Abstract:
The invention provides a technique for accurate detection of objects appearing in the image, such as single fluorescent molecules, cells, microorganisms, nucleotides, DNA strands, or stars in celestial images. The method allows to differentiate between two combined point sources even if they are closer to each other than the optical resolution of the system. The procedure involves computing several coefficients representing decomposition of the intensity data in the basis of Hermit functions. In the case when the objects are distributed randomly and higher yield of accurately detected objects is desired, the method allows for a tenfold increase in such yield.

Description:
[0001]    Deblending in Source Extractor is performed by examining an intensity profile of the objects appearing in an image and comparing that profile to a threshold. This is described in, for example, B. W. Holwerda, Source Extractor for Dummies 32-34 (Space Telescope Science Institute, Baltimore, Md.) and also in E. Bertin, SExtractor v2.3 User&#39;s Manual 20-22 (Institue d&#39;Astrophysique &amp; Observatoire de Paris). This technique in accordance with the Rayleigh criterion is generally unable to resolve individual objects that are closer than about 1.5 times their full-width half-max (FWHM). FWHM is an important measure of the quality of an imaging device and its spatial resolution. If the distribution is Gaussian, then the FWHM can be calculated as FWHM=σ2*√{square root over (−2*log(0.5))}, where σ is the standard deviation, or the square root of the variance. FWGM is also the bandwidth of the system. The Rayleigh criterion is the generally accepted criterion for the minimum optically resolvable detail. 
       SUMMARY OF THE INVENTION 
       [0002]    The invention generally relates to image processing techniques that improve the resolution of point source objects appearing in an image. The improved images can then be used in further analysis. In accordance with one aspect of the invention, images containing point source objects arranged very close together are processed, and individual point source objects are distinguished from clusters of objects. In a further aspect of the invention, the method is suggested that allows to differentiate two combined point sources from a single one even if they are closer to each other than the optical resolution of the system. For the purposes of this invention, a point source object is defined as a diffraction-limited object that is an object of a physical size less than a diffraction of light in the optical system, and which has then a size and a point spread function defined by the parameters of the optical system. 
         [0003]    In one embodiment of the invention, single molecules labeled with an optically-detectable reporter are detected. The increased accuracy and resolution provided by the invention reveals previously undetected or misdetected single objects. 
         [0004]    The present invention provides, in one aspect, methods and apparatus for facilitating the accurate detection of objects appearing in an image, such as single fluorescent molecules, or stars in astronomy related images, or fluorescing cells, or micro organisms. The invention provides resolution of closely-spaced objects without the need to perform intensive, time-consuming computations and can be applied in all areas of science and industry, such as medicine, biotechnology, microscopy, remote sensing, astronomy, defense, material science, manufacturing, security, robotics, and document processing, to name just a few. 
         [0005]    Embodiments of the invention are useful to detect single objects appearing in a dense field of objects. In particular, the method of the invention may be used in astronomy when analysis of digital images from a telescope is performed. Stars are diffraction limited objects for any possible magnification, which makes them point sources of predicted size FWHM and σ, and Gaussian point spread function. Further, the method may be successfully employed in microscopy of diffraction limited florescent entities such as florescent cells or florescent microorganisms under magnification not sufficient to see them in detail. The cells are then diffraction limited objects making them point sources of predicted size, σ, and Gaussian point spread function. 
         [0006]    Additionally, the method is highly useful in microscopy of florescent molecules. As the molecules are so small, they are diffraction-limited objects under any magnification, and they are always point sources. For example, nucleotides or DNA strands or pieces of strands with florescent dies attached to them are diffraction-limited objects. They are then point sources of predicted size, σ, and Gaussian point spread function. 
         [0007]    In one particular embodiment according to the invention, a method of image analysis includes providing a representation of a sample image that contains intensity and centroid (coordinates of object centers) information data for objects in the image. A post-processing procedure is performed on representation, which involves computing several coefficients representing decomposition of the intensity data in the basis of Hermit functions. These Hermit decomposition coefficients allow the characteristics (e.g. position and intensity) of the sample objects to be computed. The number of coefficients of decomposition that are computed depends upon the number of objects that one wishes to resolve, as taught below. 
         [0008]    In a particular alternative aspect, methods of the invention are used to detect the incorporation of single fluorescent-labeled nucleotides into a single surface-bound nucleic acid duplex in a template-directed sequencing-by-synthesis reaction. 
         [0009]    Other aspects and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating the principle&#39;s of the invention by way of example only. 
         [0010]      FIG. 1  shows the limits of resolution through a common methodology, known as Rayleigh criterion. 
