The general concept of signal–noise separation (SNS): mathematical aspects and implementation in magnetic resonance spectroscopy

Abstract

Magnetic resonance spectroscopy (MRS) and spectroscopic imaging (MRSI) are increasingly recognized as potentially key modalities in cancer diagnostics. It is, therefore, urgent to overcome the shortcomings of current applications of MRS and MRSI. We explain and substantiate why more advanced signal processing methods are needed, and demonstrate that the fast Pade transform (FPT), as the quotient of two polynomials, is the signal processing method of choice to achieve this goal. In this paper, the focus is upon distinguishing genuine from spurious (noisy and noise-like) resonances; this has been one of the thorniest challenges to MRS. The number of spurious resonances is always several times larger than the true ones. Within the FPT convergence is achieved through stabilization or constancy of the reconstructed frequencies and amplitudes. This stabilization is a veritable signature of the exact number of resonances. With any further increase of the partial signal length N, towards the full signal length N, i.e., passing the stage at which full convergence has been reached, it is found that all the fundamental frequencies and amplitudes “stay put”, i.e., they still remain constant. Moreover, machine accuracy is achieved here, proving that when the FPT is nearing convergence, it approaches straight towards the exact result with an exponential convergence rate (the spectral convergence). This proves that the FPT is an exponentially accurate representation of functions customarily encountered in spectral analysis in MRS and beyond. The mechanism by which this is achieved, i.e., the mechanism which secures the maintenance of stability of all the spectral parameters and, by implication, constancy of the estimate for the true number of resonances is provided by the so-called pole-zero cancellation, or equivalently, the Froissart doublets. This signifies that all the additional poles and zeros of the Pade spectrum will cancel each other, a remarkable feature unique to the FPT. The FPT is safe-guarded against contamination of the final results by extraneous resonances, since each pole due to spurious resonances stemming from the denominator polynomial will automatically coincide with the corresponding zero of the numerator polynomial, thus leading to the pole-zero cancellation in the polynomial quotient of the FPT. Such pole-zero cancellations can be advantageously exploited to differentiate between spurious and genuine content of the signal. Since these unphysical poles and zeros always appear as pairs in the FPT, they are viewed as doublets. Therefore, the pole-zero cancellation can be used to disentangle noise as an unphysical burden from the physical content in the considered signal, and this is the most important usage of the Froissart doublets in MRS. The general concept of signal–noise separation (SNS) is thereby introduced as a reliable procedure for separating physical from non-physical information in MRS, MRSI and beyond.