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Adaptive contrast enhancement of two-dimensional electrophoretic protein gel images facilitates visualization, orientation and alignment

Werner Van Belle1* - werner@yellowcouch.org, werner.van.belle@gmail.com
Gry Sjøholt2
Nina Ã…nensen2 - nina.anensen@med.uib.no
Kjell-Arild Høgda3 - kah@itek.norut.no
Bjørn Tore Gjertsen2,4 - bjorn.gjertsen@med.uib.no

1- Bioinformatics Group Norut IT; Research Park; 9294 Tromsø; Norway
2- Institute of Medicine Hematology Section University of Bergen Haukeland University Hospital; Bergen; Norway
3- Earth Observation Group Norut IT; Research Park; 9294 Tromsø; Norway
4- Department of Internal Medicine Hematology Section Haukeland University Hospital; Bergen; Norway
* Corresponding author

Abstract :  Two-dimensional polyacrylamide gel electrophoresis (2-DE) is a powerful technique to discriminate post-translationally modified protein isoforms. However, all steps of 2-DE preparation and gel-staining may introduce unwanted artefacts, including inconsistent variation of background intensity over the entire 2-DE gel image. Background intensity variations limit the accuracy of gel orientation, overlay alignment and spot detection methods. We present a compact and efficient denoising algorithm that adaptively enhances the image contrast and then, through thresholding and median filtering, removes the gray-scale range covering the background. Applicability of the algorithm is demonstrated on immuno-blots, isotope labeled gels, and protein stained gels. Validation is performed in contexts of i) automatic gel orientation based on Hough transformation, ii) overlay alignment based on cross correlation and iii) spot detection. In gel-stains with low background variability, e.g. Sypro Ruby, denoising will lower the spot detection sensitivity. In gel regions with high background levels denoising enhances spot detection. We propose that the denoising algorithm prepares images with high background for further automatic analysis, without requiring manual input on a gel-to-gel basis.

Keywords:  adaptive contrast enhancement, 2 dimensional electrophoretic gels, 2D gel image denoising, SDS-PAGE, background estimation hough fourier transform alignment rotation
Reference:  Werner Van Belle, Gry Sjøholt, Nina Ånensen, Kjell-Arild Høgda, Bjørn Tore Gjertsen; Adaptive contrast enhancement of two-dimensional electrophoretic protein gel images facilitates visualization, orientation and alignment; Electrophoresis; Wiley Interscience Vch; volume 27; nr 20; pages 4086-4095; October 2006
See also:
An online denoiser based on this research


Table Of Contents
Introduction
Materials and methods
Results and Discussion
Concluding Remarks
Acknowledgments
Bibliography

Introduction
Two-dimensional polyacrylamide gel electrophoresis (2-DE) separates proteins based on charge and molecular weight. It has been successfully used for protein analysis over more than three decades, and it is increasingly used in disease exploration [[1, 2, 3, 4]]. Using 2-DE to map thousands of proteins presents the researcher with valuable information about a specific biological system. In practice, due to biophysical limitations, only a fraction of a particular proteome can be examined. After digital imaging, analysis often proves difficult because noise and artefacts are generated through all steps of 2-DE [[5]].
One particular artifact are large scale, inconsistent modifications of the background intensity levels in the captured image, which can be prominent features of immuno-blots or silver stained gels. Their presence can make it hard for the human eye to distinguish spots and their large size often disturbs the correct working of standard alignment and orientation algorithms [[6]]. To remove unwanted artefacts, existing techniques are based on thresholding (which is a method useful for selecting spots with a specific strength), spot recognition based on Gaussian curve fits [[7]], wavelet transformation [[8, 9]] and Wiener filtering to remove blurs [[10]]. None of these techniques seem to handle background variations properly [[7, 11]] and their sensitivity can be dependent on the background variations itself.
In this article we present a method to isolate and remove background variations of 2-DE gel images focusing on spot positioning. The algorithmic development was guided by proteomic experiments in which protein material for 2-DE immuno-blots of p53 was isolated from characterized cryopreserved human acute myeloid leukemia (AML) cells [[12]]. We use this case to demonstrate quantitative improvements in alignment and orientation. Quantification of the background variations has been performed by measuring the variance in spot detection, the signal-to-background ratio and the amount of signal lost through denoising. A simulated denoising of Gaussian spots illustrates that the algorithm cannot be used in a quantitative setting but that it accurately preserves positional information.

