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Learning Green’s Function Efficiently Using Low-Rank Approximations

Kishan Wimalawarne 1  Taiji Suzuki 1 2  Sophie Langer 3

††footnotetext: *Equal contribution 1Department of Mathematical Informatics,
The University of Tokyo, Tokyo, Japan 2Center for Advanced Intelligence
Project (AIP), RIKEN, Tokyo, Japan 3Faculty of Electrical Engineering,
Mathematics, and Computer Science, University of Twente, Enschede, The
Netherlands. Correspondence to: Kishan Wimalawarne<EMAIL_ADDRESS>
Accepted after peer-review at the 1st workshop on Synergy of Scientific and
Machine Learning Modeling, SynS & ML ICML, Honolulu, Hawaii, USA. July, 2023.
Copyright 2023 by the author(s).

###### Abstract

Learning the Green’s function using deep learning models enables to solve
different classes of partial differential equations. A practical limitation of
using deep learning for the Green’s function is the repeated computationally
expensive Monte-Carlo integral approximations. We propose to learn the Green’s
function by low-rank decomposition, which results in a novel architecture to
remove redundant computations by separate learning with domain data for
evaluation and Monte-Carlo samples for integral approximation. Using
experiments we show that the proposed method improves computational time
compared to MOD-Net while achieving comparable accuracy compared to both PINNs
and MOD-Net.

##  1 Introduction

The Green’s function is a well-known method to solve and analyze Partial
Differential Equations (PDE) Evans (2010); Bebendorf & Hackbusch (2003).
Recently, deep learning models have been applied to learn the Green’s function
enabling to obtain solutions for both linear and nonlinear PDEs Luo et al.
(2022) as well as to learn PDEs in irregular domains Teng et al. (2022). Luo
et al. (2022) further demonstrated that deep learning models can parameterize
a class of PDEs compared to learning a specific PDE.

The MOD-Net Luo et al. (2022) uses the Green’s function approximation by using
a neural network and Monte-Carlo integration to parameterize solutions for
PDEs. A limitation of the MOD-Net is that the approximation of the Green’s
function by Monte-Carlo integration leads to high computational costs due to
their repeated evaluations for each domain element. In this research, we
propose to extend MOD-Net with low-rank decomposition of the Green’s function.
The proposed model results in a two network architecture, where one network
learns on the domain elements to be evaluated and other network learns over
all the Monte-Carlo samples, hence, avoid redundant repeated computations of
Green’s function at each domain element. Using experiments with the Poisson 2D
equation and the linear reaction-diffusion equation we show that our proposed
method is computationally feasible compared to MOD-Net, while achieving
comparable accuracy compared to PINNs Raissi et al. (2017) and MOD-Net.
Additionally, we show that our proposed method has the ability to interpolate
within the solution space as a neural operator similar to MOD-Net.

##  2 Learning Green’s Function by MOD-Net

Let us consider a domain $\Omega\subset\mathbb{R}^{d}$, a differential
operator $\mathcal{L}$, a source function $g(\cdot)$ and a boundary condition
$\phi(\cdot)$, then a partial differential equation can be represented as,

$\displaystyle\mathcal{L}[u](x)=g(x),\qquad$ $\displaystyle x\in\Omega$
$\displaystyle u(x)=\phi(x),\qquad$ $\displaystyle x\in\partial\Omega.$

For a linear PDE with the Dirichlet boundary condition $\phi(\cdot)=0$, we can
find a Green’s function
$G:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ for a fixed
$x^{\prime}\in\Omega$ as follows:

$\displaystyle\mathcal{L}[G](x)$ $\displaystyle=\delta(x-x^{\prime}),\qquad
x\in\Omega$ $\displaystyle G(x,x^{\prime})$
$\displaystyle=0,\qquad\qquad\quad\;x\in\partial\Omega,$

which leads to a solution function $u(\cdot)$ as

$u(x)=\int_{\Omega}G(x,x^{\prime})g(x^{\prime})dx^{\prime}.$ (1)

Recently developed MOD-Net Luo et al. (2022) proposes to learn the Green’s
function by using a neural network. It uses a neural network
$G_{\theta_{1}}(x,x^{\prime})$ with parameters denoted by $\theta_{1}$ to
replace the operator $G(x,x^{\prime})$ of (1). Given a set $S_{\Omega}$
consisting of random samples from $\Omega$, the Monte-Carlo approximation of
the Green’s function integral (1) results in the following:

$u_{\theta_{1}}(x;g)=\frac{|\Omega|}{|S_{\Omega}|}\sum_{x^{\prime}\in
S_{\Omega}}G_{\theta_{1}}(x,x^{\prime})g(x^{\prime}).$ (2)

MOD-Net Luo et al. (2022) also proposes of a nonlinear extension of the above
as

$u_{\theta_{1},\theta_{2}}(x;g)=F_{\theta_{2}}\left(\frac{|\Omega|}{|S_{\Omega}|}\sum_{x^{\prime}\in
S_{\Omega}}G_{\theta_{1}}(x,x^{\prime})g(x^{\prime})\right),$ (3)

where $F_{\theta_{2}}$ is a neural network ($\theta_{2}$ representing
parameters).

