WO2000043906A1 - Procede de traitement de signaux multiexponentiels et appareil - Google Patents

Procede de traitement de signaux multiexponentiels et appareil Download PDF

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WO2000043906A1
WO2000043906A1 PCT/IB2000/000212 IB0000212W WO0043906A1 WO 2000043906 A1 WO2000043906 A1 WO 2000043906A1 IB 0000212 W IB0000212 W IB 0000212W WO 0043906 A1 WO0043906 A1 WO 0043906A1
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resolution
data
function
computer
readable medium
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PCT/IB2000/000212
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Keith Cover
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University Of British Columbia
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/17—Function evaluation by approximation methods, e.g. inter- or extrapolation, smoothing, least mean square method
    • G06F17/175—Function evaluation by approximation methods, e.g. inter- or extrapolation, smoothing, least mean square method of multidimensional data

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  • This invention is concerned with signal processing of transients, and more particularly, with multiexponential signal processing.
  • U.S. Patent No. 5,517,115 issued May 14, 1996 to Manfred G. Prammer proposes a method and apparatus for efficient processing of nuclear magnetic resonance (NMR) echo trains in well logging.
  • NMR nuclear magnetic resonance
  • a priori information about the nature of the expected signals is used in an attempt to obtain an approximation of a model using a set of pre-selected basis functions.
  • a singular value decomposition (SVD) is applied to a matrix incorporating information about the basis functions, and is stored off-line in a memory.
  • the apparatus estimates a parameter related to the signal-to-noise ratio (SNR) of the received NMR echo trains and uses it to determine a signal approximation model in conjunction with the SVD of the basis function matrix. This approximation is used to determine, in real time, attributes of the earth formation being investigated.
  • SNR signal-to-noise ratio
  • An object of the present invention is to provide improved methods and apparatus for the analysis of transients and for obtaining useful information therefrom. More particularly, an object of the invention is to provide improved multiexponential signal processing.
  • One aspect of the invention involves a computer- readable medium containing a set of coefficients that define a transform operator such as a matrix. Another aspect of the invention involves a method of calculating a transform operator utilizing a plurality of resolution functions.
  • Another aspect of the invention involves a method of multiexponential signal processing in which multiexponential signals are sampled, and the above-mentioned transform operator is applied to the sampled signals.
  • Another aspect of the invention involves an apparatus for multiexponential signal processing that comprises a signal processor including the above-mentioned transform operator.
  • the present invention starts with the construction of appropriate transform operators. Once appropriate transform operators have been constructed, they are incorporated in signal processors of analytical instrumentation for processing data.
  • instrumentation includes a computer, as is well known in multiexponential analysis.
  • Multiexponential data signals are sensed or detected by conventional equipment and are input to the transform operator of the computer for signal processing.
  • the signals may be applied in real time or they may be read out from a suitable storage medium.
  • the signals are applied to the transform operator in digital form after conventional sampling and analog- to-digital conversion. For example, digital samples of multiexponential decays may be obtained at equally-spaced instants in time, beginning at or just after the start of a multiexponential signal.
  • the invention makes use of linear resolution to obtain a better estimate of an unknown model.
  • the present invention provides the very desirable property of optimal linear resolution of the unknown model when plotted against the log of the time constant of the decay curves.
  • Fig. 1 is a diagram showing a matrix with coefficients a lx . . . aj rc for transforming data d j . . . d N to produce parameters m x . . . a, of an estimate of an unknown model;
  • Fig. 2 is a flow chart showing a method for calculating the coefficients of a row of a transform matrix in accordance with the invention
  • Fig. 3 is a table showing, inter alia, coefficients for three rows of a matrix, for ⁇ ' s of interest, namely 1, 10, 100;
  • Fig. 4 is a diagram showing data functions, resolution functions, noise response, and point spread functions for a matrix constructed in accordance with the invention where the noise gain (NG) is 1.0000;
  • Fig. 5 is a similar diagram for another matrix constructed in accordance with the invention for a noise gain of 3.1623;
  • Fig. 6 is a diagram showing resolution functions for four matrices constructed in accordance with the invention with different noise gains
  • Fig. 7 is a diagram showing point spread functions associated with four matrices constructed in accordance with the invention for different noise gains
  • Fig. 8 is a block diagram showing apparatus for processing data in accordance with the invention
  • Fig. 9 is a diagram showing decay curves of an MRI
  • Fig. 10 is a diagram showing MRI relaxation distributions for four matrices constructed in accordance with the invention with different noise gains.
