Abstract:
During electronic film development, an area of conventional photographic film is scanned several times using a single scanning station, and at each subsequent time this scanned area is advanced incrementally along the film with multiple levels of overlap with previous scans. The new image scanned at each new time is aligned to an accumulating image that has been extrapolated to the image at the new time, and then the new image is added to the accumulating image in parametric summations that allow an image to be interpolated to any time free of seams where the scans overlap. The invention further teaches a method of steering the alignment by warping the leading edge of the alignment, and a registration method of aligning multiple images that takes advantage of known fixed alignments between images.

Description:
In electronic film development, conventional film is scanned electronically during development to produce a series of views of the developing image. An early scan reveals the fast developing highlight detail, while a late scan reveals slow developing shadow detail. After development, the series of views is combined into a single image in a process called stitching. In the prior art, stitching cut out the best parts of each view and merged them together. In the present invention, regression data is accumulated during development to describe a curve of density versus time of development for each pixel. After development, this regression data is used to recreate a regression curve of density versus development time for each pixel. The time at which this curve crosses a density known to give optimum grain characteristics, called the optimum density curve, is used to create the brightness for that pixel in the finished stitched image. The invention further teaches weighting regression data as a function of time and density generally following proximity to the optimum density curve. 
     BACKGROUND AND PRIOR ART 
     Recording an image at different exposures and later merging the images has been practiced since the advent of photography. A technique known to photographers for overcoming the dynamic range limit of film is to make two exposures, perhaps one for the clouds and one for the shadowed foreground, and merge the two using manual printing skill in the darkroom. A similar technique is known in astrophotography where multiple exposures reveal different features of a star cluster or nebula. In a rather flashy example, Kodak developed a film in the 1950&#39;s capable of recording the million to one brightness range of a nuclear test by making a color negative film wherein the three color layers were substituted with three monochrome layers of widely different sensitivities, each developing in color developer with a different dye color. Again, the manual skill of a darkroom printer was relied on to merge the images into one. A further example can be found in radiology where images can be made with different x-ray voltages to reveal detail in both soft and hard materials, then merging the images together. Modem color film typically uses three emulsion coatings for each color, each of a different speed. The three are merged simply by putting all three together in one film, thereby getting some benefit of a layer optimized for a particular exposure, but mixed with the grain of other layers not optimized for that particular exposure. 
     It was not until the advent of electronic film development, as taught in U.S. Pat. No. 5,519,510 issued to the present inventor, that there was a need to merge multiple exposure images using production-level speed and automation. In electronic film development, the merging of images is called stitching. The background of electronic film development in general and the prior art methods of stitching are now presented as a basis of understanding the background of the present invention. 
     Turning to FIG. 1, a scene  102 , portrayed as perceived through the wide dynamic range of the human eye, has highlights  104 , midtones  106 , and shadows  108 , with details in all areas. A camera  110  is used to project the scene onto a film inside the camera. The scene is perceived by the film to consist of points of light, each with an exposure value which may be mapped along an exposure axis  112 . 
     The film is removed from the camera after exposure and placed in a developer. In electronic film development, an electronic camera  120  views the film by nonactinic infrared light during development. As seen after a short development time of perhaps one minute, the film  122  still has a low density for shadows  124  and midtones  126 , but may optimally reveal highlights  128 . As seen by the infrared camera  120 , inverting for the negative of conventional film, the shadows and midtones  130  appear black, while the highlights  132  are seen more clearly than at any later time in development. 
     Doubling development time to two minutes, the midtones  140  have progressed to an optimum density while the highlights  142  may already be overdeveloped and the shadows  144  may still be too low in density to reveal a clear image. The film  146  would appear to have good midtone detail  148 , but the highlights  150  are already white, while the shadows  152  are still black. 
     Doubling development time again to a total of four minutes, the shadows have now reached an optimum density, but the other exposures are overdeveloped such that in image  162  they may appear white with little detail. 
