Abstract:
A method of determining presence of variations in interline spacing in a first image comprising a first plurality of parallel lines of pixels comprising: providing a second image comprising a second plurality of parallel lines; orienting the images so that the lines in the first and second pluralities are superimposed and angled with respect to each other to generate an interference image comprising a Moiré interference pattern; and using the Moiré interference pattern to determine presence of said variations.

Description:
FIELD OF THE INVENTION  
       [0001]     The invention relates to detection of banding in images printed by digital printers and in quantifying characteristics of detected banding.  
       BACKGROUND OF THE INVENTION  
       [0002]     To print an image on a substrate, a typical digital printer first forms an electrostatic copy of the image, conventionally referred to as a “latent image”, on a photosensitive surface, for example on a cylindrical roller, hereinafter referred to as a “photosensitive imaging cylinder” (PIC). First a charger deposits a substantially uniform charge density on the photosensitive surface. The latent image is then formed by discharging regions of the charged photosensitive surface to generate a pattern of charged and uncharged pixels on the photosensitive surface that replicates the image. A developer applies ink or toner of desired color to the charged or uncharged regions using an electrostatographic process.  
         [0003]     The toner on the PIC is then transferred from the PIC to a final substrate, optionally via an “intermediate transfer member” (ITM) to print the image. In single color printing, such as in black and white printing, the latent image is a copy (or inverse) of the image to be printed. In printing a multicolor image, such as in CMYK printing, the latent image is a copy (or inverse) of a color separation of a plurality of color separations required to print the image in color. A different color toner is transferred to the substrate for each of the plurality of color separations to print the image.  
         [0004]     Discharging regions of the PIC&#39;s photosensitive surface to generate the latent image is generally accomplished by illuminating the regions with a beam (or multiple beams) of light from a laser that is focused to a point on the photosensitive surface. The beam is controlled so that its focal point repeatedly scans the photosensitive surface along a line parallel to the axis of the PIC as it rotates rapidly about the axis. As the beam scans a line of the photosensitive surface, it is turned on to illuminate regions of the surface along the scan line that are to be discharged and turned off so as to not illuminate regions along the scan line that are not to be discharged. The latent image is built up line by line on the photosensitive surface as the PIC turns.  
         [0005]     To provide a latent image having accurately controlled pixel densities and consequently a printed image for which hue saturation and brightness of printed regions are accurately controlled, rotation speed of the PIC and time intervals between line scans during scanning should be substantially constant. If the rotation speed of the PIC changes, or the time interval between the onset of scans by the laser changes, spacing between scan lines will vary. As a result, the latent image will evidence bands of pixels that are parallel to the scan direction for which the pixel densities will be greater than or less than desired. When toner is applied to the PIC, the bands having greater pixel density will acquire too much toner while bands having lesser pixel density will acquire too little toner. An image printed on a substrate from the latent image will, as a result, have bands of too little or too much toner, i.e. bands of unwanted variations in optical density, that are perpendicular to the process direction of the image and quality of the printed image will be compromised. (The process direction is a direction along which the image moves during image formation or transfer.)  
         [0006]     The term “banding” is generally used to refer to bands of undesired variations in the optical density of an image. For the situation described above, these variations are substantially perpendicular to the process direction and parallel to the scan direction. Banding or other undesired variations of density in an image along the scan direction, i.e. perpendicular to the process direction, may also occur, due to variations in scanning speed. It is relatively difficult to control all the variables that can cause banding in an image printed by digital printers and digital printer images may exhibit banding of varying degrees of severity in one or both directions.  
       SUMMARY OF THE INVENTION  
       [0007]     An aspect of some embodiments of the present invention relates to providing a method of detecting banding in images printed by a digital printer.  
         [0008]     An aspect of some embodiments of the invention relates to providing a method of quantifying severity and/or other characteristics of banding in the images.  
         [0009]     In accordance with an embodiment of the invention, a reference image comprising a pattern of parallel lines is generated. Optionally, any two adjacent lines in the pattern are equally spaced. In some embodiments of the invention, the reference image comprises identical groups of equally spaced lines, wherein any two adjacent groups of lines are equally spaced. A copy of an image, hereinafter a “test image”, comprising a pattern of parallel lines optionally substantially identical to that in the reference image is printed by the digital printer with the lines substantially perpendicular to the process direction.  