Materials and methods
Protein sample preparation, two-dimensional gel electrophoresis (2-DE) and gel imaging - Protein material for 2-DE immuno-blots of p53 was collected from characterized cryopreserved human acute myeloid leukemia (AML) cells as previously described [[12]]. For Sypro Ruby gels, proteins were extracted from $2.5$ million NB4 cells by trichloroacetic acid (TCA) precipitation [[13]]. For 2-DE gels with $^{32}P$ labeled proteins, 10 million MOLM-13 cells were incubated with $^{32}P$ labeled ATP and thereafter precipitated in TCA as previously described [[14]]. The AML cell lines NB4 and MOLM-13 were cultured according to standard procedures (www.atcc.org). For all 2-DE experiments, precipitated proteins were diluted in a rehydration buffer containing 7 M urea, 2 M thiourea, 4% 3-[(3-cholamidopropyl)dimethyl-ammonio]-1-propanesulfonate (CHAPS), 2% ampholytes and 100 mM dithiothreitol (DTT). The focusing strips were incubated with protein in rehydration buffer over night at room temperature.
For immuno-blotted gels, isoelectric focusing (IEF) using 7 cm pH 3-10 strips (Zoom Strip, Invitrogen Corp., Carlsbad, CA, USA) was performed at 200 volt (V) for 40 minutes, 450 V for 30 minutes, 750 V for 30 minutes and 2000 V for 60 minutes following the manufacturers' instructions. Following IEF, the strips were equilibrated for 15 minutes in LDS sample buffer (Invitrogen Corp., Carlsbad, CA, USA) containing 100 mM DTT and then 15 minutes in LDS sample buffer containing 125 mM iodacetamide. The second dimension was run on a NuPAGE Novex 4 to 12 % Bis-Tris ZOOM Gel (Invitrogen). Thereafter, the proteins were transferred to polyvinylidene fluoride membrane (GE Healthcare Bio-sciences AB, Uppsala, Sweden) by standard electro-blotting. p53 protein was detected using primary Bp53-12 antibody (Santa Cruz Biotechnology, CA, USA) and secondary horse radish peroxidase conjugated mouse antibody (Jackson ImmunoResearch, West Grove, PA, USA). The horse radish peroxidase was excited using the Supersignal West Pico or Supersignal West Femto system (Pierce Biotechnology, Inc., Rockford, IL, USA) and visualized by standard chemiluminescence conditions on a Kodak image station 2000R (Eastman Kodak Company, Lake Avenue, Rochester, NY, USA).
For Sypro Ruby- and $^{32}P$-gels, IEF on 13 cm 3-10 NL pre-cast focusing strips (GE Healthcare Bio-sciences) was performed on an IPGphor unit (GE Healthcare Bio-sciences). IEF conditions used were an initial gradient up to 500 V for 10 minutes, a second gradient up to 4000 V for 150 minutes, and a final step at 8000 V for 130 minutes. Following IEF, equilibration was performed in two steps using an equilibration buffer (EQ) containing 6 M urea, 50 mM Tris-HCl, 30% glycerol and 2% SDS. Step one was performed in EQ containing 1.5% DTT for 12 minutes and step two in EQ containing 4.5% iodoacetamide for 5 minutes. The second dimension was run on a 10% SDS-polyacrylamide gel. 2-DE gels were stained with Sypro Ruby dye (Bio-Rad) according to the manufacturer's recommendations. These gels were imaged either with UV light as an excitation source and a 670 nm emission filter (35 nm wide band) on a Kodak image station 2000R or with a 457 nm excitation laser source and a 610 nm emission filter on a Typhoon imager 9400 (GE Healthcare Bio-sciences AB, Uppsala, Sweden). Gels with $^{32}P$-labeled proteins were dried in cellophane wrapping, thereafter exposed to a phosphoimaging plate subsequently scanned in a Fuji BAS-2500 phosphoimager (Fuji Photo Film Co., Ltd.).
Spot detection - Images were acquired with Kodak 1D software, v1.6 and exported in TIFF format with a resolution of 300 dots-per-inch (DPI). Twelve native and denoised Sypro Ruby stained 2-DE gels were analyzed with PDQuest (Bio-rad, v7.2.0) using different spot detection parameters. The default spot detection parameters were: sensitivity 5.37, size scale 3 min peak 3206, horizontal streak 33, vertical streak 33, background radius 23, median filtering, kernel size 3x3 and speckle filter sensitivity 50. These parameters were then tuned one by one, including background radius increase to 30, sensitivity decrease to 15, minimum peak decrease to 2800, and the filter used (weighted mean smoothing, power mean smoothing, contra mean smoothing, adaptive smoothing and no smoothing). The spot detection variance was calculated by averaging the absolute difference between the mean and the detected number of spots (See Table 1).
Table 1: Detected spots in native and denoised images of 12 Sypro Ruby stained gels using 9 different spot detection methods. Section A documents the number of spots detected before denoising. Section B documents the number of spots detected after denoising. Because denoising alters spot amplitudes, the number of detected spots lowers (to 76%). The variance measure reveals that denoising decreases the impact tuning of spot detection parameters has on the number of detected spots. See Material and Methods for details on the parameters used.
Simulated gels - To test the impact of denoising on Gaussian shaped spots, we created a test image based on dummy spots, defined as
\begin{displaymath} G(x,y)=a.exp(-\frac{(\frac{x-cx}{wx})^{2}+(\frac{y-cy}{wy})^{2}}{2}) \end{displaymath} (1)

