in order to collect a set of images useful to test the algorithm , we run 10 plasma samples using 2d - ge . 2d - ge was performed according to maresca et al . , with each sample run in duplicate . for the first - dimension of electrophoresis , 200  g of plasma protein samples ( approximately 3  l ) were applied to 18-cm linear ipg strips 47 ( ge healthcare , uppsala , sweden ) and focused until 72,000  v / h were reached . prior to sds  page , the ipg strips were equilibrated twice for 15min in equilibration buffer ( 50  mm tris hcl ph  8.8 , 6  m urea , 30% glycerol ( v / v ) , 2% sds ( w / v ) and traces of bromophenol blue ) containing 1% dtt ( w / v)for the first equilibration step and 2.5% iodoacetamide ( w / v ) for the second step , respectively . page was performed on 12.5% polyacrylamide gels at 60  ma / gel at 16  c and terminated when the dye front reached the lower end of the gel . after sds  page , the gels were stained overnight with rubps in fixing solution ( 30% ethanol and 10% acetic acid ) according to the manufacturer s instructions . the gel images were acquired by gel - doc - it 310 imaging system ( uvp , upland , ca ) , with the following setting exposure 7  s , gain 3 and three different apertures : 6 , 7 and 8 . aperture is expressed as f - stop ( e.g. , f2.8 or f/2.8 ) : the smaller is the f - stop number ( or f / value ) , the larger is the aperture . using a larger exposure yields a larger number of spots and , at the same time , more saturated areas . in our case , only the image acquired at the lower exposure ( figure 1c ) could be properly analysed by commercial and academic software currently available , while the saturation in some areas in the other two images ( figure 1a and b ) prevented accurate evaluation of protein expression . using the algorithm described in this work , we have been able to approximate the intensity values of the non - saturated spots within the saturated area . after image scanning , the 2d - ge images are noisy and usually affected by several artefacts such as scratches , air bubbles and spikes . scratches and air bubbles are unintentionally introduced by the operator , while spikes are caused by the presence of precipitated staining particles or imperfect polyacrylamide gel matrix . in particular , spikes are a sort of impulsive noise because of the small area of high intensity . the aim of reducing speckles in a 2d - ge image ( despeckling ) is to remove the noise without introducing any significant distortion in quantitative spot volume data . noise suppression methods employed in commercial software packages for 2d - ge image analysis are based on spatial filtering , particularly the most widely used techniques are based on local median filter . although several other methods have been developed , the advantage of using the median filter is the easy application , since it normally requires the setting of only one main parameter , the filter kernel , which is generally related to the noise size . at the same time , the size of the filter matrix also accounts for the major impact exerted on the resulting image . we evaluated the performance of filter size on a set of test images from 3    3 to 7    7 and observed remarkable deterioration for larger filter size , due to the decreased the contrast of the image . therefore , a 3    3 filter was chosen , which was capable of removing most of the spikes , while preserving the shape of the spots . the result of a 3    3 median filter on a sample image is shown in figure 5 . a portion of gel with spikes in 2d and 3d view was shown in figure 5a and c , respectively . the results of a 3    3 median filter are shown in figure 5b and d accordingly . saturation effects in 2d gel images could occur for several reasons : ( i ) intense staining , causing complete absorption of light passing through the spot , ( ii ) long exposure time during the acquisition step , ( iii ) too high protein concentration , or ( iv ) accumulation of pigment on top of high intensity spots , leading to the loss of dynamic linear range . saturation effects might also result from manipulation of image data with image processing software that is not designed for quantitative image analysis ( e.g. , photoshop ) . for instance , visually - enhanced images often show plateau areas in the high and low intensity regions . the commercial software is able to detect a saturated area when the intensity reaches the maximum value of grey - scale . however , in most cases saturated areas are very similar to plateau zones , i.e. , where the pixels have similar intensity without reaching the maximum value of grey - scale . in these cases , the software might also fail to detect the plateau regions , causing the operator to underestimate the problem . in order to identify the plateau regions , we implemented a morphological filter , inspired by the rolling - ball algorithm described by sternberg , which allows segmentation of the plateau zones . this method is based on a structural element ( se ) defined by a circle of given radius ( rd ) and a grey - scale value tolerance ( gvt ) . in particular , for each pixel ( x , y ) of the image i , the se is defined as a circular neighbourhood of rd:(1a)se=(s , t)ini/(s - x)2+(t - y)2<rdwhere(1b)i=(0,image width)[0,image height]denotes the spatial domain of the image . the gvt represents the rate of grey values of the pixel in the centre of the ses ( figure 6b ) . for instance , setting the parameter to 10 , the rd is 10  pixels and the gvt in the 10% of grey values of the pixel at position ( x , y ) . the centre of se is moved along each pixel of the image and the maximum and minimum grey values of the pixels for each point ( x , y ) within the given rd are calculated . when the difference between maximum and minimum is less than gvt , the area defined by the local operator is considered as a plateau area . it is worth noting that , when the value of the parameter is low , the rd of the ball is small and only few pixels are included in se . we examined the performance with