     For each exposure, there is an optimum density of development to reveal the clearest image. Clarity may be defined technically as the best signal to noise ratio, where signal is the incremental change in density with exposure, and noise is the RMS deviation in density across a region that has received uniform exposure, by convention scanned with a 24 micron aperture. For example at one minute of development time, the midtones  126  typically have too low a density, or are too dark, to have enough of a signal level to reveal detail through the noise of the film and capture system. On the other hand, at four minutes the midtones  164  are “washed out”, such that not only is their contrast, or image signal strength, too low, but the graininess of an overdeveloped silver halide emulsion gives a high noise. There exists a development time in between these extremes, two minutes in this example, wherein the midtones  140  have developed to an optimum density that yields the best signal to noise ratio, or image clarity, for that particular exposure value. In this example, the shadows reach optimum clarity at four minutes of development  160 , and the highlights reach optimum clarity at one minute of development  128 . In general, the optimum density will be different for different exposures, as in this example wherein the shadows  160  reveal best clarity at a lower density than the highlights  128 . 
     After the final capture of the image on the film at four minutes, electronic film development has captured optimum images for shadows, midtones, and highlights albeit at different development times. These optimum images must be combined to form a single image with clarity throughout approximating the original scene as seen by the wide dynamic range of the human eye. The process of combining these different parts of the image is called stitching. The prior art conceived this in the classic sense of merging multiple films in a darkroom by cutting out the shadows, the midtones, and the highlights, lightening and darkening each so the boundaries between regions aligned, then stitching these multiple images together into one. 
     The advantage of electronic film development is now more easily understood. In conventional development the film must be stopped and fixed at a selected development time, such as the two minute development time of this example. The detail of the highlights revealed at one minute is lost in total darkness as conventional development proceeds. Likewise, the detail that might have been revealed at four minutes never had the chance to be born in conventional development. Electronic film development turns conventional film into a “universal” film that can be used at a wide range of exposure indexes, including very high exposure indexes not currently practical. 
     In FIG. 1, the section of the density curve around the optimally developed shadows  160  is copied as segment  170 . Next, the density curve around the optimally developed midtones  140  is raised on a base value, or pedestal  172 , and copied next to curve  170  as curve  174 . The height of the pedestal  172  is adjusted so the two curves  170  and  174  align. Similarly, the curve around the optimally developed highlights  128  is adjusted and raised on pedestal  176  to produce curve  178 . The process works in theory, but in practice, development nonuniformities across the image and other spatially dependent nonlinearities made the curves difficult to match across an entire image so that the stitched image usually displayed contours at the edges of stitching regions. Obviously, an improved method of stitching was needed to realize the full benefits of electronic film development. 
     Often in electronic film development there are more than three exposures made of the film. For example, an area array camera may view the film continuously, generating hundreds of exposures. In the prior art these needed to be combined into a limited number of images to conserve memory during the capture process. For example, in FIG. 2 the exposures made at one-half and one minute, exposures  202  and  204 , respectively, could be added with function block  206  to produce a single short development image  208 . Similarly, various exposures at other development times could be added to yield a middle development image  210  and a late development image  212 . These images would then be aligned, cut, and pasted together at function block  220  to yield the finished image  222 . A problem is immediately seen if the times of capture vary, making it necessary to adjust the densities in each of the intermediate images by known time deviations. In the past the adjustments were based on estimations of development speed, and were not found to be reliable. In addition, there were difficulties if some of the capture times were missing entirely because, perhaps, a non real time operating system did not release computer resources exactly when needed. 
     Electronic film development held the promise of higher speed universal film that would work in conventional cameras. This higher speed and wider range film would enable families to record their lives beautifully in the natural light of real life, without typical problems caused by contrast light or dependence on a cold and harsh flash. However, the prior art implementations of electronic film development were plagued with problems in stitching the multiple exposure images together. Obviously, an improved stitching method is an important advance to the art. 
     OBJECTS OF THE INVENTION 
     The primary object of the invention is to merge images of differing densities into a single image which is free from the artifacts encountered in the prior art. 
     A related object is to merge images of differing densities free of edge contouring. 
     A further object is to merge images of differing densities with reduced effect from nonimage noise. 
     A further object is to merge images of differing densities while compensating for a shift in a density-affecting parameter, such as time. 
     Another object is to recover missed images in a series of images of differing densities that are to be merged. 
     SUMMARY OF THE INVENTION 
     In the present invention, a series of images are captured electronically from a developing film, each tagged with the time of capture. For each pixel of each image at each time, regression parameters are calculated, such as density times time, or density times time squared. These parameters for each time are summed into parameter accumulating arrays. As a refinement, the parameters can be weighted prior to summing by a factor sensitive to the reliability of each sample. Following film development, the regression statistics are not necessarily viewable images, rather they describe in abstract mathematical terms smooth continuous lines for each pixel that pass through the actual sampled densities for each pixel. These mathematically described smooth lines allow the development to be recreated mathematically in order to find the nonquantized time at which the density of each pixel is predicted to have attained its optimum density. A gamma correction function of this nonquantized time for each pixel is then output as the brightness for that pixel. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1 depicts the prior art of electronic film development with stitching. 