         [0010]     The reference image and the printed copy of the test image are superposed with their lines angled with respect to each other to generate an image, hereinafter an “interference image”. The interference image comprises a Moiré interference pattern generated by interference between the reference image and the copy of the test image. If there is substantially no banding in the “copy image”, the interference image exhibits an interference pattern of relatively straight Moiré interference bands. If on the other hand, banding flaws the copy image, the interference image will exhibit irregularities that perturb the straight interference bands and morph them into, generally, “zigzag” interference bands.  
         [0011]     In accordance with an embodiment of the invention, the irregularities are used as indicators of the presence and severity of banding in images printed by the printer.  
         [0012]     In accordance with an embodiment of the invention, the irregularities are used to quantify characteristics of the banding.  
         [0013]     Optionally, the irregularities are used to determine a spatial period of the banding. Optionally, the irregularities are used to determine variations in spacing between the pixel lines in the image that give rise to the banding.  
         [0014]     In some embodiments of the invention, the reference image is a carefully prepared transparency. The copy of the test image is printed by the printer and the reference image transparency is superposed on the copy image to form the interference pattern.  
         [0015]     In some embodiments of the invention the reference image is an image preprinted on a substrate and the interference image is generated by printing a copy of the test image on the substrate over the reference image.  
         [0016]     In some embodiments of the invention, a “test” PIC having an accurately configured permanent latent image of the reference image formed on its photosensitive surface is installed in the printer and the printer prints the reference image on a substrate using the test PIC. Optionally, a latent image of the test image is generated on the test PIC to superpose the latent test image with the latent reference image on the PIC photosensitive surface. The lines of the latent reference image and the latent test image are angled with respect to each other so that the superposed latent images generate a latent interference image exhibiting a Moiré pattern. The interference image is printed from the latent interference image.  
         [0017]     Whereas to test for banding along the process direction (i.e. bands that are substantially perpendicular to the process direction and substantially parallel to the scan direction), lines in a test image and a reference image are substantially perpendicular to the process direction, in some embodiments of the invention, reference and test images are used to test for bands of undesired variations in directions other than the process direction. For example, to test for variations along the scan direction, i.e. scanning “non-linearity”, a reference image and test image having lines substantially perpendicular to the scan direction and parallel to the process direction are used.  
         [0018]     In accordance with some embodiments of the invention a reference image comprises a grid formed from a first set of parallel lines crossed by a second set of parallel lines. Optionally, the lines in the first and second sets of parallel lines are substantially perpendicular to each other. Optionally, the lines in the first set of parallel lines are substantially parallel to the scan direction. Such a grid reference image and a corresponding grid test image may be used to simultaneously detect undesired “banding” or other variations along the process direction and scan directions (scanning “non-linearity). However, interference images generated by superimposing a grid reference image and a copy of a grid test image are generally more complicated and difficult to interpret than interference images generated by a simpler “one dimensional” reference image and corresponding copy of a test image.  
         [0019]     There is thus provided, in accordance with an embodiment of the invention, a method of determining presence of variations in interline spacing in a first image comprising a first plurality of parallel lines of pixels comprising:  
         [0020]     providing a second image comprising a second plurality of parallel lines;  
         [0021]     orienting the images so that the lines in the first and second pluralities are superimposed and angled with respect to each other to generate an interference image comprising a Moiré interference pattern; and  
         [0022]     using the Moiré interference pattern to determine presence of said variations.  
         [0023]     In an embodiment of the invention the variation is characterized by at least one group of lines having a plurality of consecutive lines for which interline spacing is different from interline spacing of lines outside the at least one group. Optionally, the interline spacing between lines in the at least one group is substantially the same for any pair of adjacent lines in the at least one group. Optionally, the interline spacing for lines outside the at least one group is substantially the same for any pair of adjacent lines outside the at least one group.  
         [0024]     Optionally, the at least one group comprises a plurality of groups, optionally, periodic groups.  
         [0025]     Optionally, the method includes using the interference pattern to determine a period for the groups of lines.  
         [0026]     Optionally, the method includes using the interference pattern to determine an amount by which interline spacing of lines in the group differs from interline spacing between lines outside of the at least one group.  
