Multidimensional Median Filters for Finding Bumps in Chemical Sensor Datasets

Feature detection in chemical sensors images falls under the general topic of mathematical morphology, where the goal is to detect “image objects” e.g. peaks or spots in an image. Here, we propose a novel method for object detection that can be generalized for a k-dimensional object obtained from an analogous higher-dimensional technology source. Our method is based on the smoothing decomposition, Data = Smooth + Rough, where the “rough” (i.e. residual) object from a k-dimensional cross-shaped smoother provides information for object detection. We demonstrate properties of this procedure with chemical sensor applications from various biological fields, including genetic and proteomic data analysis.


Introduction
Numerous chemical sensor platforms and technologies require image analysis techniques to isolate the signal from the associated noise in the sensor.In a one-dimensional chemical sensor setting, for example, several technologies produce spectra where scientists can gain information from associated peaks, or grayscale images where the features appear as streaks or lines.Meanwhile, in a two-dimensional setting, associated technologies produce images whose features are spots.Such image analyses usually involve methods where the goal is to identify and quantify the size of an image feature or object, i.e. feature detection and quantification.
Feature detection in multi-dimensional images is an area of great interest in a variety of applications, ranging from astronomy to proteomics [1][2][3][4][5][6][7].Proposed methods employ image segmentation techniques such as watershed methods, thresholding operators, and wavelet reconstruction methods to locate the features contained in a one-dimensional or two-dimensional image.Further, feature detection has a growing body of research in larger high-dimensional datasets, as well; see, for example, [8,9].The algorithms and methods proposed, however, usually apply solely to the application and technology of interest and may not be applicable to images of other forms or varying dimensionality.
Determining the locations and boundaries associated with various chemical sensor features has been a problem considered by computer scientists and engineers (under the guise of image analysis), as well as mathematicians and statisticians (via mathematical morphology).Mathematical morphology (MM) is the science of analyzing and processing geometric structures (e.g.local maxima) in digital images via various processing techniques (e.g.local maxima) in digital images via various processing techniques [10][11][12][13][14][15].Examples of common MM functions include opening, closing, thinning, binning, thresholding, and watershed methods, and have been employed in numerous applications including pedestrian detection [16], tumor mass detection [17], and facial feature detection [18,19].A key component in MM lies in the choice of structuring element, i.e. the shape used to interrogate the image; its two main descriptive characteristics are its shape and size.In digital images, the structuring element scans the image and alters the pixels in its window content using basic operators similar to Minkowski addition.Since the goal is commonly to smooth images by removing the statistical noise, the usual practice is to choose a window which is (hyper-) cubical or (hyper-) spherical.Since our goal is feature detection rather than data smoothing, we instead propose a MM technique with a "cross" shaped structuring element in conjunction with residual analysis to aid in bump finding in chemical sensoring images.We have found that, by choosing the window to be (hyper-) crossical (i.e.shaped like a multi-dimensional cross), the resulting residual image also contains crosses whose centers identify the locations of local maxima.This paper combines aspects of feature detection, data smoothing, and residual analysis to develop a new bump detection method for not only one-or two-dimensional images, but k-dimensional images for any .Thus, not only is this method straightforward, but it can also be applied universally to higher-dimensional images, providing researchers with a detection and quantification method for any chemical sensor technology whose features of interest are bumps.

Theoretical Model
In our method, a specialized median (referred to hereafter as an s-median) smoother is developed, where the s-median determines the median associated with the intensity values that lie spatially in the cross-shaped structuring element.Consider a k-dimensional (kD) image represented by   k I x , where x is a point location in the Cartesian coordinate system.We let I is replaced with the associated median value from the 5-pixel cross.Similarly, applying a 9-pixel cross s-median (as shown in Figure 1(b)) across 2 I produces .