It has been shown (Luo et al., 2022) that Green’s function can learn as an
operator for varying $g(\cdot)$ to parameterize a class of PDE. Following (Luo
et al., 2022), let us consider $K$ different parametarizations of a PDE class
by specifying $g^{k}(\cdot)$ for $k=1,\cdots,K$. Let $S^{\Omega,k}$ and
$S^{\partial\Omega,k}$ are domain elements from the interior and boundary,
respectively, for each $k=1,\ldots,K$. Then the objective function for MOD-Net
for the formulation in (2) is given as

$R_{S}=\\\
\frac{1}{K}\sum_{k\in[K]}\Bigg{(}\lambda_{1}\frac{1}{|S^{\Omega,k}|}\sum_{x\in
S^{\Omega,k}}\|\mathcal{L}[u_{\theta_{1}}(x;g^{k})](x)-g^{k}(x)\|_{2}^{2}\\\
+\lambda_{2}\frac{1}{|S^{\partial\Omega,k}|}\sum_{x\in
S^{\partial\Omega,k}}\|u_{\theta_{1}}(x;g^{k})\|_{2}^{2}\Bigg{)},$ (4)

where $\lambda_{1}$ and $\lambda_{2}$ are regularization parameters. Notice
that when $K=1$ the above objective function learns a specific instance of a
PDE.

A major limitation with MOD-Net is the computational bottleneck due to the
Monte-Carlo approximation of the integrals (2) and (3). The repeated summation
by elements of $S_{\Omega}$ with respect to each domain element in (4) can
become both computationally costly and redundant.

##  3 Proposed Method

We propose to extend MOD-Net with low-rank presentation of the Green’s
function to remove redundant computations and improve computational
feasibility.

First, we put forward the following approximation for low-rank decomposition
of the Green’s function from Bebendorf & Hackbusch (2003). For any
$0<\epsilon<1$ sufficiently small and $R\geq
c^{d}\lceil\log(\epsilon^{-1})\rceil^{d}+\lceil\log(\epsilon^{-1})\rceil$, and
$D_{1},D_{2}\subset\Omega$, Bebendorf & Hackbusch (2003) have given
decomposition with functions $v_{i}(\cdot)$ and $w_{i}(\cdot)$, $i=1,\ldots,R$
as

$G_{R}(x,y)=\sum_{i=1}^{R}v_{i}(x)w_{i}(y),\;\;x\in D_{1},\;y\in D_{2},$ (5)

such that

$\|G(x,\cdot)-G_{R}(x,\cdot)\|_{L^{2}(D_{1})}\leq\epsilon\|G(x,\cdot)\|_{L^{2}(\hat{D}_{1})},$
(6)

where $\hat{D}_{1}\subset\Omega$ a set slightly large than $D_{1}$.

We now apply the above low-rank decomposition (5) of the Green’s function to
(2). We propose to learn functions $u_{i}(x)$ and $v_{i}(x),\;i=1,\ldots,R$
using two neural networks
$F_{\gamma_{1}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{R}$ and
$H_{\gamma_{2}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{R}$ where $\gamma_{1}$
and $\gamma_{2}$ represent parameters. In practice, $R$ can be problem and
data dependent, hence, may require to be considered as a hyperparameter.
Further, we assume that $D_{1}=D_{2}=\hat{D}_{1}=\Omega$ for (5) and (6) and
there exist some neural network that can learn a low-rank representation.

Next, we apply the learning of low-rank decomposition to (2) which leads to
the following expansion

$\displaystyle u_{\gamma_{1},\gamma_{2}}(x;g)$
$\displaystyle=\frac{|\Omega|}{|S_{\Omega}|}\sum_{y\in S_{\Omega}}G(x,y)g(y)$
$\displaystyle\approx\frac{|\Omega|}{|S_{\Omega}|}\sum_{y\in
S_{\Omega}}\sum_{i=1}^{R}F_{\gamma_{1}}(x)_{i}H_{\gamma_{2}}(y)_{i}g(y)$
$\displaystyle=\frac{|\Omega|}{|S_{\Omega}|}\sum_{i=1}^{R}F_{\gamma_{1}}(x)_{i}\sum_{y\in
S_{\Omega}}H_{\gamma_{2}}(y)_{i}g(y)$
$\displaystyle=\frac{|\Omega|}{|S_{\Omega}|}F_{\gamma_{1}}(x)^{\top}\left[\sum_{y\in
S_{\Omega}}H_{\gamma_{2}}(y)g(y)\right].$ (7)

The last step of (7) shows the separation of the learning with Monte-Carlo
samples from the set $S_{\Omega}$ by $H_{\gamma_{2}}(\cdot)$ and learning with
each $x$ by $F_{\gamma_{1}}(\cdot)$. This separated learning helps to avoid
the computationally costly repeated evaluations in (2) and (3) since we only
need to compute the network $H_{\gamma_{2}}(\cdot)$ once for all input $x$ at
each learning iteration.

(a) Exact solution

(b) Predicted solution

(c) Error

Figure 1: Interpolation results for Poisson 2D problem by DecGreenNet-NL for the parameter $a=15$ (a) Exact solution, (b) Predicted solution by DecGreenNet-NL and (c) Error Method | Network structure | P | Loss | Time(sec)
---|---|---|---|---
PINNs | [2, 64, 64, 1] | - | 3.5e-4 | 80.14
DecGreenNet |  $F_{\gamma_{1}}$ = [2,512, 512, 512, 512, 50],$H_{\gamma_{2}}$ = [2, 16, 16, 16, 1] | 100 | 2.86e-4 | 117.95
DecGreenNet-NL |  $F_{\gamma_{1}}$ = [2,64,64,64,64,64,50],$H_{\gamma_{2}}$ = [2,64,64,64,50],$O_{\gamma_{3}}$ = [100,1] | 100 | 1.5e-3 | 550.45
MOD-Net | [4, 128, 128, 128, 128, 1] | 10 | 1.05e-3 | 721330
Table 1: Network structures, number of random samples, test loss, and
computational times for a single instance learning of Poisson 2D equation for
$a=15$ using PINNs, DecGreenNet, DecGreenNet-NL, and MOD-Net

Using the construction in (7) we propose two neural network architectures to
improve over (2) and (3). For simplicity, we omit the factor
$|\Omega|/|S_{\Omega}|$ due to its redundancy in the learning process.
Further, we assume that elements $S_{\Omega}$ are sampled once and fixed
during the learning and prediction processes.