         [0027]     Optionally, the Moiré pattern comprises a pattern of interleaved relatively light and relatively dark interference bands. In an embodiment of the invention, interference bands comprise relatively straight segments that are angled with respect to each other and using the interference pattern to determine a difference in interline spacing comprises determining an angle between the segments of an interference band. Optionally, using the angle between the segments comprises determining a ratio R of the interline spacing between lines in the at least one group of lines relative to lines outside of the at least one group in accordance with an expression of the form R=cos(θ+α/2)/cos(θ−α/2), where θ is the determined angle between segments and α is the angle between lines in the first and second images.  
         [0028]     In an embodiment of the invention, the light and dark interference bands cross the lines in the first and second images, wherein a location in the Moiré pattern is defined relative to an x-axis and a y-axis respectively parallel and perpendicular to the lines in the first image and each interference band defines a contour line that lies along a central spine of the interference band and a direction line that lies along a general direction of the contour line, and comprising estimating the interline spacing of lines in the first image at a given y-coordinate in accordance with a formula, (s 2 /L)(dΔx/dy), where s is an interline spacing in the second image, L is a distance between adjacent bright or adjacent dark interference bands at the given y-coordinate and Δx is a distance between the contour line and direction line at the given y-coordinate.  
         [0029]     Optionally, the first images comprises a third plurality of parallel lines that cross lines in the first plurality of parallel lines.  
         [0030]     Optionally, the lines in the second image comprise a fourth plurality of lines that cross the second plurality of lines. Optionally, orienting the images comprises orienting so that an angle between the lines in the first and second pluralities is substantially smaller than an angle between the lines in the first and fourth pluralities.  
         [0031]     In an embodiment of the invention, the lines in the first and third pluralities of parallel lines are substantially perpendicular. Optionally, the lines in the second and fourth pluralities are substantially perpendicular.  
         [0032]     In an embodiment of the invention, the method includes using the interference pattern to determine presence of variations in interline spacing between lines in the third plurality of lines.  
         [0033]     In an embodiment of the invention, the first image is printed on a first substrate by a printer having a process direction. Optionally, the lines in the first plurality of lines in the first image are substantially perpendicular to the process direction. Optionally, the second image is printed by the printer on a second substrate. Optionally, the first and second substrates are overlaid to generate the interference image. Optionally, the second image is printed by the printer on the first substrate. Optionally, the second image is printed simultaneously with the first image.  
         [0034]     In an embodiment of the invention, when the printer is a digital printer comprising a laser and a photosensitive imaging cylinder (PIC) having a photosensitive surface and wherein the laser scans the photosensitive surface along a scan direction to generate a latent image of an image that is printed by the printer. Optionally, the scan direction is substantially perpendicular to the process direction. Optionally, a permanent latent image of the second image is formed on the photosensitive surface. 
     
    
     BRIEF DESCRIPTION OF FIGURES  
       [0035]     Non-limiting examples of embodiments of the present invention are described below with reference to figures attached hereto, which are listed following this paragraph. In the figures, identical structures, elements or parts that appear in more than one figure are generally labeled with a same numeral in all the figures in which they appear. Dimensions of components and features shown in the figures are chosen for convenience and clarity of presentation and are not necessarily shown to scale.  
         [0036]      FIG. 1A  schematically shows a digital printing press printing an image substantially free of banding;  
         [0037]      FIG. 1B  shows the printing press shown in  FIG. 1A  printing an image flawed by banding;  
         [0038]      FIG. 2  shows a schematic reference image, in accordance with an embodiment of the present invention;  
         [0039]      FIG. 3  shows a schematic test image, in accordance with an embodiment of the invention;  
         [0040]      FIG. 4  shows an interference image generated between the reference image shown in  FIG. 2  and a “true” copy of the test image shown in  FIG. 3  that does not have banding for testing for the presence of banding in the copy image, in accordance with an embodiment of the invention;  
         [0041]      FIG. 5  shows a copy of the test image shown in  FIG. 3 , which is flawed by banding;  
         [0042]      FIG. 6  shows an interference image generated between the reference image shown in  FIG. 2  and the copy test image shown in  FIG. 5  for testing for the presence of banding in the copy image, in accordance with an embodiment of the invention;  
         [0043]      FIG. 7  shows an interference image generated between the reference image shown in  FIG. 2  and a copy of the test image shown in  FIG. 3 , in which banding is different from that in the copy image shown in  FIG. 5 , in accordance with an embodiment of the invention; and  
         [0044]      FIGS. 8A-8D  show a reference image comprising a grid of lines that cross each other at right angles and interference images generated by superposing the reference image and copies of test images characterized by different degrees of banding and/or scan direction defects. 
