR physics-driven rendering relation between given measurement data and
desired image. Effective undersampling strategies for high-quality neural
representation are investigated. The proposed method serves two benefits: (i)
The learning is based fully on single undersampled $k$-space data, not a bunch
of measured data and target image sets. It can be used potentially for
diagnostic MR imaging, such as fetal MRI, where data acquisition is relatively
rare or limited against diversity of clinical images while undersampled
reconstruction is highly demanded. (ii) A reconstructed MR image is a scan-
specific representation highly adaptive to the given $k$-space measurement.
Numerous experiments validate the feasibility and capability of the proposed
approach.

###### Index Terms:

fast MRI; accelerated MRI; undersampled MRI reconstruction; implicit neural
representation; neural radiance field.

##  I Introduction

Undersampled magnetic resonance imaging (MRI) has been gaining a great
attention to expand our capability of producing cross-sectional MR images with
high spatial resolution from optimized data acquisition. It has been
particularly anticipated to shorten a long scan time [1, 2, 3], which can
contribute to various clinical outcomes such as the increased satisfaction of
subjects through a minimized duration time in a uncomfortable narrow bow of
MRI machine and the decreased occurrence of motion artifacts induced by
deliberate or inevitable movements of subject. Specifically, the undersampled
MRI delves into a reconstruction way to minimize time-consuming $k$-space
measurements along a phase-encoding direction to the maximum extent possible
without compromising the output image quality [4].

In standard MRI reconstruction, sampling $k$-space measurements below a
certain limit, determined by Nyquist criterion [5], induces image artifacts
known as aliasing [6], which can seriously downgrade the quality of resultant
MR image. A key of undersampled MRI reconstruction is then how to overcome
such artifacts while preserving or, hopefully, restoring image information
obtainable from the standard reconstruction with a minimal or more sampling in
the sense of Nyquist.

Recently, extensive data-driven approaches with deep learning have been
proposed for undersampled MRI reconstruction [7, 8, 9]. These methods have
been showing the powerful and promising performance in various tasks such as
brain and knee MR imaging. They attempt to learn and take advantage of prior
knowledge associated with desired MR images through exploring common inter-
training-data relationship under a supervised, unsupervised, or whatever
learning framework [10]. For instance, a paper [11] leveraged the U-shaped
fully convolutional neural network to realize an all-encompassing relation
between data distributions in which aliased and corresponding high-quality MR
images lie, respectively. A reconstructed MR image is based on the learned
group knowledge rather than being highly adaptive to a given measurement. The
data-driven approaches seemingly provide the guaranteed effectiveness for
samples on or nearby a data distribution similar to a training dataset [10].
The practical performance is inevitably influenced by the quality, quantity,
and diversity of training data.

This study seeks to investigate and suggest a novel approach for data-driven
undersampled MRI reconstruction that meets the followings: (i) high
performance and robustness regardless of a bunch of training samples for
desired MR images and (ii) strong adaptiveness to a specific $k$-space
measurement. Thanks to recently emerged Neural Radiance Field (NeRF)
techniques [12, 13, 14, 15], it becomes enable to accurately represent image
rendering in an arbitrary view-point, which implicitly requires to realize an
underlying image model, from rendered data in sparse view-points even. With
radial undersampling, the corresponding MR imaging problem can be reformulated
into an image modeling task from sparse-view rendered data; therefore, a high-
quality MR image can be obtained through leveraging the concept of NeRF,
namely, implicit neural representation based on a rendering relation. This
approach could be a great fit for accomplishing the aforementioned desires.

Figure 1: Implicit neural representation for undersampled MRI reconstruction
with radial sampling.

To be precise, a multi-layer perceptron, designed to output an image intensity
from a spatial image coordinate, is learned by minimizing a rendering loss
derived from the physical relation between radially undersampled $k$-space
data and desired MR image. The learning is based fully on single $k$-space
data, not a bunch of measured data and target image sets. The network attempts
to realize and take advantage of inter-relation of image intensity over
pixels, namely, intra-relation inside a MR image. This is in contrast to
existing approaches learning and utilizing inter-relation over MR images. A
reconstructed MR image is then a scan-specific representation that is highly
adaptive to the specific $k$-space measurement.

Effective undersampling strategies are investigated for high-quality neural
representation. We try to provide some rationale somewhat explaining the
effective sampling as well as empirical examinations.

Numerous experiments were conducted to validate the feasibility and capability
of the proposed reconstruction method by using a publicly open fastMRI dataset
[16]. In our empirical experiments, a high-quality MR image could be
reconstructed from undersampled $k$-space data even with high acceleration
factors. The proposed method is likely to not only improve the image quality
but also preserve uncommon anatomical information that tends to be far from
shared patterns over MR images. We compared five undersampling strategies:
uniform, limited, random, stratified, and golden-angle samplings. The uniform,
stratified, and golden-angle tend to provide the better performance than the
random and limited under a fixed acceleration factor.

The proposed approach, in fact, is motivated to potentially tackle a challenge
in practical diagnostic MR imaging in which data acquisition is relatively
rare or limited against diversity of clinical images while undersampled
reconstruction is highly demanded. In fetal MRI, for example, the undersampled
reconstruction can be beneficial, since one of major hurdles is fetus-induced
motion artifacts because of the long scan time [17]. Unfortunately, however,
existing data-driven methods might not be very powerful as in other MRI
applications. It is because fetal MRI is typically utilized as a second-level
examination tool followed by ultrasound imaging [18]; thereby, gathering fetal
MRI data for training may not be flexible in contrast to tremendous image
variant factors such as fetal movement, fetal position, gestational age, and
etc. In this regard, the proposed method can be viewed as an evolution towards
providing an effective solution in this kind of restricted clinical
situations, not intending the superiority in all-type imaging environments.

