[Paper Review] 3D Image Reconstruction from Compton camera data

Peter Kuchment, Fatma Terzioglu|arXiv (Cornell University)|Apr 11, 2016

Medical Imaging Techniques and Applications32 references3 citations

TL;DR

This paper presents three analytical and numerical inversion methods for the 3D cone transform arising in Compton camera imaging, enabling reconstruction of 3D source distributions from conical surface integrals. It introduces an admissibility condition on detector geometry ensuring stable inversion, with numerical results showing robustness to 20% Gaussian noise for two methods, while the third method demonstrates superior noise resilience through mollifier-based regularization.

ABSTRACT

In this paper, we address analytically and numerically the inversion of the integral transform (\emph{cone} or \emph{Compton} transform) that maps a function on $\mathbb{R}^3$ to its integrals over conical surfaces. It arises in a variety of imaging techniques, e.g. in astronomy, optical imaging, and homeland security imaging, especially when the so called Compton cameras are involved. Several inversion formulas are developed and implemented numerically in $3D$ (the much simpler $2D$ case was considered in a previous publication). An admissibility condition on detectors geometry is formulated, under which all these inversion techniques will work.

Motivation & Objective

• To develop analytical and numerical inversion techniques for the 3D cone transform, which maps a function to its integrals over conical surfaces in Compton camera imaging.
• To formulate a general admissibility condition on detector geometry that ensures the invertibility of the cone transform and prevents limited data blurring artifacts.
• To compare the performance of multiple inversion formulas under noisy conditions, particularly focusing on stability and noise resilience in practical imaging scenarios.
• To demonstrate the feasibility of reconstructing 3D source distributions using synthetic data from a spherical phantom, with emphasis on the separation of Radon data recovery and subsequent inversion.

Proposed method

• The paper derives three distinct analytical inversion formulas for the 3D cone transform, based on integral geometry and the Radon transform, with one method relying on spherical harmonics expansion.
• A key component is the recovery of Radon transform data from cone data, followed by standard inversion of the Radon transform using a well-tested numerical algorithm.
• The first method uses a series expansion in spherical harmonics with truncation and regularization, while the second applies successive numerical Laplace-Beltrami operators with Tikhonov-type smoothing.
• The third method employs a mollifier-based regularization technique to stabilize the inversion process, with adjustable parameters for noise control.
• The authors generate synthetic cone data numerically for a uniform spherical phantom, simulating realistic Compton camera measurements with added Gaussian noise.
• All methods are tested on concentric and non-concentric configurations, with computational efficiency prioritized through symmetric detector and phantom geometry in initial tests.

Experimental results

Research questions

• RQ1 Can the cone transform in 3D be inverted analytically and numerically using multiple equivalent formulas, and how do they compare in stability under noise?
• RQ2 What geometric conditions on the detector array ensure the invertibility of the cone transform and prevent limited data artifacts?
• RQ3 How do different numerical inversion techniques—especially those involving second-order differential operators or mollifiers—perform when reconstructing images from noisy cone data?
• RQ4 To what extent does the symmetry of the phantom and detector configuration affect the stability and accuracy of reconstruction, and can non-concentric setups be handled effectively?
• RQ5 Can the separation of Radon data recovery from function inversion be leveraged to improve reconstruction quality and enable parameter tuning for noise robustness?

Key findings

• Method 1 and Method 3 produce visually and quantitatively acceptable reconstructions even with 20% Gaussian white noise, demonstrating strong noise resilience.
• Method 2 begins to break down at 20% noise but remains stable at 10% noise, attributed to the numerical instability of applying two successive Laplace-Beltrami operators.
• The admissibility condition on detector geometry is essential: its violation leads to limited data blurring artifacts, confirming its role in ensuring invertibility.
• All three inversion methods are effective for reconstructing a spherical phantom from synthetic cone data, with the spherical symmetry allowing clear assessment of the cone data inversion step.
• The use of a concentric spherical phantom significantly reduces computational cost for forward data generation, enabling efficient testing, while non-concentric reconstructions (e.g., phantom shifted by 20% of detector radius) remain feasible with adjusted data quality.
• The methods allow fine-tuning via parameters such as truncation level, smoothing parameter h, and mollifier settings, enabling optimization for specific noise levels and image quality requirements.