Rotating non-rotationally symmetrical halftone dots for encoding embedded data in a hyperacuity printer

Abstract: Disclosed is a hyperacuity printing system (50) for rendering halftoned image data on a recording medium (14 ; Fig. 1). The printing system includes a data source (51) for supplying grayscale input image data and halftoning circuitry (70) (Fig.27) for receiving the grayscale input image data from the data source and transforming the grayscale input image data into multi-bit value output data. The halftoning circuitry (70) includes a look-up table for providing the multi-bit value output data where the multi-bit value output data define halftone dots, the halftone dots being arranged in the look-up table as a continuum from a lowest density value to a highest density value corresponding to the grayscale input image data, the halftone dots having a reference which is a predetermined position from a defined origin in the look-up table. Further included in the printing system is addressing circuitry providing a first set of address lines and a second set of address lines defining a screen stored in the memory, the first set of address lines and the second set of address lines providing x and y address values defined by a desired screen angle. The addressing circuitry includes rotation circuitry for modifying the x and y address values thereby rotating the halftone dots, resulting in rotated halftone dots. The addressing circuitry also includes shifting circuitry for modifying the x and y address values thereby shifting the reference point of the halftone dots away from the predetermined position from the defined origin in the look-up table, the shifting and the rotating occurring while the halftone dots are being read from the look-up table. Finally there is a modulating device for receiving the multi-bit value output data from the halftoning circuitry and converting the multi-bit value output data into a form which can be directly utilized by a writing device for writing the multi-bit value output data onto the recording medium.

Description: This invention relates to halftoning, and more particularly, to a hyperacuity printing system according to the preamble of claim 1. A similar system as mantioned above is known from US-A-49 85 779. Various optical and electronic techniques have been proposed for transforming continuous tone and other types of variably shaded monochromatic and polychromatic images (collectively referred to herein as ,,toneart images") into halftone images. To simplify this disclosure, substantial portions of the following discussion focus on monochromatic halftoning, but it is to be understood that the same general teachings apply to the halftoning of the color separations of polychromatic images. The halftoning of black and white "grayscale" images is a convenient example of monochromatic halftoning, so it is noted that the term "grayscale" is used herein as a generic descriptor for the tones that can be produced by mixing any two reference colors together in any desired proportions. As is known, a halftone image is a binary image that is composed by writing "halftone dots" into a spatially periodic, two dimensional, tiled array of dimensionally identical "halftone cells." These halftone cells spatially correspond in the halftoned output image to respective small, spatially distinct areas of the source image. Furthermore, the surface area of the dot that is written into each of the halftone cells is modulated in accordance with a suitable measure of the perceived or "average" grayscale level of the spatially corresponding area of the source image (this "dot area" parameter typically is expressed by referring to the percentage of the halftone cell area that is filled by the halftone dot that is written therein). Thus, halftoning imparts an illusion of shading to the halftoned image because the halftone dots are written at a spatial frequency (usually called the "screen frequency") that exceeds the cyclical acuity of the human eye at normal viewing distances. A conservative rule of thumb is that the human eye is insensitive to cyclical contrast variations that occur at a spatial frequency in excess of about 40 cycles per degree within the field of view. As will be appreciated, halftoning is an important tool for preserving the shaded appearance of toneart images that are printed using binary printing technologies. Optically screened lithographic halftoning processes have established a challenging benchmark for the imaging fidelity that can be achieved by printing halftoned images on high gamma, photosensitive recording media (i. e., a recording medium having a steeply sloped exposure vs. contrast characteristic). Modern xerographic printers utilize high gamma photoreceptors to print lineart images that approach the fidelity of lithographically printed lineart. Heretofore, however, the electronic halftoning processes that have been available for use in xerographic printers have not had sufficient imaging fidelity to enable xerographic printers to effectively compete with lithographic printing processes in the printing of high fidelity halftoned images. Xerographic printing has made significant inroads into the lithographic printing market because of its cost advantage and the improvements in its imaging fidelity. This trend is expected to continue, but it clearly will be necessary to further improve the fidelity of electronic halftoning for xerographic printers to become a fully acceptable alternative to lithographic printers for the printing of halftoned images. Some workers have proposed electronic halftoning techniques that more or less directly emulate angularly oriented optical halftone screening functions. See, for example, US-A-3,997,911. Others have focused on modulating the size of the halftone dots that are written into tiled arrays of electronically generated halftone cells at a selected screen angle. See, for example US-A-3,688,033, US-A-4,499,489, US-A-4,680,645, US-A-5,258,849. Also showing various ways of producing halftone dots are US-A-s 4,912,568 and 5, 233,441. Various forms of encoding or embedded data into the the halftone structure can be implemented. In US-A-s 5,128,525 and 5,221,833 there are described methods to embed data with certain shape codes called "glyph codes" which allow error detection and correction. US-A-4,985,779 discloses a method and an apparatus for generating halftone images using a beam of variable intensity. A look-up table including beam intensity information may be accessed by dot set identification data and/or randomly generated numbers in order to permit formation of halftone dots with different levels of beam exposure. BE-A-1 003 657 discloses a printing system consisting of a microprocessor and a number of other units connected directly or via interfaces to a communication bus. An address generator for addressing image pixels and being associated with the microprocessor is equipped with an offset generator to reduce the effects of interference patterns on the image. A circle is an example of a rotationally symmetric object. A rotation about its center cannot be detected by a change in its form or shape. On the other hand, an angular change in the orientation of a triangle, for instance, could be detected to some extent. There are varying degrees of angular change which can be detected. Whereas a circle can be rotated any amount about its center without detection, a square's symmetry only allows angular rotations about its center modulo 90 degrees to guarantee detection, and an equilateral triangle modulo 120 degrees. The invention described herein utilizes shapes that allow detection of their shape changes through rotation (hence, non-rotationally symmetric) after being input by appropriate scanning instruments, such as an input scanner. Images are stored in a computer system as an array of pixels, each pixel being a multi-bit representation of the intensity of the original image at that position. Some pixels are represented by eight bit values, which provide a range of intensities from zero to 255, for example. Halftone dots can be defined in three regions within this range for the purposes of explaining this invention. The highlight region is the range of densities where the halftone dot is small. The shadow region is the region where the dots are so large that the absence of dot coverage is small. In between these two extremes is the midrange of densities. Since halftone dots grow from small highlight dots through the midrange densities to the dark shadow dots as the density of the image changes from high to low, and the shape of the dot cannot be controlled well in the highlight or shadow regions, it is only in the midrange of the densities that the shape has enough fidelity to be used to store data. Therefore, images that contain a lot of midrange densities can best contain embedded data. Therefore, it is the object of the present invention to provide a hyperacuity printing system according to the preamble of claim 1 allowing to embed encoded data into a halftone image. According to the present invention, this object is solved by a hyperacuity printing system having the features set out in claim 1. The present invention contrives that a transformation (or rotation) on x-axis and y-axis coordinates of the address into the addressable table look-up memory relocates the memory access inside the halftone cell to a new location for purpose of rotating the halftone dot for embedding data. The new location accessed by the rotation of the address coordinates could differ from the original location in all cases by a predetermined angle from a central point of rotation, while maintaining a constant radius from the central point of rotation. The angle would determine how much the halftone dot is rotated, where the point of rotation could be picked beforehand as the center of the halftone dot. Application of the transformation could rotate the halftone dot without affecting the screen frequency, screen angle or halftone dot density. As will be seen, the hyperacuity printer described herein or the hardware halftoners mentioned in the referenced art are an appropriate way to perform such an operation while printing due to their two dimensional address generation, the high fidelity in which the halftone dot structure is represented, and their ability to provide programmable halftone dot shapes. Alternate ways of doing this would be to precompute a bitmapped image which would suffer from lower fidelity, or to provide multiple halftone dot memories, with precomputed rotated dots, but would require an additional duplicate memory for each additional angle. The printing system includes a data source for supplying grayscale input image data and halftoning circuitry for receiving the grayscale input image data from the data source and transforming the grayscale input image data into multi-bit value output data. The halftoning circuitry includes a look-up table for providing the multi-bit value output data where the multi-bit value output data define halftone dots, the halftone dots being arranged in the look-up table as a continuum from a lowest density value to a highest density value corresponding to the grayscale input image data, the halftone dots having a reference which is a predetermined position from a defined origin in the look-up table. Further included in the printing system is addressing circuitry providing a first set of address lines and a second set of address lines defining a screen stored in the memory, the first set of address lines and the second set of address lines providing x and y address values defined by a desired screen angle. The addressing circuitry includes rotation circuitry for modifying the x and y address values thereby rotating the halftone dots, resulting in rotated halftone dots. The addressing circuitry also includes shifting circuitry for modifying the x and y address values thereby shifting the reference point of the halftone dots away from the predetermined position from the defined origin in the look-up table, the shifting and the rotating occurring while the halftone dots are being read from the look-up table. Finally, there is a modulating device for receiving the multi-bit value output data from the halftoning circuitry and converting the multi-bit value output data into a form which can be directly utilized by a writing device for writing the multi-bit value output data onto the recording medium. Embodiments of the invention will now be described, by way of example, with reference to the accompanying drawings, in which: Figure 1 is is a simplified schematic diagram of a xerographic printer that is equipped to carry out the present invention; Figure 2 is a block diagram of a single channel hyperacuity imager subsystem according to the present invention; Figure 3 is a diagram of a two channel hyperacuity imager subsystem according to the present invention; Figure 4 is a block diagram depicting the elements included in the FIFO and phase/resolution controller shown in Figures 2 and 3; Figure 5 is a block diagram depicting the elements included in the gray interlace formatter shown in Figure 3; Figure 6 is a block diagram depicting the elements included in the resampling interpolator and the phase/resolution controller shown in Figures 2 and 3; Figure 7 is a block diagram depicting the elements included in the y interpolator, x interpolator and slope interpolator shown in figure 6; Figure 8 is a diagram depicting the concept of slope thresholding in two dimensions; Figure 9 is a block diagram showing the elements of the slope thresholder; Figure 10 is a schematic representation of the full creation of a halftone dot; Figure 11 is a schematic representation of a step in the