##  II Method

For ease of explanation, we restrict ourselves to a 2D MR imaging task. A
general 3D reconstruction scheme would be discussed later. Fig. 1 illustrates
a schematic description of the proposed method.

###  II-A Inverse problem for undersampled MRI reconstruction

Let $\mathcal{X}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a nuclear spin
density distribution of 2D human body to be scanned. We assume that
$\mathcal{X}$ can be expressed as

$\mathcal{X}(\boldsymbol{x})=\sum_{\boldsymbol{v}\in\mathbb{V}}X_{\boldsymbol{v}}\phi_{\boldsymbol{v},\delta}(\boldsymbol{x})\mathrm{~{}for~{}}\boldsymbol{x}\in\mathbb{R}^{2},$
(1)

where $\mathbb{V}$ is a set of indices defined by

$\mathbb{V}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{(v_{1},v_{2})\in\mathbb{Z}\times\mathbb{Z}:0<v_{i}\leq\dfrac{V}{\delta},i=1,2~{}\right\\},$
(2)

and $\phi_{\boldsymbol{v},\delta}$ is a kernel function depending on
$\boldsymbol{v}$ and $\delta$. Here, $\frac{V}{\delta}$ is assumed to be an
even number. The distribution of coefficients, $\\{X_{\boldsymbol{v}}\\}$,
corresponds to a MR image, whose spatial resolution and pixel dimension are
associated with $\delta$ and $V$, respectively. One can use a kernel of linear
interpolation for $\\{X_{\boldsymbol{v}}\\}$, Gaussian kernel, or a square-
shaped kernel function given by

$\phi_{\boldsymbol{v},\delta}(\boldsymbol{x})=\left\\{\begin{array}[]{cl}\dfrac{1}{\delta^{2}}&\mbox{if
}\left|(x_{i}+\dfrac{V}{2})-(v_{i}-\dfrac{1}{2})\delta\right|\leq\dfrac{\delta}{2},i=1,2\vspace{0.1cm}\\\
0&\mathrm{otherwise}\end{array}.\right.$ (3)

An inverse problem for MRI reconstruction is to recover
$\\{X_{\boldsymbol{v}}\\}$ from measured $k$-space data. With radial sampling,
a measured dataset can be expressed as $\\{K_{\omega,\varphi}\\}$, where
$K_{\omega,\varphi}$ is given by

$K_{\omega,\varphi}=\mathscr{F}_{\boldsymbol{x}}[\mathcal{X}](\omega\Theta_{\varphi})$
(4)

for some $\omega\in[-\frac{1}{2\delta},\frac{1}{2\delta})\mbox{ and
}\varphi\in[0,\pi)$. Here, $\mathscr{F}_{\boldsymbol{x}}$ is the 2D Fourier
transform with respect to $\boldsymbol{x}$ defined by

$\mathscr{F}_{\boldsymbol{x}}[f](\boldsymbol{\xi})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\int_{\mathbb{R}^{2}}f(\boldsymbol{x})\exp(-2\pi
i\boldsymbol{x}\cdot\boldsymbol{\xi})d\boldsymbol{x}\mathrm{~{}for~{}}\boldsymbol{\xi}\in\mathbb{R}^{2},$
(5)

$\Theta_{\varphi}=\begin{bmatrix}\cos\varphi,\sin\varphi\end{bmatrix}$, and
$\frac{1}{\delta}$ is the minimum interval length for securing spatial
information up to the image resolution $\delta$. In standard reconstruction,
$\\{K_{\omega,\varphi}\\}$ is sampled such that the Nyquist criterion meets,
which can be roughly viewed as [6]:

$|\\{K_{\omega,\varphi}\\}|\gtrapprox|\mathbb{V}|=\left(\dfrac{V}{\delta}\right)^{2},$
(6)

where $|\cdot|$ denotes the set cardinality. Specifically, considering a
uniform sampling with respect to $\omega$ given by

$\omega\in\left\\{-\dfrac{1}{2\delta},-\dfrac{1}{2\delta}+\dfrac{1}{\sqrt{2}V},\cdots,-\dfrac{1}{2\delta}+(\dfrac{\sqrt{2}V}{\delta}-1)\dfrac{1}{\sqrt{2}V}\right\\},$
(7)

the criterion (6) yields the following condition:

$N_{\varphi}\gtrapprox\dfrac{V}{\sqrt{2}\delta},$ (8)

where $N_{\varphi}$ is the number of radial samples. We here mention that
$\frac{1}{\sqrt{2}V}$ in (7) is a sufficient sampling resolution for securing
the pixel dimension of $\\{X_{\boldsymbol{v}}\\}$, i.e., $\frac{V}{\delta}$.

In this regard, undersampled MRI reconstruction seeks to recover
$\\{X_{\boldsymbol{v}}\\}$ while considerably violating the condition (8),
i.e., minimizing $N_{\rho}$ as possible. A practical motivation comes from
that, in a typical radial sampling set-up, the number of $\varphi$-directional
measurements is proportional to a total scan time [19, 20].

The associated inverse problem is given by

$``\mbox{Recover }\\{X_{\boldsymbol{v}}\\}\mbox{ from
}\\{K_{\omega,\varphi}\\}",$ (9)

where $\omega$ is given by (7) and $\varphi$ is of $N_{\varphi}$ angles
sampled from [0,$\pi$) with $N_{\varphi}<\frac{V}{\sqrt{2}\delta}$. For
example, the following uniform undersampling of $\varphi$ is one of options:

$\varphi\in\Phi_{\mathrm{uni}}^{R}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{0,R\pi\left\lfloor\dfrac{\sqrt{2}\delta}{V}\right\rfloor,2R\pi\left\lfloor\dfrac{\sqrt{2}\delta}{V}\right\rfloor,\cdots\cdots\right\\},$
(10)

where $R>1$ denotes a undersampling (or acceleration) factor and
$\lfloor\cdot\rfloor$ denotes the floor operation.

