Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Image reconstruction from structured subsampled 2D Fourier data

Gerlind Plonka, Anahita Riahi

TL;DR

The paper addresses reconstructing images from structured undersampled 2D Fourier data, motivated by parallel MRI constraints. It shows that including a low-pass block of rows around zero is essential for stable recovery and analyzes naive approaches like zero filling and 1D filtering, which fail to capture all information. The authors develop a TV-based reconstruction via a primal-dual algorithm and augment it with a data-informed hybrid method that exploits the sampling pattern to improve results. Across diverse images and data-reduction levels, the hybrid TV method consistently outperforms low-pass only and interpolation-based approaches, highlighting the practical impact for MRI and similar applications where k-space data is inherently structured.

Abstract

In this paper we study the performance of image reconstruction methods from incomplete samples of the 2D discrete Fourier transform. Inspired by requirements in parallel MRI, we focus on a special sampling pattern with a small number of acquired rows of the Fourier transformed image. We show the importance of the low-pass set of acquired rows around zero in the Fourier space for image reconstruction. A suitable choice of the width $L$ of this index set depends on the image data and is crucial to achieve optimal reconstruction results. We prove that non-adaptive reconstruction approaches cannot lead to satisfying recovery results. We propose a new hybrid algorithm which connects the TV minimization technique based on primal-dual optimization with a recovery algorithm which exploits properties of the special sampling pattern for reconstruction. Our method shows very good performance for natural images as well as for cartoon-like images for a data reduction rate up to 8 in the complex setting and even 16 for real images.

Image reconstruction from structured subsampled 2D Fourier data

TL;DR

The paper addresses reconstructing images from structured undersampled 2D Fourier data, motivated by parallel MRI constraints. It shows that including a low-pass block of rows around zero is essential for stable recovery and analyzes naive approaches like zero filling and 1D filtering, which fail to capture all information. The authors develop a TV-based reconstruction via a primal-dual algorithm and augment it with a data-informed hybrid method that exploits the sampling pattern to improve results. Across diverse images and data-reduction levels, the hybrid TV method consistently outperforms low-pass only and interpolation-based approaches, highlighting the practical impact for MRI and similar applications where k-space data is inherently structured.

Abstract

In this paper we study the performance of image reconstruction methods from incomplete samples of the 2D discrete Fourier transform. Inspired by requirements in parallel MRI, we focus on a special sampling pattern with a small number of acquired rows of the Fourier transformed image. We show the importance of the low-pass set of acquired rows around zero in the Fourier space for image reconstruction. A suitable choice of the width of this index set depends on the image data and is crucial to achieve optimal reconstruction results. We prove that non-adaptive reconstruction approaches cannot lead to satisfying recovery results. We propose a new hybrid algorithm which connects the TV minimization technique based on primal-dual optimization with a recovery algorithm which exploits properties of the special sampling pattern for reconstruction. Our method shows very good performance for natural images as well as for cartoon-like images for a data reduction rate up to 8 in the complex setting and even 16 for real images.
Paper Structure (16 sections, 2 theorems, 63 equations, 7 figures, 5 tables, 2 algorithms)

This paper contains 16 sections, 2 theorems, 63 equations, 7 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and problem description.
  3. Outline of this paper.
  4. Challenges of the reconstruction problem
  5. Importance of the low-pass set
  6. Zero refilling and one-dimensional low-pass filtering
  7. Zero refilling.
  8. One-dimensional low-pass filtering with $\Lambda= \Lambda_L$.
  9. Non-adaptive reconstruction approach
  10. Low-rank approximation
  11. Locally adaptive interpolation in Fourier domain
  12. Reconstruction by TV functional minimization
  13. Hybrid method
  14. Numerical Results
  15. Generalizations of the sampling grid.
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let ${\mathbf A}=(a_{\mathbf k})_{\mathbf k \in \Lambda_{N,M}} \in {\mathbb C}^{N \times M}$, $\hat{\mathbf A}=(\hat{a}_{\boldsymbol \nu})_{\boldsymbol \nu \in \Lambda_{N,M}} \in {\mathbb C}^{N \times M}$ its $2$-D Fourier transform, and assume that half of the Fourier data, namely are acquired. Let ${\mathbf W} =(\mathsf{w}_{\boldsymbol \nu})_{\boldsymbol \nu \in \Lambda_{n,M}}$ be a matrix of a

Figures (7)

  • Figure 1: Masks for acquired Fourier data for an $128 \times 128$ image with width $L=11$ of the low-pass set and reduction rates $r=2,4,6,8$, where black lines illustrate the acquired rows.
  • Figure 2: Top: Reconstructions of the $512 \times 512$ pepper image from Fourier data with reduction rate $r=6$. Left: Approximation with Dirichlet window using $L=\lfloor512/r \rfloor=85$ with PSNR $28.21$; Middle: Approximation with Hamming window using $L=85$ with PSNR $25.57$; Right: Reconstruction by zero refilling with reduction rate $r=6$ and $L=43$ with PSNR $26.75$. Bottom: Corresponding representations of the windows $\check{\mathbf p}^{(\Lambda_{L})}$, $\check{\mathbf p}^{(Ham)}$ (middle), and $\check{\mathbf p}^{(\Lambda_{1})}$.
  • Figure 3: Example for different patterns ${\mathcal{P}}$ for GRAPPA interpolation using local interpolation in a $7 \times 5$ window. All gray- and green-valued pixel values are acquired. Green pixel values are employed to interpolate the red pixel value
  • Figure 4: Original image and image reconstructions for $r=6$ and $L=43$. Top: Left: "cameraman" original image $512 \times 512$; Middle: zero refilling with PSNR $27.6986$; Right: GRAPPA method with PSNR $27.7064$; Bottom: Left: low-pass reconstruction from $L=43$ rows with PSNR 25.2711; Middle: TV minimization with PSNR 31.1364; Right: hybrid method with PSNR 32.1167 (see Table \ref{['tab1']})
  • Figure 5:
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 4.2