$(cx,cy)$ is the center position, $wx$ and $wy$ are respectively the width and height and $a$ is the amplitude of the dummy spot. Based on this Gaussian bump we made different simulation images (Fig. 1).
Automated gel overlay alignment using cross correlation - The alignment between two images $M$ and $N$ was obtained by finding the translation of image $N$ which maximized $\vert M-N\vert^{2}$ (the $L^{2}$-norm). This position was found in the cross correlation image, defined as $P_{x,y}=\vert M-translate(N,x,y)\vert²$, which was calculated as $P=\mathcal{F}^{-1}(\mathcal{F}(M)\times\mathcal{F}(N)^{*})$. * is conjugation. $\mathcal{F}$ and $\mathcal{F}^{-1}$ are respectively forward and backward Fourier transforms. For more information on cross correlation see Gonzalez and Woods [[10]]. Wrap around effects have been avoided by extending both images with a mean padded border. The image acquisition region covered by the camera did not allow shifts larger than 10% of the image size, therefore we searched for the best cross-correlation only within a limited range.
Automated orientation of gels - Detection of gel orientation was performed using the Hough transform [[10, 15]] on native and denoised images. The Hough transform maps every point of an image onto the image of a sine wave. All sine wave images are added together to form the Hough transformed image. The abcis of the maximum element in the transformed image relates directly to the angle of the 2-DE gel. The Hough transform is defined as $H_{\theta,\rho}=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}A_{x,y}\delta(\rho-x\textrm{cos}\theta-y\textrm{sin}\theta)\textrm{d}x\textrm{d}y$in which $\delta $ is the Dirac delta function [[16]], $\theta$ is the angle of the investigated line and $\rho$ is the distance from that line to the origin. To increase the sensitivity of the algorithm, histogram equalization has been performed using 4 bins. Gonzales and Woods [[10]] contains more information on Hough transforms.
Image representation - We use capital letters to denote the image matrices. E.g: $A$ denotes an image, which is an element of $\mathbb{R}^{w\times h}$, where $w$ and $h$ are the width and height of the image. Pixel positions are written using subscript. E.g: $A_{x,y}$ refers to the gray value of the pixel at position $(x,y)$ in image $A$. Adhering to common interpretation, a gray value of 0 is black and a gray value of 1 is white. The algorithm requires that gel images have black spots on a white background: where the protein expression is large, the gray-value is small. Preliminary image inversion might be necessary to fulfill this requirement (step $1$ in Fig. 2).
Figure 2: Image denoising flowchart - The denoising process estimates the background intensities (step 2) and uses them to obtain the relative protein expression (step 3). All non relevant information (protein expression lower than the estimated background) is removed through thresholding (step 4) and median filtering (step 5). Input images should have black spots on a white background. If the original images have white spots on a black background (A) they should be inverted (step 1 and 6), see Results and Discussion section. Image denoising is useful for many applications, including visualization, gel orientation, gel alignment and spot.
Estimating background variations - The denoising algorithm estimates the background using a smoothing low pass filter (step 2 in Fig. 2). If $A$ is the native input image (Fig. 2B), then the smoothed image $B$ (Fig. 2C) is defined as
\begin{displaymath} B=smooth(A,s_{2})\qquad\iff\qquad B_{a,b}'=\frac{\sum_{x=a-s_{2}}^{a+s_{2}}\sum_{y=b-s_{2}}^{b+s_{2}}A_{x,y}}{4s_{2}^{2}} \end{displaymath} (2)

