Temporal and spatial analysis of potential maps via multiresolution decompositions

Journal of Electrocardiology Vol. 29 Supplement

Temporal and Spatial Analysis of Potential Maps via Multiresolution Decompositions

Dana H. Brooks, PhD,* Robert S. MacLeod, PhD,-]- Ramdas V. Chary, BS* Richard J. Gaudette, BS,* and Hamid Krim, PhD$

A b s t r a c t : Cardiac potentials recorded on the epicardium or the body surface by an array of electrodes are usually analyzed either as spatial distributions or temporal waveforms. Thus, the analysis often involves temporal descriptors (eg, max dV/dt) or spatial descriptors (eg, location of local extrema) only. The best k n o w n transform technique that has been applied to these data that combines both spatial and temporal characteristics is the Karhunen-Loeve transform, a global transform applied to temporal and/or spatial bases obtained by statistical analysis of a database. As an alternative, multiresolution decompositions and related wavelet-type transforms have recently seen great development in signal processing and related fields. They offer flexibility, employing transformations onto local (rather than global) and fixed (rather than data-dependent) databases, and allow transformation of distributions, waveforms, or both, as desired. The utility of this method as applied to temporal and spatial segmentation and analysis of map data from both epicardial plaques and body surface potentials recorded during percutaneous transluminal coronary angioplasty is illustrated. K e y w o r d s : potential mapping, wavelets, muhiresolution.

The electrophysiologic behavior of cardiac cells is strongly influenced by that of the other ceils in their vicinity. Changes in extracellular potentials and currents in one location in the myocardium, w h e n viewed on a macroscopic scale, are expected to be highly correlated with what happens in nearby locations and nearby time intervals, and dramatic changes of state such as activa-

t i o n and repolarization tend to travel through regions of

the heart in a coherent and organized fashion. Even when changes are less localized in time or space (ie, they are slower or occur over larger regions in space), there is generally strong local correlation in both space and time. Thus, one tends to conceptualize the ventricular cardiac cycle i n terms of the presence or absence of wavefronts: the depolarization wavefront associated with the QRS during activation, the lack of any wavefront behavior during the ST-segment, and the repolarization wavefront associated with the T wave. On the body surface, o n e observes a smoothed, blurred, and attenuated image of the potential distribution that is present on the epicardium; thus, the temporospatial linkage is present in body surface potential distributions as well. This linkage between space and time is reflected in subtle ways in the relationship between measurements made at varying distances from the cardiac sources. At macroscopic scales, it is well accepted that there are n o

From *Communications and Digital Signal Processing Center, ECE Department, Northeastern University, Boston, Massachusetts, y-Nora Eccles Harrison Cardiovascular Research and Training Institute, Universify of Utah, Salt Lake City, Utah, and SLIDS, Massachusetts InstiCute of Technology, Boston, Massachusetts.

Supported by The Whitaker Foundation under a Biomedical Engineering Research Grant and the Richard A. and Nora Eccles Harrison TreadwelI Fund for Cardiovascular Research and awards from the Nora Eccles Treadwell Foundation. Reprint requests: Dr. Dana H. Brooks, ECE Department, 409 DA, Northeastern University, 360 Huntington Avenue, Boston, MA 02115.

