Dither matrix producing method and apparatus, image processing method and apparatus, image forming method and apparatus, program and recording medium

• US 20050219628A1
• Filed: 12/02/2004
• Published: 10/06/2005
• Est. Priority Date: 12/02/2003
• Status: Active Grant

• Current Assignee: Ricoh Company Limited
• Original Assignee: Ricoh Company Limited
• Inventors: Ogawa, Takeshi, Takesue, Toshihiro, Yasutomi, Kei
• Granted Patent: US 7,746,502 B2
• Time in Patent Office: Days
• US Class Current: 358/3.160
• CPC Class Codes: H04N 1/4058 with details for producing a halftone screen at an oblique angle H04N1/4056 takes precedence

Abstract

A main vector: m{circumflex over ( )}=(mx, my) and a sub-vector: s{circumflex over ( )}=(sx, sy) determining cyclic structure of a basic matrix are defined, and a positional vector for a pixel configuring the basic matrix Pi{circumflex over ( )}=(Pix, Piy) is defined, where the suffix i denotes an i-th pixel configuring the basic matrix. Then, the pixels configuring the basic matrix are arranged in such a manner as to satisfy the cyclic structure, and transformation of Pi{circumflex over ( )}→Pi{circumflex over ( )}+km{circumflex over ( )}+ls{circumflex over ( )} is carried out where k and l are arbitrary integers, and may be negative values. Then a new shape of the basic matrix is obtained with Pi{circumflex over ( )}+km{circumflex over ( )}+ls{circumflex over ( )} as a new pixel positional vector Pi{circumflex over ( )}.

Claims

1. A dither matrix producing method arranging basic matrixes cyclically, comprising:

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix.

4. The dither matrix producing method as claimed in claim 1, wherein:

a growth order of the pixels of the basic matrix is determined according to a magnitude of a corresponding pixel positional vector |Pi{circumflex over ( 

)}|, wherein;

if a plurality of pixels have respective pixel positional vectors in the same magnitude, the growth order thereamong is determined according to θ

which denotes an angle of the relevant pixel positional vector Pi{circumflex over ( 

)} with respect to a reference vector.

7. The dither matrix producing method as claimed in claim 1, wherein:

a growth order of the pixels of the basic matrix is determined according to a magnitude of a corresponding pixel positional vector |Pi{circumflex over ( 

)}|, wherein;

if a plurality of pixels have respective pixel positional vectors in the same magnitude, the growth order thereamong is determined according to θ

1 or θ

2, wherein;

θ

1 denotes an angle of the relevant pixel positional vector Pi{circumflex over ( 

)} with respect to a reference vector; and

if θ

1≧

π

, θ

2 obtained from the following formula is applied instead of θ

1;

θ

2=θ

1−

π

10. The dither matrix producing method as claimed in claim 1, for producing a dither matrix based on given resolution, number of lines of screen and screen angle, wherein:

the determination of the two vectors for determining the cyclic structure of the basic vector is performed in such a manner as to achieve a number of lines of screen and a screen angle close to the given number of screen lines and screen angle, respectively, for the purpose of producing the dither matrix.

13. The dither matrix producing method as claimed in claim 1, wherein:

a multi-tone dither matrix with more than one threshold level is produced.

16. The dither matrix producing method as claimed in claim 13, wherein:

the multi-tone dither matrix to produce is a multi-tone dither matrix in which a growth order in a depth direction is determined such that, for each basic matrix, growth of another pixel starts after growth in the depth direction for one pixel is saturated.

2. A dither matrix producing method arranging basic matrixes cyclically, comprising:

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros.

5. The dither matrix producing method as claimed in claim 2, wherein:

a growth order of the pixels of the basic matrix is determined according to a magnitude of a corresponding pixel positional vector |Pi{circumflex over ( 

)}|, wherein;

if a plurality of pixels have respective pixel positional vectors in the same magnitude, the growth order thereamong is determined according to θ

which denotes an angle of the relevant pixel positional vector Pi{circumflex over ( 

)} with respect to a reference vector.

8. The dither matrix producing method as claimed in claim 2, wherein:

a growth order of the pixels of the basic matrix is determined according to a magnitude of a corresponding pixel positional vector |Pi{circumflex over ( 

)}|, wherein;

if a plurality of pixels have respective pixel positional vectors in the same magnitude, the growth order thereamong is determined according to θ

1 or θ

2, wherein;

θ

1 denotes an angle of the relevant pixel positional vector Pi{circumflex over ( 

)} with respect to a reference vector; and

if θ

1≧

π

, θ

2 obtained from the following formula is applied instead of θ

1;

θ

2=θ

1−

π

11. The dither matrix producing method as claimed in claim 2, for producing a dither matrix based on given resolution, number of lines of screen and screen angle, wherein:

the determination of the two vectors for determining the cyclic structure of the basic vector is performed in such a manner as to achieve a number of lines of screen and a screen angle close to the given number of screen lines and screen angle, respectively, for the purpose of producing the dither matrix.