rd values from 10 to 25 and for this analysis we selected a value of 15 . after localizing the plateau regions , we segmented the image to identify the isolated protein spots containing plateau areas on each gel image . the segmentation procedure yields a set of image segments , consisting of connected neighbouring pixels enclosed by a spot boundary . the image is considered as a landscape , and the segmentation is to identify all the local minima in the landscape and then find the catchment basins associated with each local minima . the boundary between several catchment basins is called a watershed . for 2d - ge each separated catchment basin is considered as an isolated protein spot . unlike the usual watershed segmentation , only the plateau areas in our method have been assigned to a catchment basin . finally , the grey values of the pixels inside the spot , excluding the region identified as a plateau area , were used in the gaussian extrapolation step to recover the distribution of the unsaturated spot . the final step consists of reconstructing the saturated spots resulting from high exposure images and approximating the unknown grey values in the plateau region . this has been done considering the unsaturated spot to be described by an analytical function , depending on a restricted set of parameters . in particular , we assumed each cross section of the spot intensity along both vertical and horizontal axes to be approximated by a generalized gaussian distribution . namely , for each value of the y - coordinates we considered a function of the form:(2)f(x , m(y),(y),x0,b)=m(y)(y)exp-|x - x0(y)|bb(y)b for b  =  2 , eq . ( 1b ) defines the kernel of a standard gaussian distribution centred in x0(y ) , where (y ) and m(y ) is the standard deviation and the maximum of intensity values , respectively . note that , unlike the other parameters , b does not depend on y , assuming that the approximating gaussian can have different maximum , center and variance in different sections . ,nyand the corresponding intensities { iij } , we determine the set of parameters ( m ,  , x0 , b ) for which the function defined in ( 2 ) fits at best the values of the intensity in the unsaturated region . in practice , we have to minimize an error function that defines how good a particular parameter set is . for example , a standard least - squared criterion can be used(4)em(yj),(yj),x0(yj),b=i=1nxfxi , m(yj),(yj),x0(yj),b - iij2 however , the error measure can be defined in different ways , e.g. , according to different norms or including different weighs for the different parameters and/or the different pixels . in particular , we modified ( 4 ) in order to control the variation of the parameters for different sections . in our case the error function ( 5 ) can be formulated as follows(5)em(yj),(yj),x0(yj),b = em(yj),(yj),x0(yj),b+m(yj)-m(yj-1)2+(yj)-(yj-1)2+x0x0(yi)-x0(yj-1)2 for positive values of the parameters ( m ,  , x0 ) , the problem is then reduced to finding the parameters yielding the minimum of the selected error function . one possibility is to perform an exhaustive search on all the values of a pre - defined parameters space . however , if the size of the parameter space is large , a more effective newton after image scanning , the 2d - ge images are noisy and usually affected by several artefacts such as scratches , air bubbles and spikes . scratches and air bubbles are unintentionally introduced by the operator , while spikes are caused by the presence of precipitated staining particles or imperfect polyacrylamide gel matrix . in particular , spikes are a sort of impulsive noise because of the small area of high intensity . the aim of reducing speckles in a 2d - ge image ( despeckling ) is to remove the noise without introducing any significant distortion in quantitative spot volume data . noise suppression methods employed in commercial software packages for 2d - ge image analysis are based on spatial filtering , particularly the most widely used techniques are based on local median filter . although several other methods have been developed , the advantage of using the median filter is the easy application , since it normally requires the setting of only one main parameter , the filter kernel , which is generally related to the noise size . at the same time , the size of the filter matrix also accounts for the major impact exerted on the resulting image . we evaluated the performance of filter size on a set of test images from 3    3 to 7    7 and observed remarkable deterioration for larger filter size , due to the decreased the contrast of the image . therefore , a 3    3 filter was chosen , which was capable of removing most of the spikes , while preserving the shape of the spots . the result of a 3    3 median filter on a sample image is shown in figure 5 . a portion of gel with spikes in 2d and 3d view was shown in figure 5a and c , respectively . the results of a 3    3 median filter are shown in figure 5b and d accordingly . saturation effects in 2d gel images could occur for several reasons : ( i ) intense staining , causing complete absorption of light passing through the spot , ( ii ) long exposure time during the acquisition step , ( iii ) too high protein concentration , or ( iv ) accumulation of pigment on top of high intensity spots , leading to the loss of dynamic linear range . saturation effects might also result from manipulation of image data with image processing software that is not designed for quantitative image analysis ( e.g. , photoshop ) . for instance , visually - enhanced images often show plateau areas in the high and low intensity regions . the commercial software is able to detect a saturated area when the intensity reaches the maximum value of grey - scale . however , in most cases saturated areas are very similar to plateau zones , i.e. , where the pixels have similar intensity without reaching the maximum value of grey - scale . in these cases , the software might also fail to detect the plateau regions , causing the operator to