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Multiresolution Analysis of Potential Maps temporal dynamics between the epicardial surface and extracardiac measurement locations, whether the measurements are made just above the epicardium or on the body surface. In other words, quasistatic assumptions are valid: the significant temporal components in electrocardiographic (ECG) potential signals are sufficiently low frequency compared to the spatial dimensions that potentials on the epicardium are sensed instantaneously by an extracardiac measurement electrode, and there is no temporal mixing, or filtering, in the intervening medium. On the other hand, it is also accepted that as one moves outward from the myocardium, one observes temporal waveforms that are increasingly smooth, or low pass in frequency. The explanation of this phenomen o n is to be found in the spatial relationships. For example, if one measures above a region where a depolarization wavefront is passing through the tissue, the measurement will reflect the average behavior of the tissue sensed by the electrode, where the average is spatially weighted via the effective current path from each location on the tissue to the electrode. Thus, close to the source one wilt see a very rapid, high-frequency change with time, but if the electrode is placed further away, the volunqe of tissue being sensed increases and the temporal change becomes increasingly more gradual and low freq u e n c y - t h e temporal and spatial behavior are linked. Patterns of variation of cardiac potential are also characterized by different behavior at different temporal and spatial scales. Scales range from microscopic behavior, to activation wavefronts, which are highly localized in space and time (ie, the wavefront exists only for a short time interval in any one spatial region, which is itself quite narrow), to somewhat less localized repolarization wavefronts (which exist over a larger region at a given time and exist for longer at any one location), to very low-frequency behavior such as injury currents, which can persist over significant regions of the myocardium for relatively long time intervals. Thus, there is a linkage not only between the spatial and temporal patterns, but also between their characteristic scale in space and time. When cardiac potential distributions are mapped by recording from an array of sensors, whether in the myocardium, on the epicardium, or on the body surface, there is a desire for conceptual and processing tools to describe the relevant features of the resulting maps and extract parameters of interest. On the one hand, there is an abundance of data with a great deal of redundancy; on the other hand, there is a scarcity of appropriate tools and models to describe and codify them. The classical models that are used in other areas of science to describe p h e n o m e n a with strong temporospatial links are wavefronts, which are usually framed in terms of sinusoids or complex exponentials with planar wavefronts, in the far field and traveling in linear media; however, none of these classical characteristics apply to cardiac potential maps. Thus, although there have been a few attempts to employ techniques based on these classical wavefront models, most researchers have either used more direct time-/space-domain descriptors, reduced the complexity by looking at only temporal or spatial descriptors, or

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used statistical techniques to focus on the most highly informative or correlated p h e n o m e n a only. In this article, we will discuss the use of a family of signal-processing techniques, k n o w n under the general heading of multiresolution decompositions, which have been developed within the last decade as the result of intensive research efforts. We will describe the basic idea behind these techniques and their connections to wavelet transforms (WT) and filter banks. We will also present a few examples of our own application of these techniques to illustrate what we see as their promise for analysis of potential maps.

Map Representations Because of the volume of data and the difficulty of distinguishing and quantifying important features, there have been a n u m b e r of techniques used to represent, display, or even compress the information in potential maps, each with its own strengths and weaknesses.

Time-/Space-domainTechniques One traditional way to study map data is as sequences of isopotential maps. Despite visualization problems, such as interpolation over saddle points, and despite their sensitivity to electrode location and noise, isopotential maps have one primary advantage: by leaving the interpretation in the eye of the h u m a n observer, they take advantage of our ability to interpolate in time and space, extract edges, etc., and thus, to "see" wavefronts and other coherent events as distinct from the background. From a processing viewpoint, isopotential maps present some problems: they implicitly impose a smoothness on the spatial distribution through the interpolant emp]oyed, but not on time, and they are notoriously difficult to quantify and compare. In response to the latter problem, researchers have generally resorted to global measures, such as relative squared error or correlation coefficients, whose inadequacies are well known, or they have extracted simple time- or spacedomain features, such as the position of the maxima or minima, or even of the angle of the zero line (despite the fact that zero potential is somewhat arbitrary due to the lack of a well-defined reference).

Spatial-domainRepresentations One approach is to reduce the sequence of potential distributions to a small n u m b e r of maps that represent an entire cardiac cycle by integral mapping; each electrode signal is integrated over predetermined time segments, producing a cumulative temporal potential at each spatial location. The endpoints of the integration are globally determined fiducial markers found by visual or algorithmic analysis of the waveforms, or sometimes subintervals between the fiducial points (eg, 3 equal