     
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS  
       [0045]      FIG. 1A  shows a schematic digital printer  20  that prints images substantially free of banding. Printer  20  is shown printing an image  22  on substrates, for example on sheets  24  of paper. For simplicity of presentation, image  22  is assumed to be a monotone solid color image printed in one color and having a constant brightness. In  FIG. 1A  image  22 , is shown formed by solid lines  60  that represent lines of closely spaced printed pixels of the color. Spacing between lines  60 , as noted below, is greatly exaggerated for convenience of presentation and would not normally be seen by the naked eye.  
         [0046]     Printer  20  optionally comprises a photosensitive imaging cylinder (PIC)  26  having a photosensitive surface  28  and axis  30 , an intermediate transfer member (ITM)  32  and an impression roller  34 . A conveyor  36  feeds unprinted sheets  24  to printer  20  in a process direction indicated by block arrow  38  and sheets printed by the printer are optionally collected at an output tray or station  40  or alternatively first printed on the opposite side of the sheet and then collected at the output tray or position. Arrows  41  indicate directions in which PIC, ITM and impression roller  34  rotate during printing of image  22 . Only elements and features of digital printer  20  that are germane to the discussion are shown in  FIG. 1A .  
         [0047]     In the printing process, as PIC rotates, a charger  42  charges photosensitive surface  28  so that it has a substantially uniform surface charge density. A laser unit  44  comprising a laser and associated optics focuses a laser beam  46  (or a plurality of laser beams) onto photosensitive surface  28  and directs the laser beam to repeatedly scan the charged photosensitive surface along a line (or a plurality of parallel lines) substantially parallel to axis  30  of the PIC. The scan direction of laser beam  46  is indicated by a block arrow  48 . During a scan of photosensitive surface  28 , as laser beam  46  moves along the scan line, laser unit  44  turns on laser beam  46  at pixels along the scan line that are to be discharged and turns off the laser beam at pixels along the scan line that are to remain charged. Thus while the image is shown as being made up of lines  60  for clarity of presentation, the image is actually made up of lines of pixels.  
         [0048]     As result of the rotation of PIC  26  and the scanning motion of laser beam  46 , a plurality of lines  50  are sequentially scanned on photosensitive surface  28  and pixels along each of the lines are selectively discharged or left charged as required to generate a latent image  23  of image  22  on photosensitive surface  28 . Regions of a scanned line  50  in latent image  23  comprising discharged pixels are represented by dashed segments of the line. It is noted that while latent image  23  is shown as being visible, it is actually an invisible “image” of charges.  
         [0049]     Regions of a scanned line  50  comprising pixels that are left charged are represented by solid segments of the line. PIC  26  rotates with a substantially constant speed of rotation such that for substantially all pairs of adjacent scanned lines  50  on photosensitive surface  28 , the scanned lines are spaced apart a same distance. In  FIG. 1A  and figures that follow, spacing between scanned lines  50  is greatly exaggerated for convenience of presentation. It should be noted that while  FIG. 1A  shows a printer in which one line is scanned at a time, a plurality of lines may be simultaneously scanned.  
         [0050]     A toner of suitable color is applied to latent image  23  as the latent image passes beneath a developer  52 . The toner is transferred from the latent image to ITM  32  and from the ITM to a sheet of paper  24  fed to printer  20  by conveyor  36  as the sheet passes through a nip  54  between ITM  32  and impression roller  34 . Because scanned lines  50  on photosensitive surface  28  are equally spaced one from the other, corresponding lines  60  of pixels in printed image  22  are also equally spaced apart. Pixel density in the printed image  22  is therefore substantially uniform and the printed image does not exhibit banding.  