creation of a halftone dot; Figure 12 is a schematic representation of a step in the creation of a halftone dot; Figure 13 is a schematic representation of a step in the creation of a halftone dot; Figure 14 is a schematic representation of a step in the creation of a halftone dot; Figure 15 is a schematic representation of a step in the creation of a halftone dot; Figure 16 is a schematic representation of a step in the creation of a halftone dot; Figure 17 is a schematic representation of a step in the creation of a halftone dot; Figure 18 is a schematic representation of a step in the creation of a halftone dot; Figure 19 is a schematic representation of a step in the creation of a halftone dot; Figure 20 is a schematic representation of a step in the creation of a halftone dot; Figure 21 is a schematic representation of a step in the creation of a halftone dot; Figure 22 is a schematic representation of a step in the creation of a halftone dot; Figure 23 is a schematic representation of a step in the creation of a halftone dot; Figure 24 is a schematic representation of a step in the creation of a halftone dot; Figure 25 is a schematic representation of a step in the creation of a halftone dot; Figure 26 is a schematic representation of a step in the creation of a halftone dot; Figure 27 is a block diagram of the halftoner; Figure 28 is a block diagram of the x and y accumulators of the halftoner in Figure 27; Figure 29 is a graphical representation of the values loaded into the x and y accumulators of Figure 28; Figure 30 is a graphical representation of a halftone dot; Figure 31 is a graphical representation of a halftone dot; Figure 32 is a graphical representation of a halftone dot; Figure 33 is a graphical representation of a halftone dot; Figure 34 is a block diagram of a variable modulus accumulator used for addressing the halftoner memory block; Figure 35 is a graphical representation depicting the rotation of a halftone dot and associated mathematics; Figure 36 is block diagram showing an alternative embodiment of the halftoner in accordance with the present invention; Figure 37 is block diagram showing another alternative embodiment of the halftoner in accordance with the present invention; Figure 38 is a graphical representation of a non-rotated elliptical halftone dot having its center at the origin; Figure 39 is a graphical representation of a rotated elliptical halftone dot having its center at the origin; Figure 40 is a graphical representation of a non-rotated elliptical halftone dot having its center offset from the origin; and Figure 41 is a graphical representation of a rotated elliptical halftone dot having its center offset from origin. I. SYSTEM OVERVIEW 1.0 Typical ROS System Turning now to the drawings, and at this point especially to Fig. 1, there is a xerographic print engine 11 (shown only in relevant part) having a more or less conventionally configured flying spot raster output scanner (ROS) 12 for scanning a data modulated light beam 13 over a xerographic photoreceptor 14 in accordance with a predetermined raster scanning pattern. To that end, the ROS 12 comprises a laser diode 15 for emitting the light beam 13 in the visible or invisible (e. g., infrared) band of the spectrum, together with a polygon scanner 17 that has a plurality of nearly identical, mirror-like exterior sidewalls or "facets" 18. In keeping with standard practices, there is a motor 21 for rotating the scanner 17 about its central axis, as indicated by the arrow 22, at a substantially constant angular velocity. The scanner 17 is optically aligned between the laser 15 and the photoreceptor 14, so its rotation causes the laser beam 13 to be intercepted by and reflected from one after another of the scanner facets 18, with the result that the beam 13 is cyclically swept across the photoreceptor 14 in a fastscan direction. The photoreceptor 14, on the other hand, is advanced (by means not shown) simultaneously in an orthogonal, process direction at a substantially constant linear velocity, as indicated by the arrow 23, so the laser beam 13 scans the photoreceptor 14 in accordance with a raster scan pattern. As shown, the photoreceptor 14 is coated on a rotating drum 24, but it will be apparent that it also could be carried by a belt or any other suitable substrate. Typically, the ROS 12 additionally includes pre-scan optics 25 and post-scan optics 26 for bringing the laser beam 13 to a generally circular focus proximate the photoreceptor 14 and for providing any optical correction that may be needed to compensate for scanner wobble and other optical irregularities. Preferably, the optical aperture of the ROS 12 is sufficiently large to avoid excessive truncation of the laser beam 13 because the beam 13 then comes to a generally circular or elliptical focus with a gaussian intensity profile. However, the broader aspects of this invention are not limited to any specific scan spot geometry or intensity profile. Accepted design principles indicate that the spatial frequency power spectrum of the scan spot profile should not have significant spatial frequency components outside the spatial frequency passband of the imaging system, but the scan spot can otherwise be tailored to satisfy a variety of system requirements. To carry out the present invention, the amplitude, duty cycle, and/or pulse width of the laser beam 13 is serially modulated (collectively referred to herein as its "intensity modulation") by intensity modulator 28 in accordance with successive multi-bit digital data values. These data values are clocked out of a buffered data source 23 serially in response to data clock pulses which are time synchronized with the scan of the scan spot from bitmap location-to-bitmap location within the raster scan pattern. Thus, the data clock frequency can be selected (by means not shown) to map the data onto the raster scan pattern at any desired resolution, using either the same or different resolutions in the fastscan and the process directions. As will be appreciated, the data may be preprocessed (such as by means herein described) for the printing of halftoned images and/or text and other types of line art, so the data source 23 generically represents any suitable source of gray scale data for intensity modulating the laser beam 13. As is known, the fastscan pixel positioning precision of the print engine 11 can be enhanced, if desired, by dynamically adjusting the frequency of the data clock to compensate for the pixel positioning errors that tend to be caused by "motor hunt" (i. e., variations in the angular velocity of the scanner 17), "polygon signature" characteristics (variations in the angular velocities at which the different facets 18 of the scanner 17 sweep the scan spot across the photoreceptor 14 from a start of scan position to an end of scan position), and "scan non-linearities" (i. e., localized variations in the linear velocity of the fastscan, which are caused by variances in the geometric relationship of the scanner 17 to spatially distinct segments of any given scanline). For more information on these sources of potential pixel positioning errors and the compensation that can be provided for those errors, reference is made to the following: US-A-4,622,593; US-A-4,639,789; US-A-4,766,560; US-A-4,860,237; US-A-4,893,136; and US-A-4,935,891. It is to be understood, however, that the increased pixel positioning precision of the existing compensation techniques listed above, strictly pertain to the spatial positioning of the pixel centers in the fastscan direction. Thus timing is used to adjust the fastscan pixel positions. This differs from the two dimensional "high addressability" or "microaddressability" utilized herein because microaddressability enables image transitions to be spatially positioned with subresolution precision in both the process direction and the fastscan direction. This feature sometimes is referred to as "two dimensional high addressability" or, in abbreviated form, as "2D high addressability" and is fully described in US-A-5,138,339. Typically, intensity is used to adjust the pixel positions in the process direction. In particular, according to US-A-5,138,339 which can be utilized by the present invention, the pitch of the scan pattern for the printer 11 is selected to be significantly finer (i. e., smaller) than the FWHM (full width half max) diameter of the scan spot that is formed from the scan beam 13. This relatively fine pitch scan pattern causes the printer 11 to operate in an "overscanned" mode because the FWHM central core of the scan spot sweeps across spatially overlapping segments of the photoreceptor 14 during the scanning of spatially adjacent (i. e., neighboring) scanlines. Overscanning slightly degrades the spatial frequency response of the printer 11 in the process direction. However, it has been found that the linearity of the addressability response of ROS printers, such as the printer 11, increases rapidly as the ratio of the scan pitch to the FWHM diameter of the scan spot is reduced to progressively smaller, sub-unity values (i. e., increasing the overscan). In other words, it has been found that overscanning is the key to providing the essentially linear position control that enables discrete exposures to map image transitions onto a scan pattern at a subresolution precision. Thus, in the printer shown in Fig. 1, a relatively small loss of process direction frequency response is accepted to achieve substantially increased process direction addressability. The use of an overscan factor (i. e., the ratio of the FWHM diameter of the scan spot to the scan pitch) in excess of about 2x only marginally improves the linearity of the addressability response of the printer 11, so designers contemplating the use of overscan factors greater than 2x or so should carefully analyze the overall performance of the system that is being designed to ensure that it is satisfactory. The reason that it is important to adjust the pixel positions with subpixel or subscan precision in both the fastscan and process directions is because of the ability of the human visual system to perceive edge placement precision from 10 to 60 times more precisely than the acuity (resolution) of the visual system. This ability of the human visual system is referred to as hyperacuity. Thus, the resolving power of the printer need extend just beyond the visual system's acuity cutoff of around 300 to 400 lines per inch (12 to 16 lines/mm) at a twelve inch view distance, the hyperacuity response of the printer needs to be able to position the edges of these relatively coarse marks with a precision of from 3,000 to 20,000 positions per inch (120 to 800 positions/mm) . It is not practical to implement high speed printers with resolutions of 3,000 to 20,000 pixels per inch (120 to 800 pixels/inch), so the alternative is to make a scanner with similar characteristics as the human visual system: its hyperacuity response is an order of magnitude or more greater than its own resolution. The linearity provided by overscanning enables this performance. It then becomes necessary to provide this scanner with enough fidelity information to drive it. Bitmaps, which are usually used, are a poor source, because they do not efficiently store hyperacuity information. The information is best provided by gray source data, which can have a limited resolution of 300 to 400 samples per inch (12 to 16 samples/mm), for instance, and a gray depth of perhaps 8 bits, which is logarithmically encoded to achieve 256 levels, which provides the hyperacuity information. This data can be converted in real time with the electronics herein described to produce the high bandwidth gray data destined for the photoreceptor or media. Thus, this information, which amounts to the equivalent of over 23 million bits per square inch (36,800 bits/mm 2 ) (at 4800 x 4800 addressability units per inch, for instance), is used only on the photoreceptor or media, not stored in an expensive page buffer. 