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subdivisions of the QRS). The integral values are presented via contour maps. Integral mapping achieves a tremendous reduction in data in exchange for the loss of temporal resolution or dynamic i n f o r m a t i o n - - i n particular, the temporospatial link is integrated out within each interval. Another technique is to define a temporal event of interest in each lead, and then draw a map by interpolating these times over space. The best k n o w n example of this technique is activation isochrone mapping, which plots activation times, usually estimated as max -dV/dt, as a function of position. This technique has been used to depict isorepolarization times as welI, and can easily be extended to any other event that can be identified on a lead-byqead basis. Thus, for each chosen event, the entire cardiac cycle is reduced to a single map. Although isochrones are a very powerful way to reveal the spatial progression of temporal events that are well defined on each lead, the detection of activation instants from measured time signals may be a n o n trivial problem w h e n the signals have complex morphologies (1), a frequent occurrence w h e n there is injury to the tissue. This can be even more of a problem for Iess temporally dramatic events such as repolarization. In fact, the reduction of such an event to a single time instant may be misleading if the true temporal scale of the underlying process is considerably broader than the sampling rate. Moreover, isochrone representations are generally confined to the epicardium and/or endocardium, where the concept of wavefronts is more clearly defined, than at the body surface.

those eigenvectors that correspond to the largest eigenvalues. The result is a reduced set of KL coefficient time series, one for each eigenvector retained. These eigenvectors are representative of the most typical distribution patterns in the data. These time series are further compressed by estimating their temporal correlation matrices and repeating the process. Compression ratios up to 320:1 have been obtained with good accuracy. These coefficients can also be used as features in classification algorithms (see Green et al. {6]). However, because the KLT is a global transform, the final coefficients are associated with an entire map sequence and cannot be localized to either time or space. Moreover, because the eigenvectors, which are the basis vectors of the transform, are dependent on the training data, they depend on a standard geometry among recordings and may be sensitive to the makeup of a particular training data set.

Best-lead Methods Another statistical approach is to perform some preprocessing of a temporal or spatial nature (for instance, integrating under subintervals of the fiducial segmentation) to extract a smaller n u m b e r of representative parameters, and then use discriminant methods to identify those leads whose features contain the most information about a particular classification problem (7). Although spatial localization is retained with this method, the chosen leads may vary from study to study and condition to condition and they also depend on the data used in the study. Moreover, any explicit linkage between temporal and spatial behavior is hard to discern in the results.

Frequency-domain Representations Multidimensional spectra can be obtained from map data w h e n the sampling grid is reasonably regular, using techniques such as the Zero-Delay Wavenumber Spectrum (ZDWS) (2). Again, each beat is reduced to one image and the results have a direct physical interpretation, but temporal and spatial localization are lost over the array. Such methods implicitly depend on Fouriertype basis representations of the spatial and temporal potential patterns, which, as mentioned earlier, do not seem particularly appropriate for potential maps. Multichannel spectral techniques have also been applied to potential maps; here, each channel is Fourier transformed separately (3). Although spatial relationships are maintained, temporal localization is lost.

Karhunen-Loeve Transform The Karhunen-Loeve Transform (KLT) is a statistical representation technique that has been used to represent body surface potential distributions (4,5). The KLT achieves a pattern representation that is optimal in a leastsquares sense with respect to a given collection of training data. An empirical spatial covariance matrix is estimated from the training database, its eigendecomposition is computed, and individual data sets are projected onto only

Issues in Representation of Potential Maps There are a n u m b e r of issues that underly the advantages and disadvantages of each representation technique. Each technique involves tradeoffs between these various desired properties. Below is listed what we believe are among the most important of these issues.

Fixed vs Signal-dependent Bases or Features Some of the methods described extract features directly from the data, while others first project the data onto a set of basis vectors and then extract information from the resulting coefficients. In either case, one can consider the method as being either signaI dependent or signal independent. Examples of the former include a direct approach, like the location and magnitude of local maxima and minima, where the n u m b e r of maxima and minima is likely to vary depending on the setting, or a basis approach, like the KLT, where the basis vectors themselves depend on the particular geometry and training data. For the latter, we can consider activation iso-

Multiresolution Analysis of Potential Maps chrone maps, where the activation time can be chosen as the max -dV/dt for any data set, or Fourier-based representations, such as the ZDWS, where the complex exponential basis functions are fixed by the method chosen.

Global vs Local Features Some of the methods extract features that are global in space, time, or both, while others retain the ability to localize their features in space and time. Isochrone or integral map methods, for instance, are local in space but global in time; the KLT applied as described is global in both space and time. Time/space descriptors, on the other hand, are local in both space and time.

Physiological Interpretability Some representations, such as time/space descriptors or isochrones, have a clear physiologic significance. Others, such as KLT coefficients, are considerably harder to associate with particular physiologic phenomena.

