Adaptive Filtering for Flow-Cytometric Particles

Transcription

1 126 Part. Part. Syst. Charact. 17 (2000) 126±133 Adaptive Filtering for Flow-Cytometric Particles alek Adjouadi*, Carlos Reyes**, John Riley***, Patricio Vidal* (Received 15 June 2000; resubmitted 26 July 2000) Abstract This paper studies the effects of FIR ltering and introduces an adaptive FIR ltering technique designed speci cally to smooth one and two dimensional accumulated owcytometric particles through frequency histograms. The adaptive smoothing technique illustrated here will be shown to be particularly useful in compensating the uneven histogram accumulation effects which take place when data is accumulated at the lower histogram channels versus the higher histogram channels. Linear smoothing techniques will not compensate for this phenomenon which is inherent to all histograms of accumulated data. In this view, a thorough analysis is provided to deal with the dilemmas imposed by the uneven accumulation of data within these histograms. 1 Introduction When cytometric data from speci c blood cell particles is accumulated into a frequency histogram with a nite number of channels (or bins) the accumulation process inherently smoothes the histogram since an averaging effect is taking place. When the cytometric data is accumulated in the higher channels of the frequency histogram, the distribution appears more spread and noisier. This is due to the fact that more channels are available and the inherent averaging effect is diminished as data is accumulated over a larger band of channels. Because of this, when data is accumulated in a histogram with a discrete number of channels, it gives rise to an uneven resolution within the histogram. The idea behind the adaptive smoothing is to take advantage of the fact that as the lter coef cients are varied over an appropriate range, less smoothing will take place. The lter coef cients can be made to vary proportionally with respect to the histogram channel where smoothing is taking place. Smoothing is usually required prior to the analysis of frequency histogrammed data to attenuate the effect of noise. Smoothing through FIR ``Finite Impulse Response'' ltering is by far the preferred method since it will not shift the position of the data distributions. A perspective on density estimators from histograms to several univariate and *. Adjouadi, Ph.D., Chairperson, P. Vidal, Ph.D. Student, Department of Electrical and Computer Engineering, Center for Engineering and Applied Sciences, EAS 3935, W. Flagler Street, Florida International University, iami, Fl (USA). ** C. Reyes, Engineer, otorola, Test Systems Engineering, 8000 West Sunrise Blvd., Fort Lauderdale, Fl (USA). *** J. Riley, Engineer, Beckman Coulter Inc., SW 147th Avenue, iami, Fl (USA). multivariate statistical data analyses can be found in studies [1±3]. Other studies using histograms bring to focus several issues including the notions of stability in the appearance of the histogram [4], the interpolations that can be achieved in kernel density estimators [5], as well as the practical but traditional aspect of enhancing images through an adaptive histogram equalization method [6]. In general, it is desirable to select a smoothing function that has the following criteria to ensure that the resulting data is not distorted 1±The area under the curve should be maintained 2±The mean of the distribution should be unchanged 3±The change in the standard deviation should be kept to a minimum These criteria for smoothing are maintained by traditional FIR ltering provided that boundary conditions are satis- ed. A modi ed ltering scheme is introduced in this study based on these mechanisms to customize the traditional FIR-type ltering schemes for discrete time signals to accommodate for data accumulated into frequency histograms. This modi ed ltering scheme is viewed as an adaptive ltering scheme that linearizes and accommodates for uneven histogram accumulation effects. To illustrate this problem a random Gaussian distribution was created using the Box-uller ethod. The same pseudo-random distribution is illustrated with three separate gain settings (1, 5, 10) as shown in Figure 1. When this data is histogrammed, as shown in Figure 2, the uneven accumulation problem is encountered and analyzed. For experimental evaluation, similar histogram accumulation problems are discussed and analyzed utilizing Phycoerythrin (PE) bead particles containing a variety of bead populations at various amounts of uorochromes. # WILEY-VCH Verlag GmbH, D Weinheim, /00/0309-page 126±133 $ =0