In Eq. 2, $s_{2}$ parametrizes the window size of the filter, which determines the accuracy by which the background variation will be estimated. The actual filter window $2s_{2}+1$ should be larger than the size of the spots we want to retain. This filter has zero-phase response, which means that spots will not be shifted over the surface.

Denoising through contrast enhancement - The third step in the algorithm relates all positions to their estimated background. Division of the native image $A$ by the background $B$ creates the relative strength image (Fig. 2D).

\begin{displaymath} C_{x,y}=\left\{ \begin{array}{cc} A_{x,y}\slash B_{x,y} & \textrm{if }B_{x,y}>0\\ 1 & \textrm{otherwise}\end{array}\right. \end{displaymath} (3)

Because intense spots are represented as black, which has a low numerical value, positions with a spot intensity larger than their background will have a smaller numerator ($A$) than denominator ($B$), resulting in values below 1. Similarly, areas with less expression than the neighboring background will have pixel values larger than 1.
Thresholding - Under the assumption that only spots more intense than the background are relevant, the image is limited to values within range $[0:c]$ (Fig. 2E). $c$ describes the threshold value which defines the maximum allowed image intensity. All positions larger than $c$ will be set to $c$. In normal conditions $c$ should be $1$. Eq. 4 formalizes thresholding.

\begin{displaymath} D_{x,y}=\left\{ \begin{array}{cc} C_{x,y} & \textrm{if }C_{x,y}<c\\ c & \textrm{otherwise}\end{array}\right. \end{displaymath} (4)

Because we might enhance 'salt and pepper' [[10]] noise during the contrast enhancement process (Eq. 3), we remove small dots using a median filter of a smaller size $s_{1}$ (Fig. 2F), defined as

\begin{displaymath} E=median(D,s_{1})\qquad\iff\qquad E_{a,b}=median\{ D_{x,y}\ \vert\ s_{1}>\vert a-x\vert\wedge s_{1}>\vert b-y\vert\} \end{displaymath} (5)

A median filter will remove all features that are smaller than half of the window size. Therefore $s_{1}$ should be much smaller than the size of the spot size we want to retain.

