Microarray Image Analysis: Background Estimation using Region and Filtering Techniques

Transcription

1 Microarray Image Analysis: Background Estimation using Region and Filtering Techniques Anders Bengtsson December 9, 2003 Abstract This report examines properties of two main methods used for local background estimation in microarray images. The first method uses samples from local regions defined such that they supposedly contain only background pixels; we call this a region method. This method is implemented in the Axon GenePix Pro software. The second method uses filtering techniques that removes the bright spots from the image. A version using morphological filters is implemented in the CSIRO Spot software. Using cdna microarray slides, a detailed study and comparison of the background estimates of the Axon GenePix and the CSIRO Spot software have been performed. We have found that both software give a bias both within and between the two channels as well as having great variability. For the region method the number of background pixels determines the variance of the estimate while the distribution determines the bias. For the morphological method the number of background pixels used as well as the distribution of these pixels determines the level of the estimate and the variance. The morphological filtering method has been extended to the generalization rank filtering. For comparison, a region method with fixed number of pixels in the local background sample has been implemented. Both the fixed region method and the rank filtering implementations have been proven to greatly improve performance over the current Axon GenePix and CSIRO Spot implementations, reducing bias as well as variability significantly. This is especially important for low intensity spots where even a small bias has great influence. Keywords: microarrays; background estimation; bias; image analysis; morphology; rank filters, percentile filters. 1

2 Contents 1 Introduction 3 2 Data 5 3 Microarray image properties Additivity of background Pixel distributions Model Background estimation - region methods Robust estimates Sample median Spot distribution Negative background-subtracted estimates Adaptive background region method implementation Fixed region method implementation Background estimation - filtering techniques Mathematical morphology Morphological filter implementations Rank filtering Block structure Bias Region methods Rank filtering Variance 23 8 Conclusion 25 9 Further work 25 2

3 1 Introduction The DNA is the template for the protein synthesis of the cell. Almost all cells in an organism have the same genetic information. However, different cell types produce different proteins and this determines the cell appearance and behavior. For instance, a human skin cell, which is part of the barrier against foreign microorganisms, looks completely different from a nerve cell transmitting signals through the body. In the cell process, mrna is formed from a nucleic DNA sequence, a gene, by the enzyme RNA polymerase and is transported out of the cell nucleus. The mrna is then translated to a linear sequence of amino acids forming a protein. This process is called the gene expression of the cell. Since proteins are translated from mrna molecules, the mrna pool from a cell reflects which proteins are synthesized by the cell at that very moment [1]. The nucleic acid microarray technique measures the amount of mrna, from one cell sample (test) in relation to another cell sample (reference). The applications for the technique are many. For example, when cells become cancer cells, some proteins are produced or produced in a higher amount than in normal cells, whereas other proteins are stopped being produced. The knowledge of which genes that differentiates a cancer cell from a normal cell, could provide information for potential drug design in cancer therapy. Of course, the technique can be applied to any disease, to elucidate the difference in gene expression between healthy and diseased cells. In the laboratory, cell culturing is a widely used method to study how drugs or signal molecules in the body affect the mrna expression and the protein synthesis in specific cell types. By adding the substance to a part of the cells, and then comparing mrna from these cells with mrna from non-stimulated cells, one can determine what effect the drug has on the gene expression of the cell. With the possibility of monitoring the whole genomic expression of an organism, including the human, in one single experiment it is no doubt that the microarray technique represents a core capability in the field of genomic research. A microarray is defined as an orded array of microscopic elements on a planar substrate that allows the specific bindings of genes or gene products [18]. In cdna microarrays double stranded DNA, with a typical length of 500 to 2500 base pairs are printed on a substrate, usually a glass slide. The printed spots have a diameter in the range of some hundred µm and every spot consists of the same DNA molecule. Oligonucleotide microarrays, with single stranded oligonucleotides consisting of 15 to 70 nucleotide molecules, are also commonly used in the spotted microarray technology. By using the enzyme reverse transcriptase, cdna is formed from both the test and the reference sample of mrna. In the transcription step (or in the following step) fluorescence molecules with different wavelengths for the test and reference are attached to the cdna. Widely used are the Cy3 and Cy5 molecules with excitation in the green and red wavelengths, respectively. Equal amount of the labeled test and reference cdna are mixed together in a solvent and placed on the slide. The test and reference cdna hybridize to the DNA printed on the slide. After a period of time the slide is washed to remove abundant molecules. To measure the amount of hybridized cdna on each spot, the slides are scanned 3