2 Part. Part. Syst. Charact. 17 (2000) 126± Fig. 1 Normally distributed data at various gains. Fig. 2 Histogram of entire data set. 2 Finite Impulse Response Filtering Normally the causal Finite Impulse Response (FIR) lter is used in signal processing, but in histogram smoothing, the noncausal FIR must be used in order to maintain symmetry and to prevent skewness in the smoothed distributions. The causal FIR lter difference equation is given by H1 i ˆ PJ jˆ0 H i j The noncausal FIR lter difference equation is given by H1 i ˆ PJ H i j The lter architectures of both FIR types are shown in Figure 3. These lter structures offer the advantage of linear phase or constant time delays. This is essential in obtaining a smoothing function, which is capable of maintaining the means of populations in the histogrammed data. A simple and effective FIR ltering scheme is achieved here with the traditional tap FIR lter. Such a ltering scheme offers the essential mechanisms required obtaining a ``good'' smoothing function. Each requirement can be met by traditional FIR ltering and maintaining the area under the curve provided that the following constraints are met Fig. 3 Causal and non-causal FIR lter structures. (a) Causal FIR lter. (b) Non-causal FIR lter Ensure that the area under the curve is maintained i.e., the sum of the FIR lter coef cients must equal 1. ˆ 1; where the number of taps in the FIR lter is 2J Prevent the data from being skewed meaning that the lter coef cients be symmetric around a 0 such that ˆ a j This requires that the number of lter taps in the FIR lter be odd. 3. aintain the area under the curve meaning that the boundary conditions speci ed by Eq. (5) is assumed to overcome the spreading effect resulting from the FIR ltering process. H i ˆ 0 for i5j and i N J; where H represents the histogrammed data and N is the number of accumulated points. 2.1 Proof of Requirements to aintaining the Area Under the Curve Let H i denote the accumulated data histogram with the channel range given by i 2 0; 1;...; N 1Š, where N is the number of channels in the histogram. The coef cients of the FIR lter are given as for j 2 J...JŠ where J is one half the number of lter taps. Using Eq. (2), the area, 1, of the output of the lter is given as 1 ˆ H1 i ˆ H i j To illustrate that the area under the histogram is maintained, we need to prove that the the sum of all points before and after the data has been ltered is maintained. Rearranging the terms from Eq. (6) we nd that 1 ˆ PJ j H i j

3 128 Part. Part. Syst. Charact. 17 (2000) 126±133 Due to the symmetry of the FIR lter coef cients, we can expand Eq. (7) as follows 1 ˆ 0 H i PJ j H i j PJ j H i j aking a change of variables and grouping similar terms we obtain 1 ˆ 0 H i PJ N 1 j P iˆj H i N 1 j P iˆ j 8 H i! 9 If we let represent the summation of the total number of point in the accumulated frequency histogram so that ˆ i 0 H i, and modify the summation limits using the boundary conditions of Eq. (5), to write! 1 ˆ a 0 P a N 1 j H i ˆ a 0 10 And using the FIR lter coef cients properties, this equation satis es the rst requirement that the area under the curve be maintained, that is 1 ˆ 11 And because the mean of the distribution is given by m ˆ 1y P N 1 ih i we satisfy the second requirement that the area under the curve be maintained, that is m1 ˆ m The Effect on Variance due to FIR Filtering FIR ltering always effects the variance of a distribution. A good smoothing function should minimize changes in variance while still eliminating unwanted noise. In this proof, we formulate an approach to measuring the changes in variance due to ltering and recovering the original variance of the un ltered distribution. Given that the variance of the ltered data is given by s 2 ˆ 1 H1 i i m 2 17 The difference in variance can be de ned, as derived in the appendix, by the expression Ds 2 ˆ 1 i 2 H1 i 1 H i j i 2 ˆ PJ j Proof of Requirement in aintaining the eans The second desirable property of a good smoothing function is to maintain the means of the data distributions. This feature is inherent to all FIR type of lters provided the boundary conditions of Eq. (5) are satis ed. Because FIR lter coef cients are symmetric, the mean, m1, of the ltered data is maintained, and is derived as m1 ˆ 1 1 H i j i ˆ 1 j ih i j 12 Due to the symmetry of the FIR lter coef cients, we can expand Eq. (12) as follows m1 ˆ 1 a 0 ih i 1 j ih i j 1 j ih i j 13 aking a change of variables and grouping similar terms we obtain m1 ˆ 1 a 0 ih i 1 N 1 j P iˆj N 1 j i j H i P iˆ j i j H i! 14 By modifying the summation limits using the boundary conditions established in Eq. (5), it can be shown that m1 ˆ 1 a 0 ˆ 1 ih i ih i 1 a 0! N 1 P i j H i i j H i ˆ 1 ih i 15 Eq. (18) provides us with the ability to track and recover changes in variances through FIR ltering. 