A MATHEMATICAL EXPLANATION OF UNET

Xue-Cheng Tai; Hao Liu; Raymond H. Chan; Lingfeng Li

doi:10.3934/mfc.2024040

A MATHEMATICAL EXPLANATION OF UNET

Xue-Cheng Tai
, Hao Liu^*
, Raymond H. Chan
, Lingfeng Li

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

10 Citations (Scopus)

Abstract

The UNet architecture has transformed image segmentation. UNet’s versatility and accuracy have driven its widespread adoption, significantly advancing fields reliant on machine learning problems with images. In this work, we gave a clear and concise mathematical explanation of UNet. We explained what is the meaning and function of each of the components of UNet. We showed that UNet was solving a control problem. We decomposed the control variables using multigrid methods. Then, operator-splitting techniques were used to solve the problem, whose architecture exactly recovered the UNet architecture. Our result showed that UNet was a one-step operator-splitting algorithm for the control problem. © (2025), (American Institute of Mathematical Sciences). All rights reserved.

Original language	English
Pages (from-to)	874-889
Number of pages	16
Journal	Mathematical Foundations of Computing
Volume	8
Issue number	5
Online published	14 Oct 2024
DOIs	https://doi.org/10.3934/mfc.2024040
Publication status	Published - Oct 2025
Externally published	Yes

Funding

2020 Mathematics Subject Classification. Primary: 68U10, 94A08. Key words and phrases. UNet, operator splitting, deep neural network, image segmentation. The first author is supported by [HKRGC-NSFC Grant N-CityU214/19, HKRGC CRF Grant C1013-21GF and NORCE Kompetanseoppbygging program]. The second author is supported by [National Natural Science Foundation of China 12201530 and HKRGC ECS 22302123]. The third author is supported by [HKRGC GRF grants CityU1101120, CityU11309922, CRF grant C1013-21GF, and HKRGC-NSFC Grant N-CityU214/19]. \u2217Corresponding author: Hao Liu.

Research Keywords

deep neural network
image segmentation
operator splitting
UNet

RGC Funding Information

RGC-funded

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title = "A MATHEMATICAL EXPLANATION OF UNET",

abstract = "The UNet architecture has transformed image segmentation. UNet{\textquoteright}s versatility and accuracy have driven its widespread adoption, significantly advancing fields reliant on machine learning problems with images. In this work, we gave a clear and concise mathematical explanation of UNet. We explained what is the meaning and function of each of the components of UNet. We showed that UNet was solving a control problem. We decomposed the control variables using multigrid methods. Then, operator-splitting techniques were used to solve the problem, whose architecture exactly recovered the UNet architecture. Our result showed that UNet was a one-step operator-splitting algorithm for the control problem. {\textcopyright} (2025), (American Institute of Mathematical Sciences). All rights reserved.",

keywords = "deep neural network, image segmentation, operator splitting, UNet",

author = "Xue-Cheng Tai and Hao Liu and Chan, \{Raymond H.\} and Lingfeng Li",

year = "2025",

month = oct,

doi = "10.3934/mfc.2024040",

language = "English",

volume = "8",

pages = "874--889",

number = "5",

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TY - JOUR

T1 - A MATHEMATICAL EXPLANATION OF UNET

AU - Tai, Xue-Cheng

AU - Liu, Hao

AU - Chan, Raymond H.

AU - Li, Lingfeng

PY - 2025/10

Y1 - 2025/10

N2 - The UNet architecture has transformed image segmentation. UNet’s versatility and accuracy have driven its widespread adoption, significantly advancing fields reliant on machine learning problems with images. In this work, we gave a clear and concise mathematical explanation of UNet. We explained what is the meaning and function of each of the components of UNet. We showed that UNet was solving a control problem. We decomposed the control variables using multigrid methods. Then, operator-splitting techniques were used to solve the problem, whose architecture exactly recovered the UNet architecture. Our result showed that UNet was a one-step operator-splitting algorithm for the control problem. © (2025), (American Institute of Mathematical Sciences). All rights reserved.

AB - The UNet architecture has transformed image segmentation. UNet’s versatility and accuracy have driven its widespread adoption, significantly advancing fields reliant on machine learning problems with images. In this work, we gave a clear and concise mathematical explanation of UNet. We explained what is the meaning and function of each of the components of UNet. We showed that UNet was solving a control problem. We decomposed the control variables using multigrid methods. Then, operator-splitting techniques were used to solve the problem, whose architecture exactly recovered the UNet architecture. Our result showed that UNet was a one-step operator-splitting algorithm for the control problem. © (2025), (American Institute of Mathematical Sciences). All rights reserved.

KW - deep neural network

KW - image segmentation

KW - operator splitting

KW - UNet

UR - https://www.scopus.com/pages/publications/105009387781

UR - https://www.scopus.com/pages/publications/105009387781#tab=impact

U2 - 10.3934/mfc.2024040

DO - 10.3934/mfc.2024040

M3 - RGC 21 - Publication in refereed journal

SN - 2577-8838

VL - 8

SP - 874

EP - 889

JO - Mathematical Foundations of Computing

JF - Mathematical Foundations of Computing

IS - 5

ER -

A MATHEMATICAL EXPLANATION OF UNET

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GRF: Efficient Mathematical and Data-driven Models for Closed-loop Adaptive Optics Systems for Ground-based Astronomy

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A MATHEMATICAL EXPLANATION OF UNET

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Research Keywords

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GRF: Efficient Mathematical and Data-driven Models for Closed-loop Adaptive Optics Systems for Ground-based Astronomy

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