2,2
After applying this s-median throughout the raw image, we examine the associated residual image, , , , to obtain information regarding bump detection and quantification.The , k c image contains k-dimensional cross features, where associated image local maxima identify the associated bump center, and local minima outline the shape of the bump.We can use this information, for example, to identify peaks and their associated area in one-dimensional applications involving spectral data, or spot detection and quantification in two-dimensional images.Sections 2.1 and 2.2 introduce the theoretical underpinnings for our method and demonstrate the procedure for continuous and discrete functions, while Section 2.3 extends these ideas to study the behavior of the smedian operator in the presence of noise.

Computation on Continuous Functions
This section develops the theoretical underpinnings for via the function mapping between input and output values.

One-Dimensional Continuous Functions
 denote the cumulative density function (cdf) and probability density function (pdf), respectively, for the a random variable X evaluated at the point x ; analogously, we denote the cdf and pdf for a random variable at point .Let be a function that maps from the support set (for the random variable (for the random variable Y).For our applications,   G x is ed from an optical device such as a charge-coupled device camera or laser scanner.Our goal is to obtain an expression for . Note that, in our notation for one dimension, x at a given location, x.Thus, for simplicity, we will denote ) is also a function of k and c.

Monotone Case
Consider the case where is strictly monotone on the interval Strict monotonicity in implies its invertibility for any , i.e.
; in particular, by definition of . Hence, in the monotone case, , where denotes the median associated with the random variable X , and "  " denotes statistical equivalence as defined in [20], p. 36.Thus, the median for is equivalent to the function evaluated at the median for Y X ; in short, .

Piecewise Monotone Case
For the piecewise monotone situation, we define is strictly monotonically increasing on

 
i .Note that continuity may not be enough for the A to be countable, where we define countability as in [21].Further, we assume strict monotonicity in the function, . We define the interval, , does not necessarily partition ; e.g., see Figure 2, where and A th decomposition of , and x is strictly monotone.Thus, for x X  , we have , implying that any function can be decomposed into the sum of its strictly monotone components, .Accordingly, we see that For all but the most simple functions, there is no closed form solution by which to define Y M .Nevertheless, the above equations will allow for the calculation of Y M and thus   S x using computational methods.

Two-Dimensional Continuous Functions
In the 2D continuous case, we introduce the function   = , Z G x y where the goal is to obtain an expression for . From standard probability theory such as in [22], we have x y is the joint pdf for X and .Note that this is the general case for obtaining the cdf of Y Z in terms of X and .For our specialized median, however, our sample space for Y X and must be defined in terms of another parameter, say , where controls the width of the smoothing window in each dimension.Figure 3 illustrates an example sample space over which to compute x y , we cannot generalize this situation to provide an explicit calculation for the median of   , Z x y .Nevertheless, computational associated with   G X  and its impact on the s-median.

One-Dimensional Discrete Functions
 denote the discrete cdf and probability mass function (pmf), respectively, for the arbitrary random variable X evaluated at the point x .Let   :  be as defined above.The calculation of the s-median proceeds by assuming X Discrete Uniform( ) as defined in [23]

  y
We can analogously represent Y  using discrete random variables X and Y as we did for the continuous case, namely where indicates probability.If we assume that X~ Discrete Uniform( ), then Pr N

Computation on Discrete Functions function
upport set  (of the discrete random variable X) to the support set (for the discrete random variable Y).This section considers computational results the s  ting the ceiling and floor functions, respectively.nc-deno For the special case of a strict monotone discrete fu does not depend on the direction of monotoni-

Functions
ple space G  .Figures 4(a)-(c) show the images from our technique pplied to a simple one-dimensional discrete piecewise monotone function.
, where Nicely, all of the above quantities can be computed we specified the distributions for X and .Note t e zero quantity in the third line is due

Extension to Larger Dimensions
In this manuscript, we directly show the calculations for one and two dimensions.However, our method can be ( 3  [24].

