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

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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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 by a computer processor according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are integers, which may be negative values or zeros.

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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(Pix,Piy)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to satisfy the following formula:
• |Pi^|≦|Pi^+km^+ls^|
• where k and l are integers, which may be negative values or zeros.

A computer-readable medium 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• and then
• e) producing the dither matrix.

A computer-readable medium 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are integers, which may be negative values or zeros.

A computer-readable medium 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(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^|≦|Pi^+km^+ls^|
• where k and l are integers, which may be negative values or zeros.

A dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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 matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• and then
• e) producing the dither matrix.

A dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are integers, which may be negative values or zeros.

A dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(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^|≦|Pi^+km^+ls^|
• where k and l are integers, which may be negative values or zeros.

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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi′^=(Pi′x,Pi′y)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to fulfill a requirement of the cyclic structure;
• then,
• d) performing a transformation according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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.

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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner as to satisfy a requirement of the cyclic structure;
• then,
• d) performing a transformation according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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.

An image processing method, comprising:

• producing a dither matrix by
• a) defining the two vectors according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(Pix,Piy)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to satisfy the following formula:
• |Pi^|≦|Pi^+km^+ls^|
• where k and l are 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.

An image forming method comprising:

• producing a dither matrix by
• a) defining two vectors determining a cyclic structure of the basic matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi′^=(Pi′x,Pi′y)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to fulfill a requirement of the cyclic structure;
• then,
• d) performing a transformation according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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 produced quantized image data.

An image forming method comprising:

• producing a dither matrix by
• a) defining two vectors determining a cyclic structure of the basic matrix according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner as to satisfy a requirement of the cyclic structure;
• then,
• d) performing a transformation according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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 produced quantized image data.

An image forming method comprising:

• producing a dither matrix by
• a) defining the two vectors according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(Pix,Piy)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to satisfy the following formula:
• |Pi^|≦|Pi^+om^+ps^|
• where k and l are 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 produced quantized image data.

An image forming method comprising:

• producing a dither matrix by
• a) defining two vectors determining a cyclic structure of the basic matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi′^=(Pix,Piy)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to fulfill a requirement of the cyclic structure;
• then,
• d) performing a transformation according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi^+km^+ls^ as a new pixel positional vector Pi^;
• 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 produced quantized image data.

An image forming method comprising:

• producing a dither matrix by
• a) defining two vectors determining a cyclic structure of the basic matrix according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner as to satisfy a requirement of the cyclic structure;
• then,
• d) performing a transformation according to the following formula:
• Pi′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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 produced quantized image data.

An image forming method comprising:

• producing a dither matrix by
• a) defining the two vectors according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(Pix,Piy)
• where the suffix i denotes an i-th pixel of the basic matrix;
• c) arranging the pixels of the basic matrix by a computer processor in such a manner to satisfy the following formula:
• |Pi^|≦|Pi^+km^+ls^|
• where k and l are 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 produced quantized image data.

An image processing apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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 matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi^=(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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi^+km^+ls^ as a new pixel positional vector Pi^;
• 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.

An image processing apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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.

An image processing apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(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^|≦|Pi^+km^+ls^|
• where k and l are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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 matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi^=(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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi^+km^+ls^ as a new pixel positional vector Pi^;
• 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(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^|≦|Pi^+km^+ls^|
• where k and l are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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 matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi^=(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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi^+km^+ls^ as a new pixel positional vector Pi^;
• 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(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^|≦|Pi^+om^+ps^|
• where k and l are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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 matrixes according to:
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi^=(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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application Pi^+km^+ls^ as a new pixel positional vector Pi^;
• 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(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^|≦|Pi^+km^+ls^|
• where k and l are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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 matrixes according to
• main vector:
• m^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) determining a positional vector of a pixel of the basic matrix according to:
• Pi^=(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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, and thus transforming to obtain a new shape of the basic matrix with application of Pi^+km^+ls^ as a new pixel positional vector Pi^;
• 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi′^=(Pi′x,Pi′y)
• 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′^→Pi′^+km^+ls^
• where k and l are integers, which may be negative values, with application of Pi′^+km^+ls^ as a new pixel positional vector Pi^;
• 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^|≦|Pi^+om^+ps^|
• where o and p are 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.

An image forming apparatus comprising:

• a dither matrix producing device comprising a computer-readable medium having instructions stored thereon which, when executed by a computer, cause the computer 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^=(mx,my);
• and
• sub-vector:
• s^=(sx,sy);
• b) defining a positional vector for a pixel of the basic matrix, according to:
• Pi^=(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^|≦|Pi^+km^+ls^|
• where k and l are which may be negative values or zeros integers,
• 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.