  • One of the principal objectives of the present invention is to provide signal processing of transients, such as multiexponential decays, producing outputs that are better estimates of an unknown model to be investigated. Such estimates permit better interpretation of data, so that a user (researcher, physician, scientist, or engineer, for example) can obtain more accurate information as to the nature of the unknown model .
  • the invention may be looked upon as a better digital lens that provides improved resolution, just as a better optical lens provides improved resolution.
  • a first step in achieving the objectives of the invention is to construct an improved transform .operator, conveniently in the form of a matrix.
  • a transform matrix may be constructed, pursuant to the invention, will be described in detail later.
  • the actual transform operator will depend upon its intended application, for example, medical imaging or well logging. For any application, several different transform operators may be constructed, to provide a user with greater flexibility.
  • the present invention requires an understanding of what is referred to in the art as estimating a solution of a linear inverse problem.
  • the linear inverse problem is one of communicating to an interpreter what is known and what is not known about an unknown model.
  • the almost universal practice in the prior art for estimating the solution of a linear inverse problem is to calculate one or more of an infinite number of estimates of the model which fit the data, i.e., which reproduce the data to within the noise that is present.
  • a priori information is used to choose which of the estimates to calculate.
  • a priori information may not be sufficiently available or may be suspect.
  • an estimate of an unknown relaxation distribution is obtained by linearly resolving each point of the unknown relaxation distribution as precisely as possible within the limits of the noise.
  • estimates do not reproduce the data. Rather, they are obtained by optimizing the linear resolution in a manner that will be described later.
  • the transform operator A is defined as "linear” if for any two real numbers a and b
  • A(a xl + b x2) a yl + b y2.
  • a transform operator which is a matrix will always have linear resolution. It is possible to construct non-matrix transform operators which behave similarly to matrices for only a restricted set of functions (or vectors) . Moreover, a matrix can be derived from a non-matrix transform operator that expresses this behavior.
  • the present invention involves the following relationship between a true model m ⁇ (y) and data d k (e.g., multiexponential decays) :
  • NMR nuclear magnetic resonance
  • d k represents the data
  • ⁇ represents the unknown model
  • -e ⁇ f */» ⁇ represents the data function, where T designates the time constant of the exponential decay.
  • ⁇ () is the Dirac delta function
  • the range of indices such as i, j and k are determined by the equation in which they are used. For example in the equation
  • the index i ranges over the rows of a transform matrix and the index j ranges over the columns of the transform matrix, which is the same range as the data functions in the previous equation.
  • m ⁇ (y) represents the unknown model
  • e ⁇ *** ⁇ * represents the data function, which may be expressed as :
  • a first step in achieving the objectives of the invention is to construct a transform operator, such as a transform matrix, which maps data (e.g. decay signals) to unknown model parameters.
  • Fig. 1 is a diagram showing a matrix with coefficients a xl . . . a i ⁇ for transforming data dj , . . . d j , to produce parameters m x . . . m k of an unknown model.
  • the transform matrix is typically a matrix in which each row corresponds to a r of interest. Selection of ⁇ 's of interest and data points for initial coefficients will be guided by available information in the particular field in which the matrix is to be used.
  • the estimate of the unknown model can be calculated by multiplying the data by a matrix.