###  II-B Reformulation into an image modeling task from sparse-view rendered
data

We transform $K_{\omega,\varphi}$ into a new data form, $P_{r,\varphi}$, as
follows:

$\displaystyle P_{r,\varphi}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathscr{F}^{-1}_{\omega}[K_{\omega,\varphi}](r),$
(11)

where $\mathscr{F}^{-1}_{\omega}$ is the 1D inverse Fourier transform with
respect to $\omega$. The Fourier slice theorem provides the following relation
between $P_{r,\varphi}$ and $\mathcal{X}$ [21]: For some $r$ and $\varphi$,

$\displaystyle P_{r,\varphi}$
$\displaystyle=\int_{\mathbb{R}}K_{\omega,\varphi}\exp(2\pi i\omega r)d\omega$
$\displaystyle=\int_{\mathbb{R}}(\int_{\mathbb{R}^{2}}\mathcal{X}(\boldsymbol{x})\exp(-2\pi
i\boldsymbol{x}\cdot\omega\Theta_{\varphi})d\boldsymbol{x})\exp(2\pi i\omega
r)d\omega$
$\displaystyle=\int_{\mathbb{R}^{2}}\mathcal{X}(\boldsymbol{x})\boldsymbol{\delta}(\boldsymbol{x}\cdot\Theta_{\varphi}-r)d\boldsymbol{x}.$
(12)

where $\boldsymbol{\delta}$ is the Dirac delta function. Accordingly, the
reconstruction problem (9) can be rewritten as

$``\mbox{Recover }\\{X_{\boldsymbol{v}}\\}\mbox{
from}~{}\\{P_{r,\varphi}\\}".$ (13)

The reconstruction problem (13) can be equivalently viewed as an image
modeling task from sparse-view rendered data, which allows us to leverage the
idea of NeRF.

A general image rendering relation comes from the radiative transfer equation
given by

$\dfrac{dI}{ds}=\epsilon(s)-\tau(s)I(s),$ (14)

where $s$ is a length along a ray, $I$ is a ray intensity, $\epsilon$ is a
light source, and $\tau$ is an extinction coefficient that can be regarded as
the image intensity, i.e., $\tau=\mathcal{X}$. A solution for (14) can be
expressed as the following form:

$\displaystyle I(r,\varphi)=$ $\displaystyle
I_{0}\exp(-\int_{\mathbb{R}^{2}}\mathcal{X}(\boldsymbol{x})\boldsymbol{\delta}(\boldsymbol{x}\cdot\Theta_{\varphi}-r)d\boldsymbol{x})$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}}\epsilon_{r,\varphi}(s)\exp(-\int_{s}^{\infty}\mathcal{X}(r\Theta_{\varphi}+\tilde{s}\Theta_{\varphi}^{\perp})d\tilde{s})ds,$
(15)

where $I_{0}$ is an initial intensity and
$\Theta_{\varphi}^{\perp}=\begin{bmatrix}\sin\varphi,-\cos\varphi\end{bmatrix}$.
A rendering relation is defined by

$\mathscr{C}[\mathcal{X}](r,\varphi)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}-\log\left(\dfrac{I(r,\varphi)}{I_{0}}\right).$
(16)

Assuming $\epsilon=0$ (no external source), we then obtain

$\mathscr{C}[\mathcal{X}](r,\varphi)=\int_{\mathbb{R}^{2}}\mathcal{X}(\boldsymbol{x})\boldsymbol{\delta}(\boldsymbol{x}\cdot\Theta_{\varphi}-r)d\boldsymbol{x}=P_{r,\varphi}.$
(17)

By reason of (1), we finally have

$\displaystyle\mathscr{C}[\\{X_{\boldsymbol{v}}\\}](r,\varphi)=P_{r,\varphi},$
(18)

where

$\mathscr{C}[\\{X_{\boldsymbol{v}}\\}](r,\varphi)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{\boldsymbol{v}\in\mathbb{V}}X_{\boldsymbol{v}}\left(\int_{\mathbb{R}^{2}}\phi_{\boldsymbol{v},\delta}(\boldsymbol{x})\boldsymbol{\delta}(\boldsymbol{x}\cdot\Theta_{\varphi}-r)d\boldsymbol{x}\right).$
(19)

As a consequence, (13) is equivalent to

$``\mbox{Recover }\\{X_{\boldsymbol{v}}\\}\mbox{
from}~{}\\{\mathscr{C}[\\{X_{\boldsymbol{v}}\\}](r,\varphi)\\}".$ (20)

###  II-C Implicit neural representation for undersampled MRI reconstruction

To solve (20), we define a complex-valued neural network
$\chi^{\boldsymbol{\vartheta}}:\mathbb{V}\rightarrow\mathbb{C}$, where
$\boldsymbol{\vartheta}$ represents a set of learnable parameters. A network
output is given by

$\displaystyle\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\chi^{\boldsymbol{\vartheta}}(\boldsymbol{v})=\mathrm{MLP}^{\boldsymbol{\vartheta}}(\boldsymbol{v},\mathrm{PE}^{L}(\boldsymbol{v})),$
(21)

where $\mathrm{MLP}^{\boldsymbol{\vartheta}}$ is a multi-layer perceptron that
generates two real values in the last layer and then assigns one as a real
part and the other as complex, and $\mathrm{PE}^{L}$ is a positional encoding
[22] given by

