Adaptive Filtering for Flow-Cytometric Particles

Abstract This paper studies the effects of FIR ﬁltering and introducesan adaptive FIR ﬁltering technique designed speciﬁcallyto smooth one and two dimensional accumulated ﬂow-cytometric particles through frequency histograms. Theadaptive smoothing technique illustrated here will beshown to be particularly useful in compensating the unevenhistogram accumulation effects which take place whendata is accumulated at the lower histogram channels versusthe higher histogram channels. Linear smoothing techni-ques will not compensate for this phenomenon which isinherent to all histograms of accumulated data. In this view,a thorough analysis is provided to deal with the dilemmasimposed by the uneven accumulation of data within thesehistograms. 1 Introduction When cytometric data from speciﬁc blood cell particles isaccumulated into a frequency histogram with a ﬁnitenumber of channels (or bins) the accumulation processinherently smoothes the histogram since an averaging effectis taking place. When the cytometric data is accumulated inthe higher channels of the frequency histogram, the dis-tribution appears more spread and noisier. This is due to thefact that more channels are available and the inherentaveraging effect is diminished as data is accumulated over alarger band of channels. Because of this, when data isaccumulated in a histogram with a discrete number ofchannels, it gives rise to an uneven resolution within thehistogram.The idea behind the adaptive smoothing is to take advan-tage of the fact that as the ﬁlter coefﬁcients are varied overan appropriate range, less smoothing will take place. Theﬁlter coefﬁcients can be made to vary proportionally withrespect to the histogram channel where smoothing is takingplace. Smoothing is usually required prior to the analysis offrequency histogrammed data to attenuate the effect ofnoise. Smoothing through FIR ‘‘Finite Impulse Response’’ﬁltering is by far the preferred method since it will not shiftthe position of the data distributions. A perspective ondensity estimators from histograms to several univariate andmultivariate statistical data analyses can be found in studies[1–3]. Other studies using histograms bring to focus severalissues including the notions of stability in the appearance ofthe histogram [4], the interpolations that can be achieved inkernel density estimators [5], as well as the practical buttraditional aspect of enhancing images through an adaptivehistogram equalization method [6].In general, it is desirable to select a smoothing function thathas the following criteria to ensure that the resulting data isnot distorted:1–The area under the curve should be maintained2–The mean of the distribution should be unchanged3–The change in the standard deviation should be kept to aminimumThese criteria for smoothing are maintained by traditionalFIR ﬁltering provided that boundary conditions are satis-ﬁed. A modiﬁed ﬁltering scheme is introduced in this studybased on these mechanisms to customize the traditionalFIR-type ﬁltering schemes for discrete time signals toaccommodate for data accumulated into frequency histo-grams. This modiﬁed ﬁltering scheme is viewed as anadaptive ﬁltering scheme that linearizes and accommodatesfor uneven histogram accumulation effects.To illustrate this problem a random Gaussian distributionwas created using the Box-Muller Method. The samepseudo-random distribution is illustrated with three separategain settings (1, 5, 10) as shown in Figure 1. When this datais histogrammed, as shown in Figure 2, the uneven accu-mulation problem is encountered and analyzed. Forexperimental evaluation, similar histogram accumulationproblems are discussed and analyzed utilizing Phycoery-thrin (PE) bead particles containing a variety of beadpopulations at various amounts of ﬂuorochromes.