Patent Publication Number: US-9904206-B2

Title: Image forming apparatus

Description:
BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The aspect of the embodiments relates to image forming apparatuses which correct distortion and density unevenness of a 2D image at a time of image formation, such as digital copiers, multifunction devices, and laser printers. 
     Description of the Related Art 
     As an electrophotographic method of an image forming apparatus, such as a laser printer or a copier, a method for forming a latent image on a photoreceptor using an optical scanning device which performs scanning using laser light is generally employed. In such an optical scanning device employing a laser scanning method, laser light formed in parallel light using a collimator lens is deflected by a rotatable polygonal mirror and an image is formed on the photoreceptor using the deflected laser light through a long fθ lens. Furthermore, such an optical scanning device employs a multi-beam scanning method for performing scanning simultaneously using a plurality of laser beams emitted from a multi-beam light source having a plurality of light emitting elements in one package. 
     On the other hand, to form an excellent image which does not include density unevenness or banding, in one embodiment, scanning lines of laser light are arranged at regular pitches on the photoreceptor. However, the pitches between the scanning lines may vary due to a plurality of reasons below. For example, the pitches between the scanning lines vary due to variation of a surface speed of the photoreceptor, variation of a rotation speed of the rotatable polygonal mirror, or the like. Furthermore, the pitches between the scanning lines also vary due to variation of an angle of a mirror surface of the rotatable polygonal mirror relative to a rotation axis of the rotatable polygonal mirror or variation of pitches between light emitting points arranged on a laser chip in a case of the multi-beam light source. In  FIG. 16A , scanning with laser light is denoted by horizontal lines and a state in which pitches between the scanning lines periodically vary is illustrated. As illustrated in  FIG. 16A , development is performed with high density in a case where a pitch between the scanning lines of the laser light is small whereas development is performed with low density in a case where a pitch between the scanning lines of the laser light is large, and accordingly, a stripe pattern (moire) is likely to be detected. To address such density unevenness and banding caused by the reasons described above, a technique of correcting banding by controlling an exposure amount of the optical scanning device has been proposed. For example, Japanese Patent Laid-Open No. 2012-098622 discloses a configuration in which a beam position detection unit for a sub scanning direction is disposed in the vicinity of a photoreceptor and an exposure amount of an optical scanning device is controlled based on scanning pitch information obtained from detected beam positions so that banding becomes unnoticeable. 
     Furthermore, an image forming apparatus performs a halftone process on image data using a dither pattern so that a halftone (intermediate gradation) is expressed. A line screen or a dot screen, for example, is used for an image which is subjected to the halftone process. 
     However, some screens used in the halftone process are affected by tilt of the mirror surface of the rotatable polygonal mirror (hereinafter simply referred to as “plane tilt” of a rotatable polygonal mirror) and others are not.  FIGS. 16B and 16C  are diagrams illustrating a phenomenon of the plane tilt of the rotatable polygonal mirror. In  FIGS. 16B and 16C , gray portions denote dither patterns. Furthermore, light gray portions (white portions) denote portions in which a pitch between scanning lines of laser light emitted from a light source is sparse, and dark gray portions (black portions) denote portions in which a pitch between scanning lines is dense. In an image using a line screen illustrated in  FIG. 16B , a stripe pattern of the line screen regularly extends across portions where dense/sparse portions of the scanning lines are generated, and therefore, moire is emphasized. On the other hand, in an image using a dot screen illustrated in  FIG. 16C , when compared with the case of the line screen, portions where dots and sparse/dense portions overlap with each other are irregularly generated, shades of gray are less generated when compared with the case of the line screen, and a degree of moire is lower when compared with the case of the line screen. 
     Furthermore, in a case where an exposure amount is controlled when the density unevenness caused by dense/sparse portions of the scanning lines is corrected, since density per a predetermined area is not stored before and after the correction, the correction may not appropriately function depending on an input image pattern, and accordingly, correction performance may be degraded. Here,  FIGS. 17C and 17D  are diagrams illustrating correction performed by extracting a portion of an image pattern (the line screen) of  FIG. 16A  using a general method for performing density adjustment using an exposure amount as disclosed in Japanese Patent Laid-Open No. 2012-098622. Specifically,  FIG. 17C  is a diagram illustrating an image pattern before the correction, and  FIG. 17D  is a diagram illustrating an image pattern after the correction. Furthermore, “A 1 ” and “A 2 ” of  FIGS. 17C and 17D  indicate correction target ranges, and “B 1 ” and “B 2 ” of  FIG. 17D  including the correction target ranges A 1  and A 2 , respectively, indicate ranges which have been subjected to the correction. In  FIG. 17D , image density is corrected in the correction target ranges A 1  and A 2  when compared with  FIG. 17C . However, in the ranges B 1  and B 2  including surrounding portions of the correction target ranges A 1  and A 2 , portions of high image density and portions of low image density are generated, that is, excessive correction occurs, since the method does not store density before and after the correction, and accordingly, correction may fail depending on a pattern of an input image. 
     To address this situation in the general method, an exposure method for correcting dense/sparse portions by shifting a center of density over a plurality of pixels as illustrated in  FIGS. 18A to 18C  is considered. However, use of the method for shifting a center of density may not obtain a correction effect if the shifted density may not be accurately reproduced in accordance with a gradation characteristic. Furthermore, environmental variation, such as aging, variation in temperature, or variation in humidity, considerably affects the gradation characteristic of electrophotography. Therefore, appropriate correction is applied when the gradation characteristic varies due to the environmental variation. 
     SUMMARY OF THE INVENTION 
     The aspect of the embodiments provides an image forming apparatus including a light source configured to emit a light beam, a photoreceptor configured to be driven for rotation on which a latent image is formed by the light beam, a rotatable polygonal mirror configured to rotate about a rotation axis and have a plurality of mirror planes which deflect the light beam so that the light beam scans the photoreceptor, a processing unit configured to perform a dither process on input image data, and a correction unit configured to correct image data which has been subjected to the dither process using correction amounts based on inclinations of the plurality of mirror planes relative to the rotation axis of the rotatable polygonal mirror. The light source emits the light beam for forming the latent image based on the corrected image data. The correction unit determines the correction amounts in accordance with a type of the dither process. 
     Further features of the disclosure will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a diagram illustrating an entire image forming apparatus according to first and second embodiments, and  FIG. 1B  is a diagram illustrating a configuration of periphery of a photoconductor drum and an optical scanning device according to the first and second embodiments. 
         FIG. 2  is a block diagram illustrating the image forming apparatus according to the first and second embodiments. 
         FIG. 3  is a diagram illustrating position shifts of scanning lines according to the first and second embodiments. 
         FIG. 4  is a block diagram illustrating a process of storing information in a memory according to the first and second embodiments. 
         FIG. 5  is a flowchart illustrating a page process according to the first embodiment. 
         FIG. 6  is a plane tilt correction table according to the first embodiment. 
         FIG. 7  is a flowchart illustrating plane tilt correction process according to the first embodiment. 
         FIGS. 8A to 8D  are diagrams illustrating position shifts of pixels for individual classifications according to the first embodiment. 
         FIGS. 9A and 9B  are diagrams illustrating coordinate conversion of pixel positions in a sub scanning direction according to the first embodiment. 
         FIGS. 10A to 10D  are diagrams illustrating the coordinate conversion of pixel positions in a sub scanning direction according to the first embodiment. 
         FIGS. 11A and 11B  are diagrams illustrating the coordinate conversion of pixel positions in a sub scanning direction according to the first embodiment. 
         FIGS. 12A to 12C  are diagrams illustrating convolution functions used in a filter process according to the first embodiment, and  FIG. 12D  is a diagram illustrating a correction value and a coefficient. 
         FIGS. 13A to 13D  are diagrams illustrating the filter process for individual classifications of the position shifts according to the first embodiment. 
         FIG. 14  is a flowchart illustrating the filter process according to the first embodiment. 
         FIGS. 15A and 15B  are plane tilt correction tables according to a second embodiment. 
         FIG. 16A  is a diagram illustrating density unevenness in the related art, and  FIGS. 16B and 16C  are diagrams illustrating influence on dithers due to plane tilt according to the related art. 
         FIGS. 17A and 17B  are diagrams illustrating the relationship between a plane tilt correction amount and a correction residual error according to the related art, and  FIGS. 17C and 17D  are diagrams illustrating plane tilt correction according to the related art. 
         FIGS. 18A to 18C  are diagrams illustrating the plane tilt correction using a shift of a center of an exposure amount according to the related art. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Hereinafter, exemplary embodiments of the disclosure will be described in detail with reference to the accompanying drawings. It is assumed that a rotation axis direction of a photoconductor drum which is a direction in which scanning is performed with laser light is referred to as a “main scanning direction” which is a second direction, and a rotation direction of the photoconductor drum which is a direction substantially orthogonal to the main scanning direction is referred to as a “sub scanning direction”, which is a first direction. Specifically, the first direction corresponds to the rotation direction of the photoconductor drum and the second direction corresponds to the direction of scanning with a light beam on the photoconductor drum. First,  FIGS. 16B, 16C, 17, and 18  described above will be further described in detail. 
     Influence of Plane Tilt to Dithers 
       FIGS. 16B and 16C  are diagrams illustrating a phenomenon of plane tilt of a rotatable polygonal mirror. It is assumed here that the rotatable polygonal mirror has five mirror planes and a light source has four light emitting elements. Laser light emitted from the light source is deflected by a mirror plane of the rotatable polygonal mirror and scanning lines are formed on a scanning target. The scanning lines formed by four laser beams deflected by the mirror plane of the rotatable polygonal mirror are represented by horizontal rectangular shapes in  FIGS. 16B and 16C . A longitudinal direction of the rectangles indicating the scanning lines corresponds to the main scanning direction and a direction orthogonal to the main scanning direction corresponds to the sub scanning direction. The photoconductor drum is exposed with the laser light of 20 lines (=4 beams×5 planes) every rotation of the rotatable polygonal mirror. Therefore, a dense/sparse portion is generated in a boundary between a scanning line of a fourth beam in a certain scanning operation and a scanning line of a first beam in a next scanning operation, and a such dense/sparse portion is repeated in a period of 20 lines. In  FIGS. 16B and 16C , gray portions denote patterns of dithers. Furthermore, light gray portions (white portions) denote portions where pitches between the scanning lines are sparse and dark gray portions (black portions) denote portions where pitches between the scanning lines are dense. 
