DE60227688C5 - Implementierung von einer transformation und einer nachfolgenden quantisierung - Google Patents

Implementierung von einer transformation und einer nachfolgenden quantisierung Download PDF

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DE60227688C5
DE60227688C5 DE60227688.8T DE60227688T DE60227688C5 DE 60227688 C5 DE60227688 C5 DE 60227688C5 DE 60227688 T DE60227688 T DE 60227688T DE 60227688 C5 DE60227688 C5 DE 60227688C5
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Antti Hallapuro
Kim Simelius
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Abstract

A method for implementing an approximation of a discrete cosine transform and a quantization operation, said method comprising:
- factorizing a discrete cosine transform matrix into a product of matrices, at least one of the matrices being a diagonal matrix and at least one of the matrices being a non-diagonal simplified transform matrix comprising elements constituting irrational numbers; and
- approximating elements of said non-diagonal simplified transform matrix constituting irrational numbers by rational numbers to form a further simplified transform matrix; said method characterized by
- adjusting elements of said diagonal matrix to compensate for said approximation of elements of said non-diagonal simplified transform matrix by rational numbers; and
- calculating elements of a quantization matrix based on a quantization parameter and on said adjusted elements of said diagonal matrix.

Description

  • Betreffend das am 16.07.2008 veröffentlichte europäische Patent EP 1 440 515 B1 mit Wirkung für die Bundesrepublik Deutschland (nationale Patentnummer DE 602 27 688 ) hat das Deutsche Patent- und Markenamt am 28.11.2019 beschlossen:
    • Das Patent wird auf den am 10.09.2019 eingegangenen Antrag der Patentinhaberin gemäß § 64 des Patentgesetzes durch die am 10.09.2019 eingegangenen Patentansprüche Nr. 1 bis 38 beschränkt.

Claims (38)

  1. A method for implementing an approximation of a discrete cosine transform and a quantization operation, said method comprising: - factorizing a discrete cosine transform matrix into a product of matrices, at least one of the matrices being a diagonal matrix and at least one of the matrices being a non-diagonal simplified transform matrix comprising elements constituting irrational numbers; and - approximating elements of said non-diagonal simplified transform matrix constituting irrational numbers by rational numbers to form a further simplified transform matrix; said method characterized by - adjusting elements of said diagonal matrix to compensate for said approximation of elements of said non-diagonal simplified transform matrix by rational numbers; and - calculating elements of a quantization matrix based on a quantization parameter and on said adjusted elements of said diagonal matrix.
  2. A method according to claim 1, wherein said elements of said diagonal matrix are adjusted such that a product of a reassembled matrix and the transpose of said reassembled matrix is equal to the identity matrix, when said reassembled matrix is formed from said further simplified transform matrix and said adjusted diagonal matrix.
  3. A method according to claim 1 or 2, wherein said rational numbers can be represented by fractions having a denominator equal to 2n, wherein n is an integer.
  4. A method according to claim 3, wherein in said transform implementation divisions are realized by bit-shifting.
  5. A method according to one of the preceding claims, the approximated discrete cosine transform is implemented with only addition, subtraction and bit-shifting operations.
  6. A method according to one of the preceding claims, wherein said discrete cosine transform matrix is a 4x4 matrix of a form: [ a a a a b c − c − b a − a − a a c − b b − c ] ,
    Figure DE000060227688C5_0001
    wherein a=1/2, b = 1 / 2 ⋅ cos ( π / 8 ) ,
    Figure DE000060227688C5_0002
     c = 1 / 2 ⋅ cos ( 3 π / 8 ) ,
    Figure DE000060227688C5_0003
    wherein ‚c‘ is substituted in said matrix according to an equation d=c/b, and wherein said discrete cosine transform matrix is factorized into a diagonal matrix comprising diagonal values {a, b, a, b} and a non-diagonal simplified transform matrix comprising only elements having absolute values of ‚1‘ and ‚d‘.
  7. A method according to claim 6, wherein in said non-diagonal simplified transform matrix said value ‚d‘ is approximated by a rational number 7/16.
  8. A method according to claim 6, wherein in said non-diagonal simplified transform matrix said value ‚d‘ is approximated by a rational number, and wherein in said diagonal matrix said value ‚b‘ is adjusted to b = 0.5 1 + d 2 ,
    Figure DE000060227688C5_0004
    ‚d‘ being in said equation said rational number.