  • the matrix is chosen so that each point of the estimate of the unknown model linearly resolves the corresponding point of the model as well as possible with an acceptable noise gain i.e., with optimal linear resolution. Since matrix multiplication is a linear operation, it yields an estimate which does not necessarily reproduce the data, but which does have linear resolution. Linear resolution is a desirable property of an estimate, because each point of the estimate resolves the corresponding point of the model in the same way, independent of any particular model.
  • a goal in constructing a transform matrix in accordance with the invention is to calculate linear combinations of the data functions which yield a resolution function that resolves as small a region of the unknown model as possible. Accordingly, it is preferred that each resolution function corresponding to a row of the matrix be characterized by optimal linear resolution. Preferred criteria for selecting coefficients of a transform matrix which yield optimal linear resolution are described later. Together, the resolution function and its noise gain give a concise formulation of the ambiguity of a point in an estimate of an unknown model from information given by data and corresponding data functions.
  • each row normally corresponds to a T of interest.
  • the rows can be ordered by increasing r from top to bottom of the matrix. A spacing of 16 r values per decade has been found to work well. As an example, r values may be calculated between 0.1t min and lOt ⁇ , where t min is the smallest time at which a decay signal is to be sampled and t max is the largest time.
  • t min is the smallest time at which a decay signal is to be sampled
  • t max is the largest time.
  • Each row of the matrix corresponds to a resolution function that is centered on a T of interest. It is presently believed that the best way to calculate the coefficients of a particular transform matrix (which, incidentally, may have only one row) is to solve a constrained minimization problem.
  • the information needed before the constrained minimization can be performed are (1) the data function,
  • the trial resolution function preferably complies substantially with the following constraints: (i) R (y) > 0 [Nonnegative constraint] (ii) R(y k ) > 1 [Peak constraint] (iii) R(y)monotonically decreases from y k [Monotonicity constraint]
  • Constraint (iii) may be satisfied automatically for the data function used for the multiexponential problem.
  • Constraint (iii) may be satisfied automatically for the data function used for the multiexponential problem.
  • Constructing a transform matrix in which each resolution function corresponding to a row of the matrix is characterized by optimal linear resolution preferably involves a process termed constrained optimization.
  • this process includes an outer loop, a middle loop and an inner loop.
  • the outer loop converts the constrained optimization problem to a series of unconstrained multidimensional minimization problems using the simple penalty method described by Fletcher (Fletcher, R. 1987 Practical Methods of Optimization; Toronto; John Wiley & Sons) incorporated herein by reference.
  • the middle loop converts the unconstrained multidimensional minimization problems into a series of one dimensional (ID) minimization problems via the conjugate gradient method described by Press et al .
  • the simple penalty method converts the constrained optimization problem to an unconstrained one via residues, r m .
  • a residue is a measure of how much a particular constraint is violated. If a particular residue is greater than zero, then the corresponding constraint is considered satisfied. The more negative the residue, the more the constraint is violated.
  • the constrained optimization problem then becomes the unconstrained one of minimizing I p in the equation:
  • I P ⁇ .+P ⁇ mm(r m ,0) 2 ⁇ "> i m
  • min(x,y) returns the minimum and x and y.
  • the value of P starts small, and is then increased by a small factor, for example, 0.1, after each unconstrained minimization.
  • the solution to one constrained minimization is used as a starting point for the next constrained minimization.
  • the value of P is increased until I p has stabilized.
  • the test for stability is whether the value of I p changes by an accumulative factor of 10 "6 over four consecutive increases in P.
  • the data functions and the resolution functions are discretized.
  • the discretized data functions, gi (y) are represented as g A1 .
  • the total number of points at which the functions are sampled is denoted L.
  • the discretized trial resolution function is represented by R ⁇ .
  • the conjugate gradient technique is explained by Press et al .
  • the termination conditions used may be an accumulative factor change in I p of no more than 10 "7 over 25 consecutive iterations.
  • the total number of iterations can be limited to 3000*N, where N is the number of data points.
  • the one dimensional minimization problems are solved using the golden section search or parabolic interpolation described by Press et al.
  • the ID optimization is terminated if the functional value does not decrease by a certain fraction each step.