$\mathrm{PE}^{L}(\boldsymbol{v})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\begin{bmatrix}\mathrm{PE}^{L}(v_{1}),\mathrm{PE}^{L}(v_{2})\end{bmatrix}.$
(22)

Here, $L$ is some positive constant and $\mathrm{PE}^{L}(v_{i})$ is defined by

$\displaystyle\mathrm{PE}^{L}(v_{i})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}[$
$\displaystyle\cos(2^{0}\pi v_{i}),\sin(2^{0}\pi v_{i}),$
$\displaystyle\cos(2^{1}\pi v_{i}),\sin(2^{1}\pi v_{i}),$
$\displaystyle~{}~{}~{}~{}~{}~{}\vdots~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}\vdots$
$\displaystyle\cos(2^{L-1}\pi v_{i}),\sin(2^{L-1}\pi
v_{i})]\in\mathbb{R}^{2L}.$ (23)

The positional encoding $\mathrm{PE}^{L}$ is a technique that can facilitate
to learn high frequency details of pixel intensity distribution [22].

The network $\chi^{\boldsymbol{\vartheta}}$ is trained such that the rendering
relation (17) is maximized as follows:

$\boldsymbol{\vartheta}=\underset{\boldsymbol{\vartheta}}{\mathrm{argmin}}~{}\sum_{\varphi}\sum_{r}~{}\|P_{r,\varphi}-\mathscr{C}[\\{\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\\}](r,\varphi)\|^{2}_{2},$
(24)

where $\|\cdot\|_{2}$ represents the Euclidean norm.

In practice, for ease of computation and training,
$\mathscr{C}[\\{\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\\}]$ is
computed as follows:

$\displaystyle\mathscr{C}[\\{\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\\}](r,\varphi)$
$\displaystyle=\sum_{\boldsymbol{v}\in\mathbb{V}}\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\left(\int_{\mathbb{R}^{2}}\phi_{\boldsymbol{v},\delta}(\boldsymbol{x})\boldsymbol{\delta}(\boldsymbol{x}\cdot\Theta_{\varphi}-r)d\boldsymbol{x}\right)$
$\displaystyle\approx\sum_{\boldsymbol{v}\in\mathbb{V}}\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\left(\sum_{j}\phi_{\boldsymbol{v},\delta}(\boldsymbol{x}_{j})\Delta_{j}\right)\mbox{with
}{\boldsymbol{x}_{j}\cdot\Theta_{\varphi}=r}$
$\displaystyle=\left(\sum_{j}(\sum_{\boldsymbol{v}\in\mathbb{V}}\chi^{\boldsymbol{\vartheta}}_{\boldsymbol{v}}\phi_{\boldsymbol{v},\delta}(\boldsymbol{x}_{j}))\Delta_{j}\right)$
$\displaystyle\approx\left(\sum_{j}\chi^{\boldsymbol{\vartheta}}(\boldsymbol{v}_{j})\Delta_{j}\right),$
(25)

where $\boldsymbol{x}_{j}$ is a $j$-th position sampled from a line
$\boldsymbol{x}\cdot\Theta_{\varphi}=r$,
$\Delta_{j}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\|\boldsymbol{x}_{j+1}-\boldsymbol{x}_{j}\|_{2}$
is the distance between adjacent samples, and $\boldsymbol{v}_{j}$ is a
relative (real-valued) voxel index corresponding to $\boldsymbol{x}_{j}$. The
last part of (II-C) can be understood as a neural network-based approximation
for the continuous distribution $\mathcal{X}$ [23]. This can reduce the
complexity associated with the kernel function. The reconstruction image is
here given by the absolute value of $\\{\chi_{\boldsymbol{v}}\\}$.

In addition, $P_{r,\varphi}$ is computed as follows: For a fixed $\varphi$,

$P_{r_{j},\varphi}\approx(\boldsymbol{P}_{\varphi})_{j}\mbox{ and
}\boldsymbol{P}_{\varphi}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathscr{F}_{\mathrm{disc}}^{-1}[\boldsymbol{K}_{\varphi}],$
(26)

where $\boldsymbol{P}_{\varphi}=[P_{r_{1},\varphi},P_{r_{2},\varphi},\cdots]$,
$\boldsymbol{K}_{\varphi}=[K_{\omega_{1},\varphi},K_{\omega_{2},\varphi},\cdots]$,
and $\mathscr{F}_{\mathrm{disc}}^{-1}$ is the discrete 1D inverse Fourier
transform. Here, $r_{j}$ and $\omega_{j}$ represent $j$-th samples with
respect to $r$ and $\omega$, respectively. Further details for training and
network architecture are described in Appendix A.

We here mention the reason why the network $\chi^{\boldsymbol{\vartheta}}$ is
designed as complex-valued. In practical MR imaging, various physical and
computational factors like noise, magnetic susceptibility, and numerical error
can generate complex-valued fluctuations on $P_{r,\varphi}$ in (24).

###  II-D Sampling strategy

This subsection examines an effective undersampling strategy for high-quality
implicit neural representation. The reconstruction problem (13) can be
equivalently viewed as, for some $R>1$,

$``\mbox{Estimate
}\\{P_{r,\varphi}\\}_{\varphi\in\Phi^{1}_{\mathrm{uni}}-\Phi^{R}}\mbox{ from
}\\{P_{r,\varphi}\\}_{\varphi\in\Phi^{R}}",$ (27)

where $\Phi^{R}\subset\Phi^{1}_{\mathrm{uni}}$,
$R|\Phi^{R}|\approx|\Phi^{1}_{\mathrm{uni}}|$, and components of $\Phi^{R}$
are assumed to be in ascending order. In this perspective, we attempt to
discuss the following two questions: (i) what choice of $\Phi^{R}$ is
advantageous to high-quality neural representation and (ii) what benefit is of
radial undersampling, compared to others like Cartesian undersampling.

