MediMath Science

Innovation and Academia: Reflections on Progress and Challenges Over four decades in academia, I have often encountered the recurring themes of innovation, reform, and the call for pioneering research and development (R&D). The mantra of "High Risk, High Return" has emphasized long-term vision over short-term gains. Yet, this relentless focus has led to widespread fatigue, as many innovative efforts remain confined to academic circles, rarely transitioning into practical, impactful industrial applications. Academics who achieve breakthroughs often lack the resources or expertise to transform them into successful commercial products. Furthermore, industry-academia collaborations frequently fall short due to subtle yet pervasive challenges, leaving promising innovations as mere line items on résumés rather than societal advancements. Lessons from History: The Role of Constraints in Driving Innovation Developing effective R&D policies that encourage both technological an...

Recently, the field of medical imaging has witnessed numerous attempts aimed at producing high-resolution images with significantly insufficient measured data. These endeavors are motivated by a variety of objectives, such as reducing data acquisition times, enhancing cost efficiency, minimizing invasiveness, and elevating patient comfort, among other factors. Nevertheless, these efforts necessitate tackling severely ill-posed inverse problems, due to the significant imbalance between the number of unknown variables (needed for desired resolution) and the number of available equations (derived from measured data). For a clearer understanding, let's examine a linear system represented by $\mathbf{A}I = \mathbf{b}_{I} + \mathbf{\epsilon}$ , where $\mathbf{A}$ represents an $m \times n$ matrix with a highly underdetermined scenario ( $m \ll n$ ). This matrix $\mathbf{A}$ serves as a linearized forward model. In this formulation, $I$ is an $n$-dimensional vector representing the imag...