     FIG. 2 further portrays prior art stitching. 
     FIG. 3 portrays density versus time of a typical development cycle. 
     FIG. 4 introduces the graphic basis of the invention. 
     FIG. 5 adds to FIG. 4 the effect of measurement noise and timebase shift. 
     FIG. 6 adds to FIG. 5 the effect of lost data. 
     FIG. 7 portrays an unweighted embodiment of the invention schematically. 
     FIG. 8 illustrates a problem with an unweighted embodiment. 
     FIG. 9 portrays weighting proportional to proximity to an optimum density curve. 
     FIG. 10 presents the preferred embodiment as a series of steps. 
     FIG. 11 presents the preferred embodiment schematically. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Sometimes a different way of looking at a problem makes it easier to recognize a new solution. The graphs of FIG. 1 described above in the background section followed the prior art photographic convention of plotting density as a function of exposure for a series of specific development times. FIG. 3 follows a much less common approach of plotting density versus development time for a series of exposures. Other than that, the points  324 ,  326 ,  328 ,  344 ,  340 ,  342 ,  360 ,  364 , and  366  in FIG. 3 intersect exactly the same triplet of density, development time, and exposure as their similarly numbered counterparts  124 ,  126 ,  128 ,  144 ,  140 ,  142 ,  160 ,  164 , and  166  of FIG.  1 . 
     The optimum density point for highlights  328 , for midtones  340 , and for shadows  360  lie on a locus of points called the optimum density curve  370 , shown by a dotted line in FIG.  3 . This curve is found empirically by measuring signal to noise ratio for varying exposures, and finding the density at which each reveals detail with the optimum clarity. 
     An image can be thought of as consisting of an array of points, or pixels, each of which receive a specific exposure. The prior art of FIG. 1 thought of this exposure as producing a specific density. Further, each development time produced images of differing density that could be cut, aligned in density, and merged together. FIG. 4 suggests thinking of the image as consisting of an array of pixels, each having received a specific exposure resulting in a specific development curve, such as the highlight curve  402 . Each specific development curve can be quantified by the time at which the density of the development curve crosses the optimum density curve  404 . In this case a specific highlight pixel produces curve  402 , which crosses curve  404  at point  406 , which can be tagged as a one minute pixel. 
     Continuing with FIG. 4, assume a pixel in a middle shadow developed to densities at one, two, and four minutes shown by solid dots  410 ,  412 , and  414 . Although none of these densities falls directly on the optimum density curve  404 , this way of seeing the problem makes it clear that a best fit line  418  could be drawn through the known points  410  to  414  to predict that the pixel crossed the optimum density curve  404  at point  416 , so that pixel can be tagged as a 2.8 minute pixel. This method requires no cutting and aligning of multiple images; rather the placement of the best fit line provides a continuum between areas of differing exposure, completely eliminating edge contouring and thereby solving a significant problem in the prior art. 
     Further, assume that there was a time base error in sampling the data. In the prior art of FIG. 2, an attempt to correct the error would have involved estimating what effect that error had on the early, middle, and late images. In the case of FIG. 4, however, the error would appear as a shift in the sample times, for example 1.5, 2.5, and 4.5 minutes shown by the x&#39;s  420 ,  422 , and  424 . It may be noted that the shifted sample times would still lie along line  418 , and therefore the best fit line to points  420 ,  422  and  424  and the best fit line to  410 ,  412  and  414  is the same line  418  which still crosses the optimum density curve at the same point  416  at 2.8 minutes. In fact, individual sample points could be added, deleted, or moved in time with minimal effect on the best fit curve or the estimate of the crossover time, thereby solving another significant problem in the prior art. 
     FIG. 5 illustrates a more typical case wherein the electronic camera adds noise to the captured images. In this case, each capture will not only lie along a line that is a function of film exposure and grain, but in addition each capture adds a random noise deviation from the line. The individual density samples for a particular pixel are illustrated in FIG. 5 as solid dots, such as dot  502  measured at a time of 2.3 minutes. The density of sample  502  differs from a theorized true curve  504  because of noise in the electronic camera arising from, typically, statistical errors in counting photons, called shot noise. Combined with several other samples, however, illustrated in FIG. 5 as the multiple solid dots, a best fit curve  504  can be estimated, and the time this best fit curve  504  crosses the optimum density curve  506  calculated, as before. 