Data Reduction Methods such as isochrones or integral maps reduce the data volume directly; transforms such as the KLT or ZDWS do so by concentrating information in the data onto a smaller n u m b e r of coefficients.

Robustness for Interbeat Alignment Often, we want to compare two different beats in space and/or time. Local space/time descriptors may make this difficult; isopotential map sequences make it even more difficult because change of status or interbeat variability can cause u n w a n t e d differences in instant-by-instant comparisons. Integral mapping significantly improves this robustness if the integration interval is long enough; however, shorter subinterval integration can be sensitive to fiducial-point determination. Global transforms avoid this problem, but localization is lost.

Quantifiability Often, we want to quantify the data representations, either directly or by computing a difference between two or more beats. Statistical measures (KLT and best-lead) are the most easily quantifiable. Isopotential maps are perhaps the least quantifiable.

Multiresolution Decompositions and the Wavelet Transform There are three somewhat different perspectives on multiresolution decompositions and the WT, corresponding to three different historical threads that have recently been connected. One approach is multiresolution theory,

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originating from work in computer vision. A second is the area of time-frequency analysis for nonstationary signals and filterbank theory, and the third is the field of basis expansions of signals and sequences in functional analysis. We will briefly describe all three perspectives in the hope of giving a global picture of the field. Our goal here is intuitive understanding rather than mathematical rigor; for more of the latter, we refer the reader to the many tutorial references n o w available (8,9). For ease of exposition, the discussion below is framed, for the most part, in terms of one-dimensional functions of time only. However, spatial signals, and temporospatial signals like potential maps, can be treated with straightforward extensions of the methods described. Thus, one can substitute space or space-time for time w h e n considering the extension to sequences of the potential maps. In those instances where this may not be the case, the spatial and temporal dimensions are treated explicitly.

The Multiresolution Approach Multiresolution methods originated from the desire to control the level of detail in the representation of an object or signal efficiently by decomposing it into separate resolution levels. This quest led to a theory of how to achieve this representation in a complete fashion, such that (1) the information contained at each level of detail or resolution is independent of that at coarser levels and (2) if we use all of the available resolution levels, we will regain all available information about the signal. The basic idea was to consider a class (technically, a subspace) of signals defined by a particular resolution, or level of detail, and find a set of fixed functions, called basis functions, that represent all signals in that subspace. The basis functions act like unit vectors along the coordinate axes of the given subspace, so one can represent each function in the subspace by a set of coefficients that represent the amplitude of the function in the direction of the associated unit vector. Thus, the level of detail of the basis functions defines the level of detail of the functions that they can represent. The simplest example is a set of piecewise constant basis functions: for a given interval length over which the functions are constant, they represent variation in a signal only if it is coarse enough, that is, longer than that interval; thus, this interval length implicitly defines a level of temporal resolution. One forms an approximation signal at a given resolution level using these bases, and the part of the signal that cannot be represented at this level of approximation is called the detail signal. The signal is separated into approximation and detail subsignals by projecting it onto the approximation and detail subspaces via the basis functions. The process can be repeated on the approximation at a coarser level of resolution; in our example, this could be done by doubling the length of the piecewise constant intervals of the basis functions. Multiresolution theory deals with how to impose conditions on the decomposition (and the basis functions) under which our original requirements of completeness and level-to-level independence (or orthogonality) are satisfied. The main relevant results are (1) that there are

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m a n y types of basis functions that can be used, (2) that it can be done using a single pair of "mother" and "father" basis functions for the detail and smooth parts of the decomposition, where the resolution level is changed by compressing or expanding (dilating) the basis functions, and (3) that one can use a set of translates of basis functions that are finite in extent, leading to algorithms to compute the coefficients of the basis expansion via a set of filters developed from the mother and father bases.