  • the fraction can be set to 10 "8 , for example.
  • a step limit of 1000 on each ID optimization can be set to prevent infinite or unproductive near infinite loops .
  • the linear transform for the trial row coefficients to the trial resolution function would usually be computed a large number of times during each one dimension optimization and would be computationally demanding. However, since the relationship between the trial row coefficients and the trial resolution function is always linear, this linearity can be used to greatly reduce the number of times the trial resolution functions have to be calculated directly from the trial coefficients.
  • a scalar parameter c is varied along a line with direction d x ; the direction being provided by the conjugate gradient method. Therefore, for the trial coefficients b , optimization occurs along a line defined by
  • the linear transform from the trial coefficients to the trial resolution function only needs to be calculated twice, to calculate R x and d 1# instead of once for every value of c considered. This results in a many fold reduction in the one dimensional optimization time.
  • the effectiveness of the minimization can be checked by how close the value of the peak of the resolution function is to unity, since the minimum area should yield a peak value of exactly unity.
  • noise gain, NG of a particular point in an estimate can be applied by the noise gain constraint stated earlier, where the area of the resolution function is included because the area of the resolution function must be normalized (set equal to 1) before the noise gain is calculated.
  • Figure 2 is a flow chart for calculating the coefficients of a row of a transform matrix pursuant to the foregoing description, using a digital computer conventionally. Coefficients for successive rows of the matrix can be calculated in the same manner for each T of interest and for each noise gain deemed appropriate. For calculating the coefficients of any row, the coefficients of the preceding row can be used as starting points..
  • Constrained optimization methods may perform better if all the data have noise with a standard deviation of 1.
  • the adjusted data functions are then supplied to the constrained optimization method along with the assumption that the standard deviations of noise at all the data points are 1.
  • the final coefficients, a i;j produced by the constrained optimization based on the adjusted data functions, need to be corrected for the adjustment.
  • the correction to the final coefficients is
  • a data set has 100' s, 1000' s, or even more evenly spaced points on a decay curve, it is much more efficient computationally for calculating the coefficients of the transform matrix (and also for applying it to data) to average adjacent data points together to create a new group data point.
  • the corresponding data functions must also be averaged together to get the grouped data function corresponding to the new grouped data points.
  • the size of the groups is important. While, in general, it is best to have larger groups at later sample times, groups which are too large will reduce the resolution of the resolution functions. Groups which are too small are inefficient. A logarithmic group size appears to be a good choice.
  • data will be measured by averaging over a window between two points in time.
  • the data function can be approximated by averaging together a larger number of sample points over the window.
  • a trigger that initiates a decay curve is not a single point in time.
  • the flash of light triggering the decay will last a finite length of time. If the intensity versus time for the flash is L(t) then each data point will be convolved with this function.
  • the corresponding data functions will also have to be convolved with the same function.
  • the next 16 data functions are evenly spaced by 30 time units between time 62 and time 512.
  • the standard deviation of the noise for the first 32 points is assumed to be 1, and the standard deviation of the noise of the next 16 is assumed to be 0.18257.
  • the standard deviation of the noise of the last 16 is a factor of l/sqrt(30) less than the first 32. This drop in the standard deviation of the noise would result if the cut-off frequency of a low pass filter before the analog- to-digital converter were dropped by a factor of 30 before the point was acquired.
  • the matrix was constructed without the use of balancing functions or data function groups.
  • Fig. 4 illustrates data functions, resolution functions, noise response, and point spread functions for a matrix constructed in accordance with the invention where the noise gain is 1.000.
  • Each resolution function corresponds to a set of data functions.
  • the abscissa in each diagram is in units of T on a log scale [ln( ⁇ )], and each resolution function is localized about a particular T value. As a result, each point spread. function tends to be localized about a particular T value.