2.0 Hyperacuity Printer Architecture High quality printing not only depends on machine noise and exposure quality, but on how much fidelity information can be passed on to the printer from the data source. It is the purpose of the electronics in the printer to convert high fidelity, sampled representation into a high bandwidth, high resolution bit stream for delivery to the photoreceptor. Figure 2 shows a block diagram of the subsystem components in a hyperacuity printing system, referred to herein as printer 50, in accordance with the present invention. Printer 50 can be implemented like the xerographic print engine 11 of Figure 1, however, it is not intended to be limited to such a print engine configuration. Printer 50 includes data sources from Image Generator 51 including Grayscale Image Generator 52, Binary Image Generator 54, and Grayscale Converter 56. Further included are FIFO (First In First Out) memory 58, Resampling Interpolator 60, segmenter 62, Phase/Resolution Controller 66, halftoner 70, D to A converter 76, laser diode 15, and Linearity and Registration Controller 78. The resampling interpolator 60, halftoner 70, thresholder 68 and digital to analog converter 76 are responsible for the conversion of the sampled representation into a high bandwidth, high resolution bitstream for delivery to the photoreceptor via laser diode 15. Figure 3 is a block diagram of a two channel hyperacuity printer whereby the printer has two laser diodes. Note that the system of Figure 3 includes the same elements as the system shown in Figure 2, with the addition of interlace formatter 80. Printer 50 will be generally described below, with detailed descriptions of each subsystem component following the general description. The image generator used in the printing system can be binary or gray, and is not restricted to a specific resolution or gray (multi-bit) depth. In general, gray depth over an image can be variable to accommodate a physical interface with the imager subsystem or to adjust the fidelity of image information sent to the printer, while sample density (the number of samples per inch) will be adjustable up to some maximum to match the resolution goal of the imager. As shown in Figure 2, image data may be provided by grayscale image generator 52 which provides multiple bit per pixel data to FIFO 58. Or, image data may be provided in binary format, (i.e. one bit per pixel), by binary image generator 54. This binary data from generator 54 can then be converted to grayscale by grayscale convertor 56 before processing. A typical operation performed by the grayscale convertor 56 is convolution or filtering, then scaling of a high resolution binary image into a coarser gray image. However, the edge information present in the binary image is encoded naturally by the convolution or filtering operation into gray depth. An exemplary description of a grayscale convertor can be found in EP-A-585,043. The image generator 51 also can download (in advance of printing) halftoning and thresholding information tuned to the special characteristics of the imager and the desired imaging response. This download file is not image data, but rather the contents of the look-up table used for halftoning or thresholding. For example, software in the image generator could determine the level to erode or dilate a specified font for the target write black or write white platform, and produce an appropriate download file. FIFO 58 provides buffering which allows data to be delayed for use in multi diode configurations and delayed for accessing two adjacent scans for interpolation. As will be seen, in a multi-diode configuration, the diodes are positioned along the process direction with some beam spacing distance. Because of this, at any moment during printing, one diode will be accessing data from a different portion of memory in the process direction than another diode. Control of the FIFO buffers in a multi-channel configuration, such as that of Figure 3, is the responsibility of interlace formatter 80 in cooperation with phase/resolution controller 66. Since the image information is sent only once from the image generator in the form of serial rasters, the FIFO buffers the number of rasters spanned by the laser diodes in the process direction. Furthermore, FIFO 58 provides delayed raster information for use with other image processing functions. In this case, these delayed rasters are used by resampling interpolator 60 in pairs. The existence of resampling interpolator 60 in the electronic pathway between the image generators and the output allows a unique opportunity to electronically map data spatial position into arbitrary misalignment of the imager and its mechanical process as well as the customary use which provides resolution conversion. The information required to conformally map the data into the imperfections of the engine and laser scanner are obtained from the linearity and registration controller 78, and processed through phase/resolution controller 66, and can be viewed as a dynamic resolution control. Any imperfection in the platform which can be measured accurately can be servoed electronically for conformal electronic registration. Resampling interpolator 60 executes a standard image processing function of interpolation, which can be linear interpolation for processing simplicity. For the hyperacuity printer 50, the spot on the photoreceptor is tracked by appropriate electronics, taking into account electronic registration requirements. Without electronic registration feedback, the phase/resolution controller 66 would start from zero and accumulate a constant value in both the fastscan and slowscan directions. With feedback, however, the start values or accumuland values are dynamically biased by adding in changing error values during printing. This information about the spot's current position is passed to resampling interpolator 60 which returns a sample intensity for that position, called a resample. To obtain a resample, the spot's current position, measured in units of addressability in both x (fastscan) and y (slowscan or process) dimensions, is not necessarily coincident with one of the integer sample positions, but instead fractionally positioned among four of the nearest neighbors. The interpolator executes an algorithm which estimates what the sample intensity is at the current position by extrapolating the intensities of these four closest neighbors. The simplest form of this calculation assumes that the intensity varies in a straight line between adjacent samples, thus the term linear interpolation. In addition, the resampling interpolator 60 will simultaneously calculate the local rate of change (slope) of the intensities and pass this information on to thresholder 68 as shown by line 85. This information can be used to predict where the exposure intensities of lineart will cross the xerographic threshold. As mentioned above, the resampling interpolator is used for resolution conversion. US-A-5,357,273 describes an alternative means and method of performing resolution conversion in the process direction in a hyperacuity printer, without the use of a resampling interpolator. US-A-5,229,760 and EP-A-461,854 also address resolution conversion in the process direction, but do not use an interpolator. Thresholder 68 converts the resampled information from interpolator 60 into a gray data stream which drives the D to A converter when lineart, such as fonts and lines, is to be rendered. The hyperacuity printer 50 has the ability to position edges of image regions very precisely in the process and fastscan directions due to the use of timing and variable intensity as described above. The thresholder, however, attempts to render the sampled data by estimating where the sampled representation crosses the xerographic threshold. One form of hardware to accomplish the thresholding function is a cache memory acting as a look-up table. Another form of hardware to perform the thresholding function could be a processor or other suitable electronics to calculate the function in real time. The thresholder 68 utilizes two dimensional slope information, line 85, generated by resampling interpolator 60, (which is generally thrown away in conventional interpolation schemes), and therefore is able to predict threshold crossings of the input file with subpixel precision. This is important because it means that using bitmaps with ever increasing resolution (beyond the human visual systems' acuity cutoff of around 300 to 400 pixels per inch (12 to 16 pixels/mm) at a 12 inch (30 cm) view distance, for instance) to position edges is no longer necessary. In addition, this edge information can be used in conjunction with prior knowledge of the printer tone reproduction curve (TRC) and spot dimensions to compensate for the width of the spot, and thus achieve printer device independence for rendering an image on charge area development (write-black) printers or exposed area development (write-white) printers. This also makes the need to predistort fonts to compensate for printing on particular printers unnecessary. Halftoner 70, including dot generator 72 and screen generator 74, converts the resampled information from resampling interpolator 60 into a binary map when pictorials are to be rendered. Pictorials require many levels of gray to be contour free, as well as a fine dot structure to make graininess imperceptible. A 4800 per inch addressability printer (1200 scans per inch (48 scans/mm) and 4 levels of gray on photoreceptor, for instance) will have the ability to produce bitmaps with over a thousand levels of gray at 150 halftone dots per inch (6 halftone dots/mm), for example (4800/150 = 32 addressability units per halftone cell side; 32 x 32 = 1024 possible levels). In addition, the tone reproduction curve can be embedded into the dot definition since only 256 levels (or less) need to be represented in the input data. In addition, the laser spot shape and size is compensated for. In order to accomplish this, a high fidelity representation of a halftone dot is required. The definition for the halftone dot, as a function of spot position on the photoreceptor and the required gray level is stored in a memory. As the spot scans, the intensity of the spot will be adjusted by the D to A converter (or any other intensity modulation scheme such as pulse width modulation or duty cycle modulation, for instance) by the multi-bit (gray) output of the halftoner to define the boundaries of the halftone dots. For the case of a halftone dot with a pitch of 1/150 inch (170 µm), 4800 addressability (4800 / 150 = 32 addressability units on each side of the halftone cell) and 256 different gray levels, the number of bits required in memory to define it is on the order of a million (32 x 32 x 256 memory locations x 4 bits per memory location = 1.05 million bits in the memory of the example.). This halftoner differs with others of prior art in that intensity information (256 values) as well as screen position (32 x 32 values) are used as an address into the memory, and the multi-bit value output, representing portions of a halftone dot is directed to a gray modulator. Prior art halftoners retrieve a value based on the screen address only, compare it with the input intensity information (with a hardware comparator), and turn a laser either on or off depending on whether the value retrieved is above or below the input intensity. In addition, halftoner 70 has very fine control over the halftone dot shape and size. For example, the halftoner will be able to produce dots with elliptical shape, or other optimized shapes to minimize dot gain in the mid tones or maximize impulse response in the highlight areas. In order to properly render printed images, all display systems utilizing binary presentation, such as xerography and offset lithography, must be able to discern the difference between gray images (toneart or pictorials), and synthetic images (lineart, including text and lines). This is because binary printers must print gray images by halftoning. There is no ability to control a sufficient gradient of gray to produce continuous tone, as in photographs or thermal transfer printers. On the other hand, it is important from a visual point of view that high contrast lineart and text be printed with smooth edges. This is accomplished through segmentation via segmenter 62. When printing the sampled representations, the samples will be passed through both halftoner 70 (for toneart) and thresholder 68 (for lineart) simultaneously. An additional bit, known as a segmentation bit or tag bit, line 63 from FIFO 58, is carried along with the multi-bit sample pixel and is delivered by the segmenter 62, after suitable time delay to the halftoner and thresholder. This bit will be used to switch between these two devices during printing, according to whether toneart from the halftoner or lineart from the thresholder is to be rendered. The resolution of information which determines this can be as high as one bit per sample, or some lower number. In order to control from where D to A converter 76 gets its input, segmenter 62 can either modify the input addresses to halftoner 70 and thresholder 68, or provide an enable signal to tri-state output devices on the outputs of halftoner 70 and thresholder 68. In any case, segmenter 62 provides a switching function between thresholder 68 and halftoner 70 via output 64. Preferably, the segmenter utilizes slope information 85 from resampling interpolator 60 in determining segmentation. It can be appreciated that other segmentation algorithms known in the art, including real-time segmentation analysis, may be used to differentiate between pixels to be halftoned and pixels to be thresholded. In addition, more than two segmentation partitions may be utilized by adding new algorithm hardware in parallel, and switching among them with extra segmentation or tag bits. Halftoner 70 and thresholder 68 provide several intensity levels at the frequency specified by the fastscan clock. The number of levels is determined by system design considerations, including tradeoffs between clock rate, number of channels, machine throughput, and xerographic noise. D to A converter 76 can use duty cycle, pulse width or current control to achieve the analog conversion for several light intensities. Note that, because of the way in which levels are specified, the number of values required to encode the levels is always one more than number of levels. For instance, if four levels are specified, the required values for intensity would be 0%, 25%, 50%, 75%, 100% = 5 values. It can be appreciated that more than four levels, such as 16 or 256, could be useful and easy to implement. An important aspect of a hyperacuity printer is the incorporation of laser diode source 15. The laser diode 15 offers a bandwidth well into the GHz range, with the ability to modulate its intensity with electronic current control. In addition, laser diode may be produced in a monolithic, closely-spaced multi-beam configuration, allowing throughput and quality improvements. In the configuration shown in Figure 3, a two laser diode system is illustrated. US-A-5,216,443 discusses embedding the digital to analog (D to A) conversion on the semiconductor substrate with the laser diode, the D to A conversion being controlled using pulse width modulation. Another useful and unique feature available in printer 50 is that of linearity and registration control. For