Gaussian Noise Setting
In this section, we examine the p perties of our procedure i setting for image 1, where i g denotes the true signal at location i , i s equals the step size at i in the monotonic sequence such that x since own in [24], this may indicate a strictly increasin decreasing sequence.In certain settings, a  the absolute maximum location as a function of p s  from noise for the top two mountains.Recall that, with the noise-free single mountain example, we clearly detected a cross in the rough operator image at the mountain's maximum.Further, the size of the observed cross was directly related to the operating window.As the window size increased, the size of the cross increased as well.Consider adding noise to the mountain in Figure 4(d).Figure 10(a     detect larger spots in the presence of noise, and (2) in the presence of noise, larger values of c are more effective for detecting spots.Collectively, Figures 6-11 illustrate the tradeoff that must be considered when determining the arm size for the s-median smoother.We see that large values of are more likely to yield positive residuals at the maximum in the I image; however, the residuals associated with large values of c are also more likely to be nonzero in the presence of noise over monotonic regions.In other words, for spot finding, large values of c improve spot detection in noisy images, however, it may cause two distinct spots to merge into one spot in the presence of noise.A balance between these two issues will be critical in choosing the optimal c value(s) for peak or spot finding (see Section 3.4).c

Results and Discussion
In this section, we present the results from applying our method to biologically motivated chemical sensor array data, including mass spectrometry, gel electrophoresis, and spotted microarray data.In mass spectrometry, the relevant data are represented as spectra where the associated peaks in the intensity plots represent proteins (or peptides) present in a sample.Obtaining the location and intensity of these peaks aides in identifying sample proteins for further study consideration.Gel electrophoresis data are represented in the form of 2D images comprised of protein spots.Again, investigators are interested in detecting these features in order to isolate their location in the image and potentially extract the associated protein sample for further analysis.Finally, spotted microarray data are represented as two-dimensional images of spots in a 2D matrix structure.Feature detection is key in order for the genetic data to be properly summarized and thus for these technologies to have utility in diagnosing disease or assessing putative biomarkers.
The code to perform our method is written using FIASCO, a collection of statistical software created in the Department of Statistics at Carnegie Mellon University that was originally designed to analyze functional magnetic resonance imaging (fMRI) data.The computer code used for this work are available upon request from the corresponding author.In the following, we demonfile protein markers from tissue or bo thus applying the s-median g median to the I image.

Mass Spectrometry
Matrix-assisted laser desorption ionization time-of-flight (MALDI-TOF) mass spectrometry is a technology that can be used to pro dily fluids, such as serum or plasma in order to compare biological samples from different patients or different conditions.The output from a MALDI-TOF experiment consists of a measured intensity for each massto-charge ratio (m/z) value; see Figure 12(a).The sets of expressed proteins are identified within each spectrum in order to ultimately determine differentially expressed proteins between conditions or samples.See [25] for further details describing the MALDI-TOF technology.
Our s-median derived R image can be used to detect peaks in MALDI-TOF images and thus locate peptides present in the sample.The spectrum for each sample consists of a single vector, I, is equivalent to applying a runnin This dataset in question was obtained from the Proteomics Core Laboratory at Roswell Park Cancer Institute.We use this real data to examine the results of applying the s-median to a MALDI-TOF spectrum.In this example, we set this dataset's bandwidth (i.e. the value of c described in [26].The ability to detect spots is crucial since missingness in this technology affects downstream analysis of detecting differential expression [27]. es, we will focus on images For our 2D-DIGE exampl representing portions of the 2D gels examining morphogenesis in Drosophila obtained from the Minden laboratory at Carnegie Mellon University [28,29].These images are obtained from a charge-coupled device (CCD) camera and the protein spots in these images allow the researchers to obtain a protein expression signature of the sample under a given condition or given time point.The images under study have been normalized according to the model described in [26].The full images are 1024 pixels 1280 pixels  and densely populated with protein spots, making it difficult to observe individual protein spots in detail.We therefore focus on a 50 mag pixel  50 pixel sub-i e to better understand the impact of applying the s-median.
Figure 13(a) shows the protein gel sub-image selected for illustration, I , with the associated perspective plot shown in Figure 13(b).Figure 13(c) displays the associated residual image, 2,6 R .From Figure 13(c), we can see the crosses associated with the protein spots shown in Figure 13(a).As well, we also see that each protein spot is outlined in black since, in the noise-free case, the R image is negative at local minima in an image.Thus, we use the black outline as a boundary identification tool to determine spot size in order to more accurately determine summary information and excise the protein sample(s) of interest from the gel.In 2D-DIGE experiments, after quantification of the protein spots under different channels or conditions, similar to gene microarrays, the spot ratios are computed and compared to assess the degree of differential expression.

