Stable Reconstruction of the T 2 Distribution by Low-Resolution NMR Measurements and the Classical Markov and Hausdorf Momentum Problem Gy. Steinbrecher,* R. Scorei,² V. M. Cimpoiasu,² and I. Petrisor‡ *Association EURATOM, International Working Group “Fusion B.F.R.,” Department of Theoretical Physics, ² Department of Biochemistry, and ‡Department of Experimental Physics, University of Craiova, 1100 Craiova, Dolj, Romania Received February 7, 2000; revised June 19, 2000 Assuming that an original distribution is a probabilistic measure and the Laplace transforms are known only for a finite number of points that are affected by errors, we develop a method forrecon- structing weak-sense mean values obtained by integrating smooth functions with the measure. Our method is useful in NMR if the NMR signal can be represented as a superposition of exponential terms. In these circumstances, we show how the data processing can be related to the classical Hausdorf momentum problem. First, we clarify the meaning of stable spectrum reconstruction, and then develop stable filtering and a mean value reconstruction algo- rithm. Ourmethod has been tested on both simulated and real sets of spin–spin relaxation curves with noise. In view of this, our method could provide an efficient and accurate reconstruction of spin–spin relaxation data. For any reader interested in applica- tions, a “practical recipe” that is almost self-consistent has been included. © 2000 Academic Press Key Words: time domain; NMR; relaxation; classical momen- tum problem; numerical Laplace transform inversion. I. INTRODUCTION In the interpretation of the spin–spin relaxation data, the first temptation is to invert the Laplace transform, in order to recover population densities and relaxation times. An almost similar mathematical problem was treated extensively in (1), where the difficulties related to the numerical inversion of the Laplace transform are solved in the special case of a known upper bound on the number of populations (exponential terms). In most situations, the restrictions imposed in (1) are not satisfied (there is no upper bound on the number of popula- tions, and, moreover, there can be an infinity of them). In this work, we solve such a generalized reconstruction problem. The final recipe is given in Section VII. In most previous works the computation of NMR parame- ters, such as relaxation times and populations, is strongly dependent on the particular details of the reconstruction algo- rithm; in other words, the reconstruction is unstable. In this paper we suppose that the dependence on time of the NMR ideal noiseless signal s ( t ) for T (spin–spin relaxation time) may be represented as (2) s  t  =  i =1 N n i exp-t / T i  , n i  0,  i =1 N n i = 1, T i  0, [1a] where n i and T i are the populations and the relaxation times, respectively. Due to lack of information on n i and T i , we are forced to consider the more general representation s  t  =  Tmin Tmax exp-t /   d     =  i =1 N n i exp-t / T i  +  Tmin Tmax exp-t /       d  . [1b] In Eq. [1b] () is a Borel measurable nondecreasing function whose continuous part of () is associated with a continuous relaxation time spectrum, typical in nonhomogeneous disperse or multiphase media, and the jump points of  (  ), ( T i ), cor- respond to the discrete spectrum. We denote by T min and T max the known bounds on relaxation times, obtained from other experiments or theoretical models. The simplest, but naı ¨ve, question that one may raise is how to compute the populations n i and the relaxation times T i , or, more generally, the distri- bution function () from the sequence of samples affected by errors ( s ' k = s ( k t ) +s k , where s k is the experimental error). As explained in Section III, the direct reconstruction of the populations n i and the relaxation times T i is impossible if N from Eq. [1a] is not known, because of the mathematical instability of the problem itself, even in the idealized noiseless case, when s ( t ) is known in a finite number of points. Never- theless, as we show in Section II, we could compute some mean values, of the form  T min T max f (  ) d(  ), for some especially Journal of Magnetic Resonance 146, 321–334 (2000) doi:10.1006/jmre.2000.2150, available online at http://www.idealibrary.com on 321 1090-7807/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.