  9. A method according to claim 6, wherein in said non-diagonal simplified transform matrix said value ‚d‘ is approximated by a rational number 7/16, and wherein said transform is implemented for a transform of a one-dimensional sequence of four values X[0],X[1],X[2],X[3] respectively with the following equations: e = X [ 0 ] + X [ 3 ] ,
    Figure DE000060227688C5_0005
     f = X [ 1 ] + X [ 2 ] ,
    Figure DE000060227688C5_0006
     Y [ 0 ] = e + f ,
    Figure DE000060227688C5_0007
     Y [ 2 ] = e − f ,
    Figure DE000060227688C5_0008
     e = X [ 0 ] − X [ 3 ] ,
    Figure DE000060227688C5_0009
     f = X [ 1 ] − X [ 2 ] ,
    Figure DE000060227688C5_0010
     Y [ 1 ] = e + ( f − f / 8 ) / 2,
    Figure DE000060227688C5_0011
    and Y [ 3 ] = ( e − e / 8 ) / 2 − f ,
    Figure DE000060227688C5_0012
    wherein Y[0],Y[1],Y[2],Y[3] is a one-dimensional sequence of four transformed values, and wherein e and f are auxiliary variables.
  10. A method according to one of claims 1 to 5, wherein said discrete cosine transform matrix is a 8x8 matrix of a form: [ a a a a a a a a b c d e − e − d − c − b f g − g − f − f − g g f c − e − b − d d b e − c a − a − a a a − a − a a d − b e c − c − e b − d g − f f − g − g f − f g e − d c − b b − c d − e ]
    Figure DE000060227688C5_0013
    wherein a = 1 / ( 2 2 ) ,
    Figure DE000060227688C5_0014
    b = 1/2 · cos(π /16), c =1/2·cos(3π/16), d = 1/2·cos(5π/16), e=1/2·cos(7π/16), f=1/2·cos(π/8), g=1/2·cos(3π/8), wherein ‚c‘ is substituted in said matrix according to an equation cb=c/b, wherein ‚d‘ is substituted in said matrix according to an equation db=d/b, wherein ‚e‘ is substituted in said matrix according to an equation eb=e/b, wherein ‚g‘ is substituted in said matrix according to an equation gf=g/f, and wherein said discrete cosine transform matrix is factorized into a diagonal matrix comprising diagonal values {a, b, f, b, a, b, f, b} and a non-diagonal simplified transform matrix comprising only elements having absolute values of ‚1‘, ‚cb‘, ‚db‘, ‚eb‘ and ‚gf‘.
  11. A method according to claim 10, wherein in said non-diagonal simplified transform matrix said value ‚cb‘ is approximated by a rational number 15/16, said value ‚db‘ is approximated by a rational number 9/16, said value ‚eb‘ is approximated by a rational number 1/4, and said value ‚gf‘ is approximated by a rational number 7/16.
  12. A method according to claim 10, wherein in said non-diagonal simplified transform matrix said values ‚cb‘, ‚db‘, ‚eb‘. and ‚gf‘ are approximated by rational numbers, and wherein in said diagonal matrix said values ‚b‘ and ‚f‘ are adjusted to b = 1 2 1 1 + c b 2 + d b 2 + e b 2
    Figure DE000060227688C5_0015
    and f = 1 2 1 1 + g f 2 ,
    Figure DE000060227688C5_0016
    values ‚cb‘, ‚db‘, ‚eb‘ and ‚gf‘ being in said equation said rational numbers.
  13. A method according to one of the preceding claims, wherein for a two-dimensional transform to be applied to two-dimensional.digital data, said non-diagonal simplified transform matrix with said approximated elements and a transpose of said non-diagonal simplified transform matrix with said approximated elements are employed as basis for implementing said transform, said quantization matrix including operations removed from both of said matrices, which operations are adjusted to compensate for said approximations in both of said matrices.
  14. A method according to one of the preceding claims, wherein before being used for quantization, said elements of said quantization matrix are converted to fixed-point values.