Spotted Microarrays
Genetic microarrays are a popular analysis tool to study eposited at that location.See [31] for a detailed description of the microarray technology, and [32] for an overview of the methods used for microarray analysis.Pin-based spotted microarrays have the probe material deposited on the glass slide via a microscopic pin tip.In the pin-tip based microarray technology, image analysis software is required to summarize the signal for a given spot on a chip.In this situation, we can examine the R image obtained from a pin microarray image for proper identification of spot locations and spots sizes to aid in spot quantitation and data summarization.Note that this technology can be extended to study proteins as in [33] and other biologically active molecules where antibodies can be developed and spotted to the chip and used as capture molecules.Thus, this technology has widespread potential as a chemical sensor panel to monitor biological activity (e.g., see [34][35][36]).
Figure 14(a) shows an example of a microarray image obtained from a cell cycle yeast experiment [37].Similar to the gel electrophoresis example, Figure 14(b) examines a subsection of the microarray chip shown in Fi- gure 14(a).Figure 14(c) shows the associated residual image ( ) obtained from applying an s-median to Fi- gure 14(b) closer inspection of Figure 14(c) reveals a black spot within the center of each microarray probe.This is an interesting phenomenon attributed to the manufacturing of the microarray.Occasionally, the impact of the pin onto the microarray chip displaces the probe material and causes a "doughnut" shape probe hybridization profile.The hybridization spot has a "hole" in the middle since there was little or no probe material deposited to hybridize.This effect is not obvious in Fi- gure 14(b) but is clearly distinguished in Figure 14(c)-(d).This kind of information can be used to improve the estimation of spot intensity in the microarray image.The spot intensity estimates are used as input for downstream processing, ultimately, yielding the expression value for each probe representing the amount of hybridized genetic material.

Discussion
The classic equation, , is well known to statisticians st ques or smoothing methods for datasets.In this manuscript, we demonstrate an application of this equation, resulting in a new operator where the residual image derived from a novel smoother can be used to locate spots or mountains ted via ou .Furth 15(c) displays the results when a median smoother with a genetic changes associated with diseases such as breast cancer [30].The laser scanner images obtained from a microarray experiment consists of a series of spots indicating the measured fluorescence of a probe (or "gene") d 2,6

