Abstract
Deciphering, or multiexponential analysis of signals in a nuclear magnetic resonance (NMR) machine, is necessary for a composite analysis (e.g., for separating oil fractions in well logging, detecting affected tissue in medicine). NMR signals are modeled by functions of the form \( f(x)=\sum {{A_i}{e^{{{\omega_i}x}}}} \), which are linear combinations of exponential terms; in these f, the parameters ω i correspond to different components, and A i characterize their relative weights in the composite. The number, and value of the parameters A i and ω i are not known, and so the multiexponential analysis consists in determining them (within some tolerance); this analysis relates to so-called inverse problems, usually characterized by a high degree of instability. This chapter contains real-life algebraic and combinatorial problems that arose in the development of a stable algorithm for multiexponential analysis.
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Notes
- 1.
Readers interested in theoretical fundamentals and application-oriented aspects of NMR may refer to Abragam (1961) and references therein.
- 2.
An exponential function is a continuous version of a discrete geometric progression: for q > 0, a⋅q k−1 = A⋅e ωk, ∀k∈ℤ, where q = e ω and a = A⋅e ω, or, equivalently, ω = ln q and A = a/q. (Actually, this holds for any nonzero complex q, using a fixed complex branch of ln q, but readers not experienced with complex numbers may consider only the case where q is a positive number.)
- 3.
In practice, the function f is known on a finite set of points (“time echoes” in NMR terminology) and, moreover, known to have an error. Usually, the error is random noise, which is assumed to be normally distributed (but a systematic error can also exist). Therefore, the correct formulation of the problem includes arguments of probability theory (Itskovich et al. 1996, 1998). We advise, however, ignoring the probabilistic aspect at this point and considering initially the resulting deterministic problem.
- 4.
The use of integrals is much more practical since it decreases the impact of uncertainties related to the assignment of F.
- 5.
A generalization for m subspaces is the principle of inclusion–exclusion
$$ \dim \sum {{A_i}} =\sum {\dim {A_i}-\sum\limits_{i<j } {\dim\,({A_i}\cap {A_j})+\ldots } } +{(-1)^{m-1 }}\dim\,\bigcap {{A_i}} . $$ - 6.
Tolerances ε 1, ε 2, … can be derived by probabilistic methods from standard deviations of F(x) (Itskovich et al. 1996). We advise, however, ignoring this subject upon first acquaintance.
- 7.
A similar principle holds for the formal power series, and smooth even (odd) functions have even (resp. odd) Taylor series at the origin (even when those series diverge). Indeed, all derivatives of a smooth even (odd) function that have odd (resp. even) orders vanish at the origin. (Why?)
- 8.
For a function f(x), this decomposition is \( f(x)=\frac{{f(x)+f\left( {-x} \right)}}{2}+\frac{{f(x)-f\left( {-x} \right)}}{2} \) and is unique. Advanced readers know that the even component is f’s average over a group of order 2, with the generator inverting the argument’s sign. An analogous decomposition into symmetric and skew-symmetric components, \( f\left( {{x_1},{x_2}} \right)=\frac{{f\left( {{x_1},{x_2}} \right)+f\left( {{x_2},{x_1}} \right)}}{2}+\frac{{f\left( {{x_1},{x_2}} \right)-f\left( {{x_2},{x_1}} \right)}}{2} \), corresponds to a group with the generator permuting the arguments (for n arguments, this group has order n!). A generalization of this concept brings analogous averaging formulas for any finite group of symmetries and, furthermore, integral formulas corresponding to any compact group of symmetries. To practice, readers can prove the following direct generalizations of the above claim about even polynomials:
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A polynomial on ℝn is invariant with respect to a reflection in the origin if and only if it contains terms of even degrees only.
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A polynomial on ℝn is invariant with respect to a group generated by reflections in the coordinate hyperplanes if and only if it is a polynomial in x 1 2,…, x n 2.
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A polynomial on ℝn is invariant with respect to an orthogonal group if and only if it is a polynomial in |x|2 = ∑x i 2 (with numerical coefficients).
(The similar claims hold for formal power series.) Readers may state and prove similar claims themselves using other groups of symmetries. Also, prove the following statement generalizing the fact that the sum of the coefficients of a polynomial p equals p(1):
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For a polynomial (or a power series of convergence radius exceeding 1) p(x) = a 0 + a 1 x + … of one variable with complex coefficients, the sum of coefficients a 0 + a m + a 2m + … equals \( {m^{-1 }}\sum\limits_{k=0}^{m-1 } {p({\xi^k})} \) , where ξ is a primitive root of degree m of 1.
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References
Abragam, A.: The Principles of Nuclear Magnetism. Oxford University Press, London/Toronto (1961)
Itskovich, G., Roytvarf, A.: Method of successive determination of parameters in the multiexponential analysis problem. Proc. Int. Seminar “Inverse problems in Geophysics”, vol. 181, Novosibirsk (1996)
Itskovich, G., Roytvarf, A.: Signal processing method for determining the number of exponential decay parameters in multiexponentially decaying signals and its application to nuclear magnetic resonance well logging. US 005764058A (1998)
Lang, S.: Algebra. Addison-Wesley, Reading/London/Amsterdam/Don Mills/Sydney/Tokyo (1965)
Van der Waerden, B.L.: Algebra II. Springer, Berlin/Heidelberg/New York (1967)
Venkataraman, C.S.: In: Problems and solutions. Math. Magazine 44(1), 55 (1971)
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Roytvarf, A.A. (2013). A Combinatorial Algorithm in Multiexponential Analysis. In: Thinking in Problems. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8406-8_3
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DOI: https://doi.org/10.1007/978-0-8176-8406-8_3
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