         [0051]      FIG. 1B  again shows printer  20  shown in  FIG. 1A  printing image  22 . However, in  FIG. 1B  as a result, for example of wear in bearings (not shown) that support PIC  26  or of vibration in printer  20 , PIC  26  exhibits a variation in its rotation speed and the printer no longer prints band free images. As a result of the variation in rotation speed, scanned lines  50  on photosensitive surface  28  are no longer equally spaced one from the other and printed image  22  exhibits bands of undesired shading. By way of example, in  FIG. 1B  the variation in rotation speed of PIC  26  is assumed to be an intermittent, recurring increase in the rotation speed. As a result, spacing between scanned lines  50  in latent image  23  is no longer uniform but periodically increases. Printed image  22  therefore exhibits bands  61  of pixel lines  60  for which interline spacing is increased and pixel and color density therefore decreased.  
         [0052]     In  FIG. 1B , variation in spacing of scanned lines  50  and consequent banding in printed image  22  are greatly exaggerated for clarity of presentation and is indicated by a recurrent absence of a scan line  50  and a corresponding image line  60 . In general, variation in spacing of scanned lines  50  is much more moderate and banding is much subtler. However, even subtle banding can affect perceived quality of a printed image and reduce the perceived quality of a high quality printed image.  
         [0053]     In accordance with an embodiment of the invention, to detect and optionally quantify characteristics of banding in an image printed by a printer, a copy of a test image printed by the printer is superposed with a reference image to generate an interference image exhibiting a Moiré interference pattern. The reference image comprises a plurality of optionally equally spaced parallel lines. The test image also comprises a plurality of parallel lines. Optionally, the parallel lines in the test image are equally spaced one from the other by a same interline spacing that separates the parallel lines of the reference image. The reference image and the copy of the test image are superposed with the lines in the images angled with respect to each other at an angle, hereinafter a “Moiré angle”, to generate the Moiré interference pattern. Features of the interference image Moiré pattern are used to determine the presence of banding and optionally to quantify characteristics of the banding.  
         [0054]     In some embodiments of the invention, the reference image is provided on a transparent sheet which is overlaid on the copy of the test image to form the interference pattern.  
         [0055]     In some embodiments of the invention, the reference image is a printed image on a sheet and a copy of the test image is overprinted on the reference image.  
         [0056]     In some embodiments of the invention, the reference image is provided using a test PIC having an accurately configured “latent” reference image permanently formed on its photosurface. The PIC is mounted to the printer and a latent image of the test image is generated over the preformed latent reference image on the test PIC to superpose the latent images and form a latent image of interference image on the PIC. The interference image is printed from its latent image. The latent image on the photoreceptor can be generated for example, by burning the pattern that characterizes the reference image on the photoreceptor using a laser beam of appropriate energy.  
         [0057]      FIG. 2  shows a schematic reference image  70  used to detect and optionally quantify characteristics of banding in images printed by a printer, in accordance with an embodiment of the invention. By way of example, reference image  70  comprises a plurality of lines  74  that are equally spaced from each other and optionally angled with respect to the horizontal, indicated by a horizontal line  76  (the horizontal is perpendicular to the process direction), by a Moiré angle, α=3.5°. For convenience of presentation reference image  70  is shown circumscribed by a circle  72 . (A reference image in accordance with an embodiment of the present invention is of course not limited to any particular shape perimeter.)  
         [0058]      FIG. 3  shows a schematic test image  80  that is printed by a printer to test if the printer generates banding in images that it prints, in accordance with an embodiment of the invention. Test image  80  comprises a plurality of parallel, optionally horizontal lines  82 . Optionally lines  82  in test image  80  are equally spaced. In some embodiments of the invention, spacing between lines  82  in test image  80  is the same as that of lines  74  in reference image  70 . In some embodiments of the invention lines  82  are configured in a pattern identical to that of lines  74  in reference image  70 . In some embodiments of the invention, spacing between lines in the test image is different from that in the reference image.  
         [0059]     By way of example, lines  82  in test image  80  are of equal length, cover a rectangular area and have a line spacing pattern identical to that of reference image  70 . If the printer that prints a copy of test image  80  does not generate banding, spacing between lines in the printed copy will not vary and will be substantially equal to that of the original test image. The printed copy of test image  80  will be a “true copy” of, and substantially identical to, the test image. An interference image generated from the superposition of reference image  70  and the printed true copy of test image  80  will exhibit an interference Moiré pattern characterized by parallel straight alternating bright and dark “interference bands”.  