         [0011]      FIG. 4  shows a dependence of yield of accurately detected objects on density of randomly distributed objects. Different curves correspond to different resolution capabilities of detection apparatus. The 4.5 pix resolution corresponds to the capabilities of the common methodology, and 1.5 pix resolution curve represents the capabilities of the embodiment of the invention. The maximum value of every curve represents the maximum possible yield of objects for the given resolution capability of detection. The observable improvement in the yield is 10 fold. 
         [0012]    As shown in the drawings of  FIG. 7  for the purposes of illustration, the invention may be embodied in methods and apparatus for analyzing images acquired during DNA sequencing. Embodiments of the invention are useful for minimizing or eliminating image artifacts that compromise the accuracy of detection. Application of methods of the invention to nucleic acid sequencing is used to demonstrate the utility of the invention. The skilled artisan understands that the principles of the invention are useful in any application in which high-resolution single object detection is desired, including applications involving diffraction limited or other symmetrical objects. 
         [0013]    In brief overview,  FIG. 5  is a flowchart depicting a method for image analysis in accordance with an embodiment of the invention. 
         [0014]      FIG. 7  shows that in the context of DNA sequencing, embodiments of the invention are used to identify the incorporation into a template/primer duplex of a single, labeled nucleotide at a discrete location on a surface. The basic process includes attaching nucleic acid duplex (comprising a template hybridized to a primer) to a surface, such as glass or fused silica (the specific type of surface is immaterial to the present invention, but should be selected to be compatible with the type of label used). The attached duplex is then exposed to an optically-labeled nucleotide that hybridizes to the next available nucleotide in the template (available meaning just 3′ of the template terminus) and a polymerizing enzyme capable of incorporating the labeled nucleotide into the primer. Incorporation is determined by observing the optically-detectable label at the known location of the duplex. For example, if the optically-detectable label is a fluorescent label, then illumination at the appropriate wavelength is used to stimulate fluorescence of the label. The invention allows one to determine whether a single optically-labeled nucleotide has been incorporated or whether there are multiple duplexes, non-specific label, dirt, etc. that overlap. 
         [0015]      FIG. 3  shows how defects in the testing apparatus and limitations on image resolution can hide or misidentify single fluorescent objects, thereby compromising the accuracy of the data. 
         [0016]      FIG. 6  shows that in embodiments of the invention, an image is acquired using, for example, a personal computer with an image capture card. The image is recorded in one or more electronic files, typically in, the SPE or TIFF or FITS format. A photometry program then operates on these files. One such program is Source Extractor, which is typically used in astronomical studies. The photometry program detects the intensities and locations of the fluorescence (i.e., the template objects) and generates a representation of the image that includes a table or catalog containing intensity data and the centroids of the objects. The intensity data generally follow a Gaussian distribution, and the centroids are typically the coordinates of the centers of the identified objects. 
         [0017]    A problem with the representation of the image ( FIG. 3 ) is that photometry programs generally have a limited ability to identify or resolve a number of closely spaced objects. For example, the photometry programs can erroneously interpret two discrete, closely spaced objects as single large object. This can occur if the objects are closer than approximately 1.5. FWHM (full-width at half-max) of the point source objects, which measures about 4 pixels on the figure. To minimize or eliminate this problem, embodiments of the invention subject the representation, of the image of  FIG. 3  to post-processing, potentially identifying cluttered objects. 
         [0018]    As shown in  FIG. 5 , this post-processing examines the intensity data (collectively, the intensity flux), and computes several coefficients of Hermit decomposition of the intensity flux. A result is a series of equations that are solved simultaneously to yield a template parameter that, in some embodiments, includes corrected values for the centroids. The corrections have the effect of revealing locations of additional objects that were unresolvable with the common technologies. 
         [0019]      FIG. 5  is a flowchart depicting one possible method for such post-processing in accordance with an embodiment of the invention. A representation of the image includes, as described above, intensity data and centroids of the fluorescing objects therein. The fluorescing objects generally appear in a constellation-like form. When the representation of the image includes many large and closely spaced fluorescing objects, as shown in illustration  FIG. 3 , the post-processing operates to minimize or eliminate artifacts that could prevent a proper analysis. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0020]      FIG. 1  shows the limits of resolution of common methodology, known as the Rayleigh criterion. 
           [0021]      FIG. 2 , example 1 and 2 each shows a point-spread function, to which two proximate objects have contributed. 
           [0022]      FIG. 3  shows a point spread function of two point source objects blended as a result of limitations on image resolution. 
           [0023]      FIG. 4  shows a dependence of yield of accurately detected objects on density of randomly distributed objects for different resolution capabilities of detection apparatus. 
           [0024]      FIG. 5  is a flowchart depicting a method for image analysis in accordance with an embodiment of the invention. 
           [0025]      FIG. 6  shows a possible software apparatus based on the embodiment of the invention used to detect point source objects present in an image. 
           [0026]      FIG. 7  shows that in the context of DNA sequencing, embodiments of the invention are used to identify the incorporation into a template/primer duplex of a single, labeled nucleotide at a discrete location on a surface. 
       