4 with two lasers (532 nm and 635 nm) that excite the fluorescent molecules and the resulting emission of light at the different wavelengths is detected with a photo multiplier tube (PMT) and quantified. This gives two high-resolution grey level images, representing the test and reference sample. Image analysis is used for locating and segmenting the spots, measuring spot intensities and local background intensity. The local background estimate is used as a zero level for the spot signal. Since it is not possible to measure the background at the position of the actual spot, it is common practice to use the area between the spots for background estimation. This relies on the assumption that the intensity between the spots is approximately equal to the zero level of the spot signal. We call this the assumption of background additivity and it is the fundamental of background estimation. The relative difference between the test and reference samples is commonly visualized in a log-ratio plot where M =log 2 (red/green) is plotted against the log-intensities A = 1 2 log 2(red green). With the base-2 logarithm a two-fold upregulated gene is equal to an M-value of +1, a four folded by +2, and so on. Similarly for a down-regulated gene but then with a negative sign. Furthermore, the lower the intensity is, the more influence an error (bias) has on the (log) ratio. Therefore, to be able to use weak spot measures that are close to the background intensity level, the background estimate must have low variability (but still in control of the trend) and a very small bias between the red and green channel. A reasonable approach could be to use only spots with intensities that significantly differs from background, for example two standard deviations above background [16]. In this report properties of two main methods for background estimation in spotted microarray images are discussed; region methods and filtering estimates. By a region method we mean a method that uses defined regions, which are set in relation to the segmented spots, for the background measures. An example is the method used by the Axon GenePix Pro software where the background regions are defined by adaptive circles. A different approach is to use a filter that removes the spots and bright outliers from the image and the background estimates are then obtained by sampling the filtered image at the spot positions. An example of this method is the CSIRO Spot software in which morphological filters are used. We have extended the morphological filtering implementation in the CSIRO Spot software to the generalization percentile filtering. We also suggest a modification to the Axon GenePix region method, where fixed regions are used instead of adaptive circles. Both the percentile filter implementation and the fixed region method have greatly improved performance over the current CSIRO Spot and the Axon GenePix software, reducing variability as well as bias. By this, spots with low intensity can be used without having to reduce the significance level. The report is organized as follows: A brief description of the origin of the images used is found in Section 2 and motivation for local background estimation in these images is given in Section 3. Elementary properties of local background estimation using region methods are discussed in Section 4. The filtering approach to background estimation is investigated in Section 5, which covers both morphological filters and their generalization to rank filters. The bias and the variability 4

5 of the background estimate of these filters are investigated in Section 6 and Section 7, respectively. We conclude the report with a brief discussion in Section 8 and suggestions for further work in Section 9. 2 Data The analysis in this report is mainly based on data from four different microarray slides from the Oncology Department at Lund University Hospital. These slides, named Slide 1 to 4, all have the same layout of 8 4 print-tip groups, each containing spots, making a total of 7680 spots per slide. The slides are replicates of each other such that the same gene is found at the same row and column. The lower 4 4 groups (right in Figure 1) are replicates of the upper 4 4 groups. The same test and reference sample as in Slide 1, but with reversed labeling, were used for the hybridization of Slide 2 and similarly for Slide 3-4 but with a different cell line. The test samples represent two different growing conditions for the cell lines, whereas all slides use the same reference samples. All slides were scanned with an Axon GenePix 4000A scanner. Image analysis was performed at the Oncology Department with the GenePix Pro (ver ) software [2] and by us with the CSIRO Spot (ver. 2.0) software [6, 23]. Moreover, the freely available slides used in [23], here referred to as the Callow slides, have in Sections 4 and 6 been used as an example of slides with very small spot separating distances. The layouts of these slides are 4 4 print-tip groups with spots in each group Figure 1: Spatial plot of Slide 1 red channel (the plot is rotated 90 degrees anticlockwise). The 8 4 print-tip groups, each containing spots, are easily identifiable. The size of the image is pixels. 5

6 3 Microarray image properties Spot intensities measured when scanning the microarray slide do not only originate from fluorescent molecules on hybridized DNA, but also from other sources referred to as background. This background is assumed to be additive to the true spot signal, such that measured spot intensity equals true intensity plus background intensity. For instance, the source of the background signal could be fluorescence from the coating of the glass or contamination effects from the hybridization and washing procedures. The scanner device can also be a major source of background intensity with different filter bandwidths, different optics or different photomultiplier tubes (PMT) [4, 5, 7, 22]. One way to measure the background is to use the area between the spots and assume that the intensity in this area is equal to the background signal of the spots. Below, the word background is restricted to mean the intensity in the area between the spots. It is important to realize that the variation of slide quality and print layout make it difficult to draw universal results from a single experiment. For example, in the comparison of different methods for image analysis [23], microarray slides of relatively low quality with small spot separating distances were used. With images of better quality and larger distances between the spots, like the images of Slide 1-4, their conclusions may have been different Figure 2: Estimated background using a percentile filter (γ B,{0.1} ζ b,{0.7} ) for Slide 1 green channel (upper image) and red channel (lower). The background trend is clearly visible. 6