       FIG. 16B  is a diagram illustrating an image in which a halftone is represented using a line screen tilted by 45 degrees relative to the sub scanning direction, and a stripe pattern of the line screen regularly extends across the dense/sparse portions of the scanning lines, and therefore, moire is emphasized. On the other hand,  FIG. 16C  is a diagram illustrating an image in which a halftone is represented using a dot screen tilted by 45 degrees relative to the sub scanning direction. In the dot screen, when compared with the case of the line screen, portions in which dots and the dense/sparse portions overlap with each other are irregularly generated, shades of gray are less generated when compared with the case of the line screen, and a degree of moire is lower when compared with the case of the line screen. 
     Relationship Between Correction Amount and Moire Degree 
     A degree of moire generated due to the plane tilt varies depending on a used dither.  FIG. 17A  is a graph illustrating the relationships between a plane tilt correction amount and a moire degree in two dithers A and B in a case where image density is fixed (D 3  in  FIG. 17A ). An axis of abscissae denotes the plane tilt correction amount and an axis of ordinates denotes the moire degree of the plane tilt. As illustrated in  FIG. 17A , the moire degree becomes the smallest when the plane tilt correction amount is equal to a plane tilt amount, and therefore, a moire degree obtained when the correction amount is equal to the plane tilt amount is in the smallest point in the graph (characteristic curves) indicating the relationship between the plane tilt correction amount and the moire degree. Furthermore, a moire degree at the smallest point varies depending on a dither, and in  FIG. 17A , a moire degree of the dither B is smaller than a moire degree of the dither A. In this graph (the characteristic curves), as the correction amount becomes far from the plane tilt amount (that is, as the correction amount becomes larger than the plane tilt amount or becomes smaller than the plane tilt amount), the moire degree becomes larger. This state is represented by a curve having a characteristic of a V shape. Furthermore, in  FIGS. 17A and 17B , a dotted line indicates a visibility limit of the moire, and H 0  and H 1  denote plane tilt correction amounts corresponding to the visibility limit when the dithers B and A are used, respectively, and have the following relationship: H 0 &lt;H 1 . If the same plane tilt correction amount is applied to different dithers, different moire degrees are obtained for the different dithers. Therefore, to obtain the same moire degree, the plane tilt correction amounts are to be corrected for the dithers to be used. 
     Furthermore, even when the same dither is used, if image densities are different, different moire degrees are obtained.  FIG. 17B  is a graph (characteristic curves) illustrating the relationships between plane tilt correction amounts and moire degrees for individual image densities when the same dither is used. An axis of abscissae denotes the plane tilt correction amount, an axis of ordinate denotes the moire degree of the plane tilt, and a dotted line denotes a visibility limit of the moire in  FIG. 17B . In  FIG. 17B , D 0  to D 6  denote image densities, and the image densities are increased in order from D 0  to D 6 . As illustrated in  FIG. 17B , as the image density is low, the moire degree is low. However, as the image density is high, the moire degree is low since shading waves crush. As a result, the relationships between the plane tilt correction amount and the moire degree in the case of the image densities D 0  and D 6  are denoted by the same characteristic curve. Similarly, the relationships between the plane tilt correction amount and the moire degree in the case of the image densities D 1  and D 5  are denoted by the same characteristic curve, and the relationships between the plane tilt correction amount and the moire degree in the case of the image densities D 2  and D 4  are denoted by the same characteristic curve. Furthermore, H 0 , H 1 , and H 2  in  FIG. 17B  denote plane tilt correction amounts for the visibility limit in the case of the image densities D 1  and D 5 , the image densities D 2  and D 4 , and the image density D 3 , respectively, and the plane tilt correction amounts H 0 , H 1 , and H 2  have the following relationship: H 0 &lt;H 1 &lt;H 2 . Note that in a case of the image densities D 0  and D 6 , the moire degree is smaller than the visibility limit irrespective of the plane tilt correction amount, and therefore, moire is not visually recognized. Even in a case where the same dither is used, if the same correction amount is applied to different image densities, different moire degrees are obtained for different image densities. Therefore, to obtain the same moire degree, the plane tilt correction amounts are to be corrected for the different image densities. 
     Correction in General Methods 
     In general, an amount of a light beam at an end in the sub scanning direction of a certain scanning operation (for example, a fourth light beam) is corrected based on a far distance or a close distance between the fourth light beam and a light beam adjacent to the fourth light beam (for example, a first light beam in a next scanning operation). A density per a predetermined area is not stored before and after the correction, and therefore, the correction may not be appropriately performed depending on an input image pattern.  FIG. 17C  is a diagram illustrating a portion of the image pattern (the line screen) of  FIG. 16B . In  FIG. 17C , plane tilt occurs in the rotatable polygonal mirror, and therefore, a region A 1  in which scanning lines adjacent to each other in the rotatable polygonal mirror is sparse is generated resulting in low density. Similarly, plane tilt occurs in the rotatable polygonal mirror, and therefore, a region A 2  in which scanning lines adjacent to each other is dense is generated resulting in high density. In this way, if the plane tilt of the rotatable polygonal mirror is generated, density unevenness is generated as a whole.  FIG. 17D  is a diagram illustrating a result of correction performed by the general method on the plane tilt of the rotatable polygonal mirror of  FIG. 17C . Also in regions B 11  and B 12  which are adjacent to the region A 1  and regions B 21  and B 22  which are adjacent to the region A 2  in  FIG. 17D , a pitch between laser beams is the same as that of the original beams. Therefore, the regions B 11  and B 12  which are positioned in opposite sides of the region A 1  which is a sparse portion in  FIG. 17C  have, as a result of the correction, high density as illustrated in  FIG. 17D . On the other hand, the regions B 21  and B 22  which are positioned in opposite sides of the region A 2  which is a dense portion in  FIG. 17C  have, as a result of the correction, low density as illustrated in  FIG. 17D . As described above, according to  FIG. 17D , the density unevenness is generated due to the correction of the plane tilt of the rotatable polygonal mirror, that is, appropriate correction may not be performed. 
     Other Correction Methods 
     As a correction method performed irrespective of an input image pattern, an exposure method illustrated in  FIGS. 18A to 18C  in which a center position of an image is shifted by combining peripheral scanning lines may be considered. In  FIGS. 18A to 18C , axes of abscissae denote a position in the sub scanning direction, axes of ordinates denote an exposure amount, and bars in bar graphs denote exposure amounts in sub scanning positions.  FIG. 18A  is a graph illustrating an exposure amount in a sub scanning position exposed based on input data.  FIGS. 18B and 18C  are graphs illustrating exposure amounts in sub scanning positions in a case where a process of shifting a center from a sparse portion to a dense portion of the scanning lines is performed when compared with  FIG. 18A . In  FIGS. 18B and 18C , as a result of a calculation for shifting a center position after image density before and after the process of shifting the center is stored, a large number of halftone (intermediate gradation) pixels are generated. Therefore, large environmental variation, such as variation in temperature or variation in humidity, affects correction performance due to an electrophotographic characteristic. Specifically, in a case where an optimum correction effect is obtained when a center of input image data is shifted in a certain direction and development is performed with a certain gradation, a gradation characteristic is changed due to the environmental variation, for example, and accordingly, a linear characteristic may be obtained or a characteristic of steeply rising at a certain exposure amount may be obtained. In this case, if the development is performed with the linear gradation characteristic, excessive correction is performed, or if development is performed in a gradation characteristic of steeply rising at a certain exposure amount, conversely, correction may not be sufficient. 
     First Embodiment 
     Configuration of Entire Image Forming Apparatus 
       FIG. 1A  is a cross sectional view schematically illustrating a digital full color printer (a color image forming apparatus) which performs image formation using a plurality of color toners. An image forming apparatus  100  according to this embodiment will be described with reference to  FIG. 1A . The image forming apparatus  100  includes four image forming sections (image forming units)  101 Y,  101 M,  101 C, and  101 Bk (denoted by dotted lines) which form images of different colors. The image forming sections  101 Y,  101 M,  101 C, and  101 Bk perform image formation using toners of yellow, magenta, cyan, and black, respectively. Here, “Y”, “M”, “C”, and “Bk” represent yellow, magenta, cyan, and black, respectively, and the indices Y, M, C, and Bk are omitted hereinafter except for a case where a specific color is described. 
     The image forming section  101  includes a photoconductor drum  102  serving as a photoreceptor. A charging device  103 , an optical scanning device  104 , and a development device  105  serving as a development unit are disposed near the photoconductor drum  102 . Furthermore, a cleaning device  106  is disposed near the photoconductor drum  102 . An intermediate transfer belt  107  as an endless belt is disposed below the photoconductor drum  102 . The intermediate transfer belt  107  is stretched by a driving roller  108  and driven rollers  109  and  110  and conveyed in a direction indicated by an arrow mark B of  FIG. 1A  (a clockwise direction) during the image formation. 
     Furthermore, a primary transfer device  111  is disposed in a position opposite to the photoconductor drum  102  through the intermediate transfer belt  107  (an intermediate transfer member). Furthermore, the image forming apparatus  100  of this embodiment further includes a secondary transfer device  112  which transfers the toner images on the intermediate transfer belt  107  on a sheet S serving as a recording medium and a fixing device  113  which fixes the toner images on the sheet S. 
     An image forming process including a charging process to a developing process of the image forming apparatus  100  will now be described. Image forming processes performed by the individual image forming sections  101  are the same, and therefore, the image forming process performed by the image forming section  101 Y is described as an example and descriptions of the image forming processes performed by the image forming sections  101 M,  101 C, and  101 Bk are omitted. The charging device  103 Y of the image forming section  101 Y charges the photoconductor drum  102 Y which is driven for rotation in a direction indicated by an arrow mark in  FIG. 1A  (an anticlockwise direction). The charged photoconductor drum  102 Y is exposed by laser light denoted by a chain line emitted from the optical scanning device  104 Y. By this, an electrostatic latent image is formed on the rotating photoconductor drum  102 Y (the photoreceptor). The electrostatic latent image formed on the photoconductor drum  102 Y is developed as a yellow toner image by the development device  105 Y. The same process is performed by the image forming sections  101 M,  101 C, and  101 Bk. 