  15. A method for implementing a dequantization and an approximation of an inverse discrete cosine transform, the method comprising: - factorizing an inverse discrete cosine transform matrix into a product of matrices, at least one of the matrices being a diagonal matrix and at least one of the matrices being a non-diagonal simplified inverse transform matrix comprising elements constituting irrational numbers; - approximating elements of said non-diagonal simplified inverse transform matrix constituting irrational numbers by rational numbers to form a further simplified inverse transform matrix; said method characterized by - adjusting elements of said diagonal matrix to compensate for said approximation of elements of said non-diagonal simplified inverse transform matrix by rational numbers; and - calculating elements of a dequantization matrix based on a quantization parameter and on said adjusted elements of said diagonal matrix.
  16. A method according to claim 15, wherein before being used for dequantization, said elements of said dequantization matrix are converted to fixed-point values.
  17. A method according to claim 15, wherein said elements of said diagonal matrix are adjusted such that a product of a reassembled matrix and the transpose of said reassembled matrix is equal to the identity matrix, when said reassembled matrix is formed from said further simplified inverse transform matrix and said adjusted diagonal matrix.
  18. A method according to one of claims 15 to 17, wherein said rational numbers can be represented by fractions having a denominator equal to 2n, wherein n is an integer.
  19. A method according to claim 18, wherein in said inverse transform implementation divisions are realized by bit-shifting.
  20. A method according to claim 18, the approximated inverse discrete cosine transform is implemented with only addition, subtraction and bit-shifting operations.
  21. A method according to one of claims 15 to 20, wherein said inverse discrete cosine transform matrix is a 4x4 matrix of a form [ a b a c a c − a − b a − c − a b a − b a − c ] ,
    Figure DE000060227688C5_0017
    wherein a=1/2, b = 1 / 2 ⋅ cos ( π / 8 ) ,
    Figure DE000060227688C5_0018
     c = 1 / 2 ⋅ cos ( 3 π / 8 ) ,
    Figure DE000060227688C5_0019
    wherein ‚c‘ is substituted in said matrix according to an equation d=c/b, and wherein said inverse discrete cosine transform matrix is factorized into a diagonal matrix comprising diagonal values {a, b, a, b} and a non-diagonal simplified inverse transform matrix comprising only elements having absolute values of ‚1‘ and ‚d‘.
  22. A method according to claim 21, wherein in said non-diagonal simplified inverse transform matrix said value ‚d‘ is approximated by a rational number 7/16.
  23. A method according to claim 21, wherein in said non-diagonal simplified inverse transform matrix said value ‚d‘ is approximated by a rational number, and wherein in said diagonal matrix said value ‚b‘ is adjusted to b = 0.5 1 + d 2 ,
    Figure DE000060227688C5_0020
    ‚d‘ being in said equation said rational number.
  24. A method according to claim 21, wherein in said non-diagonal simplified inverse transform matrix said value ‚d‘ is approximated by a rational number 7/16, and wherein said inverse transform is implemented for an inverse transform of a one-dimensional sequence of four values X[0], X[1], X[2], X[3] respectively with the following equations: e = X [ 0 ] + X [ 3 ] ,
    Figure DE000060227688C5_0021
     f = X [ 1 ] + X [ 2 ] ,
    Figure DE000060227688C5_0022
     Y [ 0 ] = e + f ,
    Figure DE000060227688C5_0023
     Y [ 2 ] = e − f ,
    Figure DE000060227688C5_0024
     e = X [ 0 ] − X [ 3 ] ,
    Figure DE000060227688C5_0025
     f = X [ 1 ] − X [ 2 ] ,
    Figure DE000060227688C5_0026
     Y [ 1 ] = e + ( f − f / 8 ) / 2,
    Figure DE000060227688C5_0027
    and Y [ 3 ] = ( e − e / 8 ) / 2 − f ,
    Figure DE000060227688C5_0028
    wherein Y[0], Y[1], Y[2], Y[3] is a one-dimensional sequence of four transformed values, and wherein e and f are auxiliary variables.