The Time-frequency Approach Fourier transforms are perhaps the most well-known technique for representing signals in terms of a fixed set of basis functions, namely single-frequency functions (ie, real or complex sinusoids). They reveal the amplitude and delay associated with the frequency content of a signal. If this frequency content changes with time (eg, speech, where frequency content changes as the speaker moves his/her mouth and vocal cords), since the Fourier transform averages the frequency content over the signal length, it cannot show which frequencies predominate at each time. (This loss of localization in traditionaI frequency-domain techniques was mentioned earlier.) One way to fix this problem is by the use of a technique k n o w n as the Short-Time Fourier transform (STFT), in which one simply truncates or windows the sinusoidal basis functions so that they cover only a smaller interval of the signal and then translates them across the signal. Thus, the result is a time-varying modified estimate of the Fourier transform. However, if we shorten the temporal interval over which we compute frequency content, we worsen the resolution at which we can distinguish different frequendes; this is a fundamental limit {in fact, an uncertainty principle) to the combined resolution in time and frequency. The STFT uses a fixed-length truncation and, thus, has a fixed resolution in both time and frequency. We can implement the STFT as a bank of narrowband filters tuned to different frequency bands; the longer the filter impulse response, the better the frequency resolution and the worse the temporal resolution. However, in many situations, we may wish to tradeoff time and frequency resolutions at different times or frequencies. For instance, if we wish to see short, high-frequency events (eg, an activation wavefront) with good localization, we may be willing to tolerate worse frequency resolution; while during the ST-segment, when the effects of injury currents last longer, we may be willing to tolerate worse temporal resolution to achieve better frequency resolution. In fact, we can, in principle, obtain in each frequency range any desired tradeoff between temporal and frequency resolution that is within the achievable limits,

The Wavelet Transform as an Expansion onto a Basis

that the Fourier bases may lack. For instance, one can find bases with either a finite extent in time that are as localized in frequency as possible, or with a fixed bandwidth in frequency that are as localized in time as possible; other time/frequency tradeoffs can also be found. One can impose orthogonality, biorthogonality, linear phase, and/or time invariance constraints. One can look for bases that enhance particular aspects of a signal, such as its first or second derivatives; thus, one can use bases that estimate derivatives at different resolutions to locate edges in an image or sharp temporal and spatial gradients in a potential map. Combinations of some of these design constraints are possible. Wavelet transforms are defined through the use of such basis functions that, as with the multiresolution bases, are derived from expansions and contractions of a pair of mother and father wavelets. The bases derived from the mother wavelet are used to compute the detail coefficients of the transform, and those from the father wavelet, called the scaling functions, are used to compute the smooth residual of the signal, to which the next rescaling of the wavelet basis is applied. Thus, the WT is a time-scale (or space-scale or even time/space-scale) transform; the two variables associated with its coefficients are the time localization and the expansion/contraction scale(s) for the basis function at which a particular coefficient was computed.

Relationship Among the Three Approaches The connection between multiresolution decompositions and the WT is through the underlying basis functions. Since the WT bases are chosen as functions of scale, they correspond to the implementation of a multiresolution decomposition of a signal; the wavelet coefficients are the detail signals at each scale and the scaling coefficients constitute the smooth approximation. The time-reversed bases of the WT can be implemented as the impulse responses of the filters in the muhiresolution decomposition. (We note that the actual situation is somewhat more complicated, as we ignore here the issues involved with sampling the WT in time and scale. Although it is not obvious at first, conditions can be imposed on either the WT or multiresolution approach to ensure that finitelength discrete filters related to the basis functions can be used to implement a discrete version of the procedure. For more information, please see Meyer [8] and Mallat [10].) The connection to time-frequency analysis is through the scaling property of the Fourier transform; expansion/contraction in time corresponds to contraction/expansion in frequency. Thus, the longer, expanded, coarser scale bases correspond to more narrow-band filters of a time-frequency analyzer, while the shorter, contracted, finer scale bases correspond to wider-band filters.