     The image forming process after the transfer process will be described. The primary transfer devices  111  to which a transfer voltage is applied transfer the toner images of yellow, magenta, cyan, and black formed on the photoconductor drums  102  of the image forming sections  101  to the intermediate transfer belt  107 . By this, the toner images of the individual colors overlap with one another on the intermediate transfer belt  107 . That is, the toner images of the four colors are transferred to the intermediate transfer belt  107  (first transfer). The toner images of the four colors transferred to the intermediate transfer belt  107  are further transferred by the secondary transfer device  112  to the sheet S conveyed to a secondary transfer section from a manual feed sheet cassette  114  or a sheet cassette  115  (secondary transfer). The unfixed toner images on the sheet S are fixed by heat by the fixing device  113  so that a full-color image is obtained on the sheet S. The sheet S having the image formed thereon is discharged to a sheet discharging unit  116 . 
     Photoconductor Drum and Optical Scanning Device 
       FIG. 1B  is a diagram illustrating configurations of the photoconductor drum  102 , the optical scanning device  104 , and a controller of the optical scanning device  104 . The optical scanning device  104  includes a multi-beam laser light source (hereinafter referred to as a “laser light source”)  201 , a collimator lens  202 , a cylindrical lens  203 , and a rotatable polygonal mirror  204 . The laser light source  201  is the multi-beam laser light source which generates laser light (light beams) by a plurality of light emitting elements. The collimator lens  202  forms laser light into parallel light. The cylindrical lens  203  collects the laser light which passes through the collimator lens  202  in the sub scanning direction. Note that, although the multi-beam light source which emits a plurality of beams is described as an example of the laser light source  201  in this embodiment, the same operation is performed in a case where a single light source is used. The laser light source  201  is driven by a multi-beam laser driving circuit (hereinafter simply referred to as a “laser driving circuit”)  304 . The rotatable polygonal mirror  204  includes a motor unit which performs rotation movement and a reflection mirror attached to a motor shaft. Hereinafter, planes of the reflection mirror of the rotatable polygonal mirror  204  are referred to as “mirror planes”. The rotatable polygonal mirror  204  is driven by a rotatable polygonal mirror driving unit  305 . The optical scanning device  104  includes fθ lenses  205  and  206  on which laser light (scanning light) deflected by the rotatable polygonal mirror  204  is incident. The optical scanning device  104  further includes a memory  302  which stores various types of information. 
     Furthermore, the optical scanning device  104  includes a beam detector  207  (hereinafter referred to as a “BD  207 ”) serving as a signal generation unit which detects the laser light deflected by the rotatable polygonal mirror  204  and which outputs a horizontal synchronization signal (hereinafter referred to as a “BD signal”) in response to the detection of the laser light. The laser light emitted from the optical scanning device  104  is used to scan the photoconductor drum  102 . The optical scanning device  104  and the photoconductor drum  102  are positioned such that scanning is performed with laser light in parallel to a rotation axis of the photoconductor drum  102 . The optical scanning device  104  shifts a spot of a light beam of the multi-beam laser in the main scanning direction (scanning) every time the mirror plane of the rotatable polygonal mirror  204  scans the photoconductor drum  102  once. In this way, scanning lines corresponding to a number of laser elements (light emitting elements) are simultaneously generated. In this embodiment, the rotatable polygonal mirror  204  has five planes and the laser light source  201  includes eight laser elements, for example. In this embodiment, image formation for eight lines is performed by one mirror plane of the rotatable polygonal mirror  204 , that is, one scanning operation with laser light. The rotatable polygonal mirror  204  performs image formation for 40 lines by performing five scanning operations with laser light per one rotation. 
     The photoconductor drum  102  includes a rotary encoder  301 , and the rotary encoder  301  detects a rotation speed of the photoconductor drum  102 . The rotary encoder  301  generates 1000 pulses during one rotation of the photoconductor drum  102 . The rotary encoder  301  includes a measurement unit, not illustrated, which measures time intervals of pluses on an internal substrate thereof. The rotary encoder  301  outputs information on the rotation speed (rotation speed data) of the photoconductor drum  102  to a CPU  303  based on the time intervals of the pulses measured by the measurement unit. Note that general speed detection techniques other than the rotary encoder may be used as long as the rotation speed of the photoconductor drum  102  may be detected. Examples of such a method other than the encoder include a method for detecting a surface speed of the photoconductor drum  102  by laser Doppler or the like. 
     Next, the CPU  303  serving as a controller will be described with reference to  FIG. 2 .  FIG. 2  is a block diagram illustrating functions as a correction unit, a conversion unit, and a filter process unit of the CPU  303  which executes a correction process of correcting distortion and density unevenness of an image described below. The CPU  303  includes a filter process unit  501 , an error diffusion process unit  502 , and a PWM signal generation unit  503 . The filter process unit  501  performs the filter process by performing a convolution calculation on input image data. The error diffusion process unit  502  performs an error diffusion process on image data which has been subjected to the filter process. The PWM signal generation unit  503  performs PWM conversion on image data which has been subjected to the error diffusion process so as to output a PWM signal to the laser driving circuit  304  of the optical scanning device  104 . 
     Furthermore, the CPU  303  includes a filter coefficient setting unit  504 , a filter function output unit  505 , and a correction value setting unit  506 . The filter function output unit  505  outputs data on a function to be used in the convolution calculation (data on a table, for example) to the filter coefficient setting unit  504 , and examples of the function to be used in the convolution calculation include an linear interpolation and bicubic interpolation. The correction value setting unit  506  calculates an amount of a position shift of a scanning line based on information on a position shift amount read from the memory  302  and a plane synchronization signal supplied from a plane specifying unit  507 . The correction value setting unit  506  calculates a correction value based on the position shift amount of a scanning line and outputs the calculated correction value to the filter coefficient setting unit  504 . The filter coefficient setting unit  504  calculates a filter coefficient based on the information on the convolution function supplied from the filter function output unit  505  and the correction value of the scanning line supplied from the correction value setting unit  506 . The filter coefficient is used in the filter process performed by the filter process unit  501 . The filter coefficient setting unit  504  sets the calculated filter coefficient to the filter process unit  501 . 
     The CPU  303  further includes the plane specifying unit  507 . The plane specifying unit  507  specifies one of the mirror planes of the rotatable polygonal mirror  204  based on an HP signal supplied from a home position sensor (HP sensor)  307  of the optical scanning device  104  and a BD signal supplied from the BD  207 . The plane specifying unit  507  outputs information on the specified mirror plane as a plane synchronization signal to the correction value setting unit  506 . 
     As illustrated in  FIG. 1B , image data is supplied to the CPU  303  from an image controller, not illustrated, which generates the image data. Furthermore, the CPU  303  is connected to the rotary encoder  301 , the BD  207 , the memory  302 , and the rotatable polygonal mirror driving unit (hereinafter referred to as a “mirror driving unit”)  305 . The CPU  303  detects a position of start of writing of the scanning line based on the BD signal supplied from the BD  207  and counts a time interval between the BD signals so as to detect a rotation speed of the rotatable polygonal mirror  204 . Furthermore, the CPU  303  outputs an acceleration/deceleration signal of an instruction for acceleration/deceleration to the mirror driving unit  305  so that the rotatable polygonal mirror  204  rotates in a predetermined speed. The mirror driving unit  305  supplies driving current to the motor unit of the rotatable polygonal mirror  204  in accordance with the acceleration/deceleration signal supplied from the CPU  303  so as to drive a motor  306 . 
     As illustrated in  FIG. 2 , the rotatable polygonal mirror  204  includes the HP sensor  307  mounted thereon which outputs an HP signal to the CPU  303  when the rotatable polygonal mirror  204  has a predetermined angle during rotation operation. The plane specifying unit  507  of the CPU  303  specifies one of the five mirror planes of the rotatable polygonal mirror  204  which has been subjected to scanning with laser light, that is, a mirror plane which has been subjected to scanning, when detecting the HP signal supplied from the HP sensor  307 . After specifying one of the mirror planes once, the plane specifying unit  507  continuously specifies one of the mirror planes based on the BD signal output from the BD  207 . The BD  207  outputs one pulse of the BD signal every time an arbitrary mirror plane of the rotatable polygonal mirror  204  is scanned once with the laser light, and therefore, the CPU  303  may count the BD signals so as to continuously specify one of the mirror planes of the rotatable polygonal mirror  204 . 
     The memory  302  stores information on positions of the mirror planes of the rotatable polygonal mirror  204  and information on a position of the multi-beam laser. The CPU  303  reads, from the memory  302 , position shift information in the sub scanning direction caused by the plane tilt of the individual mirror planes of the rotatable polygonal mirror  204  and position shift information relative to ideal positions of the multi-beam laser in resolution of 1200 dpi in the sub scanning direction. The CPU  303  calculates information on positions of the scanning lines based on the position shift information read from the memory  302 . 
     The correction value setting unit  506  calculates correction values based on the positional information of the scanning lines supplied from the memory  302  and outputs the calculated correction values to the filter coefficient setting unit  504 . The filter coefficient setting unit  504  calculates a filter coefficient using the correction values input from the correction value setting unit  506  and the filter function input from the filter function output unit  505 . The filter process unit  501  receives image data from an image controller, not illustrated, which generates image data. The filter process unit  501  performs the filter process on the image data based on the filter coefficient supplied from the filter coefficient setting unit  504  so as to calculate image data obtained taking information on correction of the positions of the scanning lines into consideration. The CPU  303  outputs a light emitting amount data to the laser driving circuit  304  based on the image data obtained taking the information on the correction of the positions of the scanning lines into consideration. Note that, in this embodiment, the laser driving circuit  304  performs light amount control by controlling a lighting time of individual pixels by pulse width modulation (PWM) control based on the light emitting amount data supplied from the CPU  303 . Note that the PWM control may not be performed when the light amount control is performed, and the light amount control may be performed by amplitude modulation (AM) control for controlling peak light amounts of the individual pixels. 
     Next, the scanning position information stored in the memory  302  will be described with reference to  FIG. 3  and Table 1.  FIG. 3  is a diagram illustrating position shifts from ideal positions of the individual scanning lines. Scanning lines of scanning with laser beams of the multi-beam laser having eight light emitting points are denoted by “LD 1 ” to “LD 8 ”. An ideal pitch (a predetermined pitch) between the scanning lines is determined based on resolution. In a case of an image forming apparatus forming an image of a resolution of 1200 dpi, an ideal pitch between the scanning lines is 21.16 μm. In a case where the scanning line LD 1  is set as a reference position, ideal distances D 2  to D 8  from the scanning line LD 1  to the individual scanning lines LD 2  to LD 8  are calculated in accordance with Expression (1).
 