We put forward DecGreenNet as a direct construction from (7) with
$|S_{\Omega}|=P$ as

$u_{\gamma_{1},\gamma_{2}}(x;g)=F_{\gamma_{1}}(x)^{\top}\sum_{i=1}^{P}H_{\gamma_{2}}(y_{i})g(y_{i}).$
(8)

Next, we construct a nonlinear extension of MOD-Net (3) with the low-rank
decomposition following (7). Here we remove the summation on the right and
arrange its elements as a concatenation. By introducing an additional neural
network $O_{\gamma_{3}}:\mathbb{R}^{P}\rightarrow\mathbb{R}$, we propose
DecGreenNet-NL as

$u_{\gamma_{1},\gamma_{2},\gamma_{3}}(x;g)=O_{\gamma_{3}}\bigg{(}F_{\gamma_{1}}(x)^{\top}\mathrm{concat}\big{[}H_{\gamma_{2}}(y_{1})g(y_{1}),\\\
H_{\gamma_{2}}(y_{2})g(y_{2}),\ldots,H_{\gamma_{2}}(y_{P})g(y_{P})\big{]}\bigg{)},$
(9)

where $\mathrm{concat}(\cdots)$ is an operation to concatenation of input
vectors. Note that when $O_{\gamma_{3}}$ is replaces by a vector of ones
($O_{\gamma_{3}}=[1,1,\ldots,1]\in\mathbb{R}^{P}$), (9) becomes equivalent to
(8).

Optimal neural architectures (layers and hidden units) of $F_{\gamma_{1}}$,
$H_{\gamma_{1}}$, and $O_{\gamma_{3}}$ need to be discovered by hyperparameter
tuning. Additionally, $R$ and $P$ should be considered as hyperparameters.
Furthermore, the activation functions for neural networks require higher-order
differential capacity in relation to the differential operators in the PDE. In
general, we use the activation function $\mathrm{ReLUK}(x):=\max\\{0,x\\}^{K}$
where $K\in\mathbb{N}_{+}$.

(a) Exact solution

(b) Predicted solution

(c) Error

Figure 2: Comparison between the exact solution and predicted solution by DecGreenNet of the reaction-diffusion equation (a) Exact solution, (b) Predicted solution by DecGreenNet and (c) Error. Method | Network structure | P | Loss | Time (sec)
---|---|---|---|---
PINNs | [2,128, 128, 128, 128, 128,1] | - | 2.6e-4 | 99.48
DecGreenNet |  $F_{\gamma_{1}}$ = [2,512, 512, 512, 512, 512,50], $H_{\gamma_{2}}$ = [2,32,32,32,50] | 300 | 7.76e-5 | 355.25
DecGreenNet-NL |  $F_{\gamma_{1}}$ = [2, 256, 256, 256, 256, 256, 50], $H_{\gamma_{2}}$ = [2, 64, 64, 64, 64, 50] | 300 | 2.70e-4 | 1075.24
|  $O_{\gamma_{3}}$ = [300,1] |  |  |
MOD-Net | [4,128,128,128,1] | 10 | 1.26e-4 | 23696
Table 2: Network structures, number of random samples, test loss, and
computational times for the linear reaction-diffusion equation using PINNs,
DecGreenNet, DecGreenNet-NL, and MOD-Net

##  4 Experiments

We experimented with PDEs used in Luo et al. (2022) and Teng et al. (2022) to
evaluate our proposed models.

###  4.1 Experimental Setup

In all our experiments we set $\lambda_{1}=\lambda_{2}=1$ in the objective
function (4) for all models. For both DecGreenNet and DecGreenNet-NL selected
the layers and hidden units of each neural network by hyperparameter tuning.
We selected layers from $1,\ldots,6$ and hidden units from
$2^{h}\;h=3,\ldots,6$ for both $F_{\gamma_{1}}(\cdot)$ and
$H_{\gamma_{2}}(\cdot)$ of (8) and (9). For $O_{\gamma_{3}}(\cdot)$ we
selected from hidden layers $0,1,2$ and hidden units $4,8,16$. We represent
the network structure of network of models by notation
$[{in},h,\ldots,h,{out}]$, ${in}$, ${out}$, and $h$ represent dimensions of
input, output, and hidden layers, respectively. We experimented with
$R\in\\{5,50,100\\}$. We used PINNs as a baseline method by performing
hyperparameter tuning for layers and hidden units varying from $1,\ldots,5$
and $2^{h}\;h=3,\ldots,6$, respectively. We also used MOD-Net as a baseline
method, however, we only used $10$ random samples to approximate the Green’s
function due to high computational cost. We used the activation function
$\mathrm{ReLU3}(\cdot)$ for all models. We used the Pytorch environment with
Adam optimization method with learning rate of $0.001$ and weight decay set to
zero. All experiments were conducted on NVIDIA A100 GPUs with CUDA 11.6 on a
Linux 5.4.0 environment. We provide the code at
https://github.com/kishanwn/DecGreenNet.

###  4.2 Poisson 2D Equation

As our first experiment, we used the Poisson 2D problem with multiple
paramterizations under the same setting used in Luo et al. (2022). The Poisson
2D problem for the the domain of $\Omega=[0,1]^{2}$ is specified as

$\displaystyle-\Delta u(x,y)$ $\displaystyle=g(x,y),\;\;(x,y)\in\Omega,$
$\displaystyle u(x,y)$ $\displaystyle=0,\qquad\quad(x,y)\in\partial\Omega.$

where $g(x,y)=-a(x^{2}-x+y^{2}-y)$. The analytical solution of the above
problem is $u(x,y)=\frac{a}{2}x(x-1)y(y-1)$. In Luo et al. (2022), multiple
parameterizations of the Poisson equation is specified by setting different
values for $a$. Following a similar setting as in Luo et al. (2022), we used
$a:=a_{k}=10k$ where $k=1,\cdots,10$ which lead to
$g(x,y):=g^{k}(x,y)=-a_{k}(x^{2}-x+y^{2}-y),\;k=1\dots,10$.