  • Fig. 5 is a diagram similar to Fig. 4 but for a matrix constructed with a noise gain of 3.1623. It should be noted that in each of Figs. 3, 4, and 5 there are 48 data points (time samples) . A comparison of resolution functions produced by matrices with different noise gains is shown in Fig. 6. Higher resolution is achieved with higher noise gains, but there is a trade-off between resolution and noise gain. Greater noise tends to make the results achieved less reliable.
  • performance of a transform matrix can be judged using the PSF's and noise response.
  • information is required as to corresponding time points on the decay curve at which data should be acquired as well as assumed noise at each data point.
  • Fig. 7 shows point spread functions associated with four matrices constructed in accordance with the invention for different noise gains.
  • the noise may be assumed to be Gaussian.
  • the standard deviation of the generated noise is scaled so that the standard deviation of the noise of the first data point is 1.
  • the sampled noise decay curves are multiplied by the transform matrix to obtain the relaxation distribution.
  • Several realizations of the relaxation distribution of the noise can be plotted to obtain a "feel" for the distribution.
  • the noise gain for each point can be defined to be the standard deviation of the point divided by the standard deviation of the noise in the data and can be calculated directly from the coefficients
  • each transform matrix is constructed so that each point of the estimate of the unknown model linearly resolves the corresponding point of the unknown model as well as possible within an acceptable noise gain. Since matrix multiplication is a linear operation, it yields an estimate which does not necessarily reproduce the data but does have linear resolution. With linear resolution, each point of the estimate resolves the corresponding point of the unknown model in the same way independent of any particular unknown model.
  • a highly desirable property of linear resolution in providing an estimate of an unknown model is that it enables a human interpreter to obtain an intuitive "feel" of what the data reveals and does not reveal about the unknown model .
  • Each resolution function gives a concise mathematical description of the linear resolution at each point of the unknown model and is independent of the unknown model and the data. Viewing the transform matrix as a digital lens, linear combinations of the data functions are calculated which yield a resolution function that resolves as small a region of the unknown model as possible.
  • a transform matrix is constructed and selected, it is incorporated in a signal processor of a computer, as software or hardware, for example, as indicated in Fig. 8.
  • the INPUT represents a source of sampled digital signals such as multiexponential decays, e.g., NMR decays obtained from well-logging.
  • the DATA PROCESSOR and STORAGE are components of a conventional digital computer.
  • the OUTPUT MODULE may have conventional DISPLAY, NETWORK INTERFACE, and PRINTER COMPONENTS, for example.
  • An estimate of an unknown model is calculated by multiplying data, e.g., multiexponential decays, by the transform matrix.
  • data e.g., multiexponential decays
  • transform matrix for different noise gains, for example, can be incorporated in the signal processor and accessed selectively to provide a user with greater flexibility.
  • an object of the present invention is to provided a better estimate of an unknown model, from which useful information can be obtained.
  • Figure 8 of the patent is a mapping of estimated NMR signal decay times into pore sizes of an investigated earth formation.
  • the curve of Fig. 8 is an estimated relaxation distribution.
  • the present invention provides a better estimate of the relaxation distribution, and also a better signal-to-noise ratio (SNR) .
  • SNR signal-to-noise ratio
  • the present invention can be used to provide better estimates of an unknown model than the prior art, estimates from which more reliable and useful information can be obtained. These improved results are achieved by emphasizing linear resolution irrespective of whether data fits a model, and, in fact, without any attempt to fit data to a model.
  • the invention is particularly useful in multiexponential signal processing, such as in the analysis of multiexponential decays.
  • Fig. 9 shows decay curves in an MRI application.
  • the decay curves represent pixels taken from a series of 32 magnetic resonance images of the brain of a multiple sclerosis patient.
  • the T 2 relaxation data were acquired with a 10ms sample time out to 320ms.
  • the strength and decay of the signals contain valuable information about the tissues.
  • Some of the major tissue types of interest in multiple sclerosis are normal appearing white matter (NAWM) , cerebral spinal fluid (CSF) and lesions, which can be classified as chronic or acute.
  • NAWM normal appearing white matter
  • CSF cerebral spinal fluid
  • lesions which can be classified as chronic or acute.