 Dn =( n− 1)×21.16 μm ( n= 2 to 8)  Expression (1)
 
     For example, the ideal distance D 4  from the scanning line LD 1  to the scanning line LD 4  is 63.48 μm (=(4−1)×21.16 μm). 
     Here, the pitch among the scanning lines has an error due to an error of an element pitch of the multi-beam laser and variation of lens magnification. It is assumed that position shift amounts of the scanning line positions of the scanning lines LD 2  to LD 8  relative to ideal positions determined in accordance with the ideal distances D 2  to D 8  are denoted by “X 1 ” to “X 7 ”. It is assumed that, in a first plane of the rotatable polygonal mirror  204 , for example, the position shift amount X 1  of the scanning line LD 2  corresponds to a difference between the ideal position of the scanning line LD 2  (hereinafter referred to as a “line  2 ” and the same is true on the other scanning lines) and an actual scanning line. Furthermore, it is assumed that the position shift amount X 3  of the scanning line LD 4  corresponds to a difference between a line  4  and an actual scanning line. 
     The rotatable polygonal mirror  204  has variation in every mirror plane since angles of mirror planes relative to the rotation axis of the rotatable polygonal mirror  204  is not completely parallel to one another due to variation of the mirror planes in fabrication. Position shift amounts relative to ideal positions of the mirror planes of the rotatable polygonal mirror  204  are represented by Y 1  to Y 5  in a case where the number of planes of the rotatable polygonal mirror  204  is 5. In  FIG. 3 , an amount of shift from an ideal position of a scanning line LD 1  in the first plane is denoted by “Y 1 ”, and an amount of shift from an ideal position of a scanning line LD 1  in a second plane is denoted by “Y 2 ”. 
     It is assumed that a mirror plane of the rotatable polygonal mirror  204  is denoted by an “m-th plane”, and a position shift amount of a scanning line (LDn) of n-th laser light of the multi-beam is denoted by “Zmn”. In this case, the position shift amounts Zmn are represented by Expression (2) using position shift amounts X 1  to X 7  of the individual scanning lines and the position shift amounts Y 1  to Y 5  of the individual mirror planes.
 
 Zmn=Ym+X ( n− 1) ( m= 1 to 5 and  n= 1 to 8)   Expression (2)
 
(Note that X(0)=0.)
 
     For example, a position shift amount Z 14  of the scanning line LD 4  of the first plane of the rotatable polygonal mirror  204  is obtained as follows in accordance with Expression (2): Z 14 =Y 1 +X 3 . Furthermore, a position shift amount Z 21  of the scanning line LD 1  of the second plane of the rotatable polygonal mirror  204  is obtained as follows in accordance with Expression (2): Z 21 =Y 2 . 
     In a case where a position shift amount Zmn is calculated by a calculation of Expression (2), a number of data corresponding to the number of mirror planes of the rotatable polygonal mirror  204  and the number of elements of the multi-beam laser are used for the calculation of the position shift amount Zmn. Here, an address map of position shift data stored in the memory  302  is illustrated. 
     
       
         
           
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 ADDRESS 
                 DATA 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 LD2 POSITIONAL INFORMATION X1 
               
               
                 1 
                 LD3 POSITIONAL INFORMATION X2 
               
               
                 2 
                 LD4 POSITIONAL INFORMATION X3 
               
               
                 3 
                 LD5 POSITIONAL INFORMATION X4 
               
               
                 4 
                 LD6 POSITIONAL INFORMATION X5 
               
               
                 5 
                 LD7 POSITIONAL INFORMATION X6 
               
               
                 6 
                 LD8 POSITIONAL INFORMATION X7 
               
               
                 7 
                 FIRST PLANE POSITIONAL INFORMATION Y1 
               
               
                 8 
                 SECOND PLANE POSITIONAL INFORMATION Y2 
               
               
                 9 
                 THIRD PLANE POSITIONAL INFORMATION Y3 
               
               
                 10 
                 FOURTH PLANE POSITIONAL INFORMATION Y4 
               
               
                 11 
                 FIFTH PLANE POSITIONAL INFORMATION Y5 
               
               
                   
               
            
           
         
       
     
     As illustrated in Table 1, information on the position shift amount (positional information) X 1  to information on the positional information X 7  from the scanning line LD 2  to the scanning line LD 8  are stored in addresses 0 to 6 in the memory  302 . Furthermore, information on the position shift amount Y 1  to information on the position shift amount Y 5  of the first to fifth mirror planes of the rotatable polygonal mirror  204  are stored in addresses 7 to 11 in the memory  302 . 
     Note that, in this embodiment, the eight scanning lines of the laser beams are uniformly shifted by the position shifts of the mirror planes of the rotatable polygonal mirror  204 . However, in a case where the position shift amounts of the individual scanning lines of the laser light vary depending on a mirror plane of the rotatable polygonal mirror  204 , a number of information on position shift amounts corresponding to a number of combinations of the mirror planes of the rotatable polygonal mirror  204  and the scanning lines of the laser light may be stored. Specifically, in this case, the number of mirror planes of the rotatable polygonal mirror  204  is 5 and the number of elements of the laser light source  201  is 8, and accordingly, 40 positional information items are stored in the memory  302 . 
     Memory Storage Operation 
     The information on the position shift amounts stored in the memory  302  corresponds to data measured in a process of adjusting the optical scanning device  104  in a factory, or the like. Furthermore, the image forming apparatus  100  may include a unit for detecting a position of a scanning line of scanning with laser light emitted from the laser light source  201  and update the information stored in the memory  302  in real time. As a unit for detecting a position of the scanning light in the sub scanning direction, a general technique may be used. For example, a method for detecting a position using a complementary metal-oxide semiconductor (CMOS) sensor or a position sensitive detector (PSD) disposed inside the optical scanning device  104  or in the vicinity of the photoconductor drum  102  may be employed. Furthermore, a method for disposing a triangle slit on a photodiode (PD) plane inside the optical scanning device  104  or in the vicinity of the photoconductor drum  102  and detecting a position from an output pulse width of a PD may be employed. 
       FIG. 4  is a block diagram illustrating a state in which the information is stored in the memory  302  of the optical scanning device  104  in a factory or the like. Note that components the same as those in  FIG. 2  are denoted by reference numerals the same as those in  FIG. 2 , and descriptions thereof are omitted. In the process of adjusting the optical scanning device  104 , a measurement tool  400  is disposed in a position corresponding to a position of the photoconductor drum  102  when the optical scanning device  104  is mounted on the image forming apparatus  100 . The measurement tool  400  includes a measurement unit  410  and a calculation unit  402  which receives a plane synchronization signal from the plane specifying unit  507  of the CPU  303  of  FIG. 2 . Note that only the plane specifying unit  507  is illustrated in the CPU  303  of  FIG. 4 . First, the optical scanning device  104  emits laser light to the measurement unit  410 . The measurement unit  410  includes a triangle slit  411  and a PD  412 . In  FIG. 4 , a light beam which is emitted from the optical scanning device  104  for scanning and which is denoted by an arrow mark of a chain line scans the triangle slit  411 . The measurement unit  410  measures positions of scanning lines in the sub scanning direction based on information on the light beams input to the PD  412  through the triangle slit  411 . The measurement unit  410  outputs information on the measured positions of the scanning lines in the sub scanning direction for each mirror plane of the rotatable polygonal mirror  204  (hereinafter referred to as “each-plane data”) to the calculation unit  402 . 
     On the other hand, an HP signal and a BD signal are input to the plane specifying unit  507  from the HP sensor  307  and the BD  207  of the optical scanning device  104 , respectively. By this, the plane specifying unit  507  specifies a mirror plane of the rotatable polygonal mirror  204  and outputs information on the specified mirror plane as a plane synchronization signal to the calculation unit  402 . The calculation unit  402  writes information on the positions of the scanning lines in the sub scanning direction measured by the measurement unit  410  in the addresses in the memory  302  of the optical scanning device  104  corresponding to the information on the mirror plane of the rotatable polygonal mirror  204  input from the plane specifying unit  507 . In this way, the information on the position shift amounts (X 1  to X 7 ) of the scanning lines generated due to variation of the eight elements of the laser light source  201  and information on the position shift amounts (Y 1  to Y 5 ) of the scanning lines caused due to the plane tilt of the mirror planes of the rotatable polygonal mirror  204  are stored in the memory  302 . 
     Page Process 
     When receiving a print job from an operation unit or an external apparatus, not illustrated, the CPU  303  performs an initial operation of resetting a timer, a counter, and the like, not illustrated, used in the following processing as a preparation for the page process in a series of processes of electrophotography. Thereafter, when the preparation for the page process is terminated and it is determined that the page process is available, the CPU  303  executes a control sequence illustrated in a flowchart of  FIG. 5 . 
     In step S 600  of  FIG. 5 , the CPU  303  determines whether a BD signal supplied from the BD  207  has been detected. When it is determined that the BD signal has been detected, the CPU  303  starts the timer for determining a period of time after the detection of the BD signal and the process proceeds to step S 601 , whereas when it is determined that the BD signal has not been detected, the process returns to step S 600 . In step S 601 , the CPU  303  determines whether a period of time T 1  has elapsed with reference to the timer. Here, the period of time T 1  is started when the BD signal is output and ended when the laser light reaches a leading end of an image region of the photoconductor drum  102  in the main scanning direction. In step S 601 , when determining that the period of time T 1  has not elapsed, the CPU  303  returns to step S 601 , and otherwise, the CPU  303  proceeds to step S 602 . 
     In step S 602 , the CPU  303  detects a feature of an image and determines a dither method so as to select an optimum dither for each region including a plurality of pixels. An image of a first page includes various types of image, such as characters and photographs, and the characters and the photographs have different image features, for example. The image feature is extracted using a general method. For example, in a case where the image forming apparatus  100  is used as a printer, an image feature is extracted in accordance with a command transmitted from a PC, and in a case where the image forming apparatus  100  is used as a copier, an image feature is extracted in accordance with a result of the filter process performed on an image read by an image reading apparatus. 
     In this embodiment, a dither (that is, a screen) which is optimum for each region including a plurality of pixels is selected. However, an optimum dither may be selected for each page or each pixel, for example. In this embodiment, one of the dither A using the line screen, the dither B using the dot screen, and the dither C using the error diffusion is selected. 
     When determining that the dither A is to be selected in step S 602 , the CPU  303  selects a correction table for the dither A (the line screen) in step S 603 , and the process proceeds to step S 606 . When determining that the dither B is to be selected in step S 602 , the CPU  303  selects a correction table for the dither B (the dot screen) in step S 604 , and the process proceeds to step S 606 . When determining that the dither C is to be selected in step S 602 , the CPU  303  selects a correction table for the dither C (the error diffusion) in step S 605 , and the process proceeds to step S 606 . In this way, the CPU  303  selects a correction table suitable for the image feature. The correction table stores information on association between a plane tilt amount and a correction amount for the plane tilt amount. 
     In step S 606 , the CPU  303  performs gradation conversion on an input image by the dither process selected in step S 603  to step S 605 . Note that it is assumed that the dither process performed in step S 606  is a general technique, and therefore, a description thereof is omitted. In step S 607 , the CPU  303  reads a plane tilt amount (a position shift amount in the sub scanning direction), corrects the plane tilt amount corresponding to the dither in accordance with the correction table of the selected dither, and performs a plane tilt correction process based on the corrected plane tilt amount. In step S 608 , the CPU  303  performs conversion into a pulse width modulation (PWM) signal suitable for laser driving so as to perform image formation on a recording member by the electrophotographic process after the plane tilt correction process is terminated. In step S 609 , the CPU  303  determines whether a process for one line has been terminated. When the determination is negative, the process returns to step S 602 , and otherwise, the process proceeds to step S 610 . In step S 610 , the CPU  303  determines whether a process for all lines, that is, one page, has been terminated. When the determination is negative, the process returns to step S 600 , and otherwise, the page process is terminated. 
     Calculation of Plane Tilt Amount Corresponding to Dither 
     Before the plane tilt correction process executed in step S 607  of  FIG. 5  is described in detail, a calculation of a plane tilt amount corresponding to a dither which is a characteristic of this embodiment will be described below. In this embodiment, the CPU  303  reads a position shift amount (which is also referred to as a “plane tilt amount”) in the sub scanning direction stored in the memory  302  in step S 3602  of  FIG. 7  described below and performs correction in accordance with the dither selected in step S 603  to step S 605 . As described above, even in a case of the same plane tilt correction amount, if selected dithers are different, different moire degrees are obtained. Therefore, to obtain the same moire degree irrespective of the selected dither when the plane tilt correction process is performed, the plane tilt amount (the position shift amount) read from the memory  302  is to be corrected depending on the selected dither. In this embodiment, a proportional relationship (a rate) of the correction amount to the plane tilt amount in the selected dither is obtained, and the plane tilt amount read from the memory  302  is multiplied by the obtained rate, so that the plane tilt amount is corrected and the plane tilt correction process is performed based on the corrected plane tilt amount. 
     The CPU  303  calculates an amplitude amount ZW which is a fluctuation range of the position shift amount (the plane tilt amount) based on the position shift amount Zmn of the scanning line (LDn) of an n-th laser light of the multi-beam and the m-th mirror plane of the rotatable polygonal mirror  204  in accordance with Expression (3) below. Here, the amplitude range ZW indicates a difference between a largest value and a smallest value of the position shift amount Zmn (Expression (2)) of the scanning line obtained while the rotatable polygonal mirror  204  rotates once.
 
