We attempt to revitalize researchers' interest in algebraic reconstruction techniques (ART) by expanding their capabilities and demonstrating their potential in speeding up the process of MRI acquisition. Using a continuous-to-discrete model, we experimentally study the application of ART into MRI reconstruction which unifies previous nonuniform-fast-Fourier-transform- (NUFFT-) based and gridding-based approaches. Under the framework of ART, we advocate the use of

There are two cultures related to medical image reconstruction: theory oriented (i.e., mentally reproducible) and application oriented (i.e., experimentally reproducible). Historically, when Wilhelm Rontgen discovered X-rays, their applications into medical imaging was immediate (he took the very first picture of his wife’s hand using X-rays). By contrast, when Johann Radon studied integral geometry and measure theory in 1910s, he could not foresee the wide application of his celebrated Radon transform in tomography. The world had to wait until 1930s—when the radiologist Alessandro Vallebona first demonstrated the potential of radiography—to appreciate the usefulness of such a mathematical tool as the Radon transform. The field of modern tomography is largely founded on two schools of researchers: one deals with the exploitation of basic physical phenomenon like Röntgen (e.g., electromagnetic interaction and electron-positron annihilation) and the other studies the mathematical abstraction of imaging modality and its inverse (i.e., tomographic reconstruction) like Radon.

The invention of digital computers in 1940s has had a profound impact on the evolution of science and engineering including tomography. On the one hand, ever-increasing computing resources have dramatically facilitated the development of numerical algorithms (e.g., the invention of fast Fourier transform (FFT) in 1960s). On the other hand, computers have played an unexpected role in disconnecting the two schools: theorists (applied mathematicians) do not need to consult practicians (e.g., practical medical physicists and engineers) for real-world data anymore because they can use computers to simulate the whole forward imaging process including the nice-looking

It is often argued that prohibitive computational complexity and memory requirements are the obstacles to the wide adoption of ART (e.g., in [

The rest of the paper is organized as follows. We first briefly review the current state-of-the-art of ART with an emphasis on its applicability to the so-called continuous-to-discrete models in Section

Input: measurement data

density compensation function

Output: reconstructed image

(i) Initialization: set

ART;

(ii) Main loop: for

filter in (

An issue fundamental to the mathematical modeling of tomographic imaging (forward scanning) is how continuous spatial information is encoded into discrete measured data. Figure

(a) An example of nonuniform sampling lattice in

The art of modeling lies in the heuristics of approximation and their computational implications. Linearized forward models are often adopted for their computational tractability; that is,

Surprisingly, experimental findings related to ART seem scarce in the open literature. Standard textbooks such as [

PSNR profiles of ART schemes from simulated MRI data (

Comparison of visual quality of reconstructed images from

Comparison of visual quality of reconstructed images from

Simulation of forwarding MRI encoding is based on the image reconstruction toolbox (available at

Figure

In this paper, we consider a new class of nonlocal regularization techniques. Unlike previous variational formulation such as [

In our recent work [

In recent years, there has been a flurry of works on nonlocal image processing (e.g., nonlocal mean denoising [

It is straightforward to incorporate nonlocal regularization into ART under the framework of alternating projections [

PSNR profiles of ART schemes from simulated MRI data (

Figures

Comparison of reconstruction errors by TV-ART (left) and NR-ART (right).

Comparison of visual quality of reconstructed images: (a) TV regularization (

Comparison of visual quality of reconstructed images: (a) TV regularization (

What is wrong with using simulated

For real-world

In this section, we want to demonstrate the potential benefit of nonlocal regularization in reducing the scanning time of MRI. Fast MRI scanning has been a hot topic in the MRI community, and various parallel imaging techniques (e.g., SMASH [

Comparison of images from undersampled spiral

Figure

For a collection of real-world

Figure

(a) The diagram of resolution-consistent NR-ART; (b) the overlap of two sampling lattices (“

Comparison of images reconstructed at two different resolutions: (a)-(b) LR and HR images obtained from the gridding algorithm [

Figures

Enhanced images of Figures

In this paper, we have experimentally studied the extensions of ART into MRI reconstruction: NR-ART for simulated

Several significant issues remain open. First, mathematical modeling of the MRI forward scanning process is the source of systematic errors causing varying interpretations of experimental results with simulated and real-world

To make sure this research is fully experimentally reproducible, we have established a homepage for this project at

This work was partially supported by Grant NSF-CCF-0914353.