Issues in the Choice of Basis Functions The third way to conceptualize the multiresolution approach is to consider the idea of replacing Fourier bases with other basis functions chosen for desirable properties

A critical issue in implementing any of these schemes is the choice of a particular basis, or wavelet. Most of the

Muitiresolution Analysis of Potential Maps

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a nonlocalized best representation. We have been developing n e w criteria particularly appropriate for ECG and map data; for details, please see Brooks et al. (I5) and Krim and Brooks (16).

development of the WT theory in signal processing has been focused on signal compression/coding/transmission/reconstruction, where reconstructability and efficiency have been key; this has led to an emphasis on orthogonal transforms, implemented with down-sampling to improve efficiency, with little concern for avoiding phase distortion in the filters. For application to potential maps, however, we are interested in being able to understand the timing of events in the coefficient domain across scale and with reference to the original data; thus, we generally desire decompositions that use linear-phase filters and that are time or shift invariant, or nearly so, and we may be willing to accept redundancy (lack of orthogonality) to retain interpretability. In particular, in the work presented here, we have adopted a nonorthogonal WT scheme proposed by Mallat and Zhong (11 ), where the wavelet coefficients calculated at a particular scale can be interpreted as estimates of the derivative of the signal after it is smoothed to that scale. Thus, peaks (local maxima) in the wavelet coefficients that persist across several scales can be interpreted as edges or regions of sharp gradient in the signal. In addition to the advantages of time/shift invariance and linear phase, the derivative interpretation enables us to link the transform coefficients directly to the physiologic processes. These wavelets have been applied by others, as well as by us, to segmentation of the standard ECG signals, in particular to QRS fiducial point determination ( 12,13). We have also recently been investigating the use of a different family of wavelet (technically, wavepacket) basis functions, k n o w n as local cosine bases, for segmentation of body surface potential maps (BSPMs) and ECG signals. The basic idea is to apply schemes that find a "best-basis" for a particular signal from this family according to some optimality criterion (14). Thus, the result is a localized best representation, as the KLT gives

S o m e Applications of Multiresolution Potential Mapping In this section, we will illustrate two applications of the multiresolution technique: (1) temporal segmentation of BSPM signals and canine epicardial electrograms to locate fiducial points, and (2) characterization of depolarization and repolarization in canine epicardial plaque data using multiresolution three-dimensional derivatives. Our discussion of each application will be brief; the intent is to illustrate areas for potential application of multiresolution analysis of maps rather than a full presentation of the results achieved.

Segmentation of Sensor Leads We used the interpretation of temporal wavelet coefficients as the first derivatives of smoothed data to find fiducial points. We will discuss its application to two segmentation problems: determination of the endpoints of the ST-segment in BSPM data recorded during percutaneous transluminaI coronary angioplasty procedures (17) and determination of recovery time in electrograms recorded on the epicardial surface by plaque electrodes. ST-Segment Endpoint Determination. The basic idea is illustrated in Figure 1. Details of the algorithm can be found in Chary and Brooks (18). The basic intuition is that the ST-segment will be characterized by

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relatively slow temporal changes in a given lead, and thus, the temporal derivative (as represented by the wavelet coefficients) will be relatively constant. A highconfidence ST region is determined from global (eg, over all of the BSPM sensors) fiducials based on examination of a root-mean-square curve. After selection of an appropriate scale (level of the decomposition), we calculate thresholds on a lead-by-lead basis from the mean and standard deviation of this high-confidence region. Starting from the middle of the high-confidence region, we search outward in both directions to find when the wavelet coefficients stay outside the thresholds for a n u m b e r of samples set adaptively by the algorithm; in parallel, we search inward from well outside the ST-segm e n t to find w h e n the wavelets stay inside the thresholds. The algorithm adapts its parameters until the inside-going and outside-going fiducials match, and these are declared as endpoints. Figure i shows results from data recorded during percutaneous translmninal coronary angioplasty procedures using a 117-lead mapping system (17). Part a shows the original BSPM signal, part b shows the smooth approximation at the third scale of the WT algorithm, and part c shows the wavelet detail coefficients at the fourth scale. The two rectangles on the latter graph show the search intervals and the

thresholds. The vertical bars show the resulting endpoints. We found that due to the tremendous variability of the BSPM waveforms, there was a small but persistent n u m b e r of leads where the algorithm failed (about 3-5 % of the 117 leads). This problem was rectified by means of a second-pass stage of spatial filtering of the resulting endpoints to remove outliers (19). Further use of the resulting space-varying ST-segments to discriminate the artery undergoing inflation can be found in Brooks et aI. (19}. An illustration of the resuIts for all of the leads on the anterior torso is shown in Figure 2. Note the stability of the results despite the variability in morphology. Determination

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We define recovery time as the m a x i m u m positive derivative during the T wave (20). The algorithm used here is illustrated for one lead in Figure 3. The data were recorded with a 16 × 16 regular grid of electrodes with 2-mm spacing embedded in a plaque and sutured to the right ventricle of an exposed canine heart during experiments carried out at the Cardiovascular Research and Training Institute as part of other studies. Here, we used a cross-scale algorithm. Starting with a relatively coarse scale, we found the last positive m a x i m u m of the wavelet coefficients, indicated by the circle on the plot grams.