 ZW =(largest value of  Zmn )−(smallest value of  Zmn ) ( m= 1 to 5 and  n= 1 to 8)  Expression (3)
 
       FIG. 6  is a graph illustrating the relationship between the plane tilt amounts of the dithers and the correction amounts for the plane tilt amounts when one of the dithers is selected and used in step S 603  to step  605  in  FIG. 5  described above. An axis of abscissae denotes a plane tilt amount, and an axis of ordinates denotes a correction amount. In  FIG. 6 , a solid line, a dotted line, and a chain line denote the relationships between the plane tilt amounts and the correction amounts in cases of the dithers A, B, and C, respectively. As illustrated in  FIG. 6 , even in a case of the same plane tilt amount, different correction amounts are obtained for different used dithers. In  FIG. 6 , in a case where the amplitude amount ZW indicating a fluctuation range of a plane tilt amount described above is set as a plane tilt amount, correction amounts for the plane tilt amounts of the dithers A, B, and C are determined as ZWa, ZWb, and ZWc, respectively. In this embodiment, a position shift amount Zmn′ after correction is calculated using a correction amount for a plane tilt amount in a case where an amplitude amount relative to the position shift amount Zmn is set as a plane tilt amount. Specifically, the position shift amount Zmn′ after correction for the position shift amount Zmn (m=1 to 5 and n=1 to 8) read in step S 3602  is calculated in accordance with Expressions (4), (5), and (6) for the dithers A, B, and C, respectively.
 
 Zmn ′=( ZWa/ZW )× Zmn   Expression (4)
 
 Zmn ′=( ZWb/ZW )× Zmn   Expression (5)
 
 Zmn ′=( ZWc/ZW )× Zmn   Expression (6)
 
     Accordingly, a correction amount Cmn for the m-th mirror plane of the rotatable polygonal mirror  204  and the scanning line (LDn) of the n-th laser light of the multi-beam is determined in accordance with Expression (7) below in this embodiment.
 
 Cmn=−Zmn′   Expression (7)
 
     Note that the CPU  303  includes a storage unit which stores tables which associate information on the plane tilt amounts with information on the correction amounts for the individual dithers A, B, and C. After selecting one of the dithers A, B, and C to be used in step S 603  to step S 605  in  FIG. 5 , the CPU  303  reads a correction amount corresponding to the amplitude amount of the plane tilt described above from the table corresponding to the selected dither. Then the CPU  303  calculates a position shift amount Zmn′ after correction based on the position shift amount Zmn, the plane tilt amplitude amount, and the read correction amount so as to obtain a correction amount Cmn. 
     Plane Tilt Correction Process 
     Next, the plane tilt correction process executed in step S 607  of  FIG. 5  will be described in detail. In this embodiment, the CPU  303  performs correction on image data based on position shift amounts in the sub scanning direction of the scanning lines of the laser light, and outputs the corrected image data to the laser driving circuit  304 . Hereinafter, a flowchart of  FIG. 7  will be described.  FIG. 7  is a flowchart of a correction process of correcting density unevenness and banding caused by position shifts in the sub scanning direction. In step S 3602 , the CPU  303  reads position shift amounts in the sub scanning direction stored in the memory  302 . Specifically, the CPU  303  reads the positional information X 1  to X 7  of the scanning lines LD 2  to LD 8  described in Table 1 and the positional information Y 1  to Y 5  of the first to fifth planes of the rotatable polygonal mirror  204  from the memory  302 . The CPU  303  adjusts the plane tilt positional information Y 1  to Y 5  of the rotatable polygonal mirror  204  based on the read correction amounts and phase amounts. In this embodiment, the position shift amounts in the sub scanning direction (X 1  to X 7  and Y 1  to Y 5  after the adjustment) are corrected in accordance with the dither selected as described above. After pixel positions in the sub scanning direction of the corrected image data are corrected, the filter process is performed so that image data, that is, density is output. 
     States of Positional Shifts of Scanning Lines 
     States of positional shifts of scanning lines may be classified into approximately four types. First, examples of the states of position shifts include (a) a case where a position of a scanning line on the photoconductor drum  102  (hereinafter referred to as a “scanning position”) shifts in an forward direction relative to an ideal scanning position and (b) a case where a scanning position on the photoconductor drum  102  shifts in a backward direction relative to the ideal scanning position. Furthermore, the examples of the states of position shifts include (c) a case where pitches between scanning positions on the photoconductor drum  102  become smaller than an ideal pitch of the scanning positions and conversely (d) a case where the pitches between scanning positions on the photoconductor drum  102  become larger than the ideal pitch of the scanning positions. The examples of the states of positional shifts in the sub scanning direction are illustrated in  FIGS. 8A to 8D . In  FIGS. 8A to 8D , dotted lines denote scanning positions and “(1)” to “(5)” denote scanning order. Eight beams are simultaneously emitted in scanning in this embodiment, and numbers are assigned to the individual beams arranged in the sub scanning direction. Left columns of  FIGS. 8A to 8D  indicate ideal scanning positions, and right columns indicate scanning positions on the photoconductor drum  102 . “S 1 ” to “S 5 ” denote position shift amounts from the ideal scanning positions corresponding to the scanning numbers (1) to (5). A unit of the position shift amounts is represented using an ideal beam pitch of 1 as a reference (21.16 μm for the resolution of 1200 dpi), and a forward direction of the light beams in the sub scanning direction (hereinafter simply referred to as a “forward direction”) corresponds to a positive value. Furthermore, a backward direction of the light beams in the sub scanning direction (hereinafter simply referred to as a “backward direction”) corresponds to a negative value. Furthermore, one pixel disposed in the sub scanning direction for description of a state of the image is denoted by a circle. Color of the circle indicates density. 
     In  FIG. 8A , the scanning positions on the photoconductor drum  102  uniformly shift from the ideal scanning positions by 0.2 in the forward direction. Hereinafter, the position shift amount illustrated in  FIG. 8A  is referred to as a shift amount of +0.2 lines. In  FIG. 8B , the scanning positions on the photoconductor drum  102  uniformly shift from the ideal scanning positions by 0.2 in the backward direction. Hereinafter, the position shift amount illustrated in  FIG. 8B  is referred to as a shift amount of −0.2 lines. In  FIGS. 8A and 8B , the scanning positions uniformly shift, and therefore, pitches between the scanning positions on the photoconductor drum  102  is 1. 
     In  FIG. 8C , a position shift amount of 0 is obtained in a certain scanning position on the photoconductor drum  102 . However, as the scanning number of the scanning position becomes smaller relative to the scanning number of the scanning position having the position shift amount of 0, a position shift amount in the forward direction becomes larger, whereas as the scanning number of the scanning position become larger relative to the scanning number of the scanning position having the position shift amount of 0, a position shift amount in the backward direction becomes larger. For example, although the position shift amount S 3  is +0 in the scanning number (3), the position shift amount S 2  is +0.2 in the scanning number (2), the position shift amount S 1  is +0.4 in the scanning number (1), the position shift amount S 4  is −0.2 in the scanning number (4), and the position shift amount S 5  is −0.4 in the scanning number (5). In  FIG. 8C , pitches between the scanning positions is 0.8 which is smaller than 1. Hereinafter, the position shift state illustrated in  FIG. 8C  is referred to as a “dense state with a pitch of (1-0.2) lines. 
     In  FIG. 8D , a position shift amount of 0 is obtained in a certain scanning position on the photoconductor drum  102 . However, as the scanning number of the scanning position becomes smaller relative to the scanning number of the scanning position having the position shift amount of 0, a position shift amount in the backward direction becomes larger, whereas as the scanning number of the scanning position becomes larger relative to the scanning number of the scanning position having the position shift amount of 0, a position shift amount in the forward direction becomes larger. For example, although the position shift amount S 3  is +0 in the scanning number (3), the position shift amount S 2  is −0.2 in the scanning number (2), the position shift amount S 1  is −0.4 in the scanning number (1), the position shift amount S 4  is +0.2 in the scanning number (4), and the position shift amount S 5  is +0.4 in the scanning number (5). In  FIG. 8D , pitches between the scanning positions is 1.2 which is larger than 1. Hereinafter, the position shift state illustrated in  FIG. 8D  is referred to as a “sparse state with a pitch of (1+0.2) lines. 
     In the dense state of  FIG. 8C , in addition to generation of position shifts, high density is obtained since pixels gather on the photoconductor drum  102  due to the small pitches between the scanning positions and a pixel value per predetermined area is increased. In the sparse state of  FIG. 8D , in addition to generation of position shifts, low density is obtained since pixels are separated from each other on the photoconductor drum  102  due to the large pitches between the scanning positions and a pixel value per predetermined area is reduced. In an electrophotographic process, a density difference may be further emphasized due to the relationship between a depth of a latent image potential and a development characteristic. Furthermore, if the dense state and the sparse state as illustrated in  FIGS. 8C and 8D , respectively, are alternately generated, cyclic gradation is seen to be moire and even if the amount of moire is generated, the moire is visually detected with ease depending on a spatial frequency. 
     The description of the flowchart of  FIG. 7  will be made again. In step S 3603 , the CPU  303  generates correction attribute information for the individual pixels of the input image using the correction value setting unit  506 . In this embodiment, pixel positions in the sub scanning direction of the input image are subjected to coordinate conversion in advance before interpolation is performed, and accordingly, correction of local gradation may also be performed while density of the input image is stored, in addition to the correction of the position shifts. Here, the correction attribute information specifically corresponds to a correction value C described below. 
     Coordinate Conversion 
     A coordinate conversion method of this embodiment will be described with reference to  FIGS. 9A and 9B ,  FIGS. 10A to 10D , and  FIGS. 11A and 11B . In graphs in  FIGS. 9A and 9B ,  FIGS. 10A to 10D , and  FIGS. 11A and 11B , axes of abscissae denote a pixel number n, axes of ordinates denote a pixel position (or a scanning position) y (or y′ after coordinate conversion), and a unit is a line.  FIGS. 9A and 9B  and  FIGS. 11A and 11B  correspond to  FIGS. 8A to 8D , respectively. The graphs on a left side in  FIGS. 9A, 9B, 11A, and 11B  are obtained before coordinate conversion is performed and the graphs on a right side are obtained after coordinate conversion is performed on y axes. Rectangles plotted in the graphs denote the scanning positions on the photoconductor drum  102 , and circles denote the ideal scanning positions. 
     Cases of Shifts in Forward Direction and Backward Direction 
     The graph on the left side in  FIG. 9A  will be described first. In the graph before the coordinate conversion, in the ideal scanning positions denoted by the circles, a pixel position y of 2 in the sub scanning direction corresponds to the pixel number 2, for example, that is, the ideal scanning positions form a straight line (denoted by a chain line) in which the pixel number n is equal to the pixel position y and an inclination is 1. The straight line of the chain line is represented by Expression (8) below.
 