Assuming that $P_{r,\varphi}$ satisfies the Lipschitz property, i.e., for
$\varphi\neq\varphi^{\prime}$,

$\exists~{}M_{r}>0~{}\mbox{such
that}~{}|P_{r,\varphi}-P_{r,\varphi^{\prime}}|\leq
M_{r}|\varphi-\varphi^{\prime}|,$ (28)

the following estimator can be derived: Let $\varphi_{j}$ denote a $j$-th
component of $\Phi^{R}$. For $\varphi\in\Phi^{1}_{\mathrm{uni}}-\Phi^{R}$,

$\displaystyle|P^{\mathrm{esti}}_{r,\varphi}-P_{r,\varphi}|\leq\underset{\varphi\in\Phi^{1}_{\mathrm{uni}}}{\mathrm{max}}|LP_{r,\varphi}-P_{r,\varphi}|=|LP_{r,\varphi_{\star}}-P_{r,\varphi_{\star}}|$
$\displaystyle\leq|LP_{r,\varphi_{\star}}-LP_{r,\varphi_{j_{\star}}}|+|P_{r,\varphi_{\star}}-P_{r,\varphi_{j_{\star}}}|\leq
C_{r}|\varphi_{\star}-\varphi_{j_{\star}}|$ $\displaystyle\leq
C_{r}\underset{j\in\\{1,\cdots,|\Phi^{R}|-1\\}}{\mathrm{max}}~{}|\varphi_{j}-\varphi_{j+1}|\mbox{
for some }C_{r}>0,$ (29)

where $j_{\star}$ is an index such that $\varphi_{j_{\star}}\in\Phi^{R}$ is
the nearest point to $\varphi_{\star}\in\Phi^{1}_{\mathrm{uni}}$. Here,
$P^{\mathrm{esti}}_{r,\varphi}$ is some reasonable approximation for
$P_{r,\varphi}$ and $LP_{r,\varphi}$ is a linear interpolation using
$\\{(\varphi,P_{r,\varphi})\\}_{\varphi\in\Phi^{R}}$. In consequence, we
obtain

$\Phi^{R}_{\mathrm{uni}}=\underset{\Phi^{R}}{\mathrm{argmin}}\underset{j\in\\{1,\cdots,|\Phi^{R}|-1\\}}{\mathrm{max}}~{}|\varphi_{j}-\varphi_{j+1}|.$
(30)

This explains that undersampling designs with the maximum length between
samples as small as possible can be effective. The uniform undersampling
$\Phi^{R}_{\mathrm{uni}}$ is one of good options.

In turn, we discuss a benefit of radial sampling. If $\mathcal{X}\in
L^{2}(\mathbb{R}^{2})$, $\mathscr{F}_{\boldsymbol{x}}\mathcal{X}$ lies on
$L^{2}(\mathbb{R}^{2})$ in the sense of distribution [6]. Here,
$L^{2}(\mathbb{R}^{2})$ denotes the Lebesgue space defined by

$L^{2}(\mathbb{R}^{2})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{~{}f:\mathbb{R}^{2}\rightarrow\mathbb{C}~{}|~{}\|f\|_{L^{2}}<\infty~{}\\},$
(31)

where

$\|f\|_{L^{2}}^{2}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\int_{\mathbb{R}^{2}}|f(\boldsymbol{x})|^{2}d\boldsymbol{x}.$
(32)

On the other hand, if $\mathcal{X}\in L^{2}(\mathbb{R}^{2})$, $P_{r,\varphi}$
lies on $H^{t}([0,\pi)\times\mathbb{R})$ for any $0<t\leq\frac{1}{2}$ [24, 25]
and satisfies (28). Here, $H^{t}([0,\pi)\times\mathbb{R})$ denotes the
fractional Sobolev space defined by

$H^{t}([0,\pi)\times\mathbb{R})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{~{}p:[0,\pi)\times\mathbb{R}\rightarrow\mathbb{C}~{}|~{}\|p\|_{H^{t}}<\infty~{}\\},$
(33)

where

$\displaystyle\|p\|_{H^{t}}^{2}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}$
$\displaystyle\int_{0}^{\pi}\int_{\mathbb{R}}|p(r,\varphi)|^{2}drd\varphi$
$\displaystyle+\int_{0}^{\pi}\iint_{\mathbb{R}\times\mathbb{R}}\dfrac{|p(r,\varphi)-p(r^{\prime},\varphi)|^{2}}{|r-r^{\prime}|^{1+2t}}dr^{\prime}drd\varphi.$
(34)

Accordingly, $P_{r,\varphi}$ possesses a higher regularity (smoothness) than
$\mathscr{F}_{\boldsymbol{x}}\mathcal{X}$; therefore, the estimation (27) with
the high regularity may be more beneficial than general $k$-space estimation
with Cartesian or other sampling strategies [26].

Experimental examination would be given in Section III-D.

(a) Reference (b) IFFT (c) CS (d) SL (e) INK (f) Ours

Figure 2: Comparison study: Reconstruction results in four different scans by using IFFT, CS, SL, INK, and our method. TABLE I: Quantitative evaluation results for comparison study. Metric | IFFT | CS | SL | INK | Ours
---|---|---|---|---|---
SSIM($\uparrow$) | 0.573$\pm$0.069 | 0.521$\pm$0.046 | 0.823$\pm$0.026 | 0.798$\pm$0.052 | 0.904$\pm$0.030
PSNR($\uparrow$) | 28.41$\pm$1.395 | 25.59$\pm$1.034 | 27.92$\pm$2.811 | 28.74$\pm$2.667 | 30.16$\pm$2.966

###  II-E Generalization to 3D MR imaging

This subsection generalizes our reconstruction framework to 3D MR imaging. The
following observation provides a base for 3D generalization.