14. The dither matrix producing method as claimed in claim 2, wherein:

a multi-tone dither matrix with more than one threshold level is produced.

17. The dither matrix producing method as claimed in claim 14, wherein:

the multi-tone dither matrix to produce is a multi-tone dither matrix in which a growth order in a depth direction is determined such that, for each basic matrix, growth of another pixel starts after growth in the depth direction for one pixel is saturated.

3. A dither matrix producing method in which a cyclic structure of a basic matrix is determined by two vectors, comprising:

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros.

6. The dither matrix producing method as claimed in claim 3, wherein:

a growth order of the pixels of the basic matrix is determined according to a magnitude of a corresponding pixel positional vector |Pi{circumflex over ( 

)}|, wherein;

if a plurality of pixels have respective pixel positional vectors in the same magnitude, the growth order thereamong is determined according to θ

which denotes an angle of the relevant pixel positional vector Pi{circumflex over ( 

)} with respect to a reference vector.

9. The dither matrix producing method as claimed in claim 3, wherein:

a growth order of the pixels of the basic matrix is determined according to a magnitude of a corresponding pixel positional vector |Pi{circumflex over ( 

)}|, wherein;

if a plurality of pixels have respective pixel positional vectors in the same magnitude, the growth order thereamong is determined according to θ

1 or θ

2, wherein;

θ

1 denotes an angle of the relevant pixel positional vector Pi{circumflex over ( 

)} with respect to a reference vector; and

if θ

1≧

π

, θ

2 obtained from the following formula is applied instead of θ

1;

θ

2=θ

1−

π

12. The dither matrix producing method as claimed in claim 3, for producing a dither matrix based on given resolution, number of lines of screen and screen angle, wherein:

the determination of the two vectors for determining the cyclic structure of the basic vector is performed in such a manner as to achieve a number of lines of screen and a screen angle close to the given number of screen lines and screen angle, respectively, for the purpose of producing the dither matrix.

15. The dither matrix producing method as claimed in claim 3, wherein:

a multi-tone dither matrix with more than one threshold level is produced.

18. The dither matrix producing method as claimed in claim 15, wherein:

the multi-tone dither matrix to produce is a multi-tone dither matrix in which a growth order in a depth direction is determined such that, for each basic matrix, growth of another pixel starts after growth in the depth direction for one pixel is saturated.

19. An article of manufacture having one or more recordable media storing instructions thereon which, when executed by a system, cause the system to perform a dither matrix producing method to arrange basic matrixes cyclically, comprising:

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then, d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix.

20. An article of manufacture having one or more recordable media storing instructions thereon which, when executed by a system, cause the system to perform a dither matrix producing method to arrange basic matrixes cyclically, comprising:

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros.

21. An article of manufacture having one or more recordable media storing instructions thereon which, when executed by a system, cause the system to perform a dither matrix producing method in which a cyclic structure of a basic matrix is determined by two vectors, comprising:

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros.

22. A dither matrix producing device output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises:

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix.

23. A dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises:

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros.

24. A dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises:

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros.

25. An image processing method, comprising:

producing a dither matrix by

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data.

28. The image processing method as claimed in claim 25,

further comprising switching the dither matrix in response to the input multi-tone image data.

31. The image processing method as claimed in claim 25,

further comprising switching the dither matrix in response to an output image mode.

26. An image processing method, comprising:

producing a dither matrix by

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data.

29. The image processing method as claimed in claim 26,

further comprising switching the dither matrix in response to the input multi-tone image data.

32. The image processing method as claimed in claim 26,

further comprising switching the dither matrix in response to an output image mode.

27. An image processing method, comprising:

producing a dither matrix by

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data.

30. The image processing method as claimed in claim 27,

further comprising switching the dither matrix in response to the input multi-tone image data.

33. The image processing method as claimed in claim 27,

further comprising switching the dither matrix in response to an output image mode.

34. An image forming method comprising:

performing dither processing on multi-tone image an image processing method to produce output image data, wherein the image producing method comprises

producing a dither matrix by

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data; and

forming an image based on the output image data.

35. An image forming method comprising:

performing dither processing on multi-tone image an image processing method to produce output image data, wherein the image producing method comprises

producing a dither matrix by

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data; and

forming an image based on the output image data.

36. An image forming method comprising:

performing dither processing on multi-tone image an image processing method to produce output image data, wherein the image producing method comprises

producing a dither matrix by

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data; and

forming an image based on the output image data.