We found that DecGreenNet-NL with $F_{\gamma_{1}}=[2,256,256,256,256,256,50]$,
$H_{\gamma_{2}}=[2,64,64,64,64,50]$, and $O_{\gamma_{3}}=[100,1]$ provided the
best solution. From the learned model, we interpolate the solution for the
Poisson 2D equation with $a=15$. The low error of the interpolated solution in
Figure 1 shows the operator learning capability of our method, hence, the
ability to learn parameterization for a class of PDE.

As our second experiment with Poisson 2D, we conducted experiments to evaluate
solutions for a single instance of the Poisson 2D equation with $a=15$ ($K=1$
in (4)). Table 1 shows the details on learning with PINNs, MOD-Net, and single
instance learning by DecGreenNet and DecGreenNet-NL. Both our proposed methods
obtained lower values for test loss compared to PINNs and MOD-Net, in addition
to the significantly small time compared to MOD-Net.

###  4.3 Reaction-Diffusion Equation

We experimented with the linear reaction-diffusion equation Teng et al. (2022)
in the domain $\Omega\in[-1,1]^{2}$ specified as

$\mathcal{L}(u)=-\nabla\cdot((1+2x^{2})\nabla)+(1+y^{2})u$ (10)

The exact solution is $u(x,y)=e^{-(x^{2}+2y^{2}+1)}$. From both Table 2 and
Figure 2, we observe that DecGreenNet obtains a good accuracy for learning the
linear reaction-diffusion function with moderate computational time.

##  5 Broader Impact

We believe that the computational advantages and operator learning capability
of our proposed method would be conducive to solving important problems in
science. A limitation of our method is the considerable amount of
hyperparameters that needed to be tuned to find the optimal neural
architectures. We do not know any negative impacts from our proposed methods.

##  6 Acknowledgement

TS was partially supported by JSPS KAKENHI (20H00576) and JST CREST. KW was
partially supported by JST CREST.

##  7 Conclusion and Future Work

We provide a computationally feasible low-rank model to learning PDEs with
Green’s function. Theoretical analysis such as convergence bounds for our
model is open for future work. Further extensions of our model to solve high-
dimensional PDEs is another future direction.

## References

  * Bebendorf & Hackbusch (2003) Bebendorf, M. and Hackbusch, W.  Existence of $\mathcal{H}$-matrix approximants to the inverse fe-matrix of elliptic operators with $L_{\infty}$-coefficients.  _Numer. Math._ , 2003.
  * Evans (2010) Evans, L. C.  _Partial differential equations_.  American Mathematical Society, 2010.
  * Luo et al. (2022) Luo, Z. L., Yaoyu, T. Z., Weinan, E., Xu, J., Zhi-Qin, and Zheng, M.  Mod-net: A machine learning approach via model-operator-data network for solving pdes.  _CiCP_ , 2022.
  * Raissi et al. (2017) Raissi, M., Perdikaris, P., and Karniadakis, G. E.  Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations, 2017.
  * Teng et al. (2022) Teng, Y., Zhang, X., Wang, Z., and Ju, L.  Learning green’s functions of linear reaction-diffusion equations with application to fast numerical solver.  In _MSML_ , 2022.


11institutetext: Department of Radiology, Washington University School of
Medicine, St. Louis, MO, USA 22institutetext: Institute for Informatics, Data
Science and Biostatistics, Washington University School of Medicine, St.
Louis, MO, USA
22email<EMAIL_ADDRESS>
# Dynamic U-Net: Adaptively Calibrate Features for Abdominal Multi-organ
Segmentation

Jin Yang 11 Daniel S. Marcus 11 Aristeidis Sotiras 1122

###### Abstract

U-Net has been widely used for segmenting abdominal organs, achieving
promising performance. However, when it is used for multi-organ segmentation,
first, it may be limited in exploiting global long-range contextual
information due to the implementation of standard convolutions. Second, the
use of spatial-wise downsampling (e.g., max pooling or strided convolutions)
in the encoding path may lead to the loss of deformable or discriminative
details. Third, features upsampled from the higher level are concatenated with
those that persevered via skip connections. However, repeated downsampling and
upsampling operations lead to misalignments between them and their
concatenation degrades segmentation performance. To address these limitations,
we propose Dynamically Calibrated Convolution (DCC), Dynamically Calibrated
Downsampling (DCD), and Dynamically Calibrated Upsampling (DCU) modules,
respectively. The DCC module can utilize global inter-dependencies between
spatial and channel features to calibrate these features adaptively. The DCD
module enables networks to adaptively preserve deformable or discriminative
features during downsampling. The DCU module can dynamically align and
calibrate upsampled features to eliminate misalignments before concatenations.
We integrated the proposed modules into a standard U-Net, resulting in a new
architecture, termed Dynamic U-Net. This architectural design enables U-Net to
dynamically adjust features for different organs. We evaluated Dynamic U-Net
in two abdominal multi-organ segmentation benchmarks. Dynamic U-Net achieved
statistically improved segmentation accuracy compared with standard U-Net. Our
code is available at https://github.com/sotiraslab/DynamicUNet.

###### Keywords:

Dynamic convolution Deformable convolution Feature Calibration Multi-organ
segmentation.

