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 [1-7]. 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 [10-15]. Examples of common MM functions include opening, closing, thinning, binning, thresholding, and watershed methods, and have been employed in numerous applications including pedestrian detection [
This paper combines aspects of feature detection, data smoothing, and residual analysis to develop a new bump detection method for not only oneor 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.
, where x is a point location in the Cartesian coordinate system. We let 
denote the kD smoothed image obtained by using an s-median operator with “arms” of length 
on
; window size examples are provided in 
, for example, refers to the smoothed image that results from applying the the 5- pixel cross (see 
. In this case, the center pixel within any 
window in 
is replaced with the associated median value from the 5-pixel cross. Similarly, applying a 9-pixel cross s-median (as shown in 
produces
.
, to obtain information regarding bump detection and quantification. The 
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.
and, subsequently, 
in the context of continuous functions. We derive 
for 1D and 2D theoretical models by characterizing the the median operator
and 
denote the cumulative density function (cdf) and probability density function (pdf), respectively, for the a random variable 
evaluated at the point
; 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
) to the support set 
(for the random variable Y). For our applications, 
is obtained from an optical device such as a charge-coupled device camera or laser scanner. Our goal is to obtain an expression for 
[and thus
] which then determines the median of
, 
, i.e. 
satisfies 
. Note that, in our notation for one dimension, 
at a given location, x. Thus, for simplicity, we will denote 
as 
(or
) with the implicit understanding that 
(or
) is also a function of k and c.
is strictly monotone on the interval
. Then, for 
increasing,
decreasing,
implies its invertibility for any
, i.e.
; in particular, by definition of
, we have
. Hence, in the monotone case, 
, where 
denotes the median associated with the random variable
, and “
” denotes statistical equivalence as defined in [
is equivalent to the function evaluated at the median for
; in short,
.
as an open interval 
with
. Let 
, be the smallest collection of disjoint open intervals such that 
is strictly monotone on each
. By definition, 
is strictly monotonically increasing on 
if, for any two values 
such that
, 
holds. Analogously, 
is strictly monotonically decreasing if 
for 
such that
. Note that continuity may not be enough for the 
to be countable, where we define countability as in [
. We define the interval,
. While the sequence 
partitions
, 
does not necessarily partition
; e.g., see 
and
.
, i.e. the 
th decomposition of
, and 
defines the indicator function of 
by 
(0) if 
is (not) in the interval
. By definition, 
is strictly monotone. Thus, for
, we have
, implying that any function 
can be decomposed into the sum of its strictly monotone components,
. Accordingly, we see that
, where, for 
,
decreasing on
,
. Nevertheless, the above equations will allow for the calculation of 
and thus 
using computational methods.
where the goal is to obtain an expression for
. From standard probability theory such as in
, where
and 
is the joint pdf for 
and
. Note that this is the general case for obtaining the cdf of 
in terms of 
and
. For our specialized median, however, our sample space for 
and 
must be defined in terms of another parameter, say
, where 
controls the width of the smoothing window in each dimension. 
. Since it is difficult to generalize
, we cannot generalize this situation to provide an explicit calculation for the median of
. Nevertheless, computational
.
denote a “discrete” function, i.e. a function with discrete/countable realizations from the continuous function
. By definition, 
is a function that maps from the support set 
(of the discrete random variable X) to the support set 
(for the discrete random variable Y). This section considers computational results associated with 
and its impact on the s-median.
and 
denote the discrete cdf and probability mass function (pmf), respectively, for the arbitrary random variable 
evaluated at the point
. Let 
be as defined above. The calculation of the s-median proceeds by assuming 
Discrete Uniform(
) as defined in [
based on the function,
. Thus, the machinery developed in Section 2.1 can be applied to compute
.
using discrete random variables 
and 
as we did for the continuous case, namely
indicates probability. If we assume that X~ Discrete Uniform(
), then
with 
and 
on the full interval
,
does not depend on the direction of monotonicity for
. Figures 4(a)-(c) show the images from our technique applied to a simple one-dimensional discrete piecewise monotone function.
be a discrete uniform on
, 
be a discrete uniform on
, and (x*, y*) be a fixed point such that x* and y*
, respectively. Then let 
be of the form,
define a mapping from the support sets 
and
, of the random variables X and Y respectively, to the support set
, for the random variable Z. We define the functions, 
and
, such that
,
, where 
is the the smallest set of disjoint open intervals such that 
is strictly monotone on each
,
; and 
is the smallest set of disjoint open intervals such that 
is strictly monotone on each
,
. In this setting,
. Nicely, all of the above quantities can be computed since we specified the distributions for 
and
. Note the zero quantity in the third line is due to the intersection of the sets containing only the single point
. The 
and 
depend on the length of 
and
, respectively.
and thus
, but the solution can be determined numerically. Figures 4(d)-(f) show the images from our technique applied to a simple two-dimensional discrete piecewise monotone function.