Software and implementation - The denoising algorithm was written in IDL (Interactive Data Language version) 6.1 as implemented by RSI Systems [[17]] and executed under Debian Linux (www.debian.org), running a 2.6.12-2 kernel. The algorithm takes four arguments. The integer $s_{1}$ parametrizes the area-size of the features to remove, the integer $s_{2}$ parametrizes the area-size of the features to retain. The optional floating point number $c$ parametrizes the threshold (Eq. 4), which by default is set to 1.0. Under normal circumstances this is a good choice. Presence of much white noise might require a decreased $c$ to e.g., 0.9. The optional white_spots boolean can be set when the image has white spots on a black background. It will then automatically invert the image before and after the actual denoising. The software can be used online at http://analysis.yellowcouch.org/2ddenoising.html
FUNCTION denoise, image, s1, s2, c = 1.0, white_spots = 0
  if white_spots then image = max(image) - image  ; invert if necessary
  bg = median(image,s2)      ; calculate the background variations
  im = image / bg            ; remove them through division
  im <= c                    ; select only areas darker than the environment
  im >= 0                    ; no -inf number when dividing by -0
  im = median(im,s1)         ; remove details and return 
  if white_spots then im = max(im)-im   ; back inversion
  return, im

Results and Discussion
Improved visualization - The main advantage of the presented algorithm is its ability to improve visualization by adaptively enhancing the image contrast. Gel intensities varied widely in our immuno-blot 2-DE experiments. Presence of speckle, white noise [[10]], a poor gray value resolution, over-saturation, air bubbles and other artefacts made automatic spot detection difficult and prompted us to do preliminary gel analysis by hand. Elimination of background variations (See Fig. 2 for overview) enabled us to acquire a better understanding of the different spot patterns. Fig. 3 shows denoised images of immuno-blots, Sypro Ruby gels and gels with $^{32}P$ labeled phosphoprotein. Spots below the visual threshold in the native image become more visible and have more distinct positioning after denoising. The first immuno-blot demonstrates sharpening of spots. The Sypro Ruby gel shows how artefacts introduced through air under the gel are removed as well. The very dark spots in the denoised $^{32}P$ phosphoprotein labeled gel are places where the phosphoimaging plate has been over-saturated by radiation. Denoising can help researchers in observing spots that are relevant but below their visual threshold.

Figure 3 - Illustration of gel denoising -Various examples of the contrast enhancement process on Sypro Ruby, gels of $^{32}P$ labeled phosphoproteins and immuno-blots. Note the high background in the 32P phosphoprotein detection image compared to Sypro Ruby.

Technique
 Native
 Denoised
Sypro Ruby

32P phosphoprotein detection by phosphoscreen imaging

Immuno-blot with background variations
Immuno-blot with low background variations
Image orientation - We tested the denoising algorithm as a pre-processor for gel orientation (see Material and Methods and Supplementary Table 'orient.xls'). Based on 73 different immuno-blots, we observed that the number of errors (angles deviating more than 3 degrees of the manual assessment) was reduced from 11% to 4%. Without denoising, the mean angle deviation was 1.11 degree and 0.65 degree with denoising, showing an accuracy increase by a factor 1.71. Fig. 4 shows how the background variations might influence the 2-DE gel orientation.

Figure 4 - Denoising Improves Automatic Orientation of 2DE Images - Two examples on how denoising helps in determining the orientation of a gel. Without denoising the background variations might influence the measured orientation. The black lines shows the measured direction(s) on the different images using a Hough transform (see Material and Methods). Equally optimal directions are all shown, possibly resulting in multiple directions for the same image. The left column shows two original images (A, C). The right column shows their denoised versions (B, D).

Image alignment - Alignment is necessary to compare two 2-DE gel images with a slightly different information content. Spot position comparison can reveal important differences between biological samples. Therefore we also relied on 2-DE gel images from slightly different biological systems. After manual alignment and orientation of 73 2-DE gel images (based on 2 known spots) we randomly selected 1050 image pairs. For every pair we calculated the overlay alignment as described in Material and Methods. An alignment was considered to be a hit when it deviated no more than 1 mm (or 12 pixels at 300 DPI) from the manual alignment. Without denoising, the alignment procedure performed correct in 48% of the cases, while denoising increased the hit rate to 61%. In the native images, the error (or misalignment) has a standard deviation of 25.7 pixels while this decreased to 18.59 pixels using the denoised images. Fig. 5 shows how the presence of strong background variations might influence overlay alignment. Supplementary Table 'align.xls' contains all couples.