##  1 Introduction

Segmentation of multiple abdominal organs in medical images plays a crucial
role in supporting clinical workflows, including diagnostic interventions and
treatment planning [4]. Manual segmentation is labor-intensive, motivating the
development of automatic tools. Specifically, U-Net and its variants have been
applied to various automatic medical image segmentation tasks with great
success [13, 15, 22]. However, most of them focused on specific single organs,
such as the liver [8, 10], pancreas [12, 20], and kidney [14]. Multi-organ
segmentation is still challenging due to the morphological complexity of the
structures, the large inter-subject variations, and the large inter-organ
variations in shape and size.

Among various multi-organ segmentation models, nnU-Net is a promising solution
[6]. However, it relies on U-Net as the foundational segmentation
architecture, which may pose three limitations in its architectural design,
potentially hindering its ability to address challenges in multi-organ
segmentation. First, its inherent limitation lies in the inability of
convolution layers to adaptively calibrate features using global contextual
information. Although attention modules can utilize global information [13,
18], these are designed as independent modules to improve the whole network
rather than improving convolution layers. The second limitation originates
from applying downsampling layers (e.g., max-pooling and average-pooling) in
the encoding path [1, 15]. These layers pool features equally by sliding a
local window. However, some features are more discriminative than neighbors
within each window. Thus, the applications of these downsampling layers cannot
preserve discriminative details of organs [3, 17]. Although strided
convolution is used for downsampling in nnU-Net [6] and other multi-organ
segmentation models [4, 21], it uses a fixed-sized convolutional window and
cannot model the deformation of input features. Thus, these models fail to
model variations in organ shape adaptively. Third, almost all multi-organ
segmentation networks fuse upsampled features and features preserved via skip
connections to facilitate information propagation [4, 6, 16, 19, 21]. However,
repeated downsampling and upsampling lead to inaccurate correspondences (i.e.,
feature misalignment and organ mismatch) between them [5, 9]. Thus, the direct
concatenation between them harms feature learning, resulting in the
degradation of segmentation accuracy.

To address the aforementioned limitations, we propose Dynamically Calibrated
Convolution (DCC), Dynamically Calibrated Downsampling (DCD), and Dynamically
Calibrated Upsampling (DCU) modules. The DCC module can model long-range
inter-dependencies and adaptively calibrate spatial-wise and channel-wise
local features by global contextual information. The other two modules are
designed based on (modulated) deformable convolutions [2, 23]. Deformable
convolutions have the capability of modeling the deformation and irregular
shape of objects. The DCD module can handle the deformation and discrimination
of features during downsampling. It dynamically determines the sampling
interval scheme rather than adapting a fixed one in strided convolutions.
Additionally, the DCD module can preserve deformation and discrimination,
since it downsamples feature maps based on a higher information level
adaptively rather than pooling features by a fixed scheme based on prior
knowledge (e.g., max-pooling and average-pooling). Third, the DCU module is
designed to adaptively align and calibrate upsampled features from lower
resolution to their reference features from skip connections. It is achieved
by adjusting each sampling location in a deformable convolutional kernel with
an offset dynamically learned from the input feature maps. We integrated the
above three modules into a standard U-Net to derive a novel Dynamic U-Net
architecture for multi-organ segmentation. The integration of our three models
enhances the capabilities of Dynamic U-Net to model organ deformations and
effectively handle large variations among different organs. This adaptability
holds promise for accurately segmenting organs with high variability, thereby
improving the overall segmentation accuracy.

Our contributions can be summarized as follows: (i) A Dynamically Calibrated
Convolution module was designed to dynamically model long-range spatial-wise
and channel-wise inter-dependencies. It exploits these interactions to
adaptively calibrate local features based on global contextual information.
(ii) A Dynamically Calibrated Downsampling module was designed to calibrate
features adaptively by preserving the deformable and discriminative details
during downsampling. This functionality enables the module to effectively
handle deformations and large variations among abdominal organs. (iii) A
Dynamically Calibrated Upsampling module was proposed to align and adaptively
calibrate upsampled features with their skip-connected reference features.
This maintains the consistency of deformation across different organs. (iv) We
proposed the Dynamic U-Net for multi-organ segmentation to effectively handle
the large variations among different organs and their complex anatomical
structures. This is achieved by capturing and adaptively calibrating spatial
and channel-wise deformable features, thereby improving segmentation
performance.

##  2 Methodology

###  2.1 Overall architecture

We proposed the Dynamic U-Net for segmenting multiple organs in abdominal CT
images (Figure 1). This approach is based on incorporating the proposed DCC,
DCD, and DCU modules into a standard 7-layer U-Net. Specifically, we used DCC
modules to replace standard convolutions at each layer. We replaced the max-
pooling layers with the DCD module and integrated the DCU module into the
decoding path. The number of input feature maps was 1, and the numbers of
feature maps in each layer were 32, 64, 128, 256, 512, and 512, respectively,
with 512 feature maps in the bottleneck.

Figure 1: The overall architecture of the Dynamic U-Net as well as the
detailed architecture of the Dynamically Calibrated Convolution (DCC) module.