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Multiresolution Analysis of Potential Maps

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labeled Detail:5. We can be confident that this corresponds to recovery time, but our temporal resolution is poor at this scale. Using the filterbank structure, we determined which coefficients at the next finer scale corresponded to this point, as shown by the vertical bars in the plot labeled Detail:4; we then searched this small window for its maximum. We iterated the process at progressively finer scales until we reached a scale where the derivative was generally dominated by noise (shown in Detail:l). As illustrated in the figure, the effect was to zoom in on a shorter and shorter window as we moved to finer scales. The results for a subarray of the plaque are shown in Figure 4. Note that the algorithm works well even w h e n recovery time is somewhat obscured by elevated ST-segments.

Spatiotemporal of Epicardial

Using the WT multiresolution approach, we can observe each dimension at its o w n appropriate scale. Recombining the derivative estimates, we can recognize wavefront b e h a v i o r - - t h e idea being that wavefronts shouId have large derivatives in both space and time, but that the i n h e r e n t scale of the wavefront may differ in space and time, and may differ b e t w e e n activation and repolarization. The resulting gradient is then a three-dimensional function of six variables, three fixing the three scales (ie, the degree of smoothing in each dimension) and three corresponding to the location of the gradient in space and time. Visualization of the result is difficult. On a computer screen, we can fix the three scale variables and view the magnitude and angle of the gradient at each location in space at a particular time, as with color mapping, and then observe changes in time as a map sequence• On paper, even this is difficult• We have tried to illustrate the results by plotting the three gradient component magnitudes and the gradient vector magnitude at a particular time as gray-scale plots and the spatial angle and residual temporal angle as quiver plots, as illustrated in Figures 5 and 6. The first compares activation at two temporal scales and the second compares

Segmentation Plaque

Data

Since the plaque data is a function of two spatial dimensions as well as time, we can calculate a threedimensional space-time derivative, or gradient vector.

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Multiresolution Analysis of Potential Maps

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recovery, as indicated by the vertical bars on the time signals plotted at the top of both figures. The format for both figures is the same. The left column shows the spatial gradient as a magnitude on top and as an angle on bottom. In the middle column, the temporal gradient component at the second scale is on top and the temporal angle, calculated as the angle created by the arctangent of the temporal component divided by the spatial magnitude, for the second finest scale is below. Thus, the temporal angle ranges from positive to negative 90°; the former means a large positive change in time compared to the spatial gradient and the latter means a large negative change in time. No change in time is at zero degrees. The right column repeats the middle column for the fourth scale. Note that activation is best seen at the finer scale and recovery at the coarser temporal scale.

Conclusion and Future Work Here we have illustrated w h y we believe that the muItiresolution approach is appropriate to analyze potential maps, taking into account their complex time-space link-

ages. Among the directions we plan to pursue in this area are improved visualization and analysis of the threedimensional gradient approach, exploration of the use of other wavelets for this analysis, and application of nonseparable wavelets and wavelets developed for irregular sampling grids.

References 1. Lander P, Berbari E: Contouring of epicardial activation using spatial autocorrelation estimates, p. 451. In Computers in cardiology. IEEE Computer Society, Los Alamitos, CA, 1992 2. Nikias CL, Raghuveer Computer Society, Siegel JH, Fabian M: The zero-delay wavenumber spectrum estimation for the analysis of array ECG signals: an alternative to isopotential mapping. IEEE Trans Biomed Eng 33:435, 1986 3. Siegel JH, Nikias CL, Fabian M et al: Advanced signal processing method for the detection, localization, and quantification of acute myocardial ischemia. Surgery 102:215, 1987