7 3.1 Additivity of background The correctness of the whole approach of background subtraction in microarray images relies on the assumption that that the local background is additive to the true signal. If the assumption of additivity holds then the number of spots with intensity below a threshold value, approximately equal to highest background level, should be greater for areas with low background than for areas with high background. As seen in figure 3 this is true for Slide 1-4. Also, if we compare the distribution of negative background corrected spots with what to be expected (see Section 4.3) the assumption of additivity seems justified. One motivation for local background estimation is the variation of background intensity within the same slide. For Slide 1-4 the background trend is clearly visible in the spatial plot of the γ B,{0.1} ζ b,{0.7} filtered image (Figure 2) Figure 3: Slide 1-4 green channel (left) and red channel (right). Fraction of spots with intensity less than a threshold equal to the maximum of the GenePix median background estimate. The spots have been sorted by their background intensity and divided in 10 groups with an equal number of spots in each group. Thus, group 1 contains the 853 spots with the lowest background and so on. Legend: Slide 1 ( ), Slide 2 (+), Slide 3 ( ), Slide 4 ( ). 3.2 Pixel distributions Commonly, when the signal level increases the variance increases proportionally [7, 17]. A direct consequence of this is that the variance of the background will not be constant but will follow the spatial trend. This is demonstrated in Figure 4, where the standard deviation (s.d.) of the background pixels is plotted against the background intensity. As we will see in Section 6.2, this has important implications on the background estimates when morphological or rank filtering methods are used. The correlation between neighboring pixels, after the trend has been subtracted, has been found to be in the order of 0.1 for Slide 1-4. The remaining correlation may be because the trend estimate is not completely accurate, which therefore induces some correlation. For the Callow slides, which has less trend (not shown), the correlation was estimated to approximately However in the discussion that follows we will assume that this effect is small and therefore also assume that the background pixels are approximately uncorrelated. 7

8 Standard deviation Background estimate Figure 4: Standard deviation for pixels inside the default background mask versus median background estimate using GenePix for Slide 1 red channel. The very noisy structure of the plot comes from using a non-robust measure such as the s.d. It seems clear that the standard deviation is not constant but increases with the background intensity. The distribution of the background pixel intensity is important. Figure 5 shows the density plots for a selected background region of Slide 1 with roughly constant level of pixel intensities. The green channel is almost symmetric whereas the red is clearly non-symmetric. Normal probability plots (not shown) give that red channel is lognormal whereas the green channel is almost normal distributed. The results are similar for Slide 2-4. density(x = Green) density(x = Red) Density Density N = Bandwidth = N = Bandwidth = Figure 5: Density plots of background pixel intensities inside a region with approximately constant background level in the green (left) and the red (right) channels of Slide 1. The green channel is almost symmetric whereas the red channel is clearly non-symmetric. The results are similar for Slide

9 3.3 Model In the models proposed in [9, 17], constant background is assumed. An extension of these models for the measured spot signal Y i of spot number i could be Y i = b global +(b local i + X i )ke εm i + ε a i where X i is the true signal, b local i is a term accounting for the trend of the image, k is a channel dependent scale factor, ε m i and ε a i are multiplicative and additive noise terms, respectively. Our initial investigation indicates that, using approximately normal distributed noise, the above model encapsulates the scanning and image analysis step fairly well. However, if also biological and slide variations are incorporated, it is much harder to fit the model to real data. An approach to this problem could be to divide the model into two separate models, the first covering the biological and physical variation prior to scanning and the second one covering the scanning and image analysis [5]. 4 Background estimation - region methods The most common method of background estimation is to select a region around every spot and compute the background from this area. GenePix [2] sets a circle around each spot and the pixels inside the circle, after the exclusion of foreground pixels, are used for background estimation, cf. Figure 6. Other software uses for instance a rectangular mask or two concentric circles [23]. The different software use different regions but all with the same goal, to identify a set of pixels containing as many true background pixels as possible and as few foreground pixels as possible. 4.1 Robust estimates The accuracy of the background estimates depends on the segmentation of foreground pixels. If the segmentation has failed and misplaced the spot mask in relation to the actual spot, this will also affect the background estimate. Moreover, artifacts such as dust particles or scratches will influence the estimate as outliers with pixel values several orders of magnitude higher. Thus, the use of a robust estimator is essential. One measure of robustness of an estimator T n (x 1,...,x n ) is the breakdown point ɛ defined as ɛ (T n ; x 1,...,x n )= 1 max{m; max n i 1,...,i m sup T n (z 1,...,z n ) < }, y 1,...,y m where (z 1,...,z n )isthesample(x 1,...,x n )with(x i1,...,x im ) replaced by arbitrary values (y 1,...,y m ) [12]. In words, the breakdown point of a sample is equal to the maximum fraction of sample values that can tend to infinity without the sample measure tends to infinity. The robustness of the commonly used sample median can be compared to the robustness of the sample mean. For the sample median the breakdown point is 9