37. An image forming method comprising:

performing dither processing on color multi-tone image having multi-color color information an image processing method in such a manner that dither processing is performed for each of the different colors to produce output image data, wherein the image producing method comprises

producing a dither matrix by

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data; and

forming a color image based on the output image data

38. An image forming method comprising:

performing dither processing on color multi-tone image having multi-color color information an image processing method in such a manner that dither processing is performed for each of the different colors to produce output image data, wherein the image processing method comprises

producing a dither matrix by

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data; and

forming a color image based on the output image data

39. An image forming method comprising:

performing dither processing on color multi-tone image having multi-color color information an image processing method in such a manner that dither processing is performed for each of the different colors to produce output image data, wherein the image producing method comprises

producing a dither matrix by

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros; and

using the dither matrix, performing dither processing on multi-tone image data, and producing quantized image data; and

forming a color image based on the output image data.

40. An image processing apparatus comprising:

a dither matrix producing device output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix; and

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device.

41. An image processing apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros; and

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device.

42. An image processing apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros; and

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device.

43. An image forming apparatus comprising:

a dither matrix producing device output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

44. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

45. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

46. An image forming apparatus comprising:

a dither matrix producing device output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises;

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix;

a determination device to perform a determination from an input multi-tone image, and determine the dither matrix to produce in the dither matrix producing device;

a dither processing device to perform a dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

47. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises;

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros;

a determination device to perform a determination from an input multi-tone image, and determine the dither matrix to produce in the dither matrix producing device;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

48. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises;

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros;

a determination device to perform a determination from an input multi-tone image, and determine the dither matrix to produce in the dither matrix producing device;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

49. An image forming apparatus comprising:

a dither matrix producing device output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises;

a) defining two vectors determining a cyclic structure of the basic matrixes according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix;

a determination device to perform a determination for an output image mode, and determine the dither matrix to produce in the dither matrix producing device;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

50. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises;

a) defining two vectors determining a cyclic structure of the basic matrix according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros;

a determination device to perform a determination for an output image mode, and determine the dither matrix to produce in the dither matrix producing device;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

51. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises;

a) defining the two vectors according to;

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros;

a determination device to perform a determination for an output image mode, and determine the dither matrix to produce in the dither matrix producing device;

a dither processing device to perform dither processing on multi-tone data with the use of the dither matrix produced by the dither matrix producing device; and

an image forming device to perform image forming based on image data having undergone the dither processing.

52. An image forming apparatus comprising:

a dither matrix producing device output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining two vectors determining a cyclic structure of the basic matrixes according to

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) determining a positional vector of a pixel of the basic matrix according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to fulfill a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)}; and

then

e) producing the dither matrix;

a dither processing device to perform dither processing on color multi-tone data having multi-color color information with the use of the dither matrix produced by the dither matrix producing device for each of the different colors; and

an image forming device to perform color image forming based on image data having undergone the dither processing.

53. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining two vectors determining a cyclic structure of the basic matrix according to

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner as to satisfy a requirement of the cyclic structure;

then,

d) performing a transformation according to the following formula;

Pi{circumflex over ( 

)}→

Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}

 

where k and l are arbitrary integers, which may be negative values, with application of Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)} as a new pixel positional vector Pi{circumflex over ( 

)};

then

e) performing a transformation or re-arrangement of the pixels of the basic matrix in such a manner to satisfy the following formula so as to determine a shape of the basic matrix;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+om{circumflex over ( 

)}+ps{circumflex over ( 

)}|

 

where o and p are arbitrary integers, which may be negative values or zeros;

a dither processing device to perform dither processing on color multi-tone data having multi-color color information with the use of the dither matrix produced by the dither matrix producing device for each of the different colors; and

an image forming device to perform color image forming based on image data having undergone the dither processing.

54. An image forming apparatus comprising:

a dither matrix producing device to output the dither matrix produced in a dither matrix producing method in a readable state for a dither processing apparatus which uses the dither matrix and performs dither processing on multi-tone image data, wherein the dither matrix producing method comprises

a) defining the two vectors according to

main vector;

m{circumflex over ( 

)}=(mx, my); and


sub-vector;

s{circumflex over ( 

)}=(sx, sy);

b) defining a positional vector for a pixel of the basic matrix, according to;

Pi{circumflex over ( 

)}=(Pix, Piy)

 

where the suffix i denotes an i-th pixel of the basic matrix;

c) arranging the pixels of the basic matrix in such a manner to satisfy the following formula;

|Pi{circumflex over ( 

)}|≦

|Pi{circumflex over ( 

)}+km{circumflex over ( 

)}+ls{circumflex over ( 

)}|

 

where k and l are arbitrary integers, which may be negative values or zeros;

a dither processing device to perform dither processing on color multi-tone data having multi-color color information with the use of the dither matrix produced by the dither matrix producing device for each of the different colors; and

an image forming device to perform color image forming based on image data having undergone the dither processing.