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4. Lux RL, Green LS, Abildskov JA: Statistical representation and classification of electrocardiographic body surface potential maps. p. 251. In Computers in cardiology, IEEE Computer Society, Los Alamitos, CA, 1984 5. Lux RL: Karhunen-Loeve representation of ECG data. J Electrocardiol 25 (suppl):l 95, 1992 6. Green LS, Lux RE, Haws CW: Detection and localization of coronary artery disease with body surface mapping in patients with normal electrocardiograms. Circulation 76:1290, 1987 7. Kornreich E Rautaharju P, Warren J et al: Identification of best electrocardiographic leads for diagnosing myocardial infarction by statistical analysis of body surface potential maps. Am J Cardiol 56:852, 1985 8. Meyer Y: Wavelets, algorithms and applications. SIAM Press, Philadelphia, 1993 9. Rioul O, Vetterli M: Wavelet and signal processing. IEEE SP Magazine 8:14, 1991 10. Mallat SG: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Machine Intell 11:7, 1989 11. Mallat S, Zhong S: Characterization of signal from multiscale edges. IEEE Trans Pattern Anal Machine Intell 14:7, 1992 12. Li C, Zheng C, Tai C: Detection of ECG characteristic points using wavelet transforms. IEEE Trans Biomed Eng 42:21, 1995 13. Murray R, Kadambe S, Boudreaux-Bartels GF: Extensive analysis of a QRS detector based on the dyadic

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wavelet transform. Proc IEEE workshop time-scale and time-frequency 540, 1994 Coifman RR, Wickerhauser MV: Entropy based algorithms for best basis selection. IEEE Trans Inf Theory 32:712, 1992 Brooks DH, Krim H, Pesquet JC, MacLeod RS: Best basis segmentation of ECG. Signals using novel optimality criteria. Presented at the ICASSP 96, Atlanta, GA, May 1996 Krim H, Brooks DH: Feature-based segmentation of ECG signals. Presented at the IEEE Workshop on Time-Scale and Time-Frequency, Paris, June 1996 MacLeod RS, Brooks DH, On H: Analysis of PTCAinduced ischemia using both an electrocardiographic inverse solution and the wavelet transform. J ElectrocardioI 27(suppl):90, 1994 Chary RV, Brooks DH: ST Segment detection and detection of inflated artery from PTCA data: current status. CDSP technical report. ECE Department, Northeastern University, June 1995 Brooks DH, Chary RV, Krim H et al: Wavelet-based temporal segmentation and analysis of body surface potential maps during PTCA-induced ischemia. Presented at the 17th Annual International Conference of the IEEE EMBS. Montreal, September 1995 Haws CW, Lux RL: Correlation between in vivo transmembrane action potential durations and activation-recovery intervals from etectrograms. Circulation 81:28I, 1990

Intracardiac Electrogram Transformation Morphometric Implications for Implantable Devices

Milton M. Morris, MSE, Janice M. Jenkins, PhD, and Lorenzo A. DiCarlo, MD Over 75,000 antitachycardia devices (ATDs) have been implanted since initial Food and Drug Administration approval in 1985 and have yielded dramatic survival rates. These devices, although life-saving, rely principally on simple measures of the heart rate for arrhythmia detection. The ATDs are highly sensitive, but their low specificity of diagnosis results in episodes of inappropri-

From the Department of Electrical Engineering and Computer Science, University of Michigan and Michigan Heart and Vascular Institute, Ann Arbor, Michigan.

Reprint requests: Milton M. Morris, University of Michigan, 442I EECS Building, Ann Arbor, MI 48109-2122.

ate therapy ranging as high as 10 to 41% of alI shocks delivered ( 1-5 ). Morphologically based algorithms have been demonstrated to improve levels of specificity dramatically while maintaining high levels of sensitivity (6-11); however, these algorithms tend to be computationally complex, placing unacceptably large demands on battery power ( 12,13). Since ATDs are small battery-operated implants, small algorithmic demands on battery power are essential for device longevity. Correlation waveform analysis (CWA), a morphologically based algorithm, has emerged as a promising technique for more specific assessment of intracardiac electrogram (IEGM) rhythms. Correlation waveform analysis