10 0.5 whereas for the sample mean it is 0. This means that it only takes one bright pixel to corrupt the sample mean, but up to half of the pixels in the sample median before the estimate becomes completely inaccurate. The drawback of the robustness of the sample median is that the relative efficiency, defined as the ratio between the variance of the sample mean and the sample median, decreases. If the sample is normal distributed then the relative efficiency is 0.63 in favor of the sample mean. An important aspect is also that the sample median is not consistent with the expected value when the distribution is non-symmetric, see Section 6.1. As microarray images almost always contains some outlier pixels, the sample mean is not appropriate. The standard deviation (s.d.) suffers from the same lack of robustness as the sample mean. With a breakdown point of 0, a more robust measure is needed. An alternative is the median absolute deviation (m.a.d.). It is defined as m.a.d. =median i=1:n x i median j=1:n {x j }, and has a breakdown point of 0.5. However, the m.a.d. estimate is not consistent with the s.d.; for normal distributions the ratio between the s.d. and the m.a.d. is Sample median The asymptotic distribution of the sample mean, x is obtained through the central limit theorem, x = 1 n σ x i AsN(m, ), n i=1 n where {x i } n 1 is a sample from an independent and identically distributed (i.i.d.) random variable X with mean m and s.d. σ. A similar theorem can be derived for the sample median. We define the sample median, x {0.5}, of the ordered sample {x (i) } n 1 as x {0.5} = { x(k), n =2k 1 (x (k) + x (k+1) )/2, n =2k. Let p X (x) be the probability density function (p.d.f.) and let F X (x) bethecumulative density function (c.d.f.) of a continuous random variable X. IfM is the median, i.e F X (M) = 1 2, then the limit distribution of the sample median is [8] ( ) 1 x {0.5} AsN µ, 2f X (M). n Not surprisingly, the variance of the sample mean and sample median decrease with the number of samples, more precisely as the inverse of the sample size. 1 Furthermore, if X is N(m, σ) we have that the s.d. is = π σ 2f X (M) (n) 2 n. Comparing this with with the s.d. of the sample mean, we indeed find that the relative efficiency is 2/π

11 4.3 Spot distribution Using the above approximations we find that the distribution of the spot and background estimates are approximately normal distributed. Using the sample mean for the spot (foreground) measure and the sample median for background measure, the background subtracted spot signal is ( ) N m fg M bg, σfg 2 /n fg +1/(4fbg 2 (M bg)n bg ), and if the sample median is used for both the spot and background measure it is ( ) N M fg M bg, 1/(4ffg 2 (M fg)n fg )+1/(4fbg 2 (M bg)n bg ), where f fg, f bg are the p.d.f. of the pixels, n bg and n fg are the number of pixels used in the background and spot samples, M fg and M bg are the medians, m fg is the mean and σ fg, σ bg is the standard deviation of the spot and background pixels, respectively. 4.4 Negative background-subtracted estimates If we assume that a spot with no expression gives a signal not different from the background, then the probability of getting a foreground estimate which is lower than the background is 0.5. Since there is a (small) probability that the foreground estimate is lower than the background estimate even for expressed spots, the number of non-expressed spots is equal to or less than twice the number of negative background-corrected spots. If the distribution of negative background subtracted spots is not in accordance with what to be expected from the distribution of background pixels, this can indicate that the background estimation is inaccurate. For a further discussion in this direction of how to model negative estimates see [15]. 4.5 Adaptive background region method implementation The GenePix background estimation method uses the segmented spot size to determine the size of the background mask [2]. The outer circle of the background mask pixels has a diameter of three times the diameter of the smallest circle covering all pixels in the corresponding segmented spot. All pixels within this background mask but outside the spot masks are used for the background sample. The spot masks are circles centered at the spots with a diameter equal to the smallest circle covering all pixels in the corresponding segmented spot plus two, cf. Figure 6. Thus, the number of background pixels inside the mask depends on the size of the individual spot and the distances between the spots. For Slide 1 the number of pixels used for the background sample varies from 40 to This causes an unnecessary change in variance between measures, which will be discussed in Section 7. 11

12 Figure 6: The background mask used by GenePix; Callow slides (left) and Slide 1-4 (right). The small grey discs are the spots. The dashed circles around each spot is a two-pixel wide safety zone around each spot. The black circle depicts the outer border of the region containing pixels used for background estimation. Pixels within this region that are neither inside a spot nor inside a safety zone are denoted background pixels. The resulting pixels used for the estimate are highlighted in black. 4.6 Fixed region method implementation For the purpose of comparing (Section 6 and 7) and understanding the different methods for background estimation, we have also implemented a region background estimation method, Fixed R, with a constant number of pixels used for the background sample. The implementation is very much similar to the GenePix implementation with a circular background mask. The radius of the outer circle is set to a fixed value R, equal for all spots. To eliminate spot segmentation variation, the diameter of the spot exclusion mask is also fixed to the diameter of the largest spot. The masks were centered for each spot using the coordinates obtained from the GenePix software. 5 Background estimation - filtering techniques The name morphology comes from the Latin word for shape, morph, and the theory of mathematical morphology presents a wide range of tools for altering shapes found in images. There are two basic operations in mathematical morphology, erosion and dilation. These can be combined to perform openings, closings, region fillings, boundary extraction, thinning, thickening and many others more or less complex operations. For further reading, see [11, 19, 20]. It is no doubt that morphological filtering and its generalization to rank filtering present an elegant set of tools for baseline estimation in microarray images. Without the need to know the exact location of the spots they are insensible to segmentation faults and furthermore to bright outliers such as dust particles and scratches on the surface of the slide. However, as shown in the proceeding sections there are some important properties of morphological and rank filters that have to be considered in order to get accurate estimates. It should be emphasized that the word filter in this report is used in the general meaning, as a synonym for operator. This is not consistent with some of the literature found in the field of mathematical morphology where the word filter is 12

