Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuous Learned Primal Dual

Christina Runkel, Ander Biguri, Carola-Bibiane Schönlieb

TL;DR

This paper addresses the ill-posed nature of CT reconstruction by integrating neural ODEs with the Learned Primal Dual framework to form a continuous learned primal dual (cLPD). By replacing discrete blocks with continuous ODE blocks, the method aims to improve noise robustness and performance under challenging acquisition conditions. Empirical results on the LIDC-IDRI dataset across clinical, reduced-dose, sparse-angle, and limited-angle scenarios show that cLPD generally matches or surpasses the standard LPD and significantly outperforms filtered backprojection (FBP), especially in high-noise or restricted-view settings. The work highlights the potential of continuous dynamics for inverse problems and points to future extensions to other imaging and reconstruction tasks.

Abstract

Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had resounding success in the deep learning literature, with direct or indirect influence in many state of the art ideas, such as diffusion models or time dependant models. Recently, a continuous version of the U-net architecture has been proposed, showing increased performance over its discrete counterpart in many imaging applications and wrapped with theoretical guarantees around its performance and robustness. In this work, we explore the use of Neural ODEs for learned inverse problems, in particular with the well-known Learned Primal Dual algorithm, and apply it to computed tomography (CT) reconstruction.

Continuous Learned Primal Dual

TL;DR

This paper addresses the ill-posed nature of CT reconstruction by integrating neural ODEs with the Learned Primal Dual framework to form a continuous learned primal dual (cLPD). By replacing discrete blocks with continuous ODE blocks, the method aims to improve noise robustness and performance under challenging acquisition conditions. Empirical results on the LIDC-IDRI dataset across clinical, reduced-dose, sparse-angle, and limited-angle scenarios show that cLPD generally matches or surpasses the standard LPD and significantly outperforms filtered backprojection (FBP), especially in high-noise or restricted-view settings. The work highlights the potential of continuous dynamics for inverse problems and points to future extensions to other imaging and reconstruction tasks.

Abstract

Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had resounding success in the deep learning literature, with direct or indirect influence in many state of the art ideas, such as diffusion models or time dependant models. Recently, a continuous version of the U-net architecture has been proposed, showing increased performance over its discrete counterpart in many imaging applications and wrapped with theoretical guarantees around its performance and robustness. In this work, we explore the use of Neural ODEs for learned inverse problems, in particular with the well-known Learned Primal Dual algorithm, and apply it to computed tomography (CT) reconstruction.
Paper Structure (10 sections, 6 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 10 sections, 6 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. Variational formulation of reconstruction
  4. Data-driven methods: Learned Primal Dual
  5. Neural Ordinary Differential Equations
  6. The Continuous Learned Primal Dual
  7. Network Architecture
  8. Experimental Setup
  9. Experimental results
  10. Discussion and Conclusions

Figures (2)

  • Figure 1: Network architecture for both the dual and primal iterates of the (continuous) learned primal dual algorithm. Each of the rectangles describes a convolution and ODE for the LPD and cLPD, respectively. The number of input channels is denoted below the box and the kernel size specified in the middle of the rectangle.
  • Figure 2: Visual results for a randomly picked image of the test set for all experimental settings. We highlight the results of our cLPD, standard LPD, FBP and the target reconstruction from left to right. With increasing noise levels, our approach (cLPD) is able to outperform the LPD more and more significantly. Both cLPD and LPD outperform FBP in all experimental settings. In the case of a limited angle geometry, cLPD reconstructs artifact free results while the standard LPD starts to blur. For high noise levels and the restricted setting especially, FBP is unsuitable as it introduces artifacts.