2.4 The 3-Tap Causal FIR Filter The standard 3-Tap FIR lter typically has lter coef cients of a 1 ˆ 025, a 0 ˆ 05, and a 1 ˆ 025 with the sum of the lter coef cients being equal to 1. This type of FIR lter is usually called the Filter. To explore different ltering possibilities, transfer functions for the coef cients given in Table 1 are shown in Figure 4. These transfer functions illustrate several key points. First, when the ltering coef cients are given as 0-1-0, no ltering occurs and the transfer function is thus a straight line. Second, as the ltering coef cients are varied from 0.1± 0.9±0.1 to the traditional 3-tap lter coef cients of 0.25± 0.5±0.25 the transfer function yields a low pass lter with the latter set of lter coef cients yielding the lowest cutoff frequency. Third, beyond the coef cients of the 3-tap FIR lter, all other coef cients yield a notch lter. To achieve variations on the traditional 3-tap FIR lter, the same data set can be applied through the 3-tap lter multiple times. If the histogrammed data is applied to the same set of lter coef cients as shown earlier in Table 1, the transfer functions illustrated in Figure 5 are obtained after 15 passes through the FIR lter. Table 1 Types of FIR lter coef cients considered. HP 0 HP 1 HP 2 HP 3 HP 4 HP 5 HP 6 HP 7 HP 8 HP

4 Part. Part. Syst. Charact. 17 (2000) 126± Fig Tap FIR ltering of histogrammed data. Fig. 4 Transfer functions for different FIR coef cients. The results illustrated in Figure 5 show how the cutoff frequency for the low pass lter is lowered as the number of iterations is increased. The band of frequencies eliminated by the notch lter is widened such that some of the coef- cients, which yielded a notch lter, appear as a low-pass lter after multiple iterations. 2.5 The Effects OF Tap FIR Filtering on Histogrammed Data Figure 6 below illustrates the smoothing effects of a typical standard tap FIR lter on the histogrammed data of Figure 1. It also illustrates the undesirable effects of smoothing the data at the low-resolution channels of the histogram. As was shown earlier in Figure 2, although the only difference in the three populations accumulated is the gain setting, the lower resolution channels of the histogram are naturally ltered by the accumulation of data into fewer channels. Because of this, the data at the low end of the histogram appears less noisy than the data accumulated in the higher histogram channels. 3 Introducing the Adaptive FIR Filter for Histogrammed Data To accommodate for the accumulation process which inherently smoothes the histogram in a linear fashion, since an averaging effect is taking place, the adaptive FIR lter is introduced. This new ltering scheme accommodates for data that is accumulated in the uneven resolution of the channels of the frequency histogram. The idea behind the adaptive smoothing is to take advantage of the fact that as the lter coef cients are varied over an appropriate range, the amount of smoothing can be altered to accommodate for the uneven accumulation of data. The adaptive FIR lter will thus adjust its weights based on its location in the histogram. This allows for less smoothing at the lower resolution channels (since the accumulation effect has already contributed a signi cant amount of smoothing) and increases the amount of smoothing at the higher resolution channels where the smoothing effect of the data accumulation is not as pronounced. 3.1 Basis of the Adaptive FIR ltering The Least eans Square (LS) algorithm is one of the simplest schemes for adaptively adjusting the weights of an adaptive lter. This is because of its ease of computation and also it does not require of ine gradient estimations or repetitions of data. The correction resulting from the adaptive lter is based on the relation y k ˆ x k w k. Since the reference signal as shown in Figure 7 is set to 1, x k ˆ 1, the correction is equal to the weight of the lter tap, y k ˆ w k. Fig. 5 ultiple iterations (n ˆ 15) of data through the 3-tap FIR lter. Fig. 7 Structure of the adaptive interference canceler.