R
. A = rough data smooth  udying regression techni r in an image.This method combines the residual operato from statistics with the structuring element (cross-shaped window) in the field of mathematical morphology.Major advantages of our method include fast running time, broad application to many image types, and universal spot detection regardless of scale.That is, irrespective of a spot's size and height, its location will be detec r method.This aspect alleviates the need to alter or change the grey scales in an image when searching for spots of varying intensities.
As demonstrated, this method uses the s-median operator to smooth images.Other window operators can be considered, however they result in different residual image implications.For example, if a mean cross (i.e."smean") smoother is used on the Gaussian mountain in Figure 15(a) rather than a median smoother, the residual image does not reveal the shape of the mountain; see, e.g., Figure 15(b) er, the shape of the smoothing window is also a critical component of consideration.grid or "box" shaped window sequence is used.Here, we now obtain a residual image that looks like a starburst instead of a cross.As a result, the spot center is now potentially more difficult to identify.The shape of the smoothing window (cross vs. box) and the summary statistic used (median versus mean) thus affect the R image and the ability to detect the mountains in an image.
The issue of rotation invariance is an important concept within mathematical morphology operators used in image detection.Rotation invariance implies that the resultant image does not change when arbitrary rotations are applied to its input argument.In general, our spot nger rotation invariant.e)-(f).Our proposed spot finding method is not rotation invariant since the images in Figure 16(b) and Figure 16(e) are clearly different.Although our proposed method is not rotation invariant, it is possible to rotate our structuring element (cross) to align with the major and minor axes of a correlated spot as in Figures 16(c) and (f).Both versions of the residual images clearly show a cross shape and provide utility in terms of locating the spots in the image.Future work will further explore the characteristics of the cross in each residual image in order to detect spots in correlated images.Note, however, in our biological applications (e.g.2D-DIGE), orrelation within a given spot.When using the s-median operator for spot finding, the major consideration is the arm-length size associated with the smoothing window, or alternativel e number of pixels included in the smoothing windo ructuring element).The s-median smoother natur removes noise from finding method is rotation invariant for the Gaussian spots with zero correlation (e.g., spots of the type shown in Figure 8).Interestingly, if we induce any nonzero orrelation in the spot, the spot finding method is no c lo R it is reasonable to assume that there is negligible correlation within a spot.For example in a DIGE image, the spots are created by electrophoresis in two dimensions where the electrophoresis for each dimension is performed separately.Similarly in pin-based microarray mages, it is reasonable to assume that there is negligible i c c y th w (st ally I , hence the size of the smoot ng window essentially des the amount of smoothi o apply to the dataset.From Figures 11 and 17, the is small will undersmooth the image and cause spurious spots due to noise to appear as real spots.Since the choice of is essentially choosing a smoothing parameter, th re several available methods to consider when choo ng an optimal value for c.The general method fo g smoothing parameters is based on cross valid algorithms described in [22].The optimal choice of c is related to the larger statistical subject of bias-variance tradeoff.Choosing c too small lead a largely variable residual image (missing small spot while choosing c too large leads to a residual im with a large bias term (too many spurious spots).Similarly, the optimal choice of c is related to se k) can determine "opt al" smoothing strategies, while other procedures det ine smoothing parameters from examining figures suc as mode trees [40] or estimates of the mean squared [41].To improve the ability of our MM operator i e presence of noise, we have explored applying stan d image smoothing techniques to the image prior to ap ying the MM filter.Future work will examine the utility of applying "pre-smoothers" to images before ap ng MM operators.In addition to examining pre-smoot rs, we will also examine data driven cross validatio mes for choosing an optimal value of c for specific image applications.In the same way we use presmoothe smooth the image prior to analysis, we will also explo smoothing the resulting residual image.
A majo ncern in proposing image analysis software algorithm volves performing the comparisons among competing methods.Unfortunately, due to the cost of these technologies and the lack of a gold standard for measuring e signal of the chemical sensor, it is difficult to design statistically appropriate benchmarks or quality tions, the success of our proposed method will be de ndent on the choice of smoothing parameter, c.It is ou e the scope of this manuscript to perform a thorough parison of competing spot finding algorithms agai a set of noise distributions.For future work, we propo erforming comparisons such as those in [42,43] to establish conditions in simulated and real datasets where our ethods are superior to competing methods.veral other problems in statistics, the optimal choice of bandwidth in kernel density estimation [38], and the amount of times to smooth a dataset [39].Various strategies that estimate error quantities (ris control studies to assess these image analysis techniques for a given chemical sensor.Although it is relatively simple to simulate "bumps" or mountains in an image, the difficulty arises in deciding the type of noise to impose upon the simulated images.In the presence of most noise distri s

Conclusion
This manuscript develops a new method for spot finding and illustrates the technique's great utility and applicability within several chemical sensor datasets such as mass spectrometry spectra, gel electrophoresis images, and microarray images.This method can be easily extended to mountains in k dimensions and can be extended to further quantify the amount of signal present in other emerging chemical sensors with Gaussian profiles.