######  Observation.

Let $\mathcal{X}:\mathbb{R}^{3}\rightarrow\mathbb{R}$ be a 3D distribution of
nuclear spin density. For $w\in\mathbb{R}$, $\varphi\in[0,\pi)$, and
$\zeta\in[0,\frac{\pi}{2}]$, we assume that measured 3D $k$-space data is
given by

$K_{\omega,\varphi,\zeta}=\mathscr{F}_{\boldsymbol{x}}[\mathcal{X}](\omega\Theta_{\varphi,\zeta}),$
(35)

where
$\Theta_{\varphi,\zeta}=\begin{bmatrix}\cos\varphi\cos\zeta,\sin\varphi\cos\zeta,\sin\zeta\end{bmatrix}$.
We define $P$ by

$P_{r,\varphi,\zeta}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\int_{\mathbb{R}}\mathcal{X}(\boldsymbol{x})\delta(\boldsymbol{x}\cdot\Theta_{\varphi,\zeta}-r)d\boldsymbol{x}.$
(36)

The following relation then holds:

$\mathscr{F}_{\omega}^{-1}[K_{\omega,\varphi,\zeta}](r)=P_{r,\varphi,\zeta}.$
(37)

The proof can be completed via similar arguments in (II-B) (3D Fourier slice
theorem). See [27, 28] for more details. The observation above provides a
rendering relationship for 3D imaging equivalent to (17). As discussed in
Section II-C, an implicit neural representation can be then learned through
minimizing the 3D rendering relation over undersampled data with respect to
either $\varphi$ or $\zeta$, or both. Alternatively, the axial extension of
the 2D imaging framework can be a simple way. It might be related to
practically feasible MR pulse sequence designs.

##  III Result

###  III-A Experimental set-up

In order to evaluate the proposed method, extensive experiments were designed
and conducted using a public fastMRI dataset [16] for brain and knee. We
obtained 3D $k$-space data and MR volume pairs, whose 2D slice dimension is
given by $320\times 320$. We simulated the radial undersampling acquisition
(4) through non-uniform fast Fourier transformation (NUFFT) to a zero-padded
image to the size of $452\times 452$.

Throughout this section, we followed the convention [29] for the number of
fully sampled angles (spokes), given by

$N_{\varphi}^{\mathrm{full}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\lfloor\frac{\pi}{2}\times
320\right\rfloor=502,$ (38)

and for the number of radial directional samples, given by

$N_{\omega}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\lfloor\sqrt{2}\times
320\right\rfloor=452.$ (39)

For some radial undersampling with $N_{\varphi}<N_{\varphi}^{\mathrm{full}}$,
we define the acceleration factor $R$ by

$R\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\dfrac{N_{\varphi}^{\mathrm{full}}}{N_{\varphi}}>1.$
(40)

For example, $R=10$ means that $N_{\varphi}=50$ and $N_{\omega}=452$, i.e.,
approximate 10 percent spokes were used for reconstruction in terms of
$N_{\varphi}^{\mathrm{full}}$.

For experiments in Section III-B and III-C, the angles of $N_{\varphi}$ spokes
are determined by the golden-angle sampling (47) with $R=8$ (62 spokes). In
Section III-D, we introduce and describe five different sampling schemes
including the golden-angle and provide their comparison.

As a quantitative metric, we computed the structural similarity index map
(SSIM) and peak signal-to-noise ratio (PSNR), which are defined as follows:
For two images $\\{\mathcal{X}_{\boldsymbol{v}}\\}$ and
$\\{\mathcal{Y}_{\boldsymbol{v}}\\}$,

$\mathrm{SSIM}(\mathcal{X},\mathcal{Y})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\dfrac{(2m_{\mathcal{X}}m_{\mathcal{Y}}+c_{1})(2\sigma_{\mathcal{X}\mathcal{Y}}+c_{2})}{(m_{\mathcal{X}}^{2}+m_{\mathcal{Y}}^{2}+c_{1})(\sigma_{\mathcal{X}}^{2}+\sigma_{\mathcal{Y}}^{2}+c_{2})},$
(41)

and

$\mathrm{PSNR}(\mathcal{X},\mathcal{Y})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}10\log_{10}\dfrac{\underset{\boldsymbol{v}\in\mathcal{V}}{\mathrm{max}}~{}\mathcal{X}_{\boldsymbol{v}}^{2}}{\displaystyle\sum_{\boldsymbol{v}\in\mathcal{V}}(\mathcal{X}_{\boldsymbol{v}}-\mathcal{Y}_{\boldsymbol{v}})^{2}},$
(42)

where $m_{\mathcal{X}}$ and $\sigma_{\mathcal{X}}^{2}$ represent the mean and
variance of $\\{\mathcal{X}_{\boldsymbol{v}}\\}$,
$\sigma_{\mathcal{X}\mathcal{Y}}$ denotes the covariance between
$\\{\mathcal{X}_{\boldsymbol{v}}\\}$ and $\\{\mathcal{Y}_{\boldsymbol{v}}\\}$,
and $c_{1}$ and $c_{2}$ are positive constants. In this work, all quantitative
values were evaluated by taking an average over 30 test data from non-
overlapped subjects.

All experiments were conducted in a computer system with two Intel Xeon CPUs
E5-2630 v4 and four NVIDIA GeForce GTX 3080ti GPUs.

###  III-B Comparison study

This subsection validates the effectiveness of the proposed method and
exhibits qualitative and quantitative comparison results to other
reconstruction approaches such as inverse fast Fourier transform with zero
filling (IFFT), compressed sensing with total variation penalty (CS),
supervised learning using TransUNet [30] (SL) and $k$-space interpolation
using neural representation [31] (INK).