5 130 Part. Part. Syst. Charact. 17 (2000) 126±133 The lter weight is adjusted using the LS algorithm as follows w k 1 ˆ w k m H k ˆ w k 2me k x k 19 Where m can be interpreted as the learning constant and e k is the error signal given by e k ˆ d k x T k w k. The sizes of these vectors in Eq. (19) are equal to the number of taps in the adaptive lter. For the particular type of adaptive interference canceler we are dealing here, the vectors become scalars. The adaptation of w k then becomes w k 1 ˆ w k 2me k x k Which yields y k 1 ˆ y k 2m d k y k Widrow in study [7] takes the z-transform of Eq. (7) to calculate the transfer function of the adaptive canceler and suggests the following relationship H z ˆE z D z ˆ z 1 z 1 2m 22 This transfer function de nes a single frequency notch lter with half the power frequency at o ˆ 2m. 3.2 Proof of Requirements to aintaining the Area under the Curve Letting again H i denote the accumulated histogram with the channel range given by i N 1, where N is the number of channels in the histogram. The coef cients of the adaptive FIR lter are given as i for j 2 J;...; JŠ where J is one half the number of lter taps. The area can then be expressed as ˆ H i i j 23 In order to maintain the area under the curve, and as derived in the Appendix, the solution for is ˆ a0 N 1 a 0 0 Š 2 N 1 PJ ( j 1 P H i 2i 2j 1Š PJ N 1 P iˆn 1 j ) H i 2i 2j 2N 1Š 3.3 The Effects of Adaptive Filtering on the ean De ning the mean, m, to be m ˆ 1 i PJ H i i j and expanding Eq. (25) yields m ˆ 1 i H i 0 PJ k i H i j PJ k i H i j 26 Using the boundary conditions of Eq. (5), the change in the mean value between the original distribution and the smoothed distribution can be expressed as Dm ˆ a0 N 1 a 0 0 Š P 2 j 2 j ih i 2 N 1 PJ The Effects of Adaptive Filtering on the Variance Just as was determined with the FIR ltering scheme, adaptive ltering also effects the variance of a data distribution. It should be a critical goal of any smoothing function to minimize changes in variance while eliminating unwanted noise. As was derived earlier in the FIR case, we formulate here an approach suitable for both measuring the changes in variance due to adaptive ltering and for recovering the original variance of the un ltered distribution. The variance of the ltered data is in this case given as s 2 ˆ 1 i PJ H i i j 28 Which, as derived in the Appendix, can be expressed as s 2 ˆ 1 i 2 H i j 2 a 0 N 1 a 0 0 Š P 3m J j 2 N 1 PJ 29 This provides the ability to track and recover changes in the variance due to adaptive ltering. 3.5 Application of the Adaptive FIR Filter Figure 8 illustrates the smoothing effects of an adaptive FIR lter on the histogram data shown earlier in Figure 2 after 20 passes. In this case a 0 was varied from a value of one (no smoothing) at channel zero to a value of 0.5 (maximum smoothing) at the last channel, which in this case is channel 255. In order to accomplish similar smoothing effects in the high resolution region of the histogram, more passes were necessary in the FIR adaptive lter (20 passes) as compared to the standard FIR lter (10 passes). Figure 9 contrasts both smoothed histograms together. In this case, there was a need for increased smoothing at the high end of the histogram while maintaining minimum smoothing at the low end. However, sometimes the opposite is desired as will be seen in the next application.