, 2 SI
k c denote the kD smoothed image obtained by using an s-median operator with "arms" of length on S k c I ; window size examples are provided in Figure 1.2,1 , for example, refers to the smoothed image that results from applying the the 5pixel cross (see Figure 1(a)) s-median structuring element across .In this case, the center pixel within any window in 2 3 3

,Figure 1 .
Figure 1.Examples of window shapes in 2D: Window shape, often called the structuring element in morphology, refers to the pixels used to compute the median for smoothing.The window shapes are shaded in grey.(a) A 3 × 3 s-median cross window consisting of five pixels.This s-median replaces the intensity in pixel location "5" with the median intensity of the grey pixels (i.e. in locations 2, 4, 5, 6, 8); (b) A 5 × 5 s-median cross window consisting of nine pixels.The s-median image in pixel location "13" is obtained by computing the median intensity of the grey pixels (locations 3, 8, 11, 12, 13, 14, 15, 18, 23).

Figure 3 .
Figure 3. 2D continuous sample space: The sample space

Figure 4 .
Figure 4. R k,c for an image I of a single mountain: (a) An image of a 1D mountain generated by I 1 (x) = -x 2 + 400, where x con-

Figure 5 Figure 5 .
Figure 5 illustrates the associated sample space.With this definition, the derivation of the s-median for Z follows analogously to the 2D continuous case (which analogous to the 1D case).Let : G      define a mapping from the support sets is

=0
i A , respectively.Similar to the 2D discrete setting, there is usually no closed form solution for Z M and thus t the be determined numerically.Fig es 4(d) ) show the imag   , S x y ur , bu solution can -(f es from our technique applied to a simple tw extended to higher dimensions ) as demonstrated in ro n light of Gaussian noise.In the 1D noise-free it can be shown (with for any o-dimensional discrete piecewise monotone function. when x is the location of the absolute maximum, and1, = 0 c Rwhen the sequence contained in each dimension of the smoo is monotone.Further, under certain circumstances associated with 1D images, when x is the location of a local minimum in our image; see[24] for details.The following examples, however, explore an , k c R image when noise is introduced in the raw image, k I .As expected, it will detection in , k c R more difficult where the signal-to-noise will be important.Consider adding independ and identically distributed (i.i.d.) Gaussian noise to the 1D monotonic sequence

Figure 7 .
Figure 7.Estimated probability of R > 0 at a local maximum in 1D and 2D: (a) Pr(R 1,c (p)) > 0, where p is the location of the absolute maximum in a 1D image; (b) Pr(R 2 ; c(p 1 ; p 2 )) > 0 for a 2D image, where (p 1 ; p 2 ) represents the location of the absolute maximum in the 2D image.
) shows a single mountain with i.i.d.N(0, = 200  ) noise added to the intensity at each pixel in Figure 4(d).Figures 10(b)-(c) show the associated 2,c R images for a = 9 c cross and a = 27 c cross, respectively.There are several interesting features to note in this example.We see the respective crosses associated with the smoothing window; however, when using the = 9 c arm, it is much harder to distinguish the cross from the remaining picture.With the = 27 c arm, the cross is more apparent, mainly due to the cross being wider tha Figure 10(b).To confirm that large values of c more effectively find s, Figures 11(a show a sequen three spots in order of increasing size with   0, = 48 N  noise.Figures 11(d)-(f) are the R 2,2 images corresponding to Figures 11(a)-(c), respectively.Figures 11(g)-(i) are the 2,3 images corresponding to Figures 11(a)-(c).Figure 11 demonstrates two important results: smoothing window in detecting spots, Figures 8(a)-(b) shows a set of four Gaussian spots with different standard deviations.

Figure 8
a)-(b) where the random (i.i.d) noise added at each pixel is distributed according to a Normal distribution with mean 0 and standard deviation 5, 15, or 50.Figures 9(a), (b), and (c) display the residual image n in spot )-(c) ce of when the smoother is applied to the aussian spots conng noise with standard de iations of 5, 15, or 50, respectively.With a fixed sm w, as the standard deviation ise increases, the ability to discern he cross decreases.Specifically rosses are nearly indistinguishable R 1) it is

Figure 8 .
Figure 8. Bump hunting: (a) A 200 pixel × 200 pixel image consisting of four Gaussian spots with different locations and scales; (b) The associated perspective plot; (c) The "rough" residual image (R 2,4 image) after running a specialized median smoother (s-median), S 2,4 .Crosses are present at the locations of their respective spot centers associated with the spots shown in (a).