13 reserved for an operator that is increasing and idempotent [13, 19, 20]. The use of morphological filters for cdna microarray background estimation is implemented in the Spot software [6]. 5.1 Mathematical morphology In the literature, there exists some confusion regarding both definitions and notations in mathematical morphology. For instance, two different definitions for dilation are used and identical symbols have different interpretations. In the following proceeding we intend to follow the notation used in [21]. A digital grey scale image can be represented by the image function f : D f T f, with domains D f Z 2,andT f R or T f Z depending on if the grey levels are continuous or discrete, respectively. That is, the image function f = f(x) isequal to the grey level at position x =(i, j). Let B be a compact subset of Z 2 that is symmetric with respect to its origin. We then define the erosion of f by the (flat) structuring element B, positioned at pixel x as [ε B (f)](x) =min{f(x y) x y D f ; y B}, and the dilation of f by the structuring element B is defined as [δ B (f)](x) =max{f(x y) x y D f ; y B}. For a composition of operators on f we will write ψφ =[ψφ](f) =ψ[φ(f)] and ψ 2 = ψψ. Erosion and dilation can be combined to perform openings, γ B,and closings, ϕ B, defined by γ B = δ B ε B, ϕ B = ε B δ B. That is, an opening is an erosion followed by a dilation and a closing is a dilation followed by an erosion, using the same structuring element B. Openings and closings are increasing, f g γ B (f) γ B (g),ϕ B (f) ϕ B (g), and idempotent, γ 2 B = γ B and ϕ 2 B = ϕ B. Openings are anti-extensive and closings are extensive, γ B (f) f, ϕ B (f) f. 13

14 Generally, an operator with these three properties is called an opening and a closing, respectively. A more complete survey of definitions and properties of morphological filters can be found in [13]. In the following filter implementations the structuring elements are square shaped. 5.2 Morphological filter implementations Two main morphological methods for background estimation are available in Spot. These are opening and dilation-erosion-dilation. The method described in [23] is the first, opening. The structuring elements used for the opening have a size of two and a half times the spot separating distance, horizontally and vertically. The reason for choosing exactly this size is unknown to us. The anti-extensive property of an opening gives that every pixel intensity in the filtered image is less than or equal to the corresponding pixel intensity in the original image. The second method, δ B ε B δ b referred to as morph.close.open in Spot, consists of the following three steps, 1. Dilation δ b = δ 3 3, 2. Erosion ε B = ε kscale (s r 2) k scale (s c 2), and 3. Dilation δ B = δ kscale s r k scale s c. where k scale is a scale factor with a default value of 2.5 and s r and s c are the spot separating distances in pixels by row and column, respectively. The name morph.close.open is somewhat misleading since it consists of neither a closing nor an opening. The reason for this is that different structuring elements are used in each step. The first dilation step is necessary to achieve a final estimate in level with mean (expected) background. Solely an opening will always give an estimate less than or equal to the original image. The maximum size of the structuring element used in the first dilation is determined by the distance between the spots and the size of the erosion in the second step. The size must be small enough to ensure that it is possible to place at least some dilation elements containing only background pixels inside the larger structuring element used in the erosion. The erosion stepremovesallspotsaswellasallotherpixelsbrighterthan background. To ensure that all spots are removed, the erosion element must be larger than any of the spots. Furthermore, the structuring element should be large enough to contain approximately the same number of background pixels independently of if it is centered over a spot or between spots. Naturally, the size of the erosion, as well as the size of the last dilation, also affects the level of the estimate. The last dilation is used to narrow the estimate after the erosion, cf. opening and closing. The use of the spot separating distances, measured from center to center, in Spot software as a parameter for the size of the erosion structuring element is not ideal. Instead the size should be determined by the number of true background pixels inside the structuring element. This will be discussed further in Section

15 Figure 7: Plot of a row 417 in Slide 1 red channel. Illustration of the three steps in Spot s morph.close.open: dilation - δ 3 3 (upper dashed line), erosion - ε (lower dashed line) and dilation - δ (upper thick line). Also included open - γ (lower thick line) with no initial dilation. Keep in mind that the plot is in one dimensions and that the morphology takes place in two, this explains why the lines do not follow the original image strictly. Inthepresenceofadark spot, that is, a spot significantly lower than the surrounding background (and if the first dilation element is small enough to fit inside), then the intensity of this spot will be the lowest local value and therefore the one remaining after the erosion. The advantage of this is doubtful for two reasons. First, we only know that this dark spot has lower intensity than the surrounding background, but we do not know that it is completely without expression. Second, if there are no unexpressed spots nearby, the estimate would still be too high. 5.3 Rank filtering A rank filter, also called order statistic or rank order filter, is defined in analogue with a morphological filter where the minimum or maximum is replaced by a rank operator, denoted rank k, which instead of the extreme chooses the kth value in ascending order. Using the same notations as in the preceding section, a grey scale image is represented by f : D f T f with the domain D f Z 2 and T f R or Z. Let B be a compact subset of Z 2 that is symmetric with respect to its origin. A rank filter of order k operating on f, using a structuring element B positioned at pixel x, is then defined by [ζ B,k (f)](x) =rank k {f(x y) x y D f ; y B}. 15

