An Overview of Biomedical Imaging

Abstract

Biomedical imaging is critically important for life science and health care. In this rapidly developing field, mathematics is one of the most powerful tools for developing image reconstruction as well as image processing theory and methods. Many of the innovations in biomedical imaging are fundamentally related to the mathematical sciences. With improvements of traditional imaging systems and emergence of novel imaging modalities such as molecular imaging towards molecular medicine, imaging equations that link measurements to original images become increasingly more complex to reflect the reality up to an ever-improving accuracy. Mathematics becomes increasingly useful and leads to a new array of interdisciplinary and challenging research opportunities. The future biomedical imaging will include advanced mathematical methods as major features.

Keywords

Biomedical imaging, Image reconstruction, Molecular imaging.

Introduction

It is a current trend that more mathematicians become engaged in biomedical imaging at all levels, from image reconstruction to image processing, and up to image understanding and various high-level applications. This special issue addresses the role of mathematics in biomedical imaging. The themes include theoretical analysis, algorithm design, system modeling and assessment, as well as various biomedical imaging applications. From 10 submissions, 7 papers are published in this special issue. Each paper was reviewed by at least two reviewers and revised according to review comments. The papers cover the following imaging modalities: X-ray computed tomography (CT), positron emission tomography (PET), magnetic resonance imaging (MRI), diffusion tensor imaging (DTI), electrical impedance tomography (EIT), and elasticity imaging using ultrasound.

The field of X-ray imaging has been expanding rapidly since historical discovery in 1895. X-ray CT, as the first noninvasive tomographic method, has revolutionized imaging technologies in general, which was also the first successful application of mathematics in biomedical imaging. The mathematics is the theory of Radon transform invented by Radon in 1917. Further research may rejuvenate this classic topic to meet modern imaging challenges such as scattering effects. In Truong et al.’s paper, the authors presented two further generalizations of the Radon transform, namely, two classes of conical Radon transforms which originate from imaging processes using Compton scattered radiation. The first class, called C1-conical Radon transform, is related to an imaging principle with a collimated gamma camera whereas the second class, called C2-conical Radon transform, contains a special subclass which models the Compton camera imaging process. They demonstrated that the inversion of C2-conical Radon transform can be achieved under a special condition.

Earlier work on total variation (TV) regularization for color (vector valued) images are naturally extended to DTI, which is composed of a symmetric positive definite (SPD) matrix at each pixel. In the last decade, a new magnetic resonance modality, DTI, has caught a lot of interest. DTI can reveal anatomical structure information. In this special issue, there are two papers on this imaging technique. In Christiansen et al.’s paper, this type of tensorvalued images is de noised using TV regularization. Recently, partial differential equation- (PDE-) based image processing methods have been very successful in many applications due to its intrinsic geometric nature. TV regularization, which can effectively remove noise while keeping sharp features, is one of the most important techniques for PDE-based image processing methods.