Figure 9 .Figure 10 .
Figure 9. Spot finding as noise increases: (a) R 2,9 image associated with Figure 8, where i.i.d.normally distributed noise with mean 0 and standard deviation 5 (i.e.N(0, σ = 5)) was added to each pixel in Figure 8; (b) R 2,9 image when the noise in Figure 8 is N(0, σ = 15); (c) R 2,9 image when the noise in Figure 8 is N(0, σ = 50).As the standard deviation of the noise increases, the ability to detect the cross at each spot decreases.

Figure 11 .
Figure 11.Choice of smoothing operator: A series of Gaussian mountains of increasing size and N(0; σ = 48) noise are shown in (a), (b) and (c).The residual images R 2,2 from applying an S 2,2 operator to (a), (b), and (c) are shown in (d), (e), and (f), respectively.The residual images R 2,3 from applying an S 2,3 operator to (a), (b), and (c) are shown in (g), (h), and (i), respectively.The spots are more easily detected using the S 2,3 operator (row 3) rather than the S 2,2 operator (row 2).
in) to 500 data points, which corresponds to ately a 95 m/z bandwidth.Figure 12(b) shows th lting s-median image using the chosen bandwidth; 12(c) shows the associated image.From examining the R image, we note that ikes in the origi ectrum are preserved, thus ai in the identifica of the peak location.Further, the "negative peaks" in dual image near the larg e serve to quank size in a MALDI-TOF im trophoresis An r application of this spot det chnique is on im ained from two-dimensi fference gel electr resis (2D-DIGE) experi such as those strate our spot detection technique on the various example sets noted above.

Figure 12 .
Figure 12.Mass spectrometry: (a) MALDI spectrum based on a tumor sample; (b) S image based on a smoothing window of 95 m/z; (c) Associated R image that contains spikes at each local maximum in the original image I.

s
50 pixel × 50 pixel image is a subimage of the Drosophilia he associated perspective plot for the data in (a); (c) The hich indicate protein spots in the image.There is also a th a black outline around several of the spots.

Figure 13 .Figure 14 .
Figure 13.2D-DIGE images: (a) Subset of 2D-DIGE image.Thi proteome gel, normalized according to the model in [26]; (b associated R 2,6 image.There are several "crosses" apparent, "speckled" black and white noise pattern present in the image ) T w wi

Figure 15 .
Figure 15.Other measures: The results from different variations on the s-median.(a) A Gaussian spot image, I, of dimension 50 × 50; (b) The R 2,2 image obtained from using an s-mean rather than s-median on the image in (a); (c) Associated residual image obtained from (a) using an s-median, where the window sequence is a 5 pixel × 5 pixel box shape containing all 25 pixels.

Figure 16 (
b) is the residual image from our proposed method.Meanwhile, Figure 16(c) is the result when employing a rotated version (45 degrees) of the structuring element used in Figure 16(b).Similarly, Figure 16(d) is the rotated version (90 degree) of Figure 16(a) with the corresponding images shown in Figures 16

Figure 16 .
Figure 16.Rotational invariance: (a) A scaled bivariate norm image using a R 2,4 operator; (c) The resulting residual image wh is rotated 45 degrees to align with the major axis of the spot in correlation of -0.50; (e) The residual image using a R 2,4 opera used in (e) is rotated 45 degrees.al e (a tor

Figure 17 .
Figure 17.Two nearby mountains: (a) Perspective plot showi associated with the image in (a).The two crosses indicate th image obtained from the image in (a).In this situation, the two ng e p m The main al of this manuscript is to establish a new method fo spot finding in images and demonstrate it choosing c too large will oversmooth the image and blend spots together, while choosing c too performance on a variety of different biological images derived from chemical sensors.