16 Obviously we have ζ B,1 ζ B,2... ζ B,n, and that morphological erosion and dilation can be written as rank filters, ε B = ζ B,1, δ B = ζ B,n ; cardinal(b) =n, where cardinal(b) is equal to the number of pixel inside B. By replacing the morphological operators in opening and closing with rank operators we get rank opening γ B,k and rank closing ϕ B,k, γ B,k = ζ B,n k ζ B,k, 1 k<n/2, ϕ B,k = ζ B,n k ζ B,k,n/2 <k n. The increasing property of morphological opening and closing holds also for rank opening and closing. However, they are in general neither extensive, antiextensive nor idempotent. For this reason, the name opening and closing is not entirely correct. An important special case of rank filtering is obtained by using a rank defined as a fraction (percentage) of the number of pixels inside the structuring element. We call this a percentile filter and use the notation ζ B,{q} with the rank defined by {q} = { [cardinal(b) q +1], 0 q<1 n, q =1, where [x] is equal to the greatest integer less than or equal to x. A percentile filtering generalization of Spot s morph.close.open is the filter ψ = γ B,{q2 }ζ b,{q1 }, 0.5 <q 1 1, 0 q 2 < 0.5, The properties of this filter are very much the same as for its morphological counterpart. The first step, ζ b,{q1 }, sets the basic level of the estimate, and the following γ B,{q2 } = ζ B,{1 q2 }ζ B,{q2 } filters out the spots and bright outliers. In addition to the discussion about the size of the structuring element for morphological filters we have that the order of the second step, γ B,{p2 }, must be sufficiently small to ensure that all spots are filtered out. 5.4 Block structure Comparing a morphological filtered image with a rank filtered one, the first thing noticed is the block structure in the morphologically filtered image, which is not seen if the less extreme rank orders are used. See Figure 8 and 9. The reason for this block structure is the use of extreme values, i.e. minimum and maximum. If 16

17 we choose the minimum (maximum) value inside a window, then the distance we have to move the window in order to find a smaller (greater) value is more likely to be long than if we for example had chosen the median value Figure 8: Slide 1 red channel (row=500:1000 column=1800:2300). Morphological opening γ (left) and rank opening γ 50 50,{0.2} (right) A B ζ 20 ζ B,(0.2) B,(0.8) Figure 9: c.d.f. for a lognormal distribution similar to the background in Slide 1 red channel. For less extreme ranks such as ζ B,{0.2} and ζ B,{0.8} the distances in intensity between pixels are much less than for the morphological minimum and maximum value. This explains the block structure in morphological filtered images. 6 Bias By the (additive) bias of an estimate is in this report meant the difference between the expected, denoted E[ ], and the estimated value. A bias between the channels is typically seen as a curvature at the low intensity end of the MA-plot. The curvature is often removed by doing background subtraction, but it can also be corrected for by using various normalization methods [3, 9, 14, 16, 24]. The locally dependent bias is harder to correct for by normalization because it is not necessarily non-symmetric between the two channels. Therefore it seems more reasonable to subtract the background before any normalization is done. 17

18 6.1 Region methods If we assume that the median region estimate is not affected by outliers or spot segmentation faults, the bias of a median region method depends entirely on the shape of the distribution. The more non-symmetric the distribution is, the more biased the median estimate will be. For a lognormal distribution LN(m, σ) the difference between the mean and median is e m+σ2 /2 e m = e m (e σ2 /2 1). When the level of the background increases, the variance also increases (see Section 3.2) and thereby also the bias. For the lognormal distributions in Figure 13, which roughly represent the background of Slide 1 red channel, the difference in bias is Rank filtering If f is a (grey scale) image, consisting of i.i.d. pixels with the c.d.f. F f (z) = P (f z) then the c.d.f. of the pixels in the rank filtered image ζ B,k (f) with cardinal(b) =n, is F ζb,k (z) = n ( n ) [F f (z)] i [1 F f (z)] n i, 1 k n. i i=k The summation term is recognized as the probability that amongst n values i of these are less than or equal to z. This is the same as the binomial probability distribution function (p.d.f.) [10]. For the rank filters ζ B,1 and ζ B,n, i.e. erosion and dilation, we get the well known c.d.f. s F ζb,1 (z) =1 (1 F f (z)) n, F ζb,n (z) =F f (z) n. Note that the pixels in the filtered image are correlated. For this reason it is much harder to derive the c.d.f. after another rank filter operation. The background pixels can be assumed to be locally i.i.d. (Section 3.2). The c.d.f. s above then gives that the mean and the variance of the background pixels in the filtered image depend both on the number of pixels inside the window (structuring element) and on the underlying distribution of the pixels. This is also true for a series of rank filters and makes an important difference to the region methods, where only the variance is affected when the size of the mask is changed. The dependence on the number of background pixels can clearly be seen in Figure 10, where the mean value of Spot s morph.close.open is plotted against the Spot scale factor for the structuring element, k scale. When the sizes of the structuring elements (B and B ) increases, the mean level of morph.close.open decrease. It is therefore, in general, not possible to use the size of the structuring element as a way to control the variance of the estimate. However, if a percentile filter, i.e. a rank filter with the rank set as a percentile of the number of pixels inside the structuring element, is used the dependence to the number of background pixels inside the structuring element is greatly reduced, cf. Figure

