Basic Tasks of Signal Processing in Spectroscopy
Challenges with quantification of time signals
The quantum-mechanical concept of resonances in scattering and spectroscopy
Resonance profiles
Why is this topic relevant to biomedical researchers and clinical practitioners?
The Role of Quantum Mechanics in Signal Processing
Direct link of quantum-mechanical spectral analysis with rational response functions
Expansion methods for signal processing
Recurrent time signals and their generating fractions as spectra with no recourse to Fourier integrals
Fast Padé transform (FPT) for quantum-mechanical spectral analysis and signal processing
Padé acceleration and analytical continuation of time series
Description of the background contribution by the off-diagonal FPT
Diagonal and para-diagonal FPT
Froissart doublets and the exact number of resonances
Harmonic Transients in Time Signals
Rational response function to generic external perturbations
The exact solution for the general harmonic inversion problem
General time series
Response or Green function
The key prior knowledge: internal structure of time signals
The Rutishauser quotient-difference recursive algorithm
The Gordon product-difference recursive algorithm
The Lanczos continued fractions
The Padé–Lanczos approximant
FPT(−) outside the unit circle
FPT(+) inside the unit circle
Signal-Noise Separation via Froissart Doublets
Critical importance of poles and zeros in generic spectra
Spectral representations via Padé poles and zeros: pFPT(±) and zFPT(±)
Padé canonical spectra
Signal-noise separation: exclusive reliance upon resonant frequencies
Model reduction problem via Padé canonical spectra
Denoising Froissart filter
Signal-noise separation: exclusive reliance upon resonant amplitudes
Padé partial fraction spectra
Model reduction problem via Padé partial fraction spectra
Disentangling genuine from spurious resonances
Padé Processing for Magnetic Resonance (MR) Total Shape Spectra from in vivo Free Induction Decays (FIDs)
Comparison of the performances of the FPT and fast Fourier transform (FFT) for total shape spectra
The FIDs, convergence regions, and absorption spectra at full signal length for 4T and 7T
Convergence patterns of the FPT(−) and FFT for absorption spectra at 4T and 7T
Error analysis
Prospects for comprehensive applications of the FPT to in vivo MR time signals for brain tumor diagnostics
Exact Reconstructions of Spectral Parameters by FPT
Tabular data
Absorption total shape spectra
Residual spectra and consecutive difference spectra
Absorption component shape spectra of individual resonances
Distributions of reconstructed spectral parameters in the complex plane
Discussion
Relevance of exact quantification in brain tumor diagnostics
Machine Accurate Padé Quantification and Exact Signal-Noise Separation
Numerical presentation of the spectral parameters
Direct comparison of the performance of the FFT and the FPT
Convergence of total shape spectra versus component spectra in the FPT
Signal-noise separation through the concept of Froissart doublets/pole-zero cancellation
Diagnostic significance of the Froissart filter for exact signal-noise separation
Magnetic Resonance Spectroscopy (MRS) and Magnetic Resonance Spectroscopic Imaging (MRSI) in Neuro-Oncology: Achievements and Challenges
MRS and MRSI as a key non-invasive diagnostic modality for neuro-oncology
Major limitations and dilemmas with MRS and MRSI in neuro-oncology
related to reliance upon conventional Fourier-based data analysis
Accurate extraction of clinically relevant metabolite concentrations for neurodiagnostics via MRS
Padé Quantification of Malignant and Benign Ovarian MRS Data
Studies to date using in vivo proton MRS to evaluate benign and malignant ovarian lesions
Insights for ovarian cancer diagnostics from in vitro MRS
Performance of the FPT for in vitro MRS data derived from benign and malignant ovarian cyst fluid, and comparisons with the FFT
Prospects for Padé-optimized MRS for ovarian cancer diagnostics
Breast Cancer and Non-Malignant Breast Data: Quantification by FPT
Current challenges in breast cancer diagnostics
In vivo MR-based modalities for breast cancer diagnostics and clinical assessment
Insights for breast cancer diagnostics from in vitro MRS
Performance of the FPT for MRS data from breast tissue
Prospects for Padé-optimized MRS for breast cancer diagnostics
Multiplet Resonances in MRS Data from Normal and Cancerous Prostate
Dilemmas in prostate cancer diagnostics and screening
Insights for prostate cancer diagnostics by means of 2D in vivo MRS and in vitro MRS
Performance of the FPT for MRS data from prostate tissue
Prospects for Padé-optimized MRSI within prostate cancer diagnostics
General Discussion
Why the FPT for signal processing?
The two variants of the FPT converging inside and outside the unit circle
Computation of the complex frequencies and amplitudes by FPT
Interpolation and extrapolation by the FPT
Determination of the exact number of metabolites
Lorentzian and non-Lorentzian spectra both computed by FPT
Validity assessment of the FPT
Error analysis
Clinical ramifications of implementing Padé-based in vivo MRS: special importance for cancer diagnostics
Conclusions and Outlooks
Prediction and extrapolation for resolution improvement
