Multidimensional Nuclear Magnetic Resonance (NMR) spectroscopy is truly a multidisciplinary research area. It is born of advances in physics, engineering, signal processing, and the demands of biomolecular structural determination applications. One of the key bottlenecks for NMR is the acquisition time -- taking up to a week even for relatively modest multidimensional experiments. Attempts to overcome this bottleneck through hardware, pulse sequence, and signal processing methods have been making steady advances for almost for the entire history of NMR. However, there appears no broadly applicable technique in common use. Compressed Sensing is a framework that exploits signal sparsity for reconstructing signals sampled well below the Nyquist rate, and has already seen application in NMR in the context of acquisition time reduction. Here the motivation of exploiting prior signal knowledge was taken a step further than assuming just sparsity with the exploitation of the structural relationship between cross and diagonal peaks observable in many NMR experiments. The approach taken was to use weighted iterative soft, and hard thresholding with statistical updates according to the structural relationship between cross and diagonal peaks performed using belief propagation. This reconstruction method was applied to one of the quintessential examples of both multidimensional NMR, and experiments exhibiting signal structure: COSY. Simulations performed on a COSY spectrum for nonactin and lactose demonstrated that imposing structure on the reconstruction yielded good improvement to the normalised mean square error of the reconstruction. It was found that IST proved more stable than IHT, but in general had slightly higher reconstruction error. In the case of IST imposition of structure consistently improved reconstruction with reconstruction possible for upwards of 85% data removal. Reconstruction influenced by structure also reduced the artefacts in areas of the spectrum not included in the signal while improving the strength of peaks degraded by general iterative thresholding algorithms. The results are a positive reflection on the exploitation of structure that is common to a whole host of NMR experiments and is able to be generalised to higher dimensional experiments. This promises improved reconstruction quality with reduced acquisition time for multidimensional NMR in the future.
Multidimensional NMR의 주요 문제점의 하나는 데이터의 획득 시간이며 이미 Compressive Sampling을 이용하여 데이터 획득 시간을 줄인 NMR application이 존재한다. 본 연구의 동기는 데이터가 sparse 하다는 가정에 데이터의 prior knowledge를 안다는 가정을 덧붙인다. 이는 NMR실험에서 cross peaks와 diagonal peaks 간의 구조적인 관계를 연구하는데 도움이 된다.
Correlated Spectroscopy (COSY) 는 multidimensional NMR 실험에서 구조를 결정하기 위해 빈번하게 사용되는 방법이다. NMR 실험들은 diagonal peak cross peak 구조를 갖는다. 예를 들면, cross peak은 두 개의 diagonal peak 사이에만 존재한다. 그렇지 않다면 magnetization transfer 가 없게 된다. 따라서 diagonal peak는 cross peak의 존재 가능한 위치를 정의하게 된다. 만약 cross peak가 diagonal peaks 사이에 존재하지 않더라도, 그것이 존재할 위치는 한정되어 있으며 그 위치 역시 diagonal peaks에 의해 결정이 된다. 만약 cross peak가 존재한다면 그에 대응하는 두 개의 diagonal peaks의 위치를 결정할 수 있다. 또한 COSY spectra는 diagonal 에 대해 대칭이다. 이는 cross peaks에 대한 정보가 두 개의 diagonal peaks의 위치를 결정할 뿐만 아니라 opposing cross peak의 위치까지 결정한다는 것을 의미한다. 이러한 구조 연구에 Compressive Sampling을 이용하여 NMR spectra의 복원을 강화한 것이 본 연구의 의미라고 하겠다.