     A particular question arises from FIG. 5 as to how to specify a best fit curve, and the answer provides another distinction in the practice of the invention over the prior art. Well known statistic practices give an array of choices. One option is to gather regression parameters on the points, and then calculate a linear best fit from the sum of densities and the sum of densities times time. By including the sum of densities times time squared as one of the parameters, a quadratic regression can be used to yield a quadratic function for the best fit curve. Best fit regression analysis of such parameters is well known in the art. For example, a linear best fit is a curve of the form density=A+Bt, where A and B are found such that the sum of the square of the distances of each sampled point from the line is minimized. 
     Now one of the distinctions in the practice of the invention over the prior art can be stated. Unlike the prior art which sought to gather actual images from the film at specific zones of development time and later merge those real images, the present invention seeks to gather more abstract regression parameters about the developing image and later use those parameters to recreate a real image for any conceivable development time. The mathematical expression of the image as a function of a time continuum then allows a seamless stitching. 
     A particularly interesting best fit function is the locus of lines representing the developed density versus time as a function of exposure level for similar film, such as curves  510 ,  512 ,  514  and  516 . The best fit curve could be described mathematically as density=F(t,tc), where t is any development time at which density is to be solved, and tc specifies the shape of the curve to cross the optimum density curve  506  when t=tc. The curve with the best fit to the sampled data, curve  504  in this example, is selected, and the crossover time read directly as the variable. 
     The best fit curve could also be described mathematically as F(t,e), where e is the exposure level yielding a particular curve. This locus of lines can be derived by actually developing a series of test films given known exposures, and storing the actual measured densities of each as a function of development time t in a lookup table where one axis of the lookup table is the known exposure, a second axis is the time since developer induction that a specific measurement is made, and the value stored in the lookup table is the empirically measured density. To distinguish such a curve repertoire from a repertoire of mathematically simple curves, such as a series of lines in linear regression or curves described by a quadratic formula in a quadratic regression, such a regression will be called an empirical curve regression after the locus of curves derived from empirically measuring film during development. Any specific curve can be read from the lookup table by selecting a particular exposure value as one axis of the lookup table, and then varying time, the other axis, while reading out density. The value of exposure yielding the curve with the best fit is the best fit curve. 
     The parameters gathered to specify one of the repertoire of available curves in the lookup table can be derived by summing the density of each sample point times functions of time. A first parameter is the product of a first function of time, a second parameter is a second function of time, and so forth. For example, in a quadratic regression, one of the functions of time is time squared. These functions of time are derivable from the empirical curves. The first function can be related to an average of the curves. The second function can be the primary mode by which the curves differ from this average, called the residue after the first function has been subtracted from each curve. The third function can be the main remaining residue after removing the components of the first and second functions, and so forth to whatever order is desired. 
     Another interesting parameter set is based on a gaussian function of time, where the gaussian function is taken to be a function of time that rises and falls smoothly in a bell shape to select a particular period of time. A series of parameters based on such overlapping gaussian functions would specify the smooth shape of a curve. The use of such curves may be found in the art, especially in spatial transforms where the human retinal neural system is found to respond to gaussian and difference of gaussian (DOG) functions. 
     Turning now to the next illustration, FIG. 6 illustrates the case of a real time operating system that terminated capture prior to reading all the data. Even though data, as represented by solid dots, was not received up to the optimum density time, nevertheless the best fit method allows a curve to be found and projected through the optimum density curve. Although this would not produce the most grain free image, it would produce an acceptable image under conditions in which the prior art would have struggled because there were no middle and late exposures to stitch. This ability to recover from a failure is again a significant advance over the known art. 
     Now that the basis of the invention is understood, a specific embodiment using quadratic regression is presented schematically in FIG.  7 . FIG. 7 inputs the same sequence of seven scanned images  702  and requires the intermediate storage of three accumulating arrays  704 ,  706 , and  708 , as shown in the prior art example of FIG.  2 . The distinction over the prior art is that in FIG. 7, the accumulating arrays sum regression statistics rather than images. This is emphasized by portraying the three accumulating arrays  704 ,  706  and  708  with crosshatching to indicate they are not meant necessarily as viewable images, but rather as statistical data. 