 y=n   Expression (8)
 
     The scanning positions denoted by the rectangles are shifted from the ideal scanning positions denoted by the circles by S (=0.2) lines in the forward direction (a positive direction in the y axis) as described with reference to  FIG. 8A . Therefore, the scanning positions denoted by the rectangle forms a straight line (denoted by a solid line) represented by Expression (9) which has an inclination of 1 and which is offset.
 
 y=n+S   Expression (9)
 
     The coordinate conversion is performed such that the actual scanning positions are converted into the ideal scanning positions in this embodiment, and accordingly, an expression below is used for the coordinate conversion in the case of  FIG. 9A . Note that “C” in Expression (10) denotes a correction amount.
 
 y′=y+C   Expression (10)
 
     Accordingly, the correction amount C is represented by Expression (11) below using a shift amount S.
 
 C=−S   Expression (11)
 
     Note that the correction amount C and the shift amount S in Expression (11) correspond to the correction amount Cmn and the position shift amount Zmn′ after the correction in the expression “Cmn=−Zmn′” in Expression (7) above. 
     Using Expression (10) for the coordinate conversion and Expression (11) for obtaining the correction amount C, Expressions (8) and (9) are converted into Expressions (12) and (13) below, respectively.
 
 y′=y+C=n +(− S )= n−S   Expression (12)
 
 y′=y+C =( n+S )+ C =( n+S )+(− S )= n   Expression (13)
 
     In  FIG. 9B , assuming that a shift amount S is −0.2, Expressions (8) to (13) are similarly satisfied, and the description of  FIG. 9A  may also be applied to  FIG. 9B . Note that, as illustrated in  FIGS. 9A and 9B , in a case of scanning lines which do not have a dense/sparse portion in the scanning lines and which are shifted in the forward direction or the backward direction, the straight lines have a constant inclination before and after the conversion. 
     Cases where Dense/Sparse Portion is Generated 
     Here, coordinate conversion applicable to cases where a dense portion and a sparse portion of the scanning positions are generated as illustrated in  FIGS. 11A and 11B  and a case of a combination of the shift and dense/sparse portions in  FIGS. 9A and 9B  and  FIGS. 11A and 11B  will be described.  FIG. 10A  is a graph illustrating the relationship between a pixel number and a scanning position. An axis of abscissae denotes a pixel number n, an axis of ordinates y denotes a scanning position in the sub scanning direction, and rectangles denote scanning positions on the photoconductor drum  102 . In  FIG. 10A , a case where the scanning lines are positioned in a dense state on the photoconductor drum  102  in a range in which the pixel number n is equal to or smaller than 2 and the scanning lines are positioned in a sparse state on the photoconductor drum  102  in a range in which the pixel number n is equal to or larger than 2 will be described. 
     As illustrated in  FIG. 10A , in the case of the dense state in the range in which the pixel number n is equal to or smaller than 2 and the sparse state in the range in which the pixel number n is equal to or larger than 2, an inclination of a straight line in the range in which the pixel number n is equal to or smaller than 2 is different from an inclination of a straight line in the range in which the pixel number n is equal to or larger than 2. Furthermore, a line is bent in a point in which the pixel number n is equal to 2. In  FIG. 10A , a function indicating change of the scanning positions indicated by the rectangles is denoted by “ft(n)” and is represented by a solid line. The function ft(n) representing the scanning positions is represented by Expression (14) below.
 
 y=ft ( n )  Expression (14)
 
     Next, when a function obtained after the coordinate conversion on the y axis which is the scanning positions in the sub scanning direction is denoted by “ft′(n)”, the functions ft′(n) representing the scanning positions after the coordinate conversion is represented by Expression (15) below.
 
 y′=ft ′( n )  Expression (15)
 
     In this embodiment, the coordinate conversion is performed by performing expansion and contraction in the y axis direction or performing shifting in the y axis direction so that the scanning positions after the coordinate conversion are uniformly positioned. Therefore, the function ft′(n) representing the scanning positions after the coordinate conversion satisfies a condition represented by Expression (16) below.
 
 ft ′( n )= n   Expression (16)
 
     Expression (16) means that, for the pixel number of 2, for example, a pixel position y′ (=ft′(2)) in the sub scanning direction after the coordinate conversion is 2. 
     Dotted lines which connect  FIG. 10A  to  FIG. 10B  indicate correspondence between original coordinate positions in the y axis and the coordinate positions in a y′ axis after the coordinate conversion from left to right, and a lower portion of the y axis (corresponding to n≦2) corresponds to expansion and an upper portion (corresponding to n≧2) corresponds to contraction. A procedure for obtaining coordinates after the coordinate conversion of the pixels of the input image data using the coordinate conversion illustrated in  FIGS. 10A and 10B  will be described with reference to  FIGS. 10C and 10D . As with  FIGS. 10A and 10B , axes of abscissae denote the pixel number n, axes of ordinates y (or y′) denote the scanning positions in the sub scanning direction in  FIGS. 10C and 10D , and the coordinate conversion has not been performed in  FIG. 10C  and has been performed in  FIG. 10D . The relationship between the pixel number n and the coordinate position y of the input image data is illustrated as below. First, a dotted line in  FIG. 10C  denotes a function fs(n) which indicates the ideal scanning positions before the coordinate conversion and which is represented by Expression (17) below.
 
 y=fs ( n )  Expression (17)
 
     Furthermore, since the pixels of the input image data are arranged with even pitches in the sub scanned direction in this embodiment, the function fs(n) is represented by Expression (18) below.
 
 fs ( n )= n   Expression (18)
 
     A scanning position in the y′ coordinate after the coordinate conversion is performed on a target pixel having a pixel number ns of the input image data is obtained by three steps below. In a first step, assuming that a y coordinate of an ideal scanning position corresponding to the pixel number ns of the input image data is denoted by “ys”, the coordinate ys may be obtained by Expression (19) below.
 
 ys=fs ( ns )  Expression (19)
 
     A pixel number nt which is the same scanning position before the coordinate conversion on the photoconductor drum  102  (a solid line) is obtained ((1) of  FIG. 10C ). Here, the scanning position on the photoconductor drum  102  is represented by a function “y=ft(n)” and the relationship “ys=ft(nt)” is satisfied. Assuming that an inverse function of the function ft(n) is denoted by ft −1 (y), the pixel number nt is represented by Expression (20) below.
 
 nt=ft   −1 ( ys )  Expression (20)
 
     In a second step, a y′ coordinate (yt) after the coordinate conversion corresponding to the pixel number nt of the scanning position on the photoconductor drum  102  is obtained by Expression (21) below using the function ft′(n) after the coordinate conversion ((2) of  FIG. 10D ).
 
 yt=ft ′( nt )  Expression (21)
 