19 110 Slide1 Red, background estimation by Spot 105 mean(morph.close.open) Spot scale factor (SE.scale) Figure 10: Slide 1 red channel. Plot of mean value for Spot s morph.close.open (δ B ε B δ b )vs. k scale. The size of the structuring element B is (k scale s r k scale s c )and the size of B is (k scale (s r 2) k scale (s c 2)), where s r, s c are the spot separating distances by row and column, respectively. There is a clear relationship between the mean level of the background estimate and the sizes of the structuring elements B and B. If ψ is an arbitrary rank filter or a series of rank filters, then every pixel in the rank filter image, ψ(f), is equal to some pixel in the original image f, chosen only by the mutual order between pixels in f. Hence, the addition of µ to f or multiplication of λ with f does not change the order of the pixels. More precisely, for the rank filtered image we have that ψ(λf)=λψ(f), λ R + or Z +, ψ(f + µ) =ψ(f)+µ, µ RorZ. For a normal and a lognormal distribution the multiplication with λ is essentially the same as changing the variance without changing the shape of the distribution. Let f i be a (local) background region in the image f where the pixels are i.i.d. with mean m. If there is a bias µ in the filter estimate, i.e. E[ψ(f i )] = m + µ and we change the variance by multiplication with λ we get E[ψ(λf)] = λ(m + µ). The bias is thus proportional to the increase in variance, as long as the shape of the distribution does not change. Furthermore, it is not easy to correct for the bias by shifting the estimate by multiplication or addition of some constant because we do not know how the variance of the background changes in the microarray image. In Section 3.2 it was shown that the variance of the background increases with the background level of Slide 1 to 4. This is confirmed by Figure 11 and 12 where the bias of the filtering estimates increase with the background level. 19

20 Figure 11: Slide 1 red channel; column 915. From bottom: (i) opening - γ 50 50, (ii) the corresponding percentile filter γ 50 50,{0.1}, (iii) the same shifted 17 unit up for visibility (dashed) and at top (iv) the γ 50 50,{0.1} ζ 3 3,{0.7} filter which is almost equal to (v) the median background (bold curve). Notice that opening and the corresponding percentile filter do not follow the median background trend well enough. Also recall that the distribution of background pixels is approximately lognormal (LN(m, σ)) in the red channel and therby the expected background is e m (e σ2 /2 1) higher than the median estimate Difference background Mean background Figure 12: Differences between γ 50 50,{0.1} ζ 3 3,{0.7} and γ 50 50,{0.1} ζ 3 3,{0.5} in the red channel of Slide 1. The difference in bias between the two filter estimates is 10 when the background is low whereas the difference is 20 when the background is high. Hence, there will be a bias both within the slide as well as between the slides (if the background is not identical in the red and green channel). 20

21 50 normal; γ B δ 3 3 normal; γ B,{0.24} ζ 3 3, sigma=70 30 sigma=70 3 mean mean n n 30 lognormal; γ B δ 3 3 lognormal; γ B,{0.14} ζ 3 3, m=5 4 m= mean mean n n 150 normal; γ B δ 3 3 normal; γ B,{0.24} ζ 3 3,6 45 sigma= sigma= variance variance n n lognormal; γ B δ 3 3 lognormal; γ B,{0.14} ζ 3 3, m=5 m=5 variance variance n n Figure 13: Mean vs. n = cardinal(b) (upper four plots) and variance vs. n (lower four plots) for filtered simulated images with i.i.d. normal, N(0,σ); σ =(30, 50, 70), and lognormal, (LN(m, 0.4) e (m+0.42/2) ); m =(3, 4, 5), pixel intensities. Two different filters are compared: a morphological filter γ B δ 3 3 and arankfilterγ B,{q} ζ 3 3,k. For a certain size of the structuring element B the filter estimates will be unbiased even if the variance of the (background) pixels change. 21