instance, color reproduction demands a greater precision in machine mechanical requirements for registration to achieve color layer to layer registration in the one to two mil range. It is extremely important that layer to layer registration be precise. With such precision, it can be possible to achieve process color for lines and text using this system. This precision is extremely difficult and expensive to achieve mechanically. Other requirements for registration in a multi station color machine can also be met, where each writing station would be calibrated to deliver a pixel map which overlaps precisely the pixel maps of the others. It is evident that, assuming the errors can be measured, that the two dimensional hyperacuity printer will be able to achieve these goals through electronic registration using linearity and registration controller 78. For example, at 4800 per inch (192 per mm) addressability, the placement precision of edges is .2 mils. Linearity and registration controller 78 does this by feeding back spot position error information to phase/resolution controller via line 81. Controller 78 also appropriately displaced samples to the decomposer electronics (halftoner 70) via line 83. In addition, x-y position registration, scan linearity correction, velocity servoing and photoreceptor skew and runout correction can be performed electronically while relaxing the specifications on the optomechanical components. Turning now to Figs 4 to 41, each subsystem in printer 50 will be described in detail. II. SYSTEM DETAILS 1.0 FIFO Figure 4 is a detailed block diagram of the elements included in FIFO 58 and phase/resolution controller 66. FIFO 58 is a memory device configured to be utilized as a two dimensional (2-D) window on the image data stored therein, having a fastscan (x) and a slowscan (y) dimension. Phase/resolution controller 66 as well as interlace formatter 80 will assist in providing addresses into that 2-D window FIFO 58 has memory 82 in which a number of rasters, or scanlines are stored. Each raster has a plurality of pixels, and each pixel has one to eight bits. The width of memory 82 is at least large enough to store an entire raster of pixels. In addition, the width includes capacity for the segmentation or tag bit or bits, as necessary. Segmentation or tag bits are delivered to the segmenter through line 63, and are stored with the pixel in the FIFO to insure correct time delay in application of the halftoning or thresholding algorithms. The height of memory 82 is at least large enough to compensate for the beam spacing in a multi-channel system, plus accommodation for one raster to be used by resampling interpolator 60, and plus one or two rasters to enable the reading of one raster while writing another. Typically four to eight rasters are stored, but the invention is not limited by such. FIFO 58 is configured such that it is a three port memory having one write port and two read ports. As memory 82 is being addressed, simultaneously the image generator loads the raster being generated into the FIFO as two rasters are being read out for processing. As will become apparent, the x address for all three ports are the same while the effective y address for each port is different by a given offset. Furthermore, for a given x address, the pixel stored at that address for each raster stored in memory 82 is available. Memory 82 is addressed in the x direction by XADDR demultiplexor 84 (demux). Phase/resolution controller 66 supplies a fourteen bit number via XADDRESS line 111 to demux 84 thereby allowing demux 84 to address 2 14 pixels along a raster stored in memory 82. Phase/resolution controller 66 supplies a four bit number via YADDRESS line 99 creating 2 4 addresses. In the y direction there need to be offsets in the addressing in order to accommodate simultaneous reading and writing of memory 82. In order to create the necessary offsets for y direction addressing, full adder 92 adds a fixed value to YADDRESS 99 which in turn is used by YADDR demux 86 for write port addressing. In a similar manner, full adder 94 supplies a fixed offset to one of the two read ports. Since all rasters stored in memory 82 are available simultaneously, mux 88 and mux 90 are used to select two of the rasters for processing at any one time. It is the use of these two muxes that creates the two read ports. Data from mux 88 is shown as line LAST 89 while data through mux 90 is shown as NEXT line 91. The use of the LAST and NEXT rasters will become apparent when discussing the resampling interpolator. Note that the addresses do not advance by more than one at a time, thus the FIFO-like nature of this memory subsystem. 2.0 Phase/Resolution Controller One purpose of phase/resolution controller 66 is to provide addressing to FIFO 58 as described above. Another purpose of phase resolution controller 66 is to provide a fractional accumulated number for use by resampling interpolator 60. Phase/resolution controller 66 has a y accumulator 96 and an x accumulator 108. Y accumulator 96 includes register 98 and full adder 100. Register 98 is a sixteen bit register whereby the leftmost four bits represent the integer portion and the remaining twelve bits represent the fractional portion of the sixteen bit number. The integer portion is used as the YADDRESS line 99 while the fractional portion is multiplication factor line YFRACT 101. As will be seen, YFRACT 101 will be used by resampling interpolator 60. Phase/resolution controller 66 also has X accumulator 108. X accumulator 108 includes register 110 and a full adder 112. Register 110 is a twenty-four bit register whereby the leftmost fourteen bits represent the integer portion and the remaining ten bits represent the fractional portion of the twenty-four bit number. The integer portion is used as the XADDRESS line 111 while the fractional portion is multiplication factor line XFRACT 113. As will be seen, XFRACT 113 will also be used by resampling interpolator 60. Each accumulator has associated with it a loadable INC register and a loadable START PHASE register. During each clock cycle controlling the accumulators, the accumulators can only increment by the number loaded in the INC registers. The value loaded in the INC registers is fractional, therefore the accumulation is fractional. As will be understood from the discussion of the resampling interpolator, the fractional increment enables the calculation of pixels values between two pixels and/or two rasters stored in the FIFO. Once the accumulators have accumulated the fractional value whereby it rolls over into the integer value, the address into the FIFO is at the next address position, or pointing to the next pixel or raster of data. In the system described herein, in order to satisfy Nyquist criteria, it may be desirable to have at least two clock cycles before changing to the next address, therefore the INC values may be restricted to less than or equal to 0.5. For x accumulator 108, the value in INC register 114 is equal to the number of samples per inch in the fastscan direction, divided by the number of fastscan clocks per inch. For instance, at 600 samples per inch (24 samples/mm), and 4800 fast scan clocks per inch (192 clocks/mm), INC=0.125. The value in START PHASE register 116 equals zero, (unless a registration correction value has been provided by the linearity and registration controller 78, which will be discussed later). The process for determining the values in the INC register 102 and START PHASE register 104 for y accumulator 96 will be explained in more detail with reference to the interlace formatter. At the beginning of a page, register 98 of y accumulator 96 is loaded with a START PHASE value from register 104. With each start of scan clock signal, SOSCLK, y accumulator 96 is incremented by the value stored in INC register 102. Since the number being accumulated is fractional, the integer part of the number in register 98 will not be incremented by more than one at a time. In this example, registers 102, 104 are each thirteen bits. At the beginning of a scan, register 110 of x accumulator 108 is loaded with START PHASE value from register 116. With each system clock signal, SYSCLK, x accumulator 108 is incremented by the value stored in INC register 114. Since the number being accumulated is fractional, the integer part of the number in register 110 will not be incremented by more than one at a time. In this example, registers 114, 116 are each eleven bits. In general, the x and y address for the FIFO are tracked by the x accumulator for the fastscan direction and with the y accumulator for the slowscan direction. Therefore, the integer portions of the accumulators should never advance by more than one since the system is systematically clocking through the memory in a FIFO-like fashion. 3.0 Interlace Formatter The phase/resolution controller along with the interlace formatter facilitate resolution conversion, registration correction, and interlacing. Referring to Figure 5, the interlace formatter is basically a duplication of the y components in the phase/resolution controller with an offset to allow each channel to access the FIFO raster data at a point in the data which matches the channel's beam position in the two channel system of this example. The offset loaded into the START PHASE registers, accounts for the beam spacing between photodiodes. As shown, there is a connection (line 99) between the channels, which maintain the write address for both channels, thus causing the write address for both channels to be the same. Since the write address is the same, the same data is being written into the memory of each channel simultaneously. However, due to the values in the START PHASE registers, the read address is staggered to compensate for the distance between the two photodiodes. Therefore, the the image generator only needs to write one raster at a time into memory before printing, thereby decreasing its complexity over prior systems which required the image generator to supply different data to each channel simultaneously. The interlace formatter facilitates accessing data in the memory in two places simultaneously, thus creating interlacing automatically. The interlace formatter described herein has the the ability to map any interlace factor into any set of rasters that are in the memory. US-A-5,233,367 describes an image forming apparatus and method for interlace scanning plural light beams by deflecting the plural beams with a rotating polygon onto a recording medium wherein the spacing between adjacent beams hold the relation I Q = Bwf V o wherein I is an interlace factor equal to the number of reference scans between adjacent beams from a single facet of the polygon on the photoreceptor. The photoreceptor velocity is V o , ω is the polygon angular velocity, f is a number of facets per revolution of the polygon, Q is the number of beams and B is the beam spacing. As will be shown, the interlace formatter described herein implements a hardware configuration which utilizes the above equation to facilitate interlacing by mapping rasters into the polygon scan structure of the system. Generally, for interlace scanning, any positive value for the interlace factor I may be selected. By selecting various values for the interlace factor I and the number of beams Q, it is possible to obtain equally spaced, fractional or multiple interlace scanning. For equally spaced, non-overlapped interlace scanning, I must be selected as a positive integer and itself and Q must not have any common integer divisors greater than 1. For equally spaced, non-overlapped, interlace scanning, the interlace factor I is the distance between adjacent scans laid down by a single facet of the polygon divided by the distance between adjacent, interlaced scans. Between any two adjacent scans laid down by a single facet of the polygon are interlaced additional scans laid down by prior or subsequent facets of the polygon. In a dual channel interlace formatter shown in Figure 5, there is a master channel (channel 0) and a slave channel (channel 1). For clarity, the device numbers in the slave channel are indicated with a prime notation. The values for START PHASE registers can be calculated by the equation START PHASE = (ch)(D)(B) where ch is the channel number, D is the target data density, or data resolution, in rasters per inch, on the photoreceptor and B is the beam spacing, in inches, between the photodiodes. B can be found from the equation above or can be measured. In Figure 5, START PHASE register 104 is loaded with a zero, since it is in channel zero. START PHASE register 104' is loaded with (D)(B) since it is channel 1. This is the amount of offset into the memory required for interlacing. The value in the INC registers 102, 102' represents how many rasters the photoreceptor moves per single facet. INC equals the density (rasters/inch) times V o /ωf in inches/facet. In other words, INC represents how far it is necessary to go through the FIFO by a certain number of rasters. This number can be fractional. Since INC can be fractional, y accumulators 96, 96' can request pixel values for a scan which falls between rasters. Therefore, y interpolators are used to interpolate the necessary data. A significant feature of this interlace formatter is in its ability to look into the FIFO, get the two rasters needed depending on where the scan is on the photoreceptor, and interpolate therebetween in order to get the correct data within a fraction of a raster precision. 