    
    
     DETAILED DESCRIPTION 
       [0027]    In the method of the current invention, a point spread function is used to characterize the intensity of the point source. In a digital image of point sources, a point spread function of a source located in (x 1 ,y 1 ) is described by Gaussian: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where F is the intensity of the source and σ is a size-parameter defined by optics that produced the digital image. For the purposes of this application, we define the optical resolution of the system, ρ, as a minimal distance between the two point sources that allow for the optical resolution of the two sources. The point sources are considered optically resolved if a point spread function has a minimum point between them. The resolved and unresolved sources are exemplified in  FIG. 1 . Let&#39;s consider Example 1 and Example 2, represented in  FIG. 2 . For Example 1, two point sources are located at a distance d, d&gt;ρ, where ρ is an optical resolution. For Example 2, two point sources are located at a distance d, d&lt;ρ. Consider F(x,y) a point spread function (pixel intensities) of the example 2 ( FIG. 2 ). It is described by a combined point spread function of the 2 sources located in (x 1 ,y 1 ) and (x 2 ,y 2 ): 
         [0000]    
       
         
           
             
               
                 
                   
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         [0028]    The method of the current invention allows to differentiate two combined point sources from a single one even if they are closer to each other than optical resolution of the system. The method accurately positions the sources by computing ƒ i  coefficients of decomposition of the point-spread function on a basis H i  of Hermit functions: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    Hermit functions are an orthogonal basis in a functional space of integrable functions and they are, 
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         [0000]    Where (x 0 ,y 0 ) is the centroid of the original and possibly combined object and h i (x,y) is a Hermit polynomial in 2 variables. 
         [0029]    If the basis of polynomials {1,x,y,x 2 ,xy,y 2 ,x 3 ,x 2 ,y,xy 2 ,y 3 , . . . } was used instead of Hermit functions, which are polynomial multiples of a Gaussian, the decomposition coefficients would have been moments: 
         [0000]      definition: Moment ( i+k,i,k ) =Mi,k=∫∫x   i   y   k   F ( x,y ) dxdy    Eq. 5
 