6 Part. Part. Syst. Charact. 17 (2000) 126± Fig. 8 Results of adaptive FIR histogram smoothing. 3.6 Adaptive FIR ltering Applied to PE Fluorescent Sensitivity Determination The ability to provide a quantitative measurement of the cellular uorescent sensitivity of a particular instrument is critical parameter in the quali cation of that instrument. To attain this, various sets of particles containing graded amounts of uorochromes have been developed which when combined with the background uorescent levels of the instrument provide a quantitative measurement of the instrument's sensitivity. Data was collected in an instrument with Phycoerythrin beads containing a mix of six-bead population at various amounts of uorochromes. The data was logarithmically transformed [8] and ltered with the adaptive FIR lter. In this case a 0 was varied from a value of 0.5 (maximum smoothing) at channel zero to a value of one (no smoothing) at the last channel. The purpose is to prepare the histogram for the automated algorithm, which nds the location of the six distributions and the background distribution in order to make the sensitivity calculations. Figure 10 shows the histogram of the transformed data and the smoothed histogram. It can be observed that the majority of the smoothing is achieved in the low end of the histogram where it is needed, Fig. 10 Histogram of uorescent particles with background. while the tight distributions at the high end of the histogram are left relatively undisturbed, which is the desired outcome. 4 Conclusion This study introduced a new adaptive FIR ltering scheme speci cally designed to deal with histogrammed data and the uneven histogram accumulation effects. A preliminary study involved the use of pseudo random Gaussian distributions that rstly illustrated the dynamic histogram resolution and illustrated how to compensate for this phenomenon by compensating for the averaging effect that histogramming has on sampled data. A mathematical framework was established to illustrate why the FIR ltering scheme is so effective as a smoothing function provided the proper boundary conditions are satis ed. However, it was also shown that for histogrammed data this might create several undesirable artifacts. The adaptive FIR ltering scheme introduced in this study provided a exible method to compensate for these undesirable artifacts while accomplishing the desired smoothing. Finally, it was shown how adjusting the initial and nal smoothing levels that are assumed could vary the smoothing function appropriately with respect to the accumulated data. It is also appropriate to note that such an adaptive ltering process could serve as a valuable preprocessing step for experimental set ups involving ow cytometric data applications as that described in study [9]. 5 Acknowledgments Fig. 9 Standard smooth histogram vs. adaptive smooth histogram. This work was supported by the National Science Foundation grants EIA and EIA , and the Of ce of Naval Research ONR Grant N with the Center for Advanced Technology & Education.

7 132 Part. Part. Syst. Charact. 17 (2000) 126±133 6 References [1] B. W. Silverman Density Estimation for Statistical Data Analysis. Chapman and Hall, New York [2] D. F. Heath Normal or Log-Normal Appropriate Distributions. Nature 213 (1967) 1159±1160. [3] P. N. Dean Data Processing, in. R. elamed, T. Lindmo,. L. endelsohn (Eds) Flow Cytometry and Sorting, 2nd edition, John Wiley & Sons, New York 1990, pp. 415±444. [4] J. S. Siminoff, F. Udinah easuring the Stability of Histogram Appearance when the Anchor Position is Changed. Computational Statistics and Data Analysis, Elsevier 1997, Vol. 23, pp. 335±353. [5] A. Kogure Effective Interpolations for Kernel Density Estimators. Nonparametrics Statistics, Overseas Publishers Association, 1998, Vol. 9, pp. 165±195. [6] J. A. Stark, W. J. Fitzgerald An Alternative Algorithm for Adaptive Histogram Equalization. Graphical odels and Image Processing 58 (1996) 180±185. [7] B. Widrow, S. D. Stearns Adaptive Signal Processing. Prentice-Hall, Inc., Englewood Cliffs, New Jersey [8] C.. David, D. Redelman, R. Vogt Computing the Central Location of Immuno- uorescence Distributions Logarithmic Data Transformations Are Not Always Appropriate. Cytometry (Communications in Clinical Cytometry) 18 (1994) 75±78. [9] C. Godefroy,. Adjouadi Particle Sizing in a Flow Environment Using Light Scattering Patterns. Part. Part. Syst. Charact. 