         [0060]      FIG. 4  shows an interference image  90  generated by superposing reference image  70  and a true copy of test image  80 . The interference image shows a regular pattern of straight bright and dark interference bands  91  and  92  respectively. The direction of an interference band  91  or  92  is tilted slightly from the vertical by an angle equal to half the Moiré angle α. (In  FIGS. 3 and 4  lines  82  of the copy of test image  80  are assumed to be horizontal, and the vertical is a direction perpendicular to lines  82 .)  
         [0061]      FIG. 5  shows a possible, “hypothetical”, example of a copy  100  of test image  80  printed by a printer that generates banding in images that it prints. Copy test image  100  appears identical to test image  80  and comprises a plurality of parallel lines  102  that appear equally spaced. However, copy  100  is flawed by banding and the copy comprises spatially periodic bands of 20 consecutive lines  102  for which spacing between the lines is about 3% greater than spacing between lines  102  in test image  80 . The bands of lines  102  having increased interline spacing are indicated by brackets  104 . Whereas the banding is substantially non-discernable in copy image  100 , such banding may reduce the perceived quality of an image printed by a printer compromised by the banding.  
         [0062]     An increased interline spacing, such as that exhibited by copy image  100 , is generally caused by a periodic increase in the rotation speed of PIC  26  and/or a periodic decrease in the frequency with which laser beam  46  scans the PIC. For example, the 3% increase in interline spacing for lines  102  in brackets  104  may be caused by a 3% increase in rotation speed of PIC  26  or a 3% decrease in scan frequency of laser beam  46 . Alternatively, the increase in interline spacing may be caused by a combination of a change in rotation speed and scan frequency. For example, a 2% increase in rotation speed and a 1% decrease in scan frequency will result in a 3% increase in interline spacing.  
         [0063]      FIG. 6  shows an interference image  110  generated by the superposition of reference image  70  and copy image  100  shown in  FIG. 5 . Whereas the presence of banding in copy  100  is substantially undetectable by the naked eye, the presence of banding in the copy is readily evidenced by the effects of the banding in interference image  110 . The banding has substantially altered the straight bright and dark interference bands in interference image  90  ( FIG. 4 ) for the case for which the copy of test image  80  is free of banding. Instead of the straight interference bands in interference image  90 , the interference pattern exhibited in interference image  110  comprises bright and dark zigzag interference bands  111  and  112  respectively. Each interference band  111  and  112  comprises substantially vertical straight segments  114  that have a direction that is the same as that of interference bands  91  and  92  shown in  FIG. 4  (i.e. angled at (90°+α/2) relative to the horizontal) alternating with and connected by relatively strongly angled (relative to the vertical) straight segments  116 . A contour line  118  is drawn along the central “spine” of one of bright zigzag bands  111  in interference image  110  to aid in visualizing the shape of the band and the angular relationship between its vertical and angled segments  114  and  116 .  
         [0064]     In an embodiment of the invention, features of the interference pattern in interference image  110  are used to quantify characteristics of banding in copy  100  ( FIG. 5 ) of test image  80  ( FIG. 3 ). For example, the width of “line-bands”  104  of “widely spaced” lines  102  in copy image  100  can be determined from the length of angled segments  116 . The spatial period of line-bands  104  is equal to the spatial period along the vertical direction in interference image  110  ( FIG. 6 ) with which angled segments  116  or vertical segments  114  repeat in a given zigzag bright interference band  111 .  
         [0065]     An angle θ between a vertical segment  114  and an angled segment  116  of a bright band  111  may be used to determine by how much spacing between widely spaced lines  102  in line-bands  104  of copy image  100  is greater than the spacing between lines  74  in reference image  70 . If the spacing between lines  74  in reference image  70  is represented by “s” and spacing between widely spaced lines  102  in copy image  100  by “S” a relative size of the interline spacing, S/s, is given by an expression: 
 
 S/s =cos(θ+α/2)/cos(θ−α/2).  1) 
 
         [0066]     In the expression for S/s, for convenience, angles are considered to be positive if “counterclockwise” and negative if “clockwise”. The angle θ is considered negative in  FIG. 6  since line segments  116  are rotated clockwise relative to vertical segments  114  and, by way of example, α is positive since test image  70  is rotated counterclockwise from the horizontal. The ratio S/s provides a measure of variation, in rotation speed of PIC  26  and/or scanning frequency of laser beam  46 . For example, for copy image  100 , the ratio S/s is equal to about 1.03 and indicates that the rotation speed of PIC  26  periodically increases by about 3%, the scan frequency periodically decreases by about 3% or that a sum of the relative changes in rotation speed and scan frequency is equal to about 3%.  