4.0 Resampling Interpolator with slope output Figure 6 and Figure 7 are detailed block diagrams of resampling interpolator 60 and associated circuitry. The outputs of the resampling interpolator are used for several image processing functions in the hyperacuity printer. First, sampled pictorial data can be reconstructed for use by the halftoner. Second, sampled text and lineart data can be reconstructed with an enhancement to the operation of the interpolator by retaining local slope information (rate of change of intensities), thereby enabling the prediction of where the sampled image will cross the xerographic threshold. The slope information will be used by the thresholder. Third, the interpolator may be used to resample the input data at different rates to achieve different resolutions, magnification and/or angular orientations of the input data. The resampling interpolator of this system is designed such that the target density or resolution can be written independent of the input resolution of the data in the memory. Finally, electronic registration, scan linearity correction, velocity servoing and photoreceptor skew correction can be performed in the printing system through suitable measurement and feedback of the errors in spot position. This information will be provided by the linearity and registration controller. Referring to Figure 6, resampling interpolator 60 receives two lines of adjacent raster data, appropriately labeled NEXT 91 and LAST 89, from the FIFO. The raster data is first processed by y interpolator 120 and then passed on to x interpolator 122 via data line 126. At the same time the raster data is being processed, slope information is being supplied by y interpolator 120 and x interpolator 122. Y slope information is processed by slope interpolator 124. Details of each interpolator will be discussed with reference to Figure 8. As part of the processing of both the raster data and the slope data, y interpolator 120 uses the fractional part of the number, YFRACT 101, from y accumulator 96. In a similar manner, x interpolator 122 and slope interpolator 124 both use the fractional part of the number, XFRACT 113, form the x accumulator 108. The outputs of resampling interpolator 60 include interpolated raster data, INTERPYINTERPX 128, with associated slope information, SLOPEX 132 and interpolated slope data SLOPEYINTERPX 134. Referring now to Figure 7, shown are detailed block diagrams of y interpolator 120, x interpolator 122 and slope interpolator 124. Raster data from the FIFO in the form of NEXT line 91 and LAST line 89 are supplied to y interpolator 120. These two rasters represent adjacent raster from an image stored in the FIFO. As shown, raster LAST 89 is subtracted from raster NEXT 91 using full adder 136. Subtraction is accomplished through twos complement arithmetic whereby bubble 137, on the B input of adder 136, denotes an inverse function and a + 1 on the C IN input adds the plus one necessary to carry out the twos complement function. The result of this subtraction is then multiplied by YFRACT 101 which is the fractional part of the number stored in y accumulator 96 (Figure 6). Also as a result of this subtraction is the associated y slope information shown as SLOPEY line 130. Multiplier 138 performs an 8x8 multiplication in which the eight most significant bits of the result, line 139, are passed on to full adder 140. Full adder 140 adds the result of the multiply with raster LAST 89 to complete the y interpolation, with the result being line INTERPY 126. INTERPY 126 represents the y interpolated data. X interpolator 122 receives its input from the output of y interpolator 120. Line INTERPY 126 becomes the A input to full adder 142. Also, line INTERPY 127 is delayed one pixel by register 141. The result of this delay, LAST INTERPY, is inverted by bubble 143 and then input into the B input of adder 143 with the carry input held high. Therefore, using twos complement arithmetic, LAST INTERPY is subtracted from INTERPY. The result of this subtraction produces associated x slope information SLOPEX 132. Also, the result of the subtraction becomes an input to multiplier 144. Multiplier 144 multiplies XFRACT 113, which is from the fractional part of the number in the x accumulator 108 (Figure 6), by the output of adder 142. The most significant eight bits of the result of the multiplication is then added to LAST INTERPY by adder 146, resulting in interpolated data INTERPYINTERPX 128. Slope interpolator 124 utilizes y slope information line SLOPEY 130 from y interpolator 120 as the A input into adder 148. SLOPEY 130 is delayed one pixel clock by register 147 becoming LAST SLOPEY. LAST SLOPEY is inverted via bubble 149 and thereby input into the B input of adder 148 with the carry input held high. The result is the subtraction of LAST SLOPEY from SLOPEY. The output of adder 148 is then multiplied by XFRACT 113 by multiplier 150. Multiplier 150 is a 5x5 bit multiplier, and the result is a five bit number being the five most significant bits resulting from the multiplication. In the final computation, the result of the multiplication, line 154, is added to LAST SLOPEY by adder 152 resulting in associated interpolated slope information denoted as SLOPEYINTERPX line 134. In summary, y interpolator 120 implements the equation: YFRACT*(NEXT-LAST)+LAST=INTERPY. X interpolator 122 uses the output of y interpolator 120 and implements the equation of XFRACT*(INTERPY-LAST INTERPY) + LAST INTERPY=INTERPYINTERPX. And finally, slope interpolator 124 implements the equation XFRACT*(SLOPEY-LAST SLOPEY) + LAST SLOPEY = SLOPEYINTERPX. Therefore, INTERPYINTERPX represents the interpolated value, SLOPEX represents the x slope of the interpolated value, and SLOPEYINTERPX represents the y slope of the interpolated value. (It should be noted that interpolation using the above equations is assumed to be linear as long as the value of YFRACT is not changing, or changing very slowly, with respect to XFRACT.) 5.0 Slope Thresholder The purpose of the slope thresholder is to render lineart or text from a sampled representation by generating linear approximations of edges, or transitions, and converting these approximations into sequences of intensity levels for the output laser diode. As will be shown, the slope thresholder takes linear interpolated data and associated two dimensional slope information from the resampling interpolator and provides a thresholding function. The two dimensional slope information is used to determine where the edge that is being rendered is located and oriented relative to the laser spot's current position. All the possible slopes and intensity input permutations are calculated by rendering software in advance of printing. Once determined, the thresholding function will provide multi-bit intensity level data stored in the thresholder look up table to be used to drive the laser diode. Figure 8 shows how the rendering software for the slope thresholder uses the intensity and slope information generated by the interpolator to position edges with high precision. The intensity 153, and the x slope and the y slope at the laser spot's current position is generated by the interpolator. The rendering software constructs an imaginary xerographic threshold as plane 165, positioned perpendicular to intensity 153, and having a nominal height of 50% of the maximum possible intensity of 153. In the x (fast) direction, the rendering software generates an imaginary ray 155 from the top of intensity 153, at an angle 154 determined by the x slope. This imaginary ray 155 will intersect the nominal xerographic threshold 165 at point 156. In the y (slowscan) direction, the rendering software generates an imaginary ray 158 from the top of intensity 153, at an angle 157 determined by the y slope. This imaginary ray 158 will intersect the nominal xerographic threshold 165 at point 159. In this way, points 156 and 159 are generated, and determine a line 166 which is an estimate of the feature edge to be rendered at the spot's current position. If the feature edge to be rendered is primarily perpendicular to the fastscan direction, timing is used to render the edge, and the rendering algorithm will utilize an on-to-off or off-to-on transition which has no intermediate gray values to achieve the steepest slope in exposure possible. The rendering algorithm must also take into account the fastscan width of the spot, the quantization of the fastscan clock, and other xerographic development transfer functions. If the feature edge to be rendered is primarily parallel with the fastscan direction, gray will be used to achieve subscan precision of the edge, and the appropriate gray value will be assigned to that memory location. The rendering algorithm must also take into account the slowscan width of the spot (which sets the linearity of the intensity to edge placement position), the number of levels being used, the scan pitch, and other xerographic development transfer functions. It should be appreciated that the method utilized here to estimate the position of the edge to be rendered is linear, that is, straight lines were used in the explanation for simplicity. However, rendering software for downloading into the look-up table is processed in advance of printing, therefore, speed in doing the calculations is of lesser importance. Therefore, more complicated methods of precalculating the position of the edges are quite appropriate, and are recommended. An example of the use of an equation to calculate the output of the thresholding function is: C = ( Sx 2 + Sy 2 / F 2 ) ½ ( Sx 2 + Sy 2 ) ( V - T ) where C represents the intensity value of the drive on the photodiode, S x is the x slope, S y is the y slope, T is the xerographic threshold and V is the magnitude of the sample being written. In accordance with the system disclosed, S x = SLOPEX 132, S y = SLOPEYINTERPX 134, and V = INTERPYINTERPX 128. T, the xerographic threshold, is typically set at the halfway point corresponding to how many bits of data are being used to modulate the intensity on the photodiode. For instance, if there are eight bits of data, then T = 128. F is a scaling factor to compensate for any difference in the slowscan and fastscan resolutions. If the resolutions are the same, F = 1. If the fastscan resolution is four times greater than the slowscan resolution, then F = 4. It should be noted that there are many equations that could be used to represent the thresholding function and the equation above is one example. Referring to Figure 9, shown is a block diagram including thresholder 68 and D to A converter 76. Included in thresholder 68 is threshold function block 160 and shift register 162. As shown, threshold function 160 receives three input lines from the resampling interpolator, INTERPYINTERPX 128, SLOPEX 132 AND SLOPEYINTERPX 134, representing interpolated data, the x slope, and the interpolated y slope, respectively. Using these input values, threshold function block 160 then renders a multi-bit intensity value to be used by the laser diode. This value can be calculated in real time or stored in a look-up table implemented with a memory block. It should be noted here that often memory access speed is limited to somewhat slower than the access requirements for retrieving data at the fastscan addressability clock, for instance 4800 clocks per inch (192 clocks/mm), or over 200 MHz in the example system. In that case, the memory fetch for the next n fetches is calculated beforehand by the rendering software, taking into account the spot's estimated position during that time, and the accesses for all n fetches are provided in one fetch in a memory which is n times bigger than otherwise. Those n m-bit values can then be loaded in parallel into an nxm bit shift register 162 and shifted out to the D to A converter, m bits at a time, n times faster. When using n fetches, the value in x INC register 114 of phase resolution controller 66 (Figure 4) must be multiplied by n. Subsequent fetches would then be made on the next SYSCLK, skipping over n memory locations to the next multi-value fetch. When threshold function block 160 is implemented as a look-up table, the resulting intensity values, C, are precalculated and stored in a memory block. During run time operation, SLOPEX 132, SLOPEYINTERPX 134, and INTERPYINTERPX 128 are used to address the memory block for the value of C corresponding to the inputs. The slope values take the place of absolute page coordinates to help form the address. Therefore, for each possible address combination, there is a precalculated intensity value corresponding to each address. As previously mentioned, the value of C may also be implemented such that C is calculated during run time operation by using a fast processor or calculating circuitry. 