Figure 5 - Denoising improves alignment algorithms- Comparison of expression profiles from different gels requires proper image alignment. Cross correlation is a common technique to align two images (see Material and Methods). This figure illustrates how background variations can lead to wrong alignments. The top row (A,B) shows the native gels participating in the alignment. Rectangle a and b mark areas with strong background variations. The bottom row shows the aligned images. (C) Without denoising, the background variations a and b superimpose, illustrating how the alignment algorithm was misguided by the background. (D) After denoising, cross correlation properly superimposes the protein spots. Denoising ran with (s1=3 and s2=23, c=0.9).

Signal-to-background ratio - The signal-to-background ratio was calculated by dividing the gel maximum intensities by the gel mean intensities. The averaged signal-to-background ratio was 4.76 for non denoised images and 147.83 for denoised imaged, indicating an increase in signal level by a factor 31. Assessment of background stability has been performed by calculating the normalized standard deviation. Without denoising the RMS was 0.1, while after denoising this became 0.06. In our experiments, denoising removed 36% of the signal variance, while still improving the analysis properties of the 2-DE gel images. Supplementary Table 'energy.xls' contains the detailed information.

Spots shrink and expression strength information is lost - We made different simulation images based on Gaussian bumps (See Material and Methods), which were then denoised. The simulations (Fig. 1) shows that Bumps larger than the filter window are quickly removed, but also that expression information is lost under the influence of background variations. Fig. 1E has a broad central spot which disappears after denoising, without influencing other spots (Fig. 1F). Whereas, denoising of Fig. 1G, which has a background spreading out over the other spots, diminishes the two lower spots (Fig. 1H). This illustrates the danger of using this method as an immediate pre-processor for quantitative analysis. Additionally, the simulation shows that spots are narrowed down to their central position because their side flanks are attenuated.

Figure 1: Denoising of simulated Gaussian spots - When spots become larger they look more as background variation. Therefore we tested the denoising algorithm on images with a stepwise increasing background spot. The top row are the native images (A, C, E, G), the bottom row are the denoised counterparts (B, D, F, H). Every image is 600x400 pixels. The central spot sizes are marked as delta. The size of the filter window is shown using straight black lines. As long as the window contains most of the spot, no attenuation occurs. When spot size delta becomes larger than the window, quick attenuation occurs (D, F) until it is removed completely (H). Denoising does not translate spots but will sharpen their positioning, providing valuable input into algorithms for alignment and orientation. The impact of background removal on the strength of nearby spots is apparent (H) and indicates that the algorithm cannot be used quantitatively. Spot alpha illustrates the threshold c. Because the spot is too weak it is removed by the denoising algorithm.





Spot detection in native and denoised images of Sypro Ruby 2-DE gels - To verify the impact of denoising on spot detection we used PDQuest to find spots in 12 Sypro Ruby stained 2-DE gels. Sypro Ruby stained gels were chosen for their relative low background variance, thereby unraveling alterations in spot detection sensitivity. A large variety of different spot detection parameters were tested (Material and Methods). In general, denoising decreased the number of detected spots. Using the same spot detection parameters, only 76% of the spots remained, with most of them common (Table 1). This was expected because most spot detection algorithms rely on size based or volume based thresholding. Our algorithm decreases spot sizes and modifies expression strengths, automatically resulting in less spots when using PDQuest with the same thresholds. To understand when denoising enhances spot detection, one needs to look at darkly stained areas (Fig. 6) where the expression strength modifications makes spots more distinct after denoising.
Figure 6 - Denoising improves spot detection in darkly stained areas - Gel numbers 2,6,8 and 9 refer to Table 1. The denoised image is shown on the right side of the 2-DE image. The same spot detection parameters were used in all cases (see Material and Methods). The denoiser ran with s1=3 and s2=23. Spots marked by yellow circles are only detected in native images. Spots marked with green circles are detected only in denoised images. Note that closely located intensely stained spots (Gels 2, 8, lower right) and regions with low signal-to-strength ratio (Gels 2, 6, 8, 9, upper part) are resolved after denoising.