###  2.2 Dynamically Calibrated Convolution (DCC) module

To equip convolutions with the ability to adaptively calibrate spatial and
channel features, we propose the Dynamically Calibrated Convolution (DCC)
module (Figure 1). Two parallel paths are adopted to generate spatial-wise
calibration information. The first path employs a $1\times 1$ convolution
layer $\mathcal{F}_{conv}^{(1)}(\cdot)$ to extract pixel-wise calibration
information $\boldsymbol{T}_{p}\in\mathbb{R}^{C_{out}\times H\times W}$ from
input feature maps
$\boldsymbol{X}=[\boldsymbol{x}_{1},\boldsymbol{x}_{2},...,\boldsymbol{x}_{C_{in}}]\in\mathbb{R}^{C_{in}\times
H\times W}$

$\boldsymbol{T}_{p}=\mathcal{F}_{conv}^{(1)}(\boldsymbol{X}).$

However, this pixel-wise information ignores the utilization of neighboring
contextual information. This is taken into account by the second path. This
path utilizes average pooling $\mathcal{F}_{AvgPool}(\cdot)$ operation with
filter $4\times 4$ and stride 4 to capture and aggregate neighboring
contextual information from input feature maps
$\boldsymbol{X}\in\mathbb{R}^{C_{in}\times H\times W}$. Subsequently, a
$3\times 3$ convolution $\mathcal{F}_{conv}(\cdot)$ operation is applied to
generate region-wise calibration information
$\boldsymbol{T}_{r}\in\mathbb{R}^{C_{out}\times\frac{H}{4}\times\frac{W}{4}}$.
Lastly, nearest neighbor interpolation $\mathcal{F}_{up}(\cdot)$ is used to
upsample the information to the original input dimension
$\boldsymbol{T}_{r}\in\mathbb{R}^{C_{out}\times H\times W}$

$\boldsymbol{T}_{r}=\mathcal{F}_{up}(\mathcal{F}_{conv}(\mathcal{F}_{AvgPool}(\boldsymbol{X}))).$

In the next step, the pixel-wise information and neighboring region-wise
information are fused, and a Sigmoid function $\mathcal{F}_{sig}(\cdot)$ is
used to generate the spatial-wise calibration information
$\boldsymbol{T}_{s}\in\mathbb{R}^{C_{out}\times H\times W}$

$\boldsymbol{T}_{s}=\mathcal{F}_{sig}(\boldsymbol{T}_{p}\oplus\boldsymbol{T}_{r}).$

Subsequently, this spatial-wise calibration information is used to calibrate
local features via element-wise multiplication to generate calibrated feature
maps $\boldsymbol{F}_{1}\in\mathbb{R}^{C_{out}\times H\times W}$. Here, local
features are extracted by a $3\times 3$ convolution
$\mathcal{F}_{conv}(\cdot)$ from input feature maps
$\boldsymbol{X}\in\mathbb{R}^{C_{in}\times H\times W}$

$\boldsymbol{F}_{1}=\mathcal{F}_{conv}(\boldsymbol{X})\otimes\boldsymbol{T}_{s}.$

In the subsequent step, another convolution layer $\mathcal{F}_{conv}(\cdot)$
is utilized to extract organ-specific spatial features and generate feature
maps $\boldsymbol{F}_{2}\in\mathbb{R}^{C_{out}\times H\times W}$:

$\boldsymbol{F}_{2}=\mathcal{F}_{conv}(\boldsymbol{F}_{1}).$

Finally, channel-wise calibration information is extracted by cascading an
adaptive average pooling $\mathcal{F}_{AvgPool}(\cdot)$, a linear layer
$\mathcal{F}_{Linear}(\cdot)$, and a Sigmoid function
$\mathcal{F}_{sig}(\cdot)$. The output of DCC module
$\boldsymbol{F}^{*}\in\mathbb{R}^{C_{out}\times H\times W}$ is generated by
using channel-wise calibration information to calibrate feature maps
$\boldsymbol{F}_{2}$

$\boldsymbol{F}^{*}=\mathcal{F}_{sig}(\mathcal{F}_{Linear}(\mathcal{F}_{AvgPool}(\boldsymbol{F}_{2})))\otimes\boldsymbol{F}_{2}.$

###  2.3 Dynamically Calibrated Downsampling (DCD) module

The extensive use of downsampling layers (i.e., pooling and strided
convolutions) prevents networks from preserving discriminative and deformable
features. This limitation impedes the networks’ ability to effectively model
organ deformation. To tackle this limitation, we propose the Dynamically
Calibrated Downsampling (DCD) module to adaptively downsample feature maps
while preserving deformations and discriminative details (Figure 2(A)). First,
the offset value $\boldsymbol{\Delta}_{i}\in\mathbb{R}^{C^{\prime}\times
H\times W}$ is learned from input features
$\boldsymbol{F}=[\boldsymbol{f}_{1},\boldsymbol{f}_{2},...,\boldsymbol{f}_{C}]\in\mathbb{R}^{C\times
H\times W}$ by a $3\times 3$ convolution $\mathcal{F}_{conv}(\cdot)$ operation

$\boldsymbol{\Delta}_{i}=\mathcal{F}_{conv}(\boldsymbol{F}).$

Subsequently, a deformable spatial feature allocation map
$\boldsymbol{M}\in\mathbb{R}^{C\times H\times W}$ is learned by cascading a
modulated deformable convolution $\mathcal{F}_{deform}(\cdot;\cdot)$, a
Sigmoid function $\mathcal{F}_{sig}(\cdot)$, and an exponential operation
$\mathcal{F}_{exp}(\cdot)$. This map describes how deformable spatial features
can be allocated during downsampling and how input features contribute to
final pooled features. The exponential operation $\mathcal{F}_{exp}(\cdot)$ is
used to make the values in this map non-negative and easy to optimize.

$\boldsymbol{M}=\mathcal{F}_{exp}(\mathcal{F}_{sig}(\mathcal{F}_{deform}(\boldsymbol{F};\boldsymbol{\Delta}_{i})))$

Given the deformable spatial feature allocation map $\boldsymbol{M}$, the
downsampled features
$\boldsymbol{F}^{*}\in\mathbb{R}^{C\times\frac{H}{2}\times\frac{W}{2}}$ can be
calculated. Specifically, features $\boldsymbol{F}$ are allocated by
$\boldsymbol{M}$ as $\boldsymbol{M}\otimes\boldsymbol{F}$. Subsequently,
allocated features are projected to a lower dimension by using the summation
of allocation maps $\boldsymbol{M}$ to divide the summation of allocated
features, where the summation is implemented by the average pooling
$\mathcal{F}_{AvgPool}(\cdot)$.