underestimate the problem . in order to identify the plateau regions , we implemented a morphological filter , inspired by the rolling - ball algorithm described by sternberg , which allows segmentation of the plateau zones . this method is based on a structural element ( se ) defined by a circle of given radius ( rd ) and a grey - scale value tolerance ( gvt ) . in particular , for each pixel ( x , y ) of the image i , the se is defined as a circular neighbourhood of rd:(1a)se=(s , t)ini/(s - x)2+(t - y)2<rdwhere(1b)i=(0,image width)[0,image height]denotes the spatial domain of the image . the se is depicted in figure 6a . the gvt represents the rate of grey values of the pixel in the centre of the ses ( figure 6b ) . for instance , setting the parameter to 10 , the rd is 10  pixels and the gvt in the 10% of grey values of the pixel at position ( x , y ) . the centre of se is moved along each pixel of the image and the maximum and minimum grey values of the pixels for each point ( x , y ) within the given rd are calculated . when the difference between maximum and minimum is less than gvt , the area defined by the local operator is considered as a plateau area . it is worth noting that , when the value of the parameter is low , the rd of the ball is small and only few pixels are included in se . we examined the performance with rd values from 10 to 25 and for this analysis we selected a value of 15 . after localizing the plateau regions , we segmented the image to identify the isolated protein spots containing plateau areas on each gel image . the segmentation procedure yields a set of image segments , consisting of connected neighbouring pixels enclosed by a spot boundary . the image is considered as a landscape , and the segmentation is to identify all the local minima in the landscape and then find the catchment basins associated with each local minima . the boundary between several catchment basins is called a watershed . for 2d - ge each separated catchment basin is considered as an isolated protein spot . unlike the usual watershed segmentation , only the plateau areas in our method have been assigned to a catchment basin . finally , the grey values of the pixels inside the spot , excluding the region identified as a plateau area , were used in the gaussian extrapolation step to recover the distribution of the unsaturated spot . the final step consists of reconstructing the saturated spots resulting from high exposure images and approximating the unknown grey values in the plateau region . this has been done considering the unsaturated spot to be described by an analytical function , depending on a restricted set of parameters . in particular , we assumed each cross section of the spot intensity along both vertical and horizontal axes to be approximated by a generalized gaussian distribution . namely , for each value of the y - coordinates we considered a function of the form:(2)f(x , m(y),(y),x0,b)=m(y)(y)exp-|x - x0(y)|bb(y)b for b  =  2 , eq . ( 1b ) defines the kernel of a standard gaussian distribution centred in x0(y ) , where (y ) and m(y ) is the standard deviation and the maximum of intensity values , respectively . note that , unlike the other parameters , b does not depend on y , assuming that the approximating gaussian can have different maximum , center and variance in different sections . the reconstruction problem can be formulated as follows . ,nyand the corresponding intensities { iij } , we determine the set of parameters ( m ,  , x0 , b ) for which the function defined in ( 2 ) fits at best the values of the intensity in the unsaturated region . in practice , we have to minimize an error function that defines how good a particular parameter set is . for example , a standard least - squared criterion can be used(4)em(yj),(yj),x0(yj),b=i=1nxfxi , m(yj),(yj),x0(yj),b - iij2 however , the error measure can be defined in different ways , e.g. , according to different norms or including different weighs for the different parameters and/or the different pixels . in particular , we modified ( 4 ) in order to control the variation of the parameters for different sections . in our case the error function ( 5 ) can be formulated as follows(5)em(yj),(yj),x0(yj),b = em(yj),(yj),x0(yj),b+m(yj)-m(yj-1)2+(yj)-(yj-1)2+x0x0(yi)-x0(yj-1)2 for positive values of the parameters ( m ,  , x0 ) , the problem is then reduced to finding the parameters yielding the minimum of the selected error function . one possibility is to perform an exhaustive search on all the values of a pre - defined parameters space . however , if the size of the parameter space is large , a more effective newton mn conceived the study , developed the algorithm for the detection of overexposed areas and plateau regions . mn and ac drafted the manuscript with the help of ef . all authors read and approved the final manuscript .
analysis of images obtained from two - dimensional gel electrophoresis ( 2d - ge ) is a topic of utmost importance in bioinformatics research , since commercial and academic software available currently has proven to be neither completely effective nor fully automatic , often requiring manual revision and refinement of computer generated matches . in this work , we present an effective technique for the detection and the reconstruction of over - saturated protein spots . 
 firstly , the algorithm reveals overexposed areas , where spots may be truncated , and plateau regions caused by smeared and overlapping spots . 
 next , it reconstructs the correct distribution of pixel values in these overexposed areas and plateau regions , using a two - dimensional least - squares fitting based on a generalized gaussian distribution . 
 pixel correction in saturated and smeared spots allows more accurate quantification , providing more reliable image analysis results . 
 the method is validated for processing highly exposed 2d - ge images , comparing reconstructed spots with the corresponding non - saturated image , demonstrating that the algorithm enables correct spot quantification .