  25. A method according to one of claims 15 to 20, wherein said inverse discrete cosine transform matrix is a 8x8 matrix of a form: [ a b f c a d g e a c g − e − a − b − f − d a d − g − b − a e f c a e − f − d a c − g − b a − e − f d a − c − g b a − d − g b − a − e f − c a − c g e − a b − f d a − b f − c a − d g − e ]
    Figure DE000060227688C5_0029
    wherein a = 1 / ( 2 2 ) ,
    Figure DE000060227688C5_0030
    b =1/2 · cos(π /16), c =1/2·cos(3π /16), d=1/2·cos(5π/16), e =1/2·cos(7π/16), f=1/2·cos(π/8), g=1/2·cos(3π/8), wherein ‚c‘ is substituted in said matrix according to an equation cb=c/b, wherein ‚d‘ is substituted in said matrix according to an equation db=d/b, wherein ‚e‘ is substituted in said matrix according to an equation eb=e/b, wherein ‚g‘ is substituted in said matrix according to an equation gf=g/f, and wherein said inverse discrete cosine transform matrix is factorized into a diagonal matrix comprising diagonal values {a, b, f, b, a, b, f, b} and a simplified inverse transform matrix comprising only elements having absolute values of ‚1‘, ‚cb‘, ‚db‘, ‚eb‘ and ‚gf‘.
  26. A method according to claim 25, wherein in said non-diagonal simplified inverse transform matrix said value ‚cb‘ is approximated by a rational number 15/16, said value ‚db‘ is approximated by a rational number 9/16, said value ‚eb‘ is approximated by a rational number 1/4, and said value ‚gf‘ is approximated by a rational number 7/16.
  27. A method according to claim 25, wherein in said non-diagonal simplified inverse transform matrix said values ‚cb‘, ‚db‘, ‚eb‘ and ‚gf‘ are approximated by rational numbers, and wherein in said diagonal matrix said values ‚b‘ and ‚f‘ are adjusted to b = 1 2 1 1 + c b 2 + d b 2 + e b 2 and f = 1 2 1 1 + g f 2 ,
    Figure DE000060227688C5_0031
    values ‚cb‘, ‚db‘, ‚eb‘ and ‚gf‘ being in said equation said rational numbers.
  28. A method according to one of claims 15 to 27, wherein for a two-dimensional inverse transform to be applied to two-dimensional dequantized digital data, said non-diagonal simplified inverse transform matrix with said approximated elements and a transpose of said non-diagonal simplified inverse transform matrix with said approximated elements are employed as basis for implementing said inverse transform, said dequantization matrix including operations removed from both of said matrices, which operations are adjusted to compensate for said approximations in both of said matrices.
  29. An encoder (4) for compressing digital data comprising: - a transformer (41) for transforming digital data by applying a simplified transform to said digital data, the simplified transform being based on a transform matrix obtained by factorizing a discrete cosine transform matrix into a product of matrices, at least one of the matrices being a diagonal matrix and at least one of the matrices being a non-diagonal simplified transform matrix comprising elements constituting irrational numbers, and by approximating elements of said simplified non-diagonal transform matrix constituting irrational numbers by rational numbers to form a further simplified transform matrix as said transform matrix for said simplified transform; and - a quantization means (42) coupled to an output of said transformer (41) for quantizing said transformed digital data with a quantization, characterized in that - said quantization is based on a quantization matrix having elements which have been calculated based on a quantization parameter and on adjusted elements of said diagonal matrix, wherein said elements of said diagonal matrix are adjusted to compensate for said approximation of elements of said non-diagonal simplified transform matrix by rational numbers.
  30. An encoder (4) according to claim 29, wherein the simplified transform is implemented with only addition, subtraction and bit-shifting operations; and wherein said discrete cosine transform matrix is a 4x4 matrix of a form [ a a a a b c − c − b a − a − a a c − b b − c ] ,
    Figure DE000060227688C5_0032
    wherein a = 1/2, b = 1 / 2 ⋅ cos ( π / 8 ) ,
    Figure DE000060227688C5_0033
     c = 1 / 2 ⋅ cos ( 3 π / 8 ) ,
    Figure DE000060227688C5_0034
    wherein ‚c‘ is substituted in said matrix according to an equation d=c/b, and wherein said discrete cosine transform matrix is factorized into a diagonal matrix comprising diagonal values {a, b, a, b} and a non-diagonal simplified transform matrix of a form [ 1 1 1 1 1 d − d − 1 1 − 1 − 1 1 d − 1 1 − d ] .
    Figure DE000060227688C5_0035
  31. An encoder (4) according to claim 29 or 30, wherein said elements of said diagonal matrix are adjusted such that a product of a reassembled matrix and the transpose of said reassembled matrix is equal to the identity matrix, when said reassembled matrix is formed from said further simplified transform matrix and said adjusted diagonal matrix.
  32. An encoder (4) according to claim 30, wherein in said non-diagonal simplified transform matrix said value ‚d‘ is approximated by a rational number to form the further simplified transform matrix, the rational number being represented by a fraction having a denominator equal to 2n, wherein n is an integer, and wherein in said diagonal matrix said value ‚b‘ is adjusted to b = 0.5 1 + d 2 ,
    Figure DE000060227688C5_0036
    ‚d‘ being in said equation said rational number.
  33. A decoder (5) comprising: - a dequantization means (52) for dequantizing compressed digital data with a dequantization; and - a transformer (51) coupled to an output of said dequantization means (52) for transforming dequantized digital data by applying a simplified inverse transform that is based on an inverse transform matrix obtained by factorizing an inverse discrete cosine transform matrix into a product of matrices, at least one of the matrices being a diagonal matrix and at least one of the matrices being a non-diagonal simplified inverse transform matrix comprising elements constituting irrational numbers, and by approximating elements of said non-diagonal simplified inverse transform matrix constituting irrational numbers by rational numbers to form a further simplified inverse transform matrix as said inverse transform matrix for said simplified inverse transform; characterized in that - said dequantization applied by said dequantization means (52) is based on a matrix having elements which have been calculated based on a dequantization parameter and on adjusted elements of said diagonal matrix, wherein said elements of said diagonal matrix are adjusted to compensate for said approximation of elements of said non-diagonal simplified inverse transform matrix by rational numbers.
  34. A decoder(5) according to claim 33, wherein the simplified inverse transform is implemented with only addition, subtraction and bit-shifting operations; and wherein said inverse discrete cosine transform matrix is a 4x4 matrix of a form [ a b a c a c − a − b a − c − a b a − b a − c ] ,
    Figure DE000060227688C5_0037
    wherein a = 1/2, b = 1 / 2 ⋅ cos ( π / 8 ) ,
    Figure DE000060227688C5_0038
     c = 1 / 2 ⋅ cos ( 3 π / 8 ) ,
    Figure DE000060227688C5_0039
    wherein ‚c‘ is substituted in said matrix according to an equation d=c/b, and wherein said inverse discrete cosine transform matrix is factorized into a diagonal matrix comprising diagonal values {a, b, a, b} and a non-diagonal simplified inverse transform matrix of a form [ 1 1 1 d 1 d − 1 − 1 1 − d − 1 1 1 − 1 1 − d ] .
    Figure DE000060227688C5_0040
  35. A decoder (5) according to claim 33 or 34, wherein said elements of said diagonal matrix are adjusted such that a product of a reassembled matrix and the transpose of said reassembled matrix is equal to the identity matrix, when said reassembled matrix is formed from said further simplified inverse transform matrix and said adjusted diagonal matrix.
  36. A decoder (5) according to claim 34, wherein in said non-diagonal simplified inverse transform matrix said value ‚d‘ is approximated by a rational number to form the further simplified transform matrix, the rational number being represented by a fraction having a denominator equal to 2n, wherein n is an integer, and wherein in said diagonal matrix said value ‚b‘ is adjusted to b = 0.5 1 + d 2 ,
    Figure DE000060227688C5_0041
    ‚d‘ being in said equation said rational number.
  37. A method of encoding, the method comprising: - applying a set of equations representing an approximated discrete cosine transform to a set of source values to obtain a corresponding set of approximated transform coefficients, wherein the approximated discrete cosine transform is based on a further simplified transform matrix that has been obtained according to the method of one of claims 1 to 14; and - quantizing said transformed digital data with a quantization, which is based on elements of a quantization matrix that have been calculated according to the method of one of claims 1 to 14.
  38. A method of decoding, the method comprising: - dequantizing compressed digital data with a dequantization, which is based on elements of a dequantization matrix that have been calculated according to the method of one of claims 15 to 28; and - applying a set of equations representing an approximated inverse discrete cosine transform to dequantized digital data, wherein the approximated inverse discrete cosine transform is based on a further simplified inverse transform matrix that has been obtained according to the method of one of claims 15 to 28.
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US09/943,241 US7082450B2 (en) 2001-08-30 2001-08-30 Implementation of a transform and of a subsequent quantization
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PCT/IB2002/003566 WO2003019787A2 (en) 2001-08-30 2002-08-27 Implementation of a transform and of a subsequent quantization

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