Tandem setup for quantitative analysis - Regarding quantitative analysis, e.g, to assess protein regulation, or measure protein expression, one can combining the spots detected in native and denoised images to create an improved master image (scout image). Subsequent spot quantification on the native gels, will provide the researcher with the required protein expression information. However, one must be careful to quantitatively interpret gels with high background variation. Background variations can be part of the investigated biological system or be introduced through technical variations in the experiment. Preferably, the experiments should be replicated to avoid misinterpretation [[11]].

Division versus subtraction to remove the background - The choice to divide the image by its background is uncommon because it might lead to division by zero and rescales the gray value distribution. A more common technique would be through subtraction. Empirical testing (see Supplementary Figures) showed that division yields the best results. This leads to the observation that 2-DE image background variations are more of a multiplicative phenomenon than they are additive. We model a 2-DE image ($P$) as a combination of signal ($S$), multiplicative background ($B_{m}$) and additive background ($B_{a}$).


\begin{displaymath} P=SB_{m}+B_{a} \end{displaymath} (6)

The calculated background signal will be $B_{m}+B_{a}$. Subtraction of this background leads to a denoised image $SB_{m}+B_{a}-B_{m}-B_{a}=B_{m}(S-1)$, which still has the full impact of the multiplicative noise. Division of the signal by its background


\begin{displaymath} \frac{SB_{m}+B_{a}}{B_{m}+B_{a}}=\frac{SB_{m}}{B_{m}+B_{a}}+\frac{B_{a}}{B_{m}+B_{a}} \end{displaymath} (7)

allows us to see a reduction in both the multiplicative factor and the additive term, especially when $B_{m}>B_{a}.$ This fact, combined with our empirical observation, leads us to conclude that background variations are mainly multiplicative.
Distribution rescaling - Dividing an image by its background modifies the gray scale. When working with black spots on a white background, the denoising algorithm will remap all spot information to the range $[0:1]$, while the background will map onto $[1:+\infty[$. If we would work with white spots on a black background the region of interest would become $[1:+\infty[$, which is difficult to normalize. The discrepancy between B/W and W/B images occurs because we have different relations: $\frac{S}{B}$ and $\frac{1-S}{1-B}$, with $S$ being the real signal and $B$ being the measured background. See Supplementary Figures for the result of different rescaling methods.
Algorithmic complexity - The complexity of an algorithm refers to the relative time the algorithm needs to compute a result when presented with a specific input size[[18]]. For fixed values of $s_{1}$, $s_{2}$ and $c$, the algorithm performs linearly to the size of the image. All steps in the algorithm: smoothing (Eq. 2), division (Eq. 3), thresholding (Eq. 4) and median filtering (Eq. 5) can be performed in $O(w.h)$, with $w$ and $h$, respectively the width and height of the image.

Concluding Remarks

We presented a denoising algorithm that requires no user input and provides an execution time linear to the image size. The algorithm relies on division of the original image by the estimated background variation and therefore only works on images with black spots on a white background. The algorithm was particular effective in removing background variance from immunoblots and radioisotopic labeled gels. Denoising improved the working of algorithms for gel orientation and overlay alignment. Spot detection on denoised Sypro Ruby images (using standard software) confirmed that the algorithm improves spot detection in areas with high background variation and in areas with closely located high-intensity spots. Because the algorithm modifies spot intensities, denoised images cannot be used for spot quantification. A future application for the algorithm could be the generation of a scout image to guide automatic spot recognition algorithms and quantify spot volumes in the original image, eliminating the manual pre- analysis processing of 2-DE images.

Acknowledgments
The expert technical assistance of Siv-Lise Bedringaas and Stein-Erik Gullaksen is highly appreciated. This study was supported by the Norwegian Research Council's Functional Genomics Program (FUGE) grant no. 151859, and the NORUT IT intramural grant. Nancy Gerits has helped substantially in editing of the manuscript.

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