     The pixel number ns may be arbitrarily selected, and therefore, an expression for obtaining the position yt on the y′ coordinate after the coordinate conversion from the pixel number ns corresponds to the function fs′(n) for obtaining the y′ coordinate as a calculation from the pixel number n of the input image data. Accordingly, a general expression represented by Expression (22) below may be obtained from Expressions (19) to (21). Note that the function indicating the ideal scanning positions denoted by the dotted line is represented as follows ((3) of  FIG. 10D ): y′=fs′(n).
 
 yt=fs ′( ns )= ft ′( nt )= ft ′( ft   −1 ( ys ))= ft ′( ft   −1 ( fs ( ns )))
 
     When “ns” is generalized to “n”, the following expression is obtained.
 
 fs ′( n )= ft ′( ft   −1 ( fs ( n )))  Expression (22)
 
     Furthermore, pitches between the pixels of the input image data and pitches between the scanning positions after the coordinate conversion are set uniform, that is, a distance of 1, and Expressions (18) and (16) are assigned to Expression (22). By this, Expression (22) is represented by Expression (23) using the inverse function ft −1 (n) of the function ft(n) which guides a scanning position from the pixel number n.
 
 fs ′( n )= ft   −1 ( n )  Expression (23)
 
     Expression (9) in which the scanning positions illustrated in  FIGS. 9A and 9B  are uniformly shifted in the forward direction and the backward direction and Expression (12) in which positions after the coordinate conversion of the input image data are obtained have also the relationship of an inverse function, and accordingly, the Expression (23) may be satisfied. Furthermore, in a case where dense/sparse portions are generated in the scanning positions illustrated in  FIGS. 11A and 11B , if a function y representing the scanning positions before the coordinate conversion is set as a straight line having an inclination k which passes a coordinate (n0, y0), the function y may be represented by Expression (24) below.
 
 fs ( n )= y=k ×( n−n 0)+ y 0  Expression (24)
 
     To obtain pixel positions in the y axis of the input image data after the coordinate conversion, an inverse function ((1/k)×(y−y0)+n0) is obtained from Expressions (22) and (23) and the pixel number n is assigned to the inverse function. In this way, Expression (25) below is obtained.
 
 y ′=(1/ k )×( n−y 0)+ n 0  Expression (25)
 
     In the state in which the pitches between the scanning lines are dense as illustrated in  FIG. 11A  and the state in which the pitches between the scanning lines are spares as illustrated in  FIG. 11B , positions of the scanning lines on the photoconductor drum  102  after the coordinate conversion may be represented by Expression (25). Furthermore, a correction value Cn of the pixel number n may be obtained from the following expression: Cn=fs′(n)−fs(n). 
     Specifically, in  FIG. 11A , the following relationships are obtained: n0=y0=3 and k=0.8.
 
 fs ′( n )=(1/0.8)×( n− 3)+3  Expression (26)
 
     For example, as for a pixel number 3, an expression “fs&#39;s(3)=3.00” is satisfied and a correction value C 3  is 0.00 (=3.00-3.00). Furthermore, as for a pixel number 5, an expression “fs′(5)=5.50” is satisfied and a correction value C 5  is +0.50 (=+5.50−5.00). Correction values C 1  to C 5  in a case where the scanning positions are dense are illustrated in  FIG. 13C . 
     Furthermore, in  FIG. 11B , the following relationships are obtained: n0=y0=3 and k=1.2.
 
 fs ′( n )=(1/1.2)×( n− 3)+3  Expression (27)
 
     For example, in the pixel number 3, fs′(3) is 3.000 and a correction value C 3  is 0.000 (=3.000−3.000). Furthermore, in the pixel number 5, fs′(5) is 4.667 and the correction value C 5  is −0.333 (=4.667−5.000). Correction values C 1  to C 5  in a case where the scanning positions are sparse are illustrated in  FIG. 13D . 
     Furthermore, even if dense/sparse portions and shifts are mixed in the scanning lines, the ideal scanning positions after the coordinate conversion may be obtained using Expressions (22) and (23). The correction value setting unit  506  performs the coordinate conversion on the ideal scanning positions based on the position shift amounts so as to obtain the correction values Cn and outputs information on the correction values Cn to the filter coefficient setting unit  504 . 
     Filter Process 
     In this embodiment, the filter process is executed to generate correction data. Note that, in this embodiment, the filter process unit  501  performs the filter process by a convolution calculation using a filter function as described below. Specifically, the filter process unit  501  performs the filter process based on the positional relationship between positions of the pixels in the sub scanning direction obtained after the correction of the scanning positions in the sub scanning direction of the pixels of the input image data and positions of the pixels in the sub scanning direction obtained by converting the pitches of the scanning lines to be uniform through the coordinate conversion. Note that the pixels which have not been subjected to the filter process are also referred to as “input pixels” and pixels which have subjected to the filter process are also referred to as “output pixels”. Furthermore, the pixels which have not been subjected to the filter process correspond to the pixels which have been subjected to the coordinate conversion described above. 
     The convolution function of this embodiment may be selected from among linear interpolation illustrated in  FIG. 12A , bicubic interpolation of  FIG. 12B , and bicubic interpolation of  FIG. 12C . The filter function output unit  505  outputs information on the convolution function used in the filter process to the filter coefficient setting unit  504  as table information, for example. In  FIGS. 12A to 12D , axes of ordinates y denote a position in the sub scanning direction and a unit is a pixel. Axes of abscissae k denote a magnitude of a coefficient. Although the unit of the ordinate axes y is a pixel, the unit may be a line since the line indicates the sub scanning direction. 
       FIG. 12A  is represented by the following expressions.
 
 k=y+ 1 (−1≦ y≦ 0)
 
 k=−y+ 1 (0&lt; y≦ 1)
 
0 ( y&lt;− 1, y&gt; 1)  Expression (28)
 
       FIGS. 12B and 12C  are represented by two expressions below. 
     Although “a” is −1, “w” is 1 in  FIG. 12B , and “w” is 1.5 in  FIG. 12C  in this embodiment, “a” and “w” may be adjusted in accordance with an electrophotographic characteristic of the image forming apparatuses. The filter coefficient setting unit  504  outputs the coefficient (“k” described below) used in the filter process based on information on the filter coefficient obtained from the filter function output unit  505  and information on the correction value C output from the correction value setting unit  506  to the filter process unit  501 . 
     
       
         
           
             
               
                 
                   
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     Here, a description will be made with reference to  FIG. 12D . In  FIG. 12D , an axis of abscissae denotes the coefficient k used in the filter process and an axis of ordinates denotes the position y in the sub scanning direction. The filter process unit  501  obtains a coefficient kn corresponding to the correction value Cn using the filter function input from the filter function output unit  505  when receiving the correction value Cn from the correction value setting unit  506 . Note that white circles in  FIG. 12D  denote coefficients before the coordinate conversion. In  FIG. 12D , a coefficient k 1  is set for a correction value C 1  and a coefficient k 2  is set for a correction value C 2  as the coefficient kn used in the filter process (black circles). In this embodiment, the same convolution function is employed irrespective of a dense/sparse state of the input image data, and sampling is performed using the ideal scanning positions so that density per a certain area of the input image data is stored. 
     Concrete Example of Filter Process 
     A concrete example of the filter process using the convolution calculation of the filter function by linear interpolation of Expression (28) based on coordinate positions obtained after the coordinate conversion of this embodiment will be described with reference to  FIGS. 13A to 13D . Note that the filter process using the convolution calculation is executed by the filter process unit  501 .  FIGS. 13A to 13D  correspond to  FIGS. 8A to 8D , respectively. Columns on left sides in  FIGS. 13A to 13D  indicate input pixels after the coordinate conversion described above. The input pixels are included in the image data subjected to the dither process in step S 606  of  FIG. 5 , and includes pixels having density values of a halftone. Furthermore, columns on right sides in  FIGS. 13A to 13D  indicate scanning positions on the photoconductor drum  102  after the coordinate conversion described above. Specifically, the scanning positions in the columns on the right sides in  FIGS. 13A to 13D  are subjected to the coordinate conversion so as to be arranged at even pitches of a distance of 1. 
     More specifically, the scanning positions in the sub scanning direction of the input pixels after the coordinate conversion are represented by straight lines (y′=fs′(n)) denoted by the chain lines in the graphs obtained after the coordinate conversion illustrated on the right sides in  FIGS. 9A, 9B, 11A, and 11B . The scanning positions on the photoconductor drum  102  after the coordinate conversion are represented by straight lines (y′=ft′(n)) denoted by the solid lines in the graphs obtained after the coordinate conversion illustrated on the right sides of  FIGS. 9A, 9B, 11A, and 11B . For example, since a shift amount is +0.2 (=S) in  FIG. 9A , the scanning positions on the photoconductor drum  102  after the coordinate conversion are represented by “fs′(n)=y−0.2=n−0.2”. 
     Furthermore, in  FIGS. 13A to 13D , magnitudes of pixel values, that is, density values, are denoted by gradation of circles. Furthermore, numbers in brackets indicate numbers of the scanning lines and are the same as the pixel numbers illustrated in  FIGS. 8A to 8D . In graphs in centers of  FIGS. 13A to 13D , axes of abscissae denote density and axes of ordinates denote a position in the sub scanning direction. In the convolution calculation, waveforms W (W 1  to W 5  corresponding to the pixels (1) to (5)) obtained by multiplying the filter function ( FIG. 12A ) having the coordinate positions of the input pixels at a center by the pixel values are developed and overlap with one another for addition. 
     A description will be made in turn from  FIG. 13A . The pixels (1) and (5) denoted by white circles have a density of 0, that is, a pixel value of 0. Therefore, W 1  and W 5  obtained by multiplying the filter function by the pixel values are both 0. Densities of the pixels (2), (3), and (4) denoted by black circles are the same, largest values of waveforms of W 2 , W 3 , and W 4  are the same, and the waveforms are obtained by developing the filter function using the pixel positions of the input pixels at centers. A total sum (EWn, n=1 to 5) of all the waveforms is obtained as a result of the convolution calculation. 
     The pixel values of the output pixels are sampled in the scanning positions on the photoconductor drum  102  after the scanning positions are subjected to the coordinate conversion. Therefore, the pixel value (1) corresponding to the scanning position on the photoconductor drum  102  intersects with the waveform W 2  at a point P 0 , for example, and therefore, a density D 1  is obtained. Furthermore, the pixel value (2) intersects with the waveform W 2  at a point P 2  and intersects with the waveform W 3  at a point P 1 , and therefore, a density of D 1 +D 2  is obtained. Thereafter, densities of the pixel values (3) to (5) are similarly obtained. Note that the pixel value (5) does not intersect with any waveform, and therefore, a pixel value is 0. Furthermore, results of calculations of the pixel values (1) to (5) in  FIGS. 13B to 13D  are denoted by gradation of the pixels in the columns on the right sides. 
     Position shifts of the input pixels correspond to the pixels in the axes of ordinates in  FIGS. 13A to 13D . The position shift amounts in the axes of ordinates in  FIGS. 13A to 13D  are information on position shift amounts obtained by the inverse function in accordance with the coordinate conversion on the scanning positions in the sub scanning direction of the pixels of the input image described above. For example, in the case of  FIG. 13A , as described with reference to  FIG. 9A , the correction amount C for the position shift amount S of the scanning lines is −0.2. Furthermore, the correction amount C is calculated using Expression (26) in the case of  FIG. 13C  and Expression (27) in the case of  FIG. 13D . 
       FIG. 13A  is a diagram illustrating a state in which although the scanning positions of the scanning lines are shifted in the forward direction in the sub scanning direction, centers of the pixel values are conversely shifted in the backward direction, and accordingly, positions of the centers of the pixel values are corrected.  FIG. 13B  is a diagram illustrating a state in which although the scanning positions of the scanning lines are shifted in the backward direction in the sub scanning direction, centers of the pixel values are conversely shifted in the forward direction, and accordingly, positions of the centers of the pixel values are corrected.  FIG. 13C  is a diagram illustrating a state in which the pitches of the scanning positions are dense, density distribution spreads due to the convolution calculation after the coordinate conversion, local concentration in density is cancelled, and local density change is corrected. Furthermore,  FIG. 13D  is a diagram illustrating a state in which the pitches of the scanning positions are conversely sparse, density distribution contracts due to the convolution calculation after the coordinate conversion, the density spread is cancelled, and local density change is corrected. The pixel value (3) of  FIG. 13D  has a density (100+α)% higher than a density of 100%. 
     Filter Process 
     Referring back to  FIG. 7 , in step S 3604 , the CPU  303  performs the filter process using the filter process unit  501  in accordance with the correction attribute information generated in step S 3603 . Specifically, the CPU  303  performs the convolution calculation and re-sampling on the input image described above. Here, the process in step S 3604  executed by the CPU  303  will be described in detail with reference to a flowchart of  FIG. 14 . 
     When starting the filter process using the convolution calculation using the filter process unit  501 , the CPU  303  executes a process in step S 3703  onwards. In step S 3703 , when it is assumed that the spread of the convolution function is denoted by “L”, the CPU  303  extracts lines of the input image included in a range ±L of the sub scanning position of a line yn (a position yn) in a target output image, that is, a range of a width of 2L (a range from (yn−L) to (yn+L)). Here, the value L is defined as a smallest value which attains a value of the convolution function of 0 out of the range ±L of the convolution function. For example, L is 1 in the linear interpolation in  FIG. 12A , L is 2 in the bicubic interpolation in  FIG. 12B , and L is 3 in the bicubic interpolation in  FIG. 12C . Using Expression (23), “ymin” and “ymax” of a range from ymin to ymax of the corresponding input image satisfy the following condition.
 