     In FIG. 7 a series of images  702  is received sequentially from an electronic camera viewing the developing film. For example, image  720  is received at two minutes of development. The density of each pixel of image  720  is summed with the density of corresponding pixels of images taken at other development times, such as image  722  at five minutes. This summation occurs in function block  724  which can either operate on all images together if they have all been accumulated and stored in memory during development, or one by one as they are captured from the electronic camera at the corresponding development times. The advantage of summing them as they are captured is that less memory is required. The resulting sum from function block  724  is stored in the accumulating array  704 . 
     Continuing the process of the present invention, the density of each pixel of image  720  is multiplied by time, two minutes in this example, and the product summed with the density times time of corresponding pixels of images representing other development times, such as image  722 , for which t=5 minutes. This summation occurs in function block  730 , and the summation from function block  730  is stored in the accumulating array  706 . Finally, the products of density times time are multiplied by time again to yield density times time squared, and the summation derived at function block  732  is stored in the accumulating array  708 . In the alternative, the process could continue to a cube or higher orders of time to support a cubic or higher order regression, or could end before the square term to support a linear regression. 
     After the last image is captured and its statistics summed to the accumulating arrays, then a best fit curve is derived for each pixel by retrieving the corresponding statistical data for that pixel from the accumulating arrays  704 ,  706  and  708 . The time of intersection of the best fit curve for each pixel with the optimum density curve is calculated in function block  740 , and a function of this time stored for the corresponding pixel in the finished image array  742 . The function stored in the final image array  742  can be the time directly; or it can be the exposure known empirically to develop to the particular time, found empirically and expressed in the computer as a function, such as through a lookup table, of the time; or it can be any other function found to have utility, such as the square root of the linear exposure value normalized and stretched to fit white level and black level, yielding a conventional 8 bit computer image representation. 
     The direct parametric embodiment just presented does have limitations that are now highlighted with reference to FIG.  8 . In this figure, a specific highlight exposure curve  802  is plotted as density versus development time. A series of noisy samples are represented by solid dots such as dot  804  representing a noisy measurement of density at one minute, and dot  806 , representing a noisy measurement of density at five minutes. The goal is to find the crossover time  808 , one minute in this example, where curve  802  crosses the optimum density curve  810 , with the added constraint that this point  808  be found without exactly knowing curve  802 , only the noisy measurements such as  804  and  806 . 
     An immediate problem which is evident is that a linear regression curve, such as straight line  812 , will be significantly affected by samples distant from the optimum density time, such as sample  806 , and as a result the best fit linear curve  812  will intersect the optimum density curve  810  at an incorrect time  814 . If this merely added a bias error, it would not be a significant problem; however, changes in the outlying samples, such as  806 , can be seen to change the crossover time  814 , and because outlying samples such as  806  bear almost no real information, this change is an unwanted noise. It should be noted that higher order regressions, such as the quadratic of FIG. 7, would put less emphasis on outlying points; however, higher order regressions require more accumulating arrays than a linear regression and correspondingly more memory and computing power, and the reduction in emphasis of outlying points is not precise or complete. 
     A solution to this problem is presented in FIG.  9 . Basically, the statistics for each sample are throttled by a weighting factor, In this example, sample  904  (equivalent to sample  804  of FIG. 8) is given a high weight, while sample  906  (equivalent to sample  806  of FIG. 8) is given a very low weight. In this specific example, weighting factors are assigned such that a linear regression yields straight line  912 , which is much closer to the true curve  902  in the region around the crossover time  922 , at the expense of deviating at outlying times, such as six minutes, for which accuracy is irrelevant. Thus, one object of this invention to reduce noise is accomplished via application of a weighting factor. 
     The weighting function should generally follow the reliability, or signal to noise ratio, of each sample. The crest of the weighting function should therefore follow the optimum density curve, shown in FIG. 9 as the 100% weighting curve  930 , and fall off in proportion to distance from this curve. To illustrate the application of this weighting process, sample points such as  904  that are close to the optimum density curve are shown large, while sample points such as  906  that are distant are drawn faintly. In this specific example, only the 100% and 50% weighting curves are shown, although normally, the falloff would be a continuous function of distance. 
     The rate of falloff with distance follows the rate of falloff of overall signal to noise ratio including both film grain and noise in the electronic capture system, and is found empirically. If the capture system is very noisy, the falloff is slower so more points are averaged. If the capture system is noise free or the film has a narrow range of optimum densities, then the weighting falloff is rapid. In practice there is a wide tolerance, and a curve such as shown in FIG. 9 will be very close to optimum for a range of films. 