$\boldsymbol{F}^{*}=\mathcal{F}_{AvgPool}(\boldsymbol{M}\otimes\boldsymbol{F})/\mathcal{F}_{AvgPool}(\boldsymbol{M})$

Figure 2: The architecture of (A) the Dynamically Calibrated Downsampling
(DCD) module and (B) the Dynamically Calibrated Upsampling (DCU) module.

###  2.4 Dynamically Calibrated Upsampling (DCU) module

In hierarchical networks, feature maps upsampled in the decoding path are
often fused with the skip-connected feature maps from the encoding path.
However, Repeated downsampling and upsampling operations result in spatial
misalignments between them. Thus, fusing them by either element-wise addition
or concatenation will harm feature learning. To tackle this limitation, we
propose the Dynamically Calibrated Upsampling (DCU) module to eliminate
misalignments by adaptively aligning and calibrating features to its skip-
connected reference before fusion (Figure 2(B)). First, the feature maps at
the level $i$ in the decoding path
$\boldsymbol{F}_{i}=[\boldsymbol{f}_{1},\boldsymbol{f}_{2},...,\boldsymbol{f}_{C}]\in\mathbb{R}^{C\times
H\times W}$ were generated from those at the higher level
$\boldsymbol{F}_{i+1}\in\mathbb{R}^{2C\times\frac{H}{2}\times\frac{W}{2}}$ by
upsampling operations $\mathcal{F}_{up}(\cdot)$ (i.e., transposed
convolutions)

$\boldsymbol{F}_{i}=\mathcal{F}_{up}(\boldsymbol{F}_{i+1}).$

Now, this upsampled feature maps $\boldsymbol{F}_{i}$ has the same dimension
as its skip-connected reference $\boldsymbol{S}_{i}\in\mathbb{R}^{C\times
H\times W}$ from the encoding path. Subsequently, feature maps (termed offset
values $\boldsymbol{\Delta_{i}}\in\mathbb{R}^{C^{\prime}\times H\times W}$
here) are extracted from both $\boldsymbol{F}_{i}$ and $\boldsymbol{S}_{i}$ by
a $3\times 3$ convolution $\mathcal{F}_{conv}(\cdot)$ operation. It is known
that convolution operations can capture the spatial location information and
present such information as feature maps. Thus, these offset values extracted
from $\boldsymbol{F}_{i}$ and $\boldsymbol{S}_{i}$ can be considered as their
spatial location information. When there are spatial misalignments between
$\boldsymbol{F}_{i}$ and $\boldsymbol{S}_{i}$, these offset values demonstrate
the shifted distances between each point in $\boldsymbol{F}_{i}$ and its
corresponding point in $\boldsymbol{S}_{i}$

$\boldsymbol{\Delta}_{i}=\mathcal{F}_{conv}([\boldsymbol{F}_{i},\boldsymbol{S}_{i}]).$

Given the learned offset $\boldsymbol{\Delta_{i}}$, a modulated deformable
convolution $\mathcal{F}_{deform}(\cdot;\cdot)$ is applied to eliminate
spatial misalignments. Specifically, this deformable convolution takes
features $\boldsymbol{F}_{i}$ as input and adds these offset values
$\boldsymbol{\Delta_{i}}$ to the regular locations of these features. Thus,
the misalignments are eliminated by adjusting and aligning spatial locations
of $\boldsymbol{F}_{i}$ to that of $\boldsymbol{S}_{i}$ using offset values
within the deformable convolutional kernel. Finally, a Leaky ReLU
$\mathcal{F}_{LReLU}(\cdot)$ is utilized to get the optimal feature maps
$\boldsymbol{F}_{i}^{*}\in\mathbb{R}^{C\times H\times W}$, which can be
concatenated with its reference $\boldsymbol{S}_{i}$

$\boldsymbol{F}_{i}^{*}=\mathcal{F}_{LReLU}(\mathcal{F}_{deform}(\boldsymbol{F}_{i};\boldsymbol{\Delta}_{i})).$

##  3 Experiments

####  3.0.1 Datasets.

We conducted experiments on two public datasets. The first one was the MICCAI
2021 FLARE Challenge dataset [11]. It consists of 361 abdominal CT images with
voxel-level annotations of four organs. The second one was the MICCAI 2022
AMOS Challenge dataset [7]. It consists of 300 abdominal CT images with voxel-
level annotations of 15 organs. Data were pre-processed and augmented by the
nnUNet pipeline [6].

####  3.0.2 Experimental details.

We used the combination of cross-entropy loss and Dice loss as the loss
function and used the Dice Similarity Coefficient (DSC) to evaluate
segmentation performance. Stochastic gradient descent (SGD) was used for
network optimization. The initial learning rate was set to 0.01 and decayed
using a poly decay strategy as nnUNet [6]. All experiments were performed
using 5-fold cross-validation without additional post-processing.

####  3.0.3 Baselines.

We compared the segmentation performance between our Dynamic U-Net and U-Net
baselines. The U-Net was implemented following the original design: max-
pooling layers were used for downsampling and transposed convolution layers
were used for upsampling. The SConv U-Net was implemented using strided
convolution layers for downsampling, and other designs are the same with
U-Net. Importantly, to demonstrate the effectiveness of each module, we
conducted ablation studies by evaluating models with modules partially
incorporated (i.e., DCC U-Net, DCD U-Net, DCU U-Net). Specifically, for DCC
U-Net, only the DCC module was incorporated into U-Net to replace convolution
blocks in each layer. For DCD U-Net, only the DCD module was used for
downsampling while the other aspects of the architecture remained the same as
U-Net. For DCU U-Net, only the DCU module was used for upsampling while the
other aspects of the architecture remained the same as U-Net.