In Fig. 2 and Table I, qualitative and quantitative comparisons are provided.
The proposed method demonstrated the superior performance rather than IFFT,
CS, and INK, while SL was comparable. As far as we have implemented, the
neural representation in $k$-space appears to be less effective in terms of
accuracy and stability. The intensity distribution over pixels in the image
domain is strongly correlated even simply in a neighborhood region. However,
it is complicatedly entangled in the $k$-space domain and fairly irrelevant in
a local region, which might cause the increased training complexity, learning
instability, performance degradation, and etc. Compared to SL, the proposed
reconstruction tends to keep anatomical details. It appears to be a
consequence of the high adaptiveness to the given data. We further elaborate
and discuss this in Section IV.

Here, CS was implemented using the open-source package, named as SigPy [32].
We trained TransUNet using supervised learning with 532 paired data, whose
input is an aliased MR image and label is the corresponding image
reconstructed from full sampling. For INK, a multi-layer perceptron (21) was
trained, which inputs a image coordinate and outputs a two dimensional vector
representing a complex-valued $k$-space intensity. The network was trained by
using observed $k$-space data and then used to interpolate unsampled values in
$k$-space.

It should be mentioned that the comparison with SL is not fair. We note that
the reconstruction approaches other than SL do not require any training data.
The use of the larger number of training data can further improve the
performance of SL and even can lead it to outperform the proposed method. Most
importantly, however, our method is based only on single $k$-space data.

Figure 3: Qualitative and quantitative results for acceleration factor
analysis. The top figure shows reconstruction results for IFFT and the
proposed method when $R=4,8,12$. Three graphs below present SSIM, PSNR, and
training time results.

(a) Reference (b) Uniform (c) Limited (d) Random (e) Stratified (f) Golden-
angle

Figure 4: Qualitative results for sampling study. The top figures in (b)-(f) present radial undersampled $k$-space data for sampling methods. The middle figures show the corresponding reconstruction results by the proposed method. The bottom figures are zoomed images. TABLE II: Quantitative evaluation results for sampling study. Metric | Uniform | Limited | Random | Stratified | Golden-angle
---|---|---|---|---|---
SSIM($\uparrow$) | 0.892$\pm$0.028 | 0.667$\pm$0.084 | 0.875$\pm$0.028 | 0.881$\pm$0.043 | 0.904$\pm$0.030
PSNR($\uparrow$) | 28.93$\pm$3.459 | 21.73$\pm$1.967 | 27.49$\pm$2.835 | 29.05$\pm$2.971 | 30.16$\pm$2.966

###  III-C Acceleration factor analysis

This subsection examines the effectiveness of the number of spokes to the
reconstruction accuracy of the proposed method. Qualitative and quantitative
evaluation results are provided in Fig. 3. In this experiment, the golden-
angle sampling (47) was used.

We observed reconstruction results by gradually increasing $R$ from 2 to 12.
As $R$ is increased (i.e., $N_{\varphi}$ is decreased), the reconstruction
ability becomes, of course, weakened. However, the results were constantly
superior than those from IFFT in both qualitative and quantitative
perspectives. Notably, even for a high acceleration factor of $R=12$ (41
spokes), the proposed method provided a high-quality MR image. Meanwhile, $R$
is associated with the training time of $\text{MLP}^{\boldsymbol{\vartheta}}$
in (21), since acquired samples are inputted for training. In other words, the
larger $R$, the shorter the training time, as shown in the bottom graph of
Fig. 3. Note that the inference time is independent to $R$, only affected by
the image resolution to be reconstructed. In our case, the inference time was
0.61 seconds approximately for any $R$.

When taking account of the compromise among the reconstruction quality,
training time, and scan acceleration, $R=8$ (62 spokes) may be regarded as an
empirically optimal choice.

###  III-D Sampling study

This subsection investigates the dependency of the proposed method on
undersampling strategies. We here examined the reconstruction performance by
varying undersampling schemes. In a fixed acceleration factor of $R=8$
($N_{\varphi}=62$), uniform, limited, random, stratified, and golden-angle
radial sampling were compared.

The uniform and limited are to uniformly sample 62 spokes as follows:

$\displaystyle\Phi_{\text{uni}}^{R}$
$\displaystyle=\left\\{0,\frac{\pi}{N_{\varphi}},\cdots,\frac{(N_{\varphi}-1)\pi}{N_{\varphi}}\right\\}\subset[0,\pi),$
(43) $\displaystyle\Phi_{\text{lim}}^{R}$
$\displaystyle=\left\\{0,\frac{\pi}{2N_{\varphi}},\cdots,\frac{(N_{\varphi}-1)\pi}{2N_{\varphi}}\right\\}\subset[0,\pi/2).$
(44)

We remark that the limited chooses angles within the restricted interval
$[0,\frac{\pi}{2})$. The random selects 62 spokes by randomly picking angles
in $[0,\pi)$, which can be expressed as

$\Phi_{\text{rand}}^{R}=\\{r_{1},r_{2}\cdots,r_{N_{\varphi}}\\}\subset[0,\pi).$
(45)

The stratified is similar to the uniform sampling, whereas there is
perturbation in sampling intervals. It can be expressed as

$\Phi_{\text{str}}^{R}=\left\\{\frac{r_{1}}{N_{\varphi}},\frac{\pi+r_{2}}{N_{\varphi}},\cdots,\frac{(N_{\varphi}-1)\pi+r_{N_{\varphi}}}{N_{\varphi}}\right\\}\subset[0,\pi),$
(46)

where each $r_{i}$ is randomly picked number in $[0,\pi)$ as in (45). The
golden-angle [33] is given by

$\Phi_{\text{gold}}^{R}=\left\\{\text{mod}\left(\frac{(n-1)\pi}{\mathrm{GR}},\pi\right):n=1,\cdots,N_{\varphi}\right\\}\subset[0,\pi),$
(47)

where $\mathrm{GR}$ stands for the golden ratio given by
$\frac{1+\sqrt{5}}{2}$.