  • FIG. 10 shows relaxation distributions yielded by applying to the decay curves of Fig. 9 transform matrices of the invention with various noise gains.
  • Estimating the noise in the estimates shown in Fig. 10 can be accomplished in several ways. The first is to estimate the noise in the data and multiply by the noise gain for each transform matrix. The ideal way to measure the noise in the data is to repeat the measurements a large number of times and calculate mean, standard deviation and covariance. These statistics can then be propagated through transform matrices using standard statistical procedures. Unfortunately, the measurement of the decay curves takes about 20 minutes to complete on a patient, so large numbers of repetitions are impractical.
  • a third way to estimate the noise is to consider that while the noise gain increases by a factor of 10 in Fig. 10, the resolution gain increases by only 1.4 for relaxation rates around five sample times. In Fig. 10, a portion of the signal between 30 and 200ms increases proportionately to the noise gain. This strongly suggests that the portion is due to random uncorrelated additive noises in the data. It is possible then to measure the standard deviation of the noise between 30 and 200ms and work back to the noise in the data.
  • decaying sinusoids are a common problem in inverse theory.
  • a decaying sinusoid can be handled if it is band pass filtered at the particular bandwidth of interest and then the magnitude of the decay curve taken. The relaxation distribution of the magnitude decay curve can then be calculated.
  • Principles of the invention can be applied to problems which data functions other than decay curves. If the data functions are cosine functions, resolution functions can be generated for low pass, band pass and high pass filters as well as windows for discrete Fourier transforms.
  • a low pass filter can be designed by requiring maximum area below the cutoff frequency, minimum area above the cutoff and an optional requirement of monotonicity to eliminate wiggles. The bounds on all parts of the filter would be 0 and 1.
  • a limit on broadband noise gain could also be imposed to improve the robustness of the filter.

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Abstract

Un opérateur de transformée (notamment matricielle) est réalisé dans le cadre du traitement de signaux de transitoires, comme les décroissances multiexponentielles, par la mise en évidence de la résolution linéaire au détriment de routines d'adaptation qui essaient de trouver une estimation d'un modèle inconnu qui s'adapte aux données. L'utilisation de l'opérateur de transformée pour traiter les signaux multiexponentiels produits des sorties qui sont une meilleure estimation du modèle inconnu ou d'un segment de ce modèle inconnu.
PCT/IB2000/000212 1999-01-19 2000-01-19 Procede de traitement de signaux multiexponentiels et appareil WO2000043906A1 (fr)

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Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5764058A (en) * 1996-09-26 1998-06-09 Western Atlas International, Inc. Signal processing method for determining the number of exponential decay parameters in multiexponentially decaying signals and its application to nuclear magnetic resonance well logging

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5764058A (en) * 1996-09-26 1998-06-09 Western Atlas International, Inc. Signal processing method for determining the number of exponential decay parameters in multiexponentially decaying signals and its application to nuclear magnetic resonance well logging

Non-Patent Citations (4)

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Title
KNISLEY J R ET AL: "A linear method for the curve fitting of multiexponentials neurophysiology application", JOURNAL OF NEUROSCIENCE METHODS, AUG. 1996, ELSEVIER, NETHERLANDS, vol. 67, no. 2, pages 177 - 183, XP000672459, ISSN: 0165-0270 *
NAJFELD I ET AL: "A robust method for estimating cross-relaxation rates from simultaneous fits to build-up and decay curves", JOURNAL OF MAGNETIC RESONANCE, FEB. 1997, ACADEMIC PRESS, USA, vol. 124, no. 2, pages 372 - 382, XP000672642, ISSN: 1090-7807 *
SUN T ET AL: "Analysis of double exponential fluorescence decay behavior for optical temperature sensing", REVIEW OF SCIENTIFIC INSTRUMENTS, JAN. 1997, AIP, USA, vol. 68, no. 1, pt.1, pages 58 - 63, XP002139860, ISSN: 0034-6748 *
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