6.0 Halftoner The purpose of the halftoner is to convert the resampled information from the interpolator into a binary map when pictorials are to be rendered. Pictorials require many levels of gray to be contour free, as well as a fine dot structure to make graininess imperceptible. When representing toneart, that is, pictorials or "natural" images intended for halftoning, sampled data, like that used from the resampling interpolator, can be utilized. Figure 10 is a diagram representing the full creation of a halftone dot in a hyperacuity printer. Shown is halftone cell 170, which for this example, is a 1/150 inch (170 µm) on a side, and a halftone dot 172, which is to be rendered. In this example, spot 182 size is 1/600 of an inch (43 µm) with addressability within cell 170 of 1/4800th of an inch (5 µm) in both the fastscan and slowscan directions as indicated by addressability unit markers 176 and 178. Waveforms above and to the left of cell 170 represent the fastscan and slowscan exposure profiles of the laser diode as it writes halftone dot 172. The fastscan exposure profile is taken through section A-A, and the process direction, or slowscan, exposure profile is taken through section B-B. Lines 174 represent scan trajectories and are spaced 1/1200 of an inch (21 µm) apart. This means that the scanner is overscanned, which delivers a more linear response of edge placement to intensity adjustment than if the spot were 1/1200 of an inch (21 µm), however, the system can be tuned to any spot size or any scan spacing. The resolution of the spot is determined by the spot size 182. Because of this the diagram showing the different spot sizes 175, 175', 175" may be misleading to the casual reader in implying that the resolution has been increased because the spot size is smaller. On the contrary, the diagram is for explanation convenience only, and the size of the spots 175, 175', 175" helps represent where in the process direction the exposure edge will be rendered with the given intensity. In other words, the smaller spots represent the effect the boundary scan has on the rendered edge. In all cases, the spot size 182 is unchanged because the spot size is measured as the "full width at half max", (FWHM), a measure which is intensity invariant. In order to render halftone dot 172, scan spots are placed using high addressability such that the edges of the exposure produced by the scanning spot are as close as possible to the target edge of the halftone dot 172. In the fastscan direction, high addressability is accomplished by turning the laser on and off at appropriate places along the scan with 1/4800th of an inch (5 µm) accuracy, which best renders edges which are more or less perpendicular to the scanning direction. In the slowscan direction, intensity adjustment of the boundary scans can be used to gain the high addressability at 1/4800th of an inch (5 µm) as explained in US-A-5,138,339, and best renders edges which are more or less parallel to the scanning direction. Figures 11 through 26 show a step-by-step creation of halftone dot 172. Referring to Figures 11 through 26, inclusive, shown in each Figure is a step in the creation of a halftone dot 172 with an associated intensity profile 180. As previously mentioned, the purpose is to try to keep the exposures as close as possible to the target boundary of the halftone dot 172. Beginning with Figure 11, the first scanline, a first boundary scan of intensity two is shown, and exists for two addressability units in the fastscan direction. Next Figures 12 and 13 show the boundary scan increasing to intensity three for several fastscan clocks, and then returning to a scan of intensity two for several clocks as shown in Figure 14 to finish scanline one. Going on to the next scanline and Figure 15, the intensity of the boundary scan starts off at level four, or full on, and remains on until the end of the scanline as shown in Figure 16. Scanlines three and four are written like scanline two, and are represented by Figures 17, 18, 19, and 20. Referring next to Figure 21, scanline five begins with an intensity level three, going to full on (Figure 22 and 23) then finishes with a level three intensity (Figure 24). Finally, the halftone dot 172 is completed with scanline six writing intensity level one values as shown in Figures 25 and 26. Referring now to Figure 27, shown is block diagram of a high addressability halftoner 70. Halftoner 70 has a memory block 190 (which is equivalent to dot generator 72) and accumulators 194 and 196 (which are equivalent to screen generator 74). Memory block 190 is divided into an array of addressability units 192 which for this example are 1/4800 of an inch (5 µm) square. Memory block 190 is also divided into 256 different dot planes, one for each grayscale data level. Addressability in the fastscan direction, as designated by arrow 194, is 1/4800th of an inch (5 µm) and scan spacing in the slowscan direction is 1/1200th of an inch (21 µm) as designated by arrow 204. The use of four levels of intensity (created by five values 0%, 25%, 50%, 75%, 100%) instead of binary intensity allows addressability in the slowscan direction to also be 1/4800th of an inch (5 µm). Memory 190 is presented as a slice through the intensity dimension of a three dimensional object by dividing its address lines into three sets, one for the x dimension, one for the y dimension and one for the intensity dimension. These addresses are provided by x address accumulator 194, y address accumulator 196 and grayscale data input line 197. The x and y address accumulators provide the x and y component of the screen address used for rotating the screen, such as shown in angle 202. This address information represents the position of laser diode spot 200 in the x,y plane. For the intensity dimension, the grayscale data input 197, from the interpolator, determines which one of the 256 dot planes, such as 206 or 207, for instance, is being accessed. Typically, as the value for the grayscale data 197 changes, the radius (or other density determining parameter) of the halftone dot changes, as represented by the three dimensional "bullet" shape 199 in the diagram. One of the 256 halftone dots, this one in the midrange where the memory has been sliced, is shown as halftone dot profile 198 in dot plane 206. If a highlight dot is to be rendered, the grayscale data in 197 will select a dot plane with a dot profile that has a small radius, such as might be in plane 207. A shadow dot would be on the other side of plane 206 from 207 and is not shown. The halftone screen is a regular rectangular array of dot positions covering the entire image. Accumulators 194 and 196 track the trajectory of laser spot 200 as shown by line 195. The trajectory moves through the memory at a screen angle  shown as arrow 202. The five bits from each of the accumulators provide a thirty-two by thirty-two position halftone tile for defining a halftone dot for each of the 256 halftone dots addressed by grayscale input value 197. Therefore, there are a total of 1024 positions within each tile defining a given halftone dot. As will be discussed in reference to Figure 28, accumulators 194 and 196 increment fractionally, but provide their five most significant bits as address YADDR and XADDR to memory 190. As they increment they will periodically rollover in x and y, thereby replicating the halftone cell , into a regular two dimensional array over the entire image area. While this is occurring, the intensity is varying, causing different dot layers to be accessed, thereby modifying the density of the image. The data fetched at each memory location is one of the five values (0%, 25%, 50%, 75%, or 100%) required by the digital to analog converter to cause the laser spot to have the intensity required to render the halftone dot as shown in Figures 10 through 26. There are two notable characteristics of this halftone generator 70. First, the value obtained at each memory location is a value meant to go directly to the D to A converter 76. This differs from prior art where numbers accessed are generally compared with the input intensity value 197 by a binary comparator which outputs either a zero or one to drive the laser diode, and which has only two dimensions in the memory array since the grayscale data in 197 is sent to the comparator instead of the third dimension of a memory array. The prior art saves memory but this method is more flexible and allows gray output. Secondly, the halftone cell is accessed scanline by scanline, which is a number of addressability units (gray levels) apart. For instance, in this example, there are four gray levels, allowing four subscanline edge position opportunities per scan, and there are 1200 scans per inch (48 scans/mm), giving 4800 per inch (192/mm) addressability in the slowscan direction. The scans are four addressability units apart. Because of this, the scans only sample one fourth of the memory locations available in the halftoner as the scan structure passes through. This means there are four phases which the scan structure can pass through. Since the halftoner accumulators are loaded with numbers which have fractional values, (because there is a requirement for halftone screen at any frequency and any angle), the scan access will typically roll through all four phases periodically as the image is made. This creates an irrational halftone generator. It is the intent of the rendering data loaded into the halftoner to achieve the same exposure on the photoreceptor regardless of the phase in which the scan structure passes through the cell. This will be helped by (1) increasing the addressability in both the fastscan and slowscan direction to minimize quantization error; and (2) overscanning as in US-A-5,138,339, which allows better control of exposure edge placement in the process direction. It should be noted here that often memory access speed is limited to somewhat slower than the access requirements for retrieving data at the fastscan addressability clock, for instance 4800 clocks per inch (192 clocks/mm), or over 200 MHz in the example system. In that case, the memory fetch for the next n fetches is calculated beforehand by the rendering software, taking into account the screen angle and frequency, and the accesses for all n fetches are provided in one fetch in a memory which is n times bigger than otherwise. Those n m-bit values can then be loaded in parallel into an nxm bit shift register (like shift register 162 of Figure 9) and shifted out to the D to A converter m bits at a time, n times faster. Referring to Figure 28, shown is detailed block diagram of both the x address accumulator 194 and y address accumulator 196. X address accumulator has register 210 which accumulates a value having five bits of integer and eleven bits of fraction. The five bit integer portion supplies signal XADDR, or the x address, for memory block 190 of the halftoner. Accumulator 194 includes full adder 212 and multiplexer 214. Multiplexer 214 selects either INC register 216 or register 218 to be an input to full adder 212. Register 218 accumulates a value having five bits of integer and eleven bits of fraction. Accumulator 194 further includes full adder 220 and multiplexer 224. Multiplexer 224 selects either INC register 226 or First register 228 as an input into full adder 224. The hardware of Y accumulator 196 is identical to that of x accumulator 194. As will be seen, the differences are in the values loaded into the registers, as the registers represent the x and y component of the screen angle. Y address accumulator 196 has register 230 which accumulates a value having five bits of integer and eleven bits of fraction. The five bit integer portion supplies signal YADDR, or the y address, for memory block 190 of the halftoner. Accumulator 196 includes full adder 232 and multiplexer 234. Multiplexer 234 selects either INC register 236 or register 238 to be an input to full adder 232. Register 238 accumulates a value having five bits of integer and eleven bits of fraction. Accumulator 196 further includes full adder 240 and multiplexer 242. Multiplexer 242 selects either INC register 244 or First register 246 as an input into full adder 242. In operation, There are six registers that must be pre-loaded, three in x accumulator 194 and three in y accumulator 196. The values in the registers are variable and must be calculated prior to printing a page. Calculations will be discussed below. The following are assignments of variables to their respective registers: XFastSteplNC is loaded into INC register 216; XSlowBoLINC is loaded into INC register 226; XSlowBoLFIRST is loaded into FIRST register 228; YFastSteplNC is loaded into INC register 236; YSlowBoLINC is loaded into INC register 244; and YSlowBoLFIRST is loaded into FIRST register 246. In order to scale the locations of boundaries to the halftoner memory array, the number of memory locations per SYSCLK (the number of memory locations traversed during one SYSCLK) must be determined in the fastscan direction, and the number of memory locations per facet (the number of memory locations traversed while the polygon rotates through one facet) must be determined in the slowscan direction. Equation 1 Memory Locations SYSCLK = Memory Locations CELL × CELLS INCH × INCHES SYSCLK Equation 2 Memory Locations FACET = Memory Locations CELL × CELLS INCH × INCHES FACET Once this scaling information is calculated, it can be used in turn to calculate the x and y components for the six registers. Referring also to Figure 29 to help understand the relationship among the registers and their load values, the spot can be visualized as traveling over the face of a cross-section of the halftone memory, where block 192 represents an addressability unit. The values in the registers track the movement of the spot and the modulo of the number of memory locations per cell over the face. The x and y FastSteplNC registers together track the spot's fastscan progress, and have units of memory locations per SYSCLK, since the registers are clocked by the SYSCLK. The x and y SlowBoLINC registers together track the spot's slowscan progress, and have units of memory locations per facet, since the registers are clocked once per facet by the start of scan signal. Both the fast and slow increments above are the same for all channels, regardless of the number of sources, however, the SlowBoLFIRST is dependent on the channel number (ch). I is the interlace factor in scans per beam and Q is the number of scans per facet (i.e., the number of beams in a multi-beam system), so I/Q has units of facets per beam. Therefore, the units for SlowBoLFIRST is (memory locations per facet) times (facets per beam) = (memory locations per beam). The slow beginning of line first value (SlowBoLFirst), then, is the number of memory locations to stagger each channel in a multi-beam system at the start of