These structures arise not only from the noise associated with encoding, but are also produced by the highly over-determined system of linear equations, as well as from numerical computations themselves whenever the finite arithmetic precision is used, as in our work. This over-determination is, in fact, imposed by both noise and a high spectral density, such as those associated with the MRS data encoded from the brain.

### Keywords/Phrases

For benchmarking purposes, the present study extensively applies the iterative averaging procedure, all the way to the 9th iteration. With regard to the total shape spectra or envelopes, the progressive disappearance of all spurious structures is shown. The 7th, 8th and 9th iterations generate converged envelopes, with the corresponding average envelopes being practically indistinguishable. The large spikes seen on Fig. This was examined both within the interval [, ] of the K values and for K above that interval.

The latter case for K further confirms the extrapolation capabilities of the FPT, as a rational polynomial. It should be noted that, in sharp contradistinction to the FPT, when ordinary polynomials are employed as in Fourier processing to approximate a function in a given interval, they generally perform inadequately outside that interval. This multiple cross-validation is the prerequisite for estimation reliability which is critical to clinical MRS-based diagnostics.

All the signal processors are vexed with pronounced sensitivity to changes in the otherwise unknown model order K , which is the number of the component metabolites in the spectral analysis quantification. Via spectra averaging, this sensitivity can be fully suppressed, such that, at convergence, the shape estimations are stabilized.

Since our SRI is between 0. Thereby, the residual and still giant water resonance is automatically bypassed. In contrast, for the non-parametric FPT, it is not possible to pre-select the frequency region of interest without a windowing procedure which is needed to suppress the water residual.

In our earlier study [ 14 ], we confirmed that within the non-parametric FPT, the windowing procedure with a step function fully preserves the spectral information, excepting the non-consequential part at the edges of the selected window. All told, the advantages of the parametric FPT compensate for the extra processing time needed. Not only is the water suppression problem completely obviated, but the most abiding aim of MRS, namely, quantitative analysis of the metabolic content of the scanned tissue can be accomplished forthwith through the parametric FPT.

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## Dimitris Giannakis

Moreover, by avoiding the water and other noisy content, the dimensionality of the problem is diminished, i. Going beyond total shape spectra to analyze the components, as achieved through the parametric FPT, is the most important goal, especially given the high density of MR spectra from the brain. Fitting procedures cannot even ascertain the number of peaks underlying, e.

The relative abundance of the resonances in these crowded spectral regions would, therefore, be entirely equivocal if one relied upon the conventional Fourier analysis with post-processing via fitting.

## Uncertainty principle

As noted, the chemical shift region around 1. However, in vivo proton MRS studies [ 24 , 52 ] indicate that much uncertainty arises in attempts to assess this chemical shift region. Not only were each of the resonances in the chemical shift region of 1. The latter case was verified by detailed examination of convergence of all four spectral parameters.

Phosphocholine, PC, another indicator of hypoxia [ 12 ], and which is also a biomarker of malignancy [ 15 , 16 , 17 , 18 ], was likewise identified and its peak parameters accurately assessed via the FPT. This was the case despite the fact that PC overlaps extremely closely with GPC and the much more prominent free Cho resonance centered at 3.

Wide-ranging possibilities emerge for multivariate exploration to find metabolite patterns that best characterize various types and grades of cancer versus diverse benign pathology which cause differential diagnostic dilemmas. Comparisons with normal, non-infiltrated tissue are also essential. This perspective holds promise for improved clinical diagnostics through in vivo MRS.

In Ref. The FIDs were zero-filled 8 times to a total signal length of points. Analysis of the computed FFT spectra was also carried out in the interval between 0. Further, fitting in Ref. Therein a high level of precision was attained, as seen in very small standard deviations obtained from the analysis of the reconstructed parameters from several consecutive values of the model order or rank K , after convergence had been reached.

A similar approach is applied in the present paper, for the first time, on in vivo MRS time signals encoded from human brain. By scrutinizing the nine consecutive iterations, progressively fewer and smaller discrepancies appear, until these reach the minimal level which is consistent with stochasticity contained in the encoded time signals. As to the rate of convergence of the spectral parameters, consistent with our previous study [ 21 ] on synthesized time signals associated with in vitro data encoded from ovarian lesions [ 50 ] and with controlled levels of added noise, the chemical shifts converged most rapidly, whereas the peak widths were the slowest to achieve convergence.

These slowly converging peak widths are noted particularly, though not exclusively, as being located deep in the complex frequency plane.

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Broadly speaking, we have herein addressed a type of inverse problem, whose basic characteristic is mathematical ill-conditioning or ill-posedness. This difficulty is due to the lack of continual dependence of observables on the independent variables time in the FID for the studied harmonic inversion problem. Most troublesome is that the solution to an inverse problem is, in principle, non-unique. In other words, this ill-conditioning is seen in that some markedly different solutions are obtained for even a minimal change of the input data e. With regard to total shape spectra, this ill-conditioning is reflected in the sensitivity of lineshapes to changes in model order K.

The procedure of iterative averaging of envelopes through the FPT computed for a sequence of values of K is shown to be capable of fully stabilizing the total shape spectra. In other words, the set of stabilized fundamental spectral parameters complex frequencies and complex amplitudes represent the unique solution to the harmonic inversion problem. The fundamental frequencies and amplitudes are the parameters in the partial fractions that contain the complete information about the examined system.

Thereby, quantum mechanics, through its completeness relation, fully and exactly parametrizes any system. Stable dynamical parameters are the key to the stability of a given system, and this is the hallmark of a robust system. This stability can be achieved through the iterative averaging procedure, as has been thoroughly demonstrated herein. Skip to main content Skip to sections.

Advertisement Hide. Download PDF. Iterative averaging of spectra as a powerful way of suppressing spurious resonances in signal processing. Open Access. First Online: 19 October However, the tightly packed, harmonically oscillating and exponentially attenuated waveforms are difficult to interpret directly from the FID. By mapping the FID into the frequency domain via mathematical transforms, a spectrum is generated which is more amenable to interpretation.

We will present the component spectra in two different modes. Consequently, the interference effects are eradicated via the said external suppression. This generates pure absorptive Lorentzians, that can be helpful for visualization purposes. The real part of the encoded FID is depicted on the top left panel a , with the imaginary part on the top right panel d. Moreover, the waveforms are asymmetric around the abscissae because the residual water peak is still about times more abundant than all the other metabolites. Open image in new window. The first set of iterates is seen in Fig.

Therein, many large noise-like spikes are observed.

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In panel b of Fig. Thus, in this average spectrum, the stable structures remain, while the spikes are markedly attenuated or have practically disappeared. As a reminder, the real and imaginary parts of the originally encoded FID from Fig.

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We now proceed to further iterations and testing the robustness of the stability of the average envelopes. The 2nd iteration with the 31 newly obtained envelopes from panel b has notably fewer spikes, and these are of much smaller heights compared to the corresponding spurious structures from the 1st iteration, shown in panel a.

Averaging is performed once again, and this time for the 31 complex envelopes whose real parts are from panel b of Fig. This is the 3rd iteration, whose spurious structures are further attenuated and much sparser compared to the 2nd iteration on panel b. In panel d , the real parts of the complex 1st, 2nd and 3rd arithmetic averages are overlain, in the respective colors of green, magenta and blue, following the like colors on panels a , b and c. It is seen therein that these three curves closely coincide, with just a few scattered small deviations.

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The entire procedure displayed in Fig. The spikes have practically disappeared in panels a , b and c , for iterations 4, 5 and 6, respectively. The 4th, 5th and 6th averages presented in panel d of Fig.