 ft   −1 ( y min)= yn−L,ft   −1 ( y max)= yn+L   Expression (31)
 
     By deforming Expression (31), “ymin” and “ymax” are obtained using Expression (32) below.
 
 y min= ft ( yn−L ), y max= ft ( yn+L )  Expression (32)
 
     Accordingly, lines of the input image extracted for the lines yn of the target output image are all integer lines in the range from ymin to ymax. 
     When the lines of the target output image are denoted by “yn” and the lines of the input image which is a target of the convolution calculation which are denoted by “ym”, distances dnm are represented by Expression (33) below.
 
 dnm=yn−ft   −1 ( ym )  Expression (33)
 
     Accordingly, in step S 3704 , the CPU  303  obtains a coefficient knm as a convolution function g(y) using the filter coefficient setting unit  504  in accordance with Expression (34) below.
 
 knm=g ( dnm )  Expression (34)
 
     In step S 3707 , the CPU  303  obtains a position n in the sub scanning direction of the input image extracted in step S 3703  and image data in a target position N in the main scanning direction. Here, it is assumed that pixel data corresponds to input pixel data Pinm. In step S 3708 , the CPU  303  calculates the convolution calculation using the filter process unit  501  and terminates the process. More specifically, the filter process unit  501  performs a product-sum operation using the corresponding coefficient knm obtained in step S 3704  and the input pixel data Pinm obtained in step S 3707  so as to obtain a value Poutn of a target pixel. Note that the input pixel data Pinm indicates density of the target pixel before the filter process, and the value Poutn of the target pixel indicates density of the target pixel after the filter process which is output pixel data. 
     
       
         
           
             
               
                 
                   
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     Here, Expression (35) corresponds to  FIGS. 13A to 13D , color strength (density) of the circles on the left sides in  FIGS. 13A to 13D  correspond to the input pixel data Pinm, D 1  and D 2  in  FIG. 13A  correspond to knm×Pinm, and color strength (density) of the circles on the right sides in  FIGS. 13A to 13D  correspond to Poutn. 
     In this way, according to this embodiment, distortion and density unevenness of an image caused by shifts of irradiation positions due to variation of positions of the multi-beam and the plane tilt of the rotatable polygonal mirror are corrected by performing the coordinate conversion on the pixel positions of the input image based on a selected dither and a profile of the position shifts in the sub scanning direction of the input image. Thereafter, by performing the filter process and the sampling, local density concentration, such as position shifts and banding, may be cancelled while density of the input pixels is stored, and accordingly, an excellent image may be obtained. 
     As described above, according to this embodiment, the plane tilt correction may be appropriately performed irrespective of an image pattern. 
     Second Embodiment 
     In the first embodiment, the correction amount for the position shift amount is determined depending on a type of dither to be used. In a second embodiment, a correction amount for a position shift amount is determined in accordance with a type of a dither to be used and image density. Although a configuration of an image forming apparatus and basic configurations of a control sequence of a page process are the same as those in the first embodiment, the process in step S 3602  in  FIG. 7  of the first embodiment for calculating a correction amount based on a position shift amount is different. Components the same as those of the first embodiment are denoted by reference numerals the same as those of the first embodiment, and detailed descriptions thereof are omitted. 
     Plane Tilt Correction Process 
     In  FIG. 5  described above, a CPU  303  executes the control sequence illustrated in  FIG. 7  so as to perform a plane tilt correction process in step S 607  after terminating the dither process in step S 606 .  FIG. 7  is a control sequence for performing the plane tilt correction process. The CPU  303  reads position shift amounts in a sub scanning direction, that is, position shift amounts Zmn of scanning lines (LDn) of n-th laser light of multi-beam and m-th mirror plane of a rotatable polygonal mirror  204  in step S 3602 . Subsequently, the CPU  303  selects a plane tilt correction table corresponding to a dither selected in step S 603  to step S 605  of  FIG. 5  described above. The correction table stores information on association between density of an image and a correction amount for a plane tilt amount for each plane tilt amount. The CPU  303  includes a storage unit which stores correction tables corresponding to dithers A to C. 
       FIGS. 15A and 15B  are diagrams illustrating content of the plane tilt correction tables for individual dithers of this embodiment as one graph for description.  FIGS. 15A and 15B  are examples of the correction tables of the dithers in a case of plane tilt amounts of 5 μm and 3 μm, respectively. In  FIGS. 15A and 15B , axes of abscissae denote density and axes of ordinates denote a correction amount. Furthermore, in  FIGS. 15A and 15B , graphs denoted by solid lines indicate the content of the correction table of the dither A, graphs denoted by dotted lines indicate the content of the correction table of the dither B, and graphs denoted by chain lines indicate the content of the correction table of the dither C. Furthermore, “D 0 ” to “D 6 ” in the axes of abscissae indicate density of an image, and the density becomes higher (a density value is increased) in order from D 0  to D 6 . In a case where a plane tilt amplitude amount ZW calculated by Expression (23) described above is 5 μm, for example, the CPU  303  selects a correction table for the plane tilt amount of 5 μm from among a plurality of correction tables and selects a curve for a selected dither for each pixel. In this case, the CPU  303  selects the graph of  FIG. 15A , and if the dither A is selected as a dither to be used, a correction amount for the plane tilt amount is determined in accordance with image density of a target pixel. The image density of the pixel in this case is obtained before the dither process is executed in step S 606  of  FIG. 5 . 
     According to the graph of  FIG. 15A , when the dither A is selected, a correction value for the plane tilt amount of 5 μm in a case where the image density is D 0  or D 6  is determined to 0 and a correction value for the plane tilt amount of 5 μm in a case where the image density is D 1  or D 5  is determined to ZWa 0 . Furthermore, according to the graph of  FIG. 15A , when the dither A is selected, a correction value for the plane tilt amount of 5 μm in a case where the image density is D 2  or D 4  is determined to ZWa 1  and a correction value for the plane tilt amount of 5 μm in a case where the image density is D 3  is determined to ZWa 2 . In this embodiment, the CPU  303  determines, as with the first embodiment, the plane tilt amplitude amount ZW in accordance with Expression (23) described above. Then the CPU  303  calculates a position shift amount Zmn′ after correction using Expression (24) described above when the dither A is selected and a correction amount in a case where the plane tilt amount is ZW and image density is D 1  is ZWa 0 . Then the CPU  303  obtains a correction amount Cmn serving as correction attribute information based on the position shift amount Zmn′ after the correction in step S 3603 . Thereafter, in step S 3604 , the CPU  303  performs the filter process using the filter process unit  501  based on the correction attribution information generated by the process described above. 
     As described above, a correction amount for a position shift amount is determined using a combination between a type of dither and image density in this embodiment. Accordingly, moire caused by the plane tilt may be corrected and large electrophotographic environmental variation, such as temperature variation and humidity variation, may be coped with. 
     As described above, according to this embodiment, the plane tilt correction may be appropriately performed irrespective of an image pattern. 
     According to the aspect of the embodiments, the plane tilt correction may be appropriately performed irrespective of an image pattern. 
     While the disclosure has been described with reference to exemplary embodiments, it is to be understood that the disclosure is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions. 
     This application claims the benefit of Japanese Patent Application No. 2015-228744, filed in Nov. 24, 2015, which is hereby incorporated by reference herein in its entirety.