an image to accomplish interlacing. For example, in the fastscan direction, memory locations per cell = 32, halftone cells per inch = 150 and inches per SYSCLK = 1/4800, therefore memory locations per SYSCLK = 1. In the slowscan direction, memory locations per cell = 32, cells per inch = 150 and inches per facet = 600, therefore, memory locations per facet = 8. And finally, Θ = 45 degrees. Substituting in to the above equations, values for XFastSteplNC = 0.707, XSlowBoLINC = 5.667, XSlowBoLFIRST = 0 for channel 0, 8.485 for channel 1, etc., YFastSteplNC = 0.707, YSlowBoLINC = 5.667, and YSlowBoLFIRST = 0 for channel 0, 8.485 for channel 1, etc. and can be loaded into their respective registers. In Figure 29, arrow 250 represents both the XFastSteplNC and the YFastSteplNC vector. In a similar manner, arrow 252 represents both the XSlowBoLINC and YSlowBoLINC vectors. Arrow 254 represents both the XSlowBoLFIRST and YSlowBoLFIRST vector for a channel greater than zero, in this case channel 1. Arrow 256 represents an extension to arrow 250 by a magnitude n, where n is the number of values fetched on one memory access as described above. Note that for a sixteen bit register, five bits above the heximal point leaves eleven bits below the heximal point for the fractional part as shown in Figure 28. These values are found by normal twos compliment arithmetic by converting the decimal value to its twos compliment hexadecimal value. In operation, registers 216, 226, 228, 236, 244, and 246 are pre-loaded. At the start of printing a page, the value stored in FIRST register 228 becomes an input to full adder 220 through MUX 224. In a similar manner, the value stored in FIRST register 246 becomes an input to full adder 240 through MUX 242. Once the page has begun, MUX 224 switches so that the input to full adder 220 is the increment value stored in INC register 226. In a similar manner, MUX 242 switches to INC register 244. Register 218 is incremented by the value from INC register 226 every SOS, (start of scan) clock. Register 238 is also incremented by the value in INC register 244 every SOS clock. In this way, the starting position for the scanlines within the halftone screen are calculated as printing progresses. At the start of each scan, the value in register 218 becomes an input into adder 212 through mux 214 and the value in register 238 becomes an input into adder 232 through mux 234. Mux 214 then switches to connect INC register 216 to adder 212 and mux 234 switches to connect INC register 236 to adder 232. Register 210 is then incremented by the value from INC register 216 upon every SYSCLK (system clock). Register 236 is also incremented by the value in INC register 236 upon every SYSCLK. As defined using the example values, SYSCLK occurs ever 1/4800th of an inch. In this way, the incremental positions of the spot within the halftone screen are calculated as printing progresses. These values are then used as the memory address for a memory fetch operation. For the case where the memory accesses would occur too fast for a physical memory fetch as mentioned above, the equation for the memory access address in the fastscan direction is multiplied by n where n is the number of MicroCLKs per SYSCLK, and n simultaneous fetches are made while making the memory data word n times bigger. Subsequent fetches would then be made on next SYSCLKs, skipping over n memory locations to the next multi-value fetch. In this case, distance in the fastscan direction would be measured in MicroCLKs instead of SYSCLKs. Therefore, it would be necessary to modify equation 1 above to determine the number of memory locations per MicroCLKs as follows: For example, if eight memory fetches were to be made in one SYSCLK, the number of SYSCLKs per MicroCLK would be 1/8, or in other words, there would be 8 MicroCLKs per SYSCLK. Memory Locations MicroCLK = Memory Locations CELL × CELLS INCH × INCHES SYSCLK × SYSCLK MicroCLK The information stored in memory 190 of Figure 27 must also be precalculated before printing a page. In general, for each defined intensity 197 input to the memory, a corresponding halftone dot must be defined and stored in memory planes 206 or 207, for example. Once the shape and size of the halftone dot is defined with respect to the size of the memory array, each dot needs to be scaled, thereby defining it in the fastscan direction as number of SYSCLKs, and the slowscan direction as number of levels. Finally, each halftone dot has to be rendered such that each memory location defining the halftone dot is assigned a value corresponding to the intensity of the signal being written by the laser diode. In order to calculate values for the halftone look-up table, several parameters must be defined. These parameter are screen angle and screen frequency in both the fastscan direction and slowscan direction, modulus and addressability in the fastscan direction, number of facets per inch, number of scans per facet (Q), and number of levels per scan in the slowscan direction. In keeping with the exemplary values used throughout this disclosure, screen angle is 45 degrees, screen frequency is 150 halftone dots per inch, modulus is 32 (corresponding to the number of bits used for the x and y addressing), fastscan addressability is 4800 SYSCLKs per inch (192/mm), facets per inch equal 600 (24/mm), scans per facet equal 2 and levels per scan equal 4, for a slowscan addressability of 4800 levels per inch (192 levels/mm) also. Referring to Figures 30 through 33, inclusive, shown are halftone dot planes with examples of halftone dots. As previously mentioned, the first step in determining values for the halftone memory is to grow, or define, each of the 265 halftone dots. The dots can be grown utilizing an algorithm that takes into account the TRC (Tonal Reproduction Curve) and shape of the dots. Figures 30 through 33 give four examples of such dots. It can be appreciated that there are many ways to grow halftone dots as well as many shapes that halftone dots can have. The following dots are examples and are not meant to be limiting. Beginning with Figure 30, halftone dot plane 260 represents a 32 x 32 bit address space defining halftone dot 261. As can be seen halftone dot 261 is essentially circular and represents a certain density. As the densities increase, the halftone dots grow as shown in Figure 31. Figure 31 shows halftone dot plane 262 with corresponding dot 263. It is to be understood that there are many levels and sizes of halftone dot between Figures 30 and 31. Also shown is diamond region 264. As the halftone dots are grown, they remain circular until they hit the boundary of diamond region 264. Then the dots are grown radially to fill in the diamond region, until the entire region is filled as shown in Figure 32. Figure 32 shows dot plane 266 with corresponding dot 267. Dot 267 can represent a density level near 128, since it conveys an attempt to render fifty percent of the halftone cell as black, however, when taking into account the tone reproduction curve of the printer, this fifty percent coverage may not exactly coincide with intensity level 128. Figure 33 shows halftone dot plane 268 with dot 269 grown outside of diamond region 264, again radially. However, the centers of radius not are at the corners of the dot cell 268, The blank semi-circular spaces at the corners of cell 268 join with other cells to make round spots with diminishing size. As can be envisioned, the halftone dots would continue to grow until a dot plane is completely filled at density level 255 (not shown). The next step in determining the values for the halftone memory is to scale the grown halftone dots into printer defined space of SYSCLKs in the fastscan direction and intensity levels in the slowscan direction. The distance to any boundary from any memory location is available in units of memory locations, as these were the units used to grow the halftone dot in the previous paragraph. The component of distance in the fastscan direction is multiplied by the number of SYSCLKs per memory location, as previously calculated in equation 1 above, to determined the distance in SYSCLKs. Likewise, the component of distance in the slowscan direction is multiplied by the number of levels per memory location to yield the distance in levels. To obtain the number of levels per memory location, two additional pieces of information are required in addition to equation 2 above. Equation 2 gives the number of memory locations per facet, to which the number of facets per scan (e.g. 1/2), and the number of scans per level (e.g. 1/4) are multiplied to obtain the number of memory locations per level as follows: Memory Locations LEVEL = Memory Locations CELL × CELLS INCH × INCHES FACET × FACET SCAN × SCAN LEVEL Finally, each edge in each of the 256 halftone dot planes needs to be rendered by assigning an intensity value to the memory locations. If the edge to be rendered is primarily perpendicular to the fastscan direction, timing is used to render the edge, and the rendering algorithm will utilize an on-to-off or off-to-on transition which has no intermediate gray values to achieve the steepest slope in exposure possible. The rendering algorithm must also take into account the fastscan width of the spot, the quantization of the fastscan clock, and other xerographic development transfer functions. If the edge to be rendered is primarily parallel to the slowscan direction, gray will be used to achieve subscan precision of the edge, and the appropriate gray value will be assigned to that memory location. The rendering algorithm must also take into account the slowscan width of the spot (which sets the linearity of the intensity to edge placement position), the number of levels being used, the scan pitch, and other xerographic development transfer functions. This can be seen in Figure 10, and Figures 11 through 26, where the fastscan is using timing to render and the slowscan is using gray to render. As previously mentioned, these intensity values go directly to a D to A converter to be written by the laser diodes. One way of rendering the halftone dot is to grow gaussians of the shape of the spot at every possible position for the laser spot in the immediate past and future with respect to a memory location. The gaussians could be grown such that in the center of the halftone dot they would be full on and outside the halftone dot they would be full off (for a write black system, for instance), and at the edge of the halftone dot they would be grown just enough to move the exposure edge, (which is the composite sum of all the gaussians), over to coincide with the boundary of the dot being rendered. Once that is done, the level needed for that particular memory location can be written. So far, the halftoner has been described using a fixed number of memory locations per halftone cell dictated by a fixed addressing scheme. However, there may be cases where a 32x32 bit halftone cell is not optimum for a particular application. Therefore, a variable modulus addressing scheme would be useful. Such a scheme could allow any sized halftone cell to be used without changing the hardware. Referring to Figure 34, shown is a block diagram of a variable modulus address accumulator 270. Address accumulator 270 may be substituted for either or both x address accumulator 194 and y address accumulator 196 (Figures 27 and 28). The use of address accumulator 270 provides the opportunity to use a smaller memory block or different sized halftoned cells. It can be appreciated that this scheme as described can accommodate any modulus up to 32, including a fractional modulus. However, by adding more bits to the addressing, a modulus of any size can be envisioned. A variable modulus accumulator can be substituted for the x accumulator, the y accumulator, or both. Variable address accumulator 270 includes register 272 which provides a five bit integer value, ADDRESS line 274, to be used in addressing the halftoner memory block. ADDRESS line 274 can be substituted for the XADDR line or YADDR line. Address accumulator 270 also includes mux 276, start block 278, mux 280, full adder 282 INC register 284 and subtracter 286. The components in START block 278 are equivalent to components described in Figure 28 and are operated accordingly. In particular, register 292 is equivalent to register 218 and 238. Full adder 294 is equivalent to adder 220 and 240. In a similar manner, mux 296 is like mux 224 and mux 242, and INC register 298 is equivalent to INC register 226 and INC register 244. INC register 298 is loaded with a SlowBoLINC value, a XSlowBoLINC value if it is being used as an x address and a YSlowBoLINC value if it is being used as a y address. Finally, FIRST register 299 is like that of FIRST register 228 and FIRST register 246, and are loaded with x and y SlowBoLFIRST values. Comparing other components of address accumulator 270 with address accumulators 194 and 196, register 272 is like register 210 and register 230. Mux 276 is equivalent to mux 214 and mux 234. And, adder 282 is like adder 212 and adder 232. Returning to the variable modulus operation of address accumulator 270, a desired modulus number is supplied to subtracter 286. INC register 284 is pre-loaded with a FastSteplNC value. As adder 282 and register 272 accumulate a value in FastSteplNC increments, subtracter 286 is evaluating the accumulated value to determine if it is larger than the value of modulus line 288. If the accumulated value is larger than the modulus value, carry out line 290 of subtracter 286 switches mux 280 to allow the value in subtracter 286 to be loaded into register 272. This switching essentially subtracts out the modulus value in register 272 clearing the five integer address bits, which is equivalent to a roll over. The value in register 272 can then begin to accumulate again. Therefore, the value on address line 274 is never larger than the modulus value and the halftone memory is address at the set modulus. Note that the extra circuitry needed to implement variable modulus addressing may cause a timing delay which can be remedied by adding pipelined stages. 