         [0000]    However, the representation or decomposition in Hermit basis is more powerful then a polynomial representation. It delivers a less noise dependent solution due to fast convergence of the Gaussian to 0 with increasing distance from origin. The solution based on Hermit function decomposition would also address some other problems besides the resolution. For instance, the polynomial moments augment the dependency of the data on the random noise present in digital images while Hermit decomposition coefficients suppress it and thus make the method more resilient to noise. Also Hermit decomposition coefficients allow for automatic scaling of relative intensities of blended objects. It is also called in literature a product with Gaussian measure or weighted product (these coefficients of decomposition arc a scalar product of a point spread function F(x,y) and the functions of a basis such as polynomial, Hermit functions etc). 
         [0030]    Resilience to noise is an important advantage of Hermit functions representation that allows dispensing with a computationally intense verification step such as fitting (Non-Linear Least Square fitting). This leads to a significant improvement in speed and potentially cost of the processing of the method of the invention over any method that has to employ fitting. In a situation when speed of the processing is a limiting factor it can lead to significant savings in cost due to less hardware equipment (CPU) needed in the processing apparatus. One of the examples of the processing time limited application is image processing done in real time without saving the images. The range of savings in processing time or hardware cost may differ from 10 times to 1000 times depending on object density and image quality. 
         [0031]    If F(x,y) is a combined point-spread function of 2 point sources (x 1 ,y 1 ) and (x 2 ,y 2 ), of intensities F 1  and F 2  correspondingly, then F(x,y) is a sum of point spread !functions of single sources (Eq. 2). F(x,y) can be decomposed over the Hermit. function basis, and the coefficients of that decomposition ƒ i  will be defined by F 1 , F 1 , x 1 ,y 1 ,x 2 ,y 2  and σ. However, σ is defined by the optical system and is a known value in that system of equations. 
         [0032]    Therefore, to define a two-point-source combination, 6 unknowns need to be found, namely, 2 intensities and 4 (x,y) coordinates. Six equations are described below that define the coefficients of decomposition through F(x,y), the pixel intensities in the image. These values of ƒ i -are set equal to their predicted value defined through the theoretical form of F(x,y) as combination of two point sources (Eq. 2) decomposed over Hi basis. The system of equations then is solved to obtain numerical values for F 1 , F 1 , x 1 ,y 1 ,x 2 ,y 2 . If the values of intensities and coordinates thus defined are physically meaningful, that constitutes a solution for two closely standing separate sources, otherwise the source is called a single point source. 
         [0033]    We will choose the first 6 Hermitian functions to get the equations to connect the specific data from the image to a theoretical formula expressing that value through the unknown F 1  and F 2  and x 1 , x 2 , y 1 , y 2 . The solution of this system of equations for F-s and xy-s will provide the coordinates and intensities of the point sources of interest. The first of the basic Hermit functions are: 
         [0000]      7)  H   0   =e   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2     2      Eq. 6
 
         [0000]      8)  H   1,0   =xe   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2     2      Eq. 7
 
         [0000]      9) and for  y: H   0,1   =ye   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2     2      Eq. 8
 
         [0000]      10)  H   2,0 =( x   2 −1) e   (−(x−x     0     σ)     2     −(y−y     0      )     2     )/2    Eq. 9
 
         [0000]      11)  H   1,1 =( xy ) e   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2      2      Eq. 10
 
         [0000]      12 )  H   0,2 =( y   2 −1) e   (−(x−x     0     σ)     2     −(y−y     0     )     2     /2     2      Eq. 11
 
         [0034]    Therefore the first decomposition coefficients for a point spread function F(x,y) are: 
         [0000]      7) ƒ 0   =∫∫F ( x,y * e   (−(−x−x     0     σ)     2     −(y−y     0     )     2     )/2     2     dxdy    Eq. 12
 
         [0000]      8) ƒ 1,0   =∫∫F ( x,y ) xe   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2     2     dxdy    Eq. 13
 
         [0000]      9) and for  y: ƒ   0,1   =∫∫F ( x,y ) ye   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2     2     dxdy    Eq. 14
 
         [0000]      10) ƒf 2,0   =∫∫F ( x,y )( x   2 −1) e   (−(x−x     0     σ)     2     −(y−y     0     )     2     /2     2     dxdy    Eq. 15
 
         [0000]      11) ƒ 1,1   =∫∫F ( x,y )( xy ) e   (−(x−x     0     σ)     2     −(y−y     0     )     2     )/2     2     dxdy    Eq. 16
 
         [0000]      12) ƒ 0,2   =∫∫F ( x,y )( y   2 −1) e   (−(x−x     0     σ)     2     −(y−y     0     )     2     /2     2     dxdy    Eq. 17
 
         [0035]    These are values of coefficients ƒ i  computed from the image data. 
         [0036]    From the functional characteristics of F(x,y) (Eq. 2) the same ƒ i  values can be computed as follows: 
         [0000]      7) ƒ 0   F   1   e   −x     1       2     4σ     2     +F   2   e   −x     2       2     /4σ     2      Eq. 19
 
         [0000]      8) ƒ 1,0   =F   1 ( x   1 /2) e   −x     1     σ3/4     2     +F   2 ( x   2 /2) x   e   −     2     σ3/4     2      Eq. 20
 
         [0000]      9) ƒ 0,1   =F   1 ( y   1 /2) e   −x     1     σ3/4     2     +F   2 ( x   2 /2) x   e   −     2     σ3/4     2      Eq. 21
 