22 When estimating the background with the above rank filters, only those pixels that are true background pixels should be counted. Assume that there are no artifacts and that the distances between spots are greater than the size of the first dilation. Then the number n of background pixels inside the structuring element B is approximately equal to n cardinal(b) n spots (r spot + µ) 2 π, where n spots is the number of spots inside the structuring element and r spot is the radius of the average spot. The addition of µ to the radius is needed because of a boundary effect around each spot, which depends on the rank and size of the initial dilation. If the first step is the rank filter ζ b,k, then there can be a maximum of cardinal(b) k spot pixels inside the structuring element, if the filtered pixel is to be regarded as a background pixel in the proceeding erosion, cf. Figure 14. Morphological dilation δ 3 3 requires µ =1. A A A C C B B Figure 14: For the rank filter γ B,{q2 }ζ b,k there will be a boundary effect around each spot. The size of this effect depends on the rank k. If the initial filtered pixel is to be considered as a background pixel in the proceeding step, the maximum number of spot pixels inside the structuring element b is equal to cardinal(b) k. For δ 3 3 = ζ 3 3, 1 this is zero (A), for ζ 3 3, 8 there can be one spot pixel inside b =3 3(B),forζ 3 3, 7 two(c)andsoon. If the same structuring element is used for the whole image there will be a negative bias in the background estimate for areas with absent spots, i.e. spots not different from the background, as the number of background pixels inside the structuring element increases. For spots located at the border to the alleys, which separates different print tip groups, the effect is the same. For these spots the number of background pixels inside the structuring element is greater than at spots away from the alleys. Close examination of the estimated background images in Figure 2 reveals that the intensity is somewhat lower in the alleys, making them appear as darker lines in the images. The effect is stronger if the spot separating distance is small. In Figure 15 the density of the estimated background 22

23 for both alley spots and non-alley spots are plotted for two slides with different spot separating distances. A possible solution to the problem could be to use a structuring element with a variable size such that the number of background pixels inside the structuring element is independent of where on the image it is placed alley spots other spots alley spots other spots Figure 15: Spot morph.close.open background estimate. Density plots for spots at the border to alleys compared to not alley spots for a Callow slide (left) and Slide 3 green channel (right). It is clear that the filter estimates is lower for spots with few neighbors, i.e. alley spots, than for spots with eight neighbors. The effect is greater for the Callow slide because on this slide the spots are printed closer to each other within the print-tip group. 7 Variance When local background subtraction is done, noise is added to the spot intensity estimate. Since two sample measures are used, one for the spot intensity and one for the local background, this is inevitable. For this reason it is desirable to have a smooth background estimate with small variance and at the same time control for the overall trend of the slide. To measure the variability of the background estimate we use the squared nearest neighbor deviation (s.n.n.d.) defined as n i n j s.n.n.d. = [[(φ(i, j) φ(i +1,j)) 2 +(φ(i, j) φ(i, j +1)) 2 ]/(2n i n j )]. i=1 j=1 Measured pixel by pixel, φ(i, j) is equal to the value of the pixel located at x =(i, j) in the filtered image and n i n j is the size of the image. Measured spot by spot, φ(i, j) is the background estimate at spot number (i, j) wherei and j are the spot row and spot column, respectively. In Section 4.2 we concluded that the variance of the sample median estimate decrease in proportion to the inverse of the number of pixels used. By default the background mask in GenePix is a disc with a diameter of three times the spot diameter, with the exclusion of spot pixels. For Slide 1, the number of pixels used for the background estimate varies from about 40 to 1500 on the same slide, and 23

24 this gives an estimate with an unnecessary large variance. Just because of variable number of background pixels, the variance of the background estimate varies with a scale factor of 1/40 to 1/ Figure 16: Estimated background for Slide 1 green channel spot column number 40. Fixed 35 (thick line), γ B,{0.1} ζ b,{0.7} +3 (thick dashed line) and GenePix median (thin line). The background estimates of Fixed 35 region method and the rank filter follow each other closely whereas the GenePix median estimate have much greater variability. green/red Slide 1 Slide 2 Slide 3 Slide 4 γ B,{0.1} ζ b,{0.7} 6.61 / / / / 1.92 Fixed / / / / 1.88 Fixed / / / / 5.03 γ B δ b 20.9 / / / / 6.01 GenePix 64.5 / / / / 22.8 Table 1: Squared nearest neighbor deviation (in the green and red channels) measured spot by spot for a percentile filter, γ B,{0.1} ζ b,{0.7}, a morphological filter γ B δ b (almost equal to morph.close.open) and three region based methods, Fixed 35, Fixed 18 and GenePix. The structuring elements are B =50 50, b =3 3and the subscript of Fixed R is the radius of the background circle mask. By measuring spot by spot the effect of the block structure in the morphological filter estimate is to some extent reduced, cf. Table 2. For comparison, a background estimation method with fixed size of the background mask has been implemented, see Section 4.6. The results are presented in Table 1 and Figure 16. With a circle diameter equal to three times the average spot diameter the s.n.n.d. is considerably lower than for the GenePix estimate, and with a circle diameter comparable to the size of the structuring element B in γ B,{0.1} ζ b,{0.7}, the s.n.n.d. is roughly equal to the s.n.n.d. of the percentile filter. Apart from having the same s.n.n.d., Figure 16 shows that the percentile filter γ B,{0.1} ζ b,{0.7} and Fixed 35 give almost equal estimates, except for a bias of 3 in 24