         [0067]     In some embodiments of the invention, a displacement of a zigzag Moiré interference band, such as a bright interference band  111 , from a straight line along a general direction of the interference band is used to estimate an error in interline spacing of lines that generates the zigzag interference band. The straight line is optionally a regression line determined from a contour line of the interference band or a line determined visually that connects recurrent features of the contour line.  
         [0068]     For example, let the y-axis and x-axis respectively of a coordinate system  117  shown in  FIG. 6  indicate the process direction and scan direction respectively of the printer that printed copy image  100 . A location along the process direction of a particular given line  102  labeled  119  in the test image is given by its y-coordinate. A line segment  121  optionally between intersections of vertical and angled segments  114  and  116  along a same side of contour line  118  indicates an average slope of the bright interference band associated with the contour line.  
         [0069]     Assume, that were there no banding errors in copy image  100 , line  119  would have a y-coordinate equal to y o , but because of banding, the actual y-coordinate of the line is displaced by Δy from y o , i.e. Δy=(y−y o ). In accordance with an embodiment of the invention, Δy is estimated from an expression 
 
Δ y =( s/L )Δ x.   2) 
 
 In the expression, s is the spacing noted above between lines in reference image  70 , L is a distance along the scan direction between adjacent bright interference bands  111  (or dark interference bands  112 ), and Δx is a distance along line  119  between contour line  118  and direction line segment  121 . L and Δx are indicated in  FIG. 6 . 
 
         [0070]     If S(y) is the interline spacing between lines  102  at the y coordinate of a line  102 , such as line  119 , then S(y)≈s(1+dΔy/dy)≈s(1+(s/L)dΔx/dy), where dΔy/dy and dΔx/dy are the first derivatives respectively of Δy and Δx with respect to y and dependence of L on y is, optionally, ignored. An error in interline spacing at coordinate y is equal to 
 
( S ( y )− s )≈ dΔy/dy )≈( s/L ) dΔx/dy ).  3) 
 
         [0071]     It is noted that bands of unwanted changes in optical density (OD), are generated by changes in interline spacing of printed lines of pixels. Optical density as a function of y, OD(y), is proportional to 1/S(y) and an unwanted change in optical density ΔOD(y) is proportional to [1/S(y)−1/s] so that ΔOD(y)≈[(s−S(y)]/s 2 ≈(1/sL)dΔx/dy.  
         [0072]      FIG. 7  shows an interference image  120  between reference image  70  and a copy  130  of test image  80  ( FIG. 3 ) characterized by banding in which lines  132  in bands indicated by brackets  134  of copy  130  have interline spacing about 15% greater than that of lines  74  in the reference image. Interference image  120  exhibits a Moiré pattern of bright and dark interference bands  141  and  142  respectively that is easily recognized as different from the Moiré pattern exhibited in interference image  110  ( FIG. 6 ). A contour line  144  traces the spine of one of bright bands  141 .  
         [0073]     Whereas, the spatial frequency of line-bands  134  in copy image  130  is the same as that of line-bands  104  in copy image  100  ( FIGS. 5 and 6 ), the angle θ between angled and vertical segments  151  and  152  of a bright band  141  in interference image  120  is equal to about 66.73°. From expression 1 given above, the relative interline spacing between lines  132  in line-bands  134  of copy image  22  and line spacing in reference image  70  is about 15%.  
         [0074]     It is noted that whereas in the above examples of exemplary embodiments of the invention undesired bands of shading substantially perpendicular to the process direction are detected, a band or bands of undesired shading along substantially any direction in a printed image may be similarly detected. Bands along a given direction may be detected in accordance with an embodiment of the invention using a reference image comprising lines oriented substantially parallel to the given direction and a corresponding test image having lines angled by a Moiré angle with respect to the reference image (or by suitably rotating a transparent reference image overlay). For example, spacing variations in the scan direction cause bands parallel to the process direction, which may be detected in accordance with an embodiment of the invention using a reference image and a corresponding test image having lines substantially parallel to the process direction. Such a measurement can be used to determine variations in scan velocity of laser beam  46  and can be used to control when the laser beam is turned on and off to compensate for such variations.  