17 (2000) 47±55. [10] C. Reyes,. Adjouadi A Directional Clustering Technique for Random Data Classi cation. J. Cytometry 27 (1997). Appendix athematical Derivations ean Value The area can be expressed as ˆ H i i j A1 Recall that in order to maintain the area under the curve, as was established in the FIR case, we require that a 0 i i ˆ1 where i ˆ 0 k i A2 By allowing ˆ 0, k i can be de ned as k i ˆ 1 a 0 i A3 With these conditions set, it can be shown that one possible expression for the adaptive lter coef cients is i a 0 i ˆa 0 0 a 0 N 1 a 0 0 Š N 1 A4 In this case, the lter coef cients are adjusted based on the location of the events on the frequency histogram. In this way, the lower resolution frequency bins are ltered less than the higher frequency bins at the upper end of the histogram. Due to symmetry of the adaptive FIR lter coef cients Eq. (A1) can be rewritten as " # ˆ a 0 H i i PJ H i i j PJ H i i j A5 And substituting Eqs. (A2) and (A3) into (A5) yields 2 3 ˆ a 0 H i i PJ 1 a o i P H i j J 1 a o i 6 H i j A6 Assuming a common denominator, and grouping like terms yields ˆ 1 8 >< > PJ N 1 j P iˆj H i 2a 0 i a 0 i j a 0 i j PJ N 1 j P iˆn j If a 0 is assumed linear, then N 1 j P iˆj and ˆ 1 " # Pj 1 a 0 i H i >= a 0 i H i iˆn j a 0 i j H i PJ j 1 7 P 5 a 0 i j H i >; iˆ j 39 A7 H i 2a 0 i a 0 i j a 0 i j ˆ 0 A8 8 >< > P N 1 P iˆn j H i 2a 0 i a 0 j 1 i a 0 i j Š H i 2a 0 i a 0 i j a 0 2N j i 1 Š 9 >= >; A9 Substituting Eq. (A4) into Eq. (A9) yields the solution for as ˆ a0 N 1 a 0 0 Š 2 N 1 PJ PJ j 1 P H i 2i 2j 1Š PJ j H i 2i 2j 2N 1Š iˆn 1 j The Variance A10 The variance of the ltered data which is in this case given by s 2 ˆ i PJ H i i j A11

8 Part. Part. Syst. Charact. 17 (2000) 126± which can be expressed as i 2 a 0 i H i i H i i j i H i i j s 2 ˆ And substituting the following expression 0 1 i ˆ 1 a 0 i B A a 0 i into Eq. (A12) yields; 8 i 1 a 0 i H i j >< s 2 ˆ or, s 2 ˆ > 8 >< > i A12 A a 0 i H i j >= j 2i 2 a 0 i H i PJ j i 2 1 a 0 i H i j PJ i 2 a 0 i H i >; A14 9 >= j i 2 1 a 0 i H i j >; A15 Rewriting the summation limits and adjusting the histogram data subscripts yields ( s 2 1 ˆ j 2i 2 a 0 i H i Thus, s 2 ˆ 1 PJ j i j 2 1 a 0 i j H i PJ i 2 H i j i j 2 1 a 0 i j H i ) 16 j 2 a 0 N 1 a 0 0 Š P 3m j 2 N 1 PJ easuring the Changes in Variance 17 derived. Recall that the variance of the ltered data is given by s 2 ˆ 1 H1 i i m 2 A18 Rewriting this expression in terms of the original un ltered data we nd that 1 H1 i i m 2ˆ 1 H i j i m 2 A19 Leaving the term momentarily out of the derivation we expand Eq. (A18) as i 2 N 1 j H1 i 2m P i H1 i m 2 H1 i ˆ ˆ " # H i j i 2 2m i PJ H i j m 2 H i j A20 Noting that the area under the curve remains unchanged and assuming that the mean remains constant, we have i 2 H1 i ˆ H i j i 2 Rearranging terms of Eq. (A21) yields i 2 H1 i ˆ PJ N 1 j P iˆj H i j A21 A22 Also, making use again of the boundary conditions of Eq. (A5), we establish that i 2 H1 i ˆ PJ j H i i j 2 and by expansion, we obtain i 2 H1 i ˆ PJ N 1 P i 2 H i 2j ih i j 2 H i A23 A24 Again, due to the symmetry of the FIR lter coef cients; we note that 2j P N 1 ih i ˆ 0. And thus, i 2 H1 i ˆ H i j i 2 PJ j 2 H i A25 Reintroducing the term to facilitate the last step of this formulation and de ning a difference in variance we nd that Due to ltering and recovering the original variance of the un ltered distribution, the changes in variance can now be Ds 2 ˆ 1 i 2 H1 i 1 H i j i 2 ˆ PJ j 2 A26