         [0000]      10) ƒ 2,0   =σF   1 ( 2   +x 3/4 e   1   2 ) −x     1     σ3/4   +F   2   σ   2 ( 2   +x 3/4 e   2   2 ) x−     2     σ3/4     2      Eq. 22
 
         [0000]      11) ƒ 1,1   =σF   1 ( 2   +x 3/4 e   1   2 ) −     1     σ3/4   +F   2   σ   2 ( 2   +x 3/ y 4 e   2 2 ) x−     2     σ3/4     2      Eq. 23
 
         [0000]      12) ƒ 0,2   σF   1 ( 2   +y 3/4 e   1   2 ) −x     1     σ3/4   +F   2 σ 2 ( 2   +y 3/4 e   2   2 ) x−     2     σ3/4      2      Eq. 24
 
         [0037]    Additionally, flux and centroid information can be used: 
         [0000]      flux( F ( x,y ))= F ( x,y ) dxdy=F   1   +F   2   =M   0    Eq. 25
 
         [0000]    and centroid (coordinate of center of mass) 
         [0000]      0= x   1   *F   1   +x   2   *F   2    Eq. 26
 
         [0000]    in a system of coordinates (x,y) with the origin of x 0 ,y 0 , the center of mass of the point spread function F(x,y). 
         [0038]    When the system of equations (1 to 6), is solved in algebraically it can be assumed that 
         [0000]        e   σ   −x     2     σ     2     /2     2   ≈1−( 2   2 /4)   Eq. 27,
 
         [0000]    because, as we are considering the case of optically unresolved and thus sufficiently close pair of sources, it is implied, that the sources distances x i   x     i    from the center of mass are less than √2σ. Therefore, x i   2 /2σ 2 &lt;½, and the approximation applies with enough accuracy. 
         [0039]    The system of equations 19-27 can be solved as follows: from Eq. 19 and Eq. 27: 
         [0000]      ƒ 0   Fq 1≠ Fr   1   2 /4 2   x )σ 2 (1 =F   2   2/4   +F () F   1   x   2 )= F ( x   1 σ 1   2 σ 2 * 2   2 )4 2  
 
         [0000]      since 
         [0000]    
       
      
       F 
       1 
       +F 
       2 
       =M 
       0  
      
     
         [0000]      and 
         [0000]      ( F   1   +F   2 )−( F*             *Fc   2   2 ) x /4 x   2x   0 σ 1 * 1 ( 1 2 )/4 2    Eq. 28
 
         [0000]      and 
         [0000]      ( M   0 +σƒ 0 )=4 F   x   2   c   1           ( 1 2 )   Eq. 29
 
         [0000]    from Eq. 20 and Eq. 27: 
         [0000]      ƒ 1,0   =F   1   *x   1 /2 *E   (−x     1     94 3/4     2   ) +F   2   *x   2 /2 *e   x(−     2     σ3/4     2     )   =F   1   *x   1 *( e   x(−     1     σ3/4     2     )   −e   x(−     2     σ3/4     2     ) )2ƒ 1,0 ≈( x   1   −σx   2 )( x   1   +x   2 )* F   1   *x   1 /4 2    Eq. 30
 
         [0000]    Combining Eq. 28 and Eq. 30, it can be observed that: 
         [0000]      2ƒ 1,0 ≈( x   1   σ−x   2 )(ƒ 0   +M   0 )/4 2    Eq. 31
 
         [0000]      and 
         [0000]        x   1   −xσ   2 =2ƒ 1,0   +*             2 /( 0 0 )   Eq. 32;
 
         [0000]      therefore 
         [0000]        x   1   +x   2   =x   2   −             /(   0 0 )2 1    Eq. 33
 
         [0000]      and 
         [0000]        x   1   +x   2   =x   2   −             /(   0 0 )2 1    Eq. 34
 
         [0000]      and 
         [0000]      2 F   1   x   1   2   +Fσ   1 *2 ƒ 1 ƒ 0           /( x   0  σ 0 )*ƒ 1             2 (= 0 0 )0   Eq. 35
 
         [0000]    Eq. 35 is a quadratic equation in x 1  and allows for a solution 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
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                   36 
                 
               
             
           
         
       
     
