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Methods based on Radon transform for non-affine deformable image registration of noisy images

Daniel E. Hurtado, Axel Osses, Rodrigo Quezada

TL;DR

This study introduces two new DIR methods designed to capture non-affine deformations using Radon transform-based similarity measures and a classical regularizer based on linear elastic deformation energy.

Abstract

Deformable image registration is a standard engineering problem used to determine the distortion experienced by a body by comparing two images of it in different states. This study introduces two new DIR methods designed to capture non-affine deformations using Radon transform-based similarity measures and a classical regularizer based on linear elastic deformation energy. It establishes conditions for the existence and uniqueness of solutions for both methods and presents synthetic experimental results comparing them with a standard method based on the sum of squared differences similarity measure. These methods have been tested to capture various non-affine deformations in images, both with and without noise, and their convergence rates have been analyzed. Furthermore, the effectiveness of these methods was also evaluated in a lung image registration scenario.

Methods based on Radon transform for non-affine deformable image registration of noisy images

TL;DR

This study introduces two new DIR methods designed to capture non-affine deformations using Radon transform-based similarity measures and a classical regularizer based on linear elastic deformation energy.

Abstract

Deformable image registration is a standard engineering problem used to determine the distortion experienced by a body by comparing two images of it in different states. This study introduces two new DIR methods designed to capture non-affine deformations using Radon transform-based similarity measures and a classical regularizer based on linear elastic deformation energy. It establishes conditions for the existence and uniqueness of solutions for both methods and presents synthetic experimental results comparing them with a standard method based on the sum of squared differences similarity measure. These methods have been tested to capture various non-affine deformations in images, both with and without noise, and their convergence rates have been analyzed. Furthermore, the effectiveness of these methods was also evaluated in a lung image registration scenario.
Paper Structure (12 sections, 6 theorems, 64 equations, 11 figures)

This paper contains 12 sections, 6 theorems, 64 equations, 11 figures.

Table of Contents

  1. Introduction
  2. DIR Problem
  3. Similarity measures based on the Radon transform
  4. Conditions for the existence and uniqueness of the solution
  5. Numerical implementation
  6. Numerical Tests
  7. Test A: Registration of non-affine deformations in noise-free images
  8. Test B: Registration of a non-affine deformation in images with random noise
  9. Test C: Convergence rates in the non-affine registration of noise-free images
  10. Test D: Convergence rates in the non-affine registration of noisy images
  11. Example E: Application to lung deformations
  12. Discussion

Key Result

Theorem 1

Let $\,\mathcal{T}: H \rightarrow H$ be an operator with $\mathcal{T}(z) = u$, where for each $z\in H$, $u$ is the solution of the problem: Find $u \in H$ such that where $\hat{\alpha}$ is the regularization parameter associated to the DIR problem, $F_z\in H',$ and $a$ is the continuous and non-negative bilinear form in $H^1(\Omega)$, given by where $\mathbb C$ was defined in eq:ElasticityTenso

Figures (11)

  • Figure 1: The figure illustrates the $\Omega$ and $\tilde{\Omega}$ domains of the images $R$ and $T,$ respectively. In addition, it is shown how a point $x \in \Omega$ could be transported by the vector field $u$ into the $\Omega$ domain.
  • Figure 2: Delaunay meshes utilized in this work.
  • Figure 3: Sampling of images considered in Test A: Reference R and a sample of five of the random Template images T
  • Figure 4: Results of the registration of thirty random deformations in noise-free images. The best regularization parameters for each method were as follows: RSSD: $\alpha= 0.02,$ SSD: $\alpha= 0.003,$ y R$^\#$SSD: $\alpha= 0.007$.
  • Figure 5: Sampling of images contained in Test B - Low Noise: The first row displays the template T images from a sampling of thirty high-noise images. The second row displays their respective six reference images with different seed noise. The third, fourth, and fifth rows displays the difference between the reference image R and the registration image from the RSSD, SSD, and R$^\#$SSD methods, respectively, with the noise removed for comprehension.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1: Barnafi et al. (2018) barnafi2018primal
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 2 more