Table 1: Comparison of multi-organ segmentation between Dynamic U-Net and
baselines on the FLARE 2021. (Bold represents the best segmentation results).

Tasks |  U-Net† |  SConv U-Net† |  DCC U-Net |  DCD U-Net |  DCU U-Net |  Dynamic U-Net
---|---|---|---|---|---|---
Liver |  97.33 |  97.36 |  $\boldsymbol{98.46}$ |  98.39 |  98.40 |  98.40
Kidney |  95.50 |  95.58 |  96.66 |  96.70 |  96.62 |  $\boldsymbol{97.22}$
Spleen |  95.81 |  95.85 |  98.12 |  98.02 |  98.11 |  $\boldsymbol{98.14}$
Pancreas |  79.91 |  80.66 |  81.83 |  81.38 |  81.23 |  $\boldsymbol{83.54}$
Average |  92.14 |  92.36 |  93.77 |  93.62 |  93.59 |  $\boldsymbol{94.33}^{*}$

†: we implemented nnUNet with different architectures here.
∗: $p<0.01$ with Wilcoxon signed-rank test between Dynamic U-Net, and U-Net,
SConv U-Net.

####  3.0.4 Main Results.

Table 1 and Table 2 show the result comparison of Dynamic U-Net with the
baselines on multi-organ segmentation on the FLARE 2021 and AMOS 2022
benchmarks, respectively. The integration of the proposed DCC, DCD, and DCU
modules in the proposed Dynamic U-Net resulted in a significant improvement in
segmentation performance (Dice score) of both average foreground values and
other organ-specific segmentation tasks compared with U-Net and SConv U-Net.
To be specific, DCC U-Net achieved higher Dice scores across all tasks than
U-Net, which demonstrates the effectiveness of DCC modules in the improvement
of segmentation performance. This suggests that utilizing global contextual
information may improve the representation capabilities of Dynamic U-Net.
Furthermore, DCD U-Net and DCU U-Net achieved higher segmentation Dice scores
than U-Net and SConv U-Net, demonstrating the improvement in the segmentation
performance due to the proposed DCD and DCU modules. The DCD and DCU modules
enable the Dynamic U-Net to adaptively preserve and calibrate deformable and
discriminative features, thereby effectively addressing large variations among
different organs.

Table 2: Comparison of multi-organ segmentation between Dynamic U-Net and
baselines on the AMOS 2022. (Bold represents the best segmentation results).

Tasks |  U-Net† |  SConv U-Net† |  DCC U-Net |  DCD U-Net |  DCU U-Net |  Dynamic U-Net
---|---|---|---|---|---|---
Spleen |  96.31 |  96.60 |  96.83 |  97.05 |  97.12 |  $\boldsymbol{97.64}$
R. kidney |  96.45 |  96.15 |  96.61 |  97.18 |  97.03 |  $\boldsymbol{97.70}$
L. kidney |  96.36 |  96.23 |  96.68 |  96.81 |  $\boldsymbol{96.99}$ |  96.33
Gall bladder |  78.58 |  79.56 |  79.77 |  80.50 |  80.22 |  $\boldsymbol{83.83}$
Esophagus |  84.13 |  83.61 |  84.84 |  85.37 |  85.25 |  $\boldsymbol{86.17}$
Liver |  96.74 |  96.76 |  96.97 |  97.53 |  97.36 |  $\boldsymbol{98.52}$
Stomach |  87.39 |  88.65 |  89.44 |  89.32 |  89.62 |  $\boldsymbol{90.29}$
Arota |  94.54 |  94.46 |  94.71 |  95.24 |  95.27 |  $\boldsymbol{96.65}$
Postcava |  89.02 |  88.78 |  89.90 |  90.25 |  90.06 |  $\boldsymbol{90.36}$
Pancreas |  83.93 |  84.50 |  85.52 |  85.67 |  86.38 |  $\boldsymbol{87.06}$
R. adrenal gland |  77.71 |  78.27 |  78.74 |  78.65 |  79.50 |  $\boldsymbol{80.47}$
L. adrenal gland |  79.65 |  79.58 |  81.29 |  81.32 |  81.16 |  $\boldsymbol{81.44}$
Duodenum |  77.49 |  77.87 |  79.04 |  78.79 |  80.22 |  $\boldsymbol{81.11}$
Bladder |  86.40 |  85.40 |  88.12 |  89.04 |  88.90 |  $\boldsymbol{89.06}$
Prostate/uterus |  80.56 |  80.97 |  83.30 |  $\boldsymbol{84.52}$ |  84.49 |  83.81
Average |  87.02 |  87.16 |  88.12 |  88.48 |  88.64 |  $\boldsymbol{89.36}^{*}$

†: we implemented nnUNet with different architectures here.
∗: $p<0.01$ with Wilcoxon signed-rank test between Dynamic U-Net, and U-Net,
SConv U-Net.

##  4 Conclusion

We introduce a novel Dynamic U-Net for abdominal multi-organ segmentation from
CT images, and it is proposed to handle the morphological complexity and large
variation among different organs by incorporating Dynamically Calibrated
Convolution, Dynamically Calibrated Downsampling, and Dynamically Calibrated
Upsampling modules. It was empirically shown that the incorporation of these
modules may continuously improve segmentation accuracy.

Figure 3: Qualitative representations of multi-organ segmentation on AMOS 2022
dataset. Dynamic U-Net shows better segmentation quality than U-Net.

####  4.0.1 Acknowledgements

This research was funded by 5 U24 CA258483. Computations were performed using
the facilities of the Washington University Center for High Performance
Computing (CHPC), which was partially funded by National Institutes of Health
(NIH) grants S10OD025200, 1S10RR022984-01A1, and 1S10OD018091-01.

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