         [0000]    and, by symmetry, the other root of Eq. 35 is x 2 : 
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         [0000]    Now when X 1  and x 2  have been found, Eq. 26 allows to find: 
         [0000]        F   1   x   1   x   2 ( M   0   −F   1 ) 
         [0000]        F   1   =M   0   x   2 /( x   1   x+x   2 ) 
         [0000]        F   2   =−F   1   x   1   /x   2    Eq. 38-40
 
         [0040]    Thus x 1 , X 2 , F 1 , F 2  have all been found. 
         [0041]    A list of detected objects in the final representation of the image lacks many of the image artifacts present in the initial representation of the image. In particular, blended pairs of two fluorescing objects appear resolved as singles, and their positions and intensities arc accurately detected. Before applying the method that is the embodiment the invention, many closely spaced pairs of fluorescing objects would be erroneously rendered as single large objects, and their coordinates would be called as centers of mass of blended objects. These centers of mass of blended objects may be far from the centers of the actual objects the blended object consists of. The error in centroid (position of the object) determination may result in an erroneous answer to a question “was there an object located in certain location or vicinity?”, thereby preventing a proper analysis of, for example, chemical incorporations in DNA sequencing. 
         [0042]    The post-processing of a blended object after subtracting the effect of detected objects from the image data, and computing new centroids by the calculation of decomposition coefficients can be performed several times. In theory, repeating the process will refine the image data, thereby reducing artifacts and allowing for the resolution of more (e.g., three or greater) closely spaced objects. 
         [0043]    In brief overview; the image analysis apparatus in accordance with of the invention includes an image capture subsystem that acquires images of fluorescing objects (i.e., molecules or cells or stars), digitizes them, and generates corresponding optical data that can be stored in computer files, typically in the SPE or TIFF or FITS format. A software code processes the optical data and generates field pattern data that includes original centroids of the fluorescing objects. 
         [0044]    Further, software code based on the algorithm of the current invention processes the optical data, or the field pattern data, or both, computes the coefficients of Hermit decomposition of the intensity data corresponding to each fluorescing object, and generates a replacement field data pattern. From the computation of the coefficients of Hermit decomposition, the software code based on the algorithm of the current invention also calculates replacement centroids. The apparatus of the invention can repeat this process any number of times to refine the data. 
         [0045]    Additionally, the software code based on the algorithm of the current invention determines if any of the original centroids should be replaced by two or more replacement centroids. This can occur when, for example, the coefficients of. Hermit decomposition suggest that what was thought to be a single fluorescing object is actually, two (or more) closely spaced fluorescing objects, each having its awn centroid. For example, compare the residual, light of the image with a two centroid configuration with a residual of the image with a single centroid configuration. Choose the hypothesis, on a number of objects (one or two) that leads to less residual. 
         [0046]    Further, the method and apparatus of the current invention may be advantageously used for the purposes of experiment optimization in any type of assay involving fluorescent entities, such as microbiological, cellular, nucleic acid or any other assay. By allowing much better resolution of the point source objects, it becomes possible to considerably increase the density and the output of such assays. As shown on  FIG. 4 . application of the method of the embodiment of the invention increased the maximum possible output of accurately detected objects by a factor of 10 while resolution of that particular optical system improved from 4.5 pixels to 1.5 pixels. That is an example of the effect of the application of the method compared to the common methodology. 
         [0047]    Though the components of the apparatus of the current invention are described throughout this application as separate entities, in actual implementations of the invention, however, they may be inseparable components of other electronic devices such as a digital computer. Thus, actions described above may be implemented in software that may be embodied in an article of manufacture that includes a program storage medium. The program storage medium includes, for example, data signals embodied in one or more of a carrier wave, a computer disk (magnetic, or optical (e.g., CD or DVD), or both), non-volatile memory, tape, a system memory, and a computer hard drive. 
         [0048]    From the foregoing, it will be appreciated that methods and apparatus according to the invention afford a simple and effective way to analyze images used in. DNA sequencing or in any other application where images must be examined or compared with accuracy and can be difficult to obtain due to, for example, defects in the testing apparatus and/or limitations on image resolution. 
         [0049]    The invention may be embodied in other specific forms than what is particularly disclosed herein without departing from the spirit or scope of the invention. The foregoing disclosed embodiments are in all respects illustrative rather than limiting on the invention.