) as demonstrated in [
, it can be shown (with our proposed methods) that 
for any 
when 
is the location of the absolute maximum, and 
when the sequence contained in each dimension of the smoothing window is monotone. Further, under certain circumstances associated with 1D images, 
when 
is the location of a local minimum in our image; see [
image when noise is introduced in the raw image,
. As expected, it will make spot detection in 
more difficult where the signal-to-noise will be important.
, where
. Let
, where 
denotes the true signal at location
, 
equals the step size at 
in the monotonic sequence such that
, and 
denotes normally distributed noise of mean zero and standard deviation
. We fix 
for our examples such that the signal-to-noise ratio (
) remains constant within each simulation.
at an arbitrary location 
since, as shown in [
image is zero (
) over a monotone sequence with i.i.d. noise. In most cases, however, it is not possible to compute a closed-form solution for
; for notational simplicity, we use 
rather than
. 
over a monotone sequence for different values of 
in the 1D situation. Note that the 
-axis is the ratio of stepsize to standard deviation of the noise, and that each curve begins at 1/(2c + 1) and asymptotes at 1. The signal-to-noise ratio, 
, is the critical value in evaluating how 
operates in a noisy environment. If the step size is 0, then each point in the window is equally likely to be the median, hence
for any 
when
. As the step size increases relative to the standard deviation of the noise, naturally for any
, we expect the probability to converge to one. Hence, for noise-free monotone images, 
for all
.
decreases as 
increases. Intuitively, as 
increases, the number of points in the smoothing window increases hence there are more “opportunities” for other points to be the median, thus making the residual nonzero at that location.
in a noise-free 1D spot (mountain),
. In the presence of noise, we can estimate (via simulation) the probability that the 1D residual image intensity value at the local maximum location is positive; i.e., we can estimate 
when 
and
as a function of 
for different values of 
in a 1D image. Analogously, 
for a maximum location 
when 
is a 2D image. For all
, the simulation results show that 
monotonically as the signal-tonoise ratio increases,
. Further, as shown in Figures 7(a)-(b), as 
increases, the rate in which 
converges to 1 also increases, k = 1, 2.
image resulting from the smoother being applied to 
) noise added to the intensity at each pixel in 
images for a 
cross and a 
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 
arm, it is much harder to distinguish the cross from the remaining picture. With the 
arm, the cross is more apparent, mainly due to the cross being wider than in 
noise. Figures 11(d)-(f) are the R2,2 images corresponding to Figures 11(a)-(c), respectively. Figures 11(g)-(i) are the 
images corresponding to Figures 11(a)-(c). 
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).
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, thus applying the s-median is equivalent to applying a running median to the I image. 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 
in
) to 500 data points, which corresponds to approximately a 95 m/z bandwidth. 
image. From examining the R image, we note that the spikes in the original spectrum are preserved, thus aiding in the identification of the peak location. Further, the “negative peaks” in the residual image near the large spike serve to quantify the peak size in a MALDI-TOF image.
and densely populated with protein spots, making it difficult to observe individual protein spots in detail. We therefore focus on a 
sub-image to better understand the impact of applying the s-median.
, with the associated perspective plot shown in 
. From 
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.
) obtained from applying an s-median to 
, is well known to statisticians studying regression techniques 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 in an image. This method combines the residual operator 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 detected via our method. This aspect alleviates the need to alter or change the grey scales in an image when searching for spots of varying intensities.
images shown in Figures 16(e)-(f). Our proposed spot finding method is not rotation invariant since the images in 
associated with the smoothing window, or alternatively the number of pixels included in the smoothing window (structuring element). The s-median smoother naturally removes noise from
, hence the size of the smoothing window essentially decides the amount of smoothing to apply to the dataset. From Figures 11 and 17, the choice of 
is critical, since choosing 
too large will oversmooth the image and blend spots together, while choosing 
too 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, there are several available methods to consider when choosing an optimal value for c. The general method for choosing smoothing parameters is based on cross validation algorithms described in [