Fig. 4 and Table II demonstrates qualitative and quantitative comparison
results. The golden-angle was the empirical best, complying with reports in
conventional radial undersampling MRI [34]. The stratified and uniform were
comparable, but less effective than the golden-angle sampling. The limited has
the densest spokes between 0 and $\frac{\pi}{2}$, but its result exhibited the
lowest performance. The random further improves the reconstruction quality,
but it does not still surpass the performance of the uniform, stratified, or
golden-angle. This observation agrees with the discussion in Section II-D. We
note that angle densities in the golden-angle, stratified, and uniform
sampling are more even than in the remaining.

##  IV Conclusion and Discussion

This paper seeks to pave a new data-driven imaging way for undersampled MRI
reconstruction that can potentially provide an effective solution in clinical
applications in which gathering data is restrictive in spite of large
variability of target images or tracing anatomical information highly lying on
scan or subject is critical. To achieve this, the proposed approach harnesses
the power of NeRF that can accurately infer image rendering in an arbitrary
view even from sparse-view rendering information. We demonstrated, with radial
sampling, the undersampled MR imaging is equivalent to an image modeling from
sparse-view rendered data and, therefore, attempted to deal with it by
borrowing the idea of NeRF: rendering relation-induced implicit neural
representation. One remark is that the representation is, in NeRF, for image
rendering implicitly combined with an underlying image model, whereas, in
ours, for the direct realization of the model. Numerous experiments validated
the feasibility and capability of the proposed method successfully.

We should mention an approach proposed in [35] that does not follow the
concept of NeRF but shares the utilization of implicit neural representation.
They used a subject-specific priori embedding come from a scan record history,
whereas the proposed method is not subjected to the embedding. Since ours
follows the formulation of NeRF, in addition, efficient strategies developed
along with the advance in the NeRF field can be simply incorporated.

(a) Reference (b) IFFT (c) SL (d) Ours

Figure 5: Results for a case of involved anomalies and high acceleration
factor of $R=12$ (41 spokes).

Fig. 5 highlights the advantage of the proposed method. It is an extreme case
of involved anomalies and high acceleration factor of $R=12$ (41 spokes). Just
only with single radially undersampled $k$-space data, we can obtain a highly
improved image in (d), compared to that of the standard IFFT reconstruction
with zero-filling in (b). In addition, it is competitive to an image obtained
by the TransUnet with supervised leaning of 532 paired data in (c). The
proposed approach seems to better capture small variations and uncommon
patterns (red boxes) and tends to less produce fake structures (yellow arrows)
that do not exist in the reference MR image in (a). It appears to be because
the proposed method finds a highly adaptive solution to the given measurement,
in contrast, SL attempts to find common patterns over training data and
provides an output in some sense of nonlinear averaging [10], possibly
resulting in the generation of fake structures and the blurring or distortion
of anatomical structures.

(a) reference (b) $[0,\pi)$ (c) $\left[0,\frac{\pi}{2}\right)$ (d)
$\left[\frac{\pi}{4},\frac{\pi}{2}\right)$

Figure 6: Results obtained from uniform undersampling in different intervals:
(b) $[0,\pi)$, (c) $\left[0,\frac{\pi}{2}\right)$, and (d)
$\left[\frac{\pi}{4},\frac{\pi}{2}\right)$. The acceleration factor was $R=12$
(41 spokes).

However, this characteristic can be a double-edged sword. When compared to
existing data-driven approaches like SL, our method can be less powerful to
produce a plausible (MR image-like) output, as measurement data is less
informative in terms of target MR image. Fig. 6 shows this limitation. Under a
fixed and high acceleration factor of $R=12$ (41 spokes), three reconstruction
results were compared, which were obtained by the proposed method with uniform
undersampling in the full angle interval ($[0,\pi)$ in (b)) and limited
intervals ($[0,\frac{\pi}{2})$ in (c) and $[\frac{\pi}{4},\frac{\pi}{2})$ in
(d)). As the interval range is more limited, that is, measurement data is less
informative, the reconstructed image becomes worse. In contrast, SL can
provide a more realistic output as the number of training data and the
learning capacity become larger, namely, the better realization on an
underlying data distribution over MR images is allowed. The other difficulty
is the involved optimization per each reconstruction task. We expect that
these aspects might be improved by appropriately integrating both
reconstruction ways so that playing complementary roles. This is in our future
research direction.

##  Appendix A Network and Training Details

This appendix provides network and training details. For the implementation of
the proposed method, we set the parameter $L$ in the positional encoding in
(22) as $L=20$ and used a multi-layer perceptron described in Table III. For
training, we used learning strategies in the vanilla NeRF [13] and Adam
optimizer. The number of steps was constantly set as 500. Fig. 7 shows a
convergence history for one case.

Figure 7: Convergence history of the proposed method when the golden-angle subsampling with $R$=8 (62 spokes) was used. TABLE III: Network architectures Layer | Input Dim | Output Dim | Activation
---|---|---|---
Input | 82(=2+20$\times$4) | -
Linear | 82 | 256 | sin
Linear | 256 | 256 | sin
Linear | 256 | 256 | sin
Linear | 256 | 256 | sin
Concat | 82 | 338 | -
Linear | 338 | 256 | sin
Linear | 256 | 256 | sin
Linear | 256 | 256 | sin
Linear | 256 | 2 | -
Output | 2 | -

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