         [0075]     In some embodiments of the invention, a reference image comprising a grid of crossed lines is used to simultaneously detect banding in more than one direction. Optionally, the crossed lines comprise a first plurality of equally spaced parallel lines perpendicular to a second plurality of equally spaced parallel lines. Optionally the first plurality of lines is tilted with respect to the horizontal (perpendicular to the process direction) by a “horizontal” Moiré angle and the second plurality of lines is tilted with respect to the vertical (parallel to the process direction) by a “vertical” Moiré angle. Optionally, the horizontal and vertical Moiré angles are equal. A reference image comprising horizontal and vertical crossed lines may be used to simultaneously detect banding along the process direction and along the scan direction.  
         [0076]     By way of example,  FIG. 8A  shows a reference image  200  suitable for simultaneously detecting banding in both scan and process directions. Reference image  200  optionally comprises a first plurality of parallel lines  199  tilted with respect to the horizontal by a Moiré angle equal to about 3.5° and a second plurality of lines  201  tilted with respect to the vertical by Moiré angle also equal to about 3.5°.  
         [0077]      FIG. 8B  shows reference image  200  superimposed with a copy  202  of a test image comprising a grid of horizontal and vertical lines  203  and  204  respectively to generate an interference image  206  in accordance with an embodiment of the invention. Copy  202  of the test image is not degraded by banding in either the process direction or the scanning direction. Interference image  206  therefore exhibits a pattern of straight horizontal and vertical bright interference bands  208  and  209  that are perpendicular to each other and form a regular pattern of relatively dark regions that appear square.  
         [0078]      FIG. 8C  shows a superposition of reference image  200  with a copy  210  of the test image that suffers from scan direction spacing variations to form an interference image  212 . By way of example, the banding in copy  210  of the test image is characterized by vertical line-bands indicated by brackets  211  comprising vertical lines  203  for which the interline spacing is about 3% greater than for lines  203  outside of the line-bands. Interference image  212  is characterized by relatively bright straight vertical interference bands  214  similar to interference bands  209  in interference image  206  ( FIG. 8B ) but does not comprise straight horizontal interference bands. Because of the scan variations, the bright horizontal interference bands  208  characteristic of interference image  206  are morphed in interference image  212  into zigzag bright interference bands  215 .  
         [0079]     In  FIG. 8D  reference image  200  is superimposed with a copy  220  of the test image that suffers from spacing variations in both the scan and process directions to form an interference image  222 . By way of example the banding and scan variations in copy  220  comprises vertical line-bands indicated by brackets  225  comprising vertical lines  227  and horizontal line bands  226  comprising horizontal lines  228  for which interline spacing is about 3% greater than interline spacing outside of the line bands. Interference image  222  exhibits zigzag bright interference bands along both the horizontal and vertical directions.  
         [0080]     It is noted that for convenience of presentation, banding in copy images  100 ,  130 ,  210  ( FIGS. 5-7 ) and  220  ( FIGS. 8C and 8D ) is characterized by line-bands having a same width, constant spatial period, same uniform interline spacing and a discontinuous change in interline spacing between lines in the line-bands and lines outside the line-bands. However, banding is of course not limited to such “regular” and “convenient” patterns and discontinuities. Banding may be characterized by interline spacing changes that are gradual or continuous and/or by line-bands having different widths and/or line-bands that are not periodic with constant period. Irregular and more “plastic” line-band patterns may be especially characteristic of scan direction defects caused by changes in the scan velocity with which for example laser beam  46  ( FIGS. 1A and 1B ) scans a line on PIC  26 .  
         [0081]     In the description and claims of the present application, each of the verbs, “comprise” “include” and “have”, and conjugates thereof, are used to indicate that the object or objects of the verb are not necessarily a complete listing of members, components, elements or parts of the subject or subjects of the verb.  
         [0082]     The present invention has been described using detailed descriptions of embodiments thereof that are provided by way of example and are not intended to limit the scope of the invention. The described embodiments comprise different features, not all of which are required in all embodiments of the invention. Some embodiments of the present invention utilize only some of the features or possible combinations of the features. Variations of embodiments of the present invention that are described and embodiments of the present invention comprising different combinations of features noted in the described embodiments will occur to persons of the art. The scope of the invention is limited only by the following claims.