     FIG. 9 further portrays the true curve  940  for a shadow exposure with the actual sample points from which the true curve is to be mathematically estimated. Again, sample points such as  942  that are proximate to the optimum density curve  930  are portrayed large to indicate a high weighting, while outlying sample points, such as  944 , are drawn faintly. Again it may be visualized that a best fit linear line through the large points around curve  940  will closely approximate curve  940  around five minutes at which time the curve crosses the optimum density point. Curve  940  is included to point out that a weighting factor proportional to distance from the optimum density curve will tend to emphasize samples at short development times with highlight exposures, and samples with long development times for shadow exposures. 
     Finally, it should be noted that the weighting factor should not drop to zero at very low densities or short times, but should maintain some finite weight. This insures that even if the system fails to capture samples at later times, there will be some regression data in the accumulating arrays from which a reasonable guess at the best fit curve can be extracted. Therefore, sample point  944  would be given a small but non-zero weight. The weight can, however, go to zero for high densities and long times. 
     FIG. 10 portrays the embodiment just described as a series of steps such as would be implemented in a computer program. The specific example given is one of linear regression. 
     Initially, there are available to the process three arrays to hold the regression statistics. All the elements of these regression arrays are initially set to zero. Three functions are also available: a first function receives a density and time as parameters and returns a weight factor, a second function receives time as a variable and returns an optimum density, and a third function receives a crossover time as a parameter and returns a brightness value artistically representative of the light the crossover time represents, which is typically the square root of linear brightness, called a gamma correction. 
     During development, an electronic camera views the film. The image from the camera is digitized into pixels. As the process continues, each point from the film is sensed at several times. The process receives these pixels one at a time, receiving for each pixel four numbers: a density, x and y coordinates of the pixel, and a time at which the specific density was measured. 
     Based on these four numbers for each pixel, regression parameters are calculated and collected. Specifically, regression parameters are calculated based on density and time, and array elements in the regression summation arrays pointed to by x and y are incremented by the regression parameters. 
     After the last pixel is received for the last time, the summation arrays are complete, and the data they hold can be used. For each x, y location, regression data is read from the arrays and used to calculate a linear best fit curve. The time at which this best fit curve crosses the optimum density curve is calculated, or read from a lookup table, or solved by iteration. Finally, the equivalent x,y element of a final image is set according to a gamma function of this crossover time. 
     FIG. 11 covers the procedure of FIG. 10 schematically. A series of images are received from an electronic camera viewing a film at specific times since a development has been induced. For example, image  1102  is received for two minutes of development. A function such as  1104  labeled “weight” receives the time and also receives the numeric density of each pixel in the image, returning for each pixel a first value that sums to the corresponding pixel in a “summation weight” array  1106  via conduit  1108  terminating in  1110 . This first value is also sent to multiplication block  1112  along with the numeric density of each pixel to produce an output product “summation density” for each pixel. “Summation density” sums to the corresponding pixel in “summation density” array  1114  via conduit  1116  and  1118 . Another multiplication block  1120  also receives this “summation density” for each pixel, further receives the corresponding time, and outputs a product called “summation time” for each pixel that is the product of weight, density, and time. “Summation time” sums to the corresponding pixel in the “summation time” array  1122  via conduit  1124  and  1126 . The process is repeated for each new image, such as image  1128 , that is received by the electronic camera captured at a different development time. 
     Following development, the three parametric arrays  1106 ,  1114 , and  1122  are used to calculate a brightness value for each pixel. The three values received for each pixel from the three arrays can be used to regress to a linear equation of the form density=A+Bt, and t allowed to vary iteratively to solve for the time at which a function optimum_density(t)=A+Bt. A faster method divides the two values from the summation weight array and summation time array by the corresponding value from the summation density array, and the resulting two numerical values are used as a pointer into a two dimensional lookup table that holds the precalculated time, or a gamma function of the time. Finally, the value so found is placed into the corresponding pixel of the finished image  1128 , which now, with the full brightness perception range of the human eye, is seen to be Shirley and little Albert frolicking in the bluebonnets. 
     While the present invention has been described herein with reference to particular embodiments thereof, a latitude of modification, various changes and substitutions are intended in the foregoing disclosure, and it will be appreciated that in some instances some features of the invention will be employed without a corresponding use of other features without departing from the spirit and scope of the invention