7.0 Transformation of Halftone Dots The purpose of transformation of halftone dots is to provide a means for embedding data into the halftoned image via the rendered halftone dot. Rotation of halftone dots will be described below with reference to Figures 35 through 41, building on the halftoner configuration described above. As discussed in the background, by applying different angles of rotation to the non-rotationally symmetric halftone dot while printing, various forms of encoded data can be embedded into the the halftone structure. Furthermore, this application of transformation will rotate the halftone dot without effecting the screen frequency, screen angle or halftone dot density. In practice, data used to determine the halftone dot rotation angle must accompany the image data. As image data is presented to the halftone generator, so is rotation angle data presented to the address transformation electronics. It is not the desire here to present an encoding algorithm for embedding data, but instead to present how to successfully rotate a halftone dot by any angle without effecting the screen angle or other attributes which effect image viewing quality. The highlight region is the range of densities where the halftone dot is small. The shadow region is the region where the dots are so large that the absence of dot coverage is small. In between these two extremes is the midrange of densities. This method of presenting encoded information in the halftone dot works only in the midranges of tone density, when the dot structure is neither too small to have a shape nor so large as to have its shape distorted by adjacent dots. First, the mathematics of the transformation will be presented with reference to Figure 35, then block diagrams for exemplary hardware configurations will be presented with reference to Figure 36 and Figure 37. To best depict the concept, complex numbers are used to present the transformation since they offer simplification in presenting it. Figure 35 shows a complex plane with horizontal Real and vertical Imaginary axis, Re and Im respectively, and a point a + ib subtending an angle  o and having a radius vector r o . By inspection, a = r o ·cos  o , and b = r o ·sin  o . Let r o = 1 so that the point is on the unit circle, therefore a = cos  o , and b = sin  o . Let x + iy be a second point anyplace in the complex plane, so that it subtends an angle  with a radius r. The object is to find the point x' + iy' after rotating the point x + iy by  o around the origin. This is done by observing that x + iy may be represented by r·e i , and a + ib can be represented by e i o , so that the rotated point is represented by r·e i( o + ) . Then: r·e i( o + ) = = r·(e i o )·(e i ) r·(a + ib)·(x + iy)=(ax-bx) + i(ay-bx) = x' + iy'. So x' = ax-by, and y' = ay + bx, and the new point will be rotated a total of  +  o , and will have the same radius r as before. Therefore, the purpose of the hardware to implement this transformation would be to provide a transformed x address (x') into a memory by implementing the equation x' = ax-by. In a similar manner, the hardware would also provide a transformed y address (y') into a memory by implementing the equation y' = ay + bx. Figure 36 shows a block diagram of exemplary additional addressing circuitry or hardware needed to rotate an elliptical halftone dot 300 stored in memory 190. Note that memory 190 is the same as dot generator 72 (Figures 2 and 3) and that all the remaining components shown in Figure 36 make up an alternate embodiment of screen generator 74 (Figures 2 and 3) of halftoner 70. Also, components which are the same as those described with reference to the halftoner configuration of Figures 27 and 28 are numbered the same. In general, the first operation transforms the center of rotation from the origin 301, to the center of the halftone dot which is at a predetermined position from the origin , by subtracting an offset constant from the x and y inputs. The y offset coordinate is stored in register 302 and the x offset coordinate is stored in register 304. The multipliers and adders are then arranged to provide the transformed outputs (ax-by) for the x coordinate and (ay + bx) for the y coordinate, as in the above calculations. Finally, the coordinates are transformed back to the origin by adding the same constant which was originally subtracted. Although shifting about the center of the halftone dot is disclosed, it can be appreciated that the rotation point does not have to be the center of the halftone dot. For the purpose of simplifying this example, assume that the values in registers 302 and 304 are equal to zero. (Non-zero values will be discussed later in this disclosure.) To solve for y', the value in register 302 is subtracted from variable y, (which is in y address accumulator 196), by subtracter 306. Since the value in register 302 is zero for this example, the result of the subtraction is simply y. The value y is then multiplied by the value a in multiplier 308 resulting in the value ay. To calculate the bx component of the equation, the value in register 304 is subtracted from variable x, (which is in the x address accumulator 194), by subtracter 310. Since the value in register 304 is zero for this example, the result of the subtraction is simply x. Variable x is then multiplied by variable b in multiplier 312 resulting in bx. Adder 314 then adds ay and bx together resulting in ay + bx. Adder 316 then adds the value stored in register 302 back into the ay + bx equation. Since this value in register 302 is zero, the output of adder 316 is ay + bx which is equal to transformed y address, y'. To solve for x', the value in register 304 is subtracted from variable x, by subtracter 306 as previously stated. The result of this subtraction, which is x, is then multiplied by variable a in multiplier 318 resulting in value ax. To calculate the by component of the equation, variable b is multiplied by y in multiplier 320. Subtracter 322 then subtracts the output of multiplier 320 from the output of multiplier 318 resulting in ax-by. Adder 324 then adds the value stored in register 304 back into the ax-by equation. Since this value in register 304 is zero, the output of adder 324 is ax-by which is equal to transformed x address, x'. It can be appreciated that rotations by 90 degrees, where a equals zero and b equals one, for instance, can be accomplished easily without full implementations of expensive multipliers. From the equations, the new coordinates would be y' equals x and x' equals minus y. When register 302 and register 304 contain zero, then the rotation takes place about the origin, 301. When these registers contain non-zero values, they then serve to transform the position of the center of rotation. In order to rotate the halftone dot about its center, registers 302 and 304 would contain the coordinates of the center of the halftone dot and would figure into the equations described above accordingly. Therefore, given the values for a, b, x, and y, the halftone dot is rotated while being read out of the memory using the rotational addressing circuitry described above. Figure 37 shows another embodiment of the addressing circuitry or hardware needed to rotate an elliptical halftone dot 300 and then shift the center of the dot to a new location. In essence, the center of the halftone dot is at some predetermined position from the origin when it is stored in memory. Once the dot is rotated, it can then be shifted away from that predetermined position using the additional shifting circuitry that is part of the addressing circuitry. As shown, registers 332 and 334 have been added to be used in shifting the location of the halftone dot after it has been rotated. Offset values can be placed in registers 302 and 304 to shift the center of rotation to the center of the dot. Then, shifting values stored in registers 332 and 334 could be used to shift the center of the rotated dot to a new location just prior to accessing memory 190. One advantage to shifting the center position of the dot is to eliminate image distortion such as moire. White noise can be used to generate random repositioning offset values for registers 332 and 334 to help mask the effects of systematic errors introduced by the halftoner's positioning and gray quantization roundoff. As can be appreciated, shifting the position of the halftone dot could be used with circular, or symmetrical dots as well as non-rotationally symmetrical dots. Figures 38 through 41 show various exemplary rotations and positions of an elliptical dot 342 in a dot plane 340. Figure 38 is an example of a typical elliptical dot 342 centered at the center of dot plane 340. Figure 39 shows dot 342 rotated about its center point. Figure 40 shows a non-rotated elliptical dot 41 shifted from the center of dot plane 340. And finally, Figure 41 shows a rotated elliptical dot 342 shifted from center of dot plane 340. It should be noted that variables, a = sin Θ and b = cos Θ can be continuously varied by an outside process to vary the dot rotation angle. This process must be synchronized with the halftone dot rate so that anytime the halftone generator is rendering a particular halftone dot, the same rotation angle is presented to the generator for the entire rendering process for that dot. However, the variables a and b can change for the next dot to be rendered. In summary, the present invention allows rotation of non-rotationally symmetrical halftone dots. By utilizing such rotations, embedded data could be incorporated within the halftone dots.

Claims: A hyperacuity printing system for rendering halftoned image data on a recording medium (14), comprising: a data source (23) for supplying a grayscale input image data; halftoning circuitry (70) for receiving said grayscale input image data from said data source and transforming said grayscale input image data into multi-bit value output data, said halftoning circuitry including a memory containing a look-up table (190) for providing said multi-bit value output data, wherein said multi-bit value output data are arranged in said look-up table as a quasi-continuum for rendering halftone dots from a lowest density value to a highest density value corresponding to said grayscale input image data; addressing circuitry providing a first set of address lines and a second set of address lines defining a screen stored in said memory, said first set of address lines and said second set of address lines providing x and y address values defined by a desired screen angle, said addressing circuitry including rotating circuitry (308,312,...,324) for modifying said x and y address values; and a modulating device for receiving said multi-bit value output data from said halftoning circuitry and converting said multi-bit value output data into a form which can be directly utilized by a writing device for writing said multi-bit value output data onto said recording medium, characterized in that said rotating circuitry is adapted to receive external rotation angle data individually for each pixel in order to embed data into a halftone image by individually rotating said halftone dots around predefined angles corresponding to various forms of encoded data to be embedded, while said halftone dots are being read from said look-up table. A hyperacuity printing system according to claim 1, wherein said halftone dots have a reference point which is a predetermined position away from a predefined origin in said look-up table, said addressing circuitry further including shifting circuitry (306,310) for modifying said x and y address values thereby shifting said reference point of said halftone dots away from said predetermined position from said defined origin in said look-up table, said shifting and said rotating occurring while said halftone dots are being read from said look-up table. A printing system according to any of claims 1 or 2, wherein said modulating device transforms said multi-bit value output data from said halftoning circuitry into a form which can be directly utilized by said writing device by intensity modulating scan spots in accordance with said multi-bit value output data. A printing system according to any of claims 1 to 3, wherein said look-up table (190) is a three-dimensional memory device. A printing system according to any of claims 1 to 4, wherein said addressing circuitry further includes variable modulus circuitry allowing said first set of address lines and said second set of address lines to operate at variable modulii. A printing system according to any of the preceding claims, wherein said addressing circuitry provides for a multiple number of said multi-bit value output data values to be accessed from said three-dimensional memory device simultaneously. A printing system according to any of the preceding claims, wherein said rotating of said halftone dots is about a center of said halftone dots, or about a non-center of said halftone dots. A printing system according to any of the preceding claims, wherein said halftone dots are non-rotationally symmetrical. A printing system according to any of the preceding claims, wherein said halftone dots are elliptic.

Inventors: CURRY DOUGLAS N, CURRY, DOUGLAS N.
Applicants: XEROX CORP, XEROX CORPORATION
Application year: 1995
Main classification: H04N1
IPC codes: H04N1, B41J2, G03G21