Reduction of errors during computation of inverse discrete cosine transform

• US 8,606,023 B2
• Filed: 06/22/2007
• Issued: 12/10/2013
• Est. Priority Date: 06/26/2006
• Status: Expired due to Fees

• Current Assignee: Qualcomm, Inc.
• Original Assignee: Qualcomm, Inc.
• Inventors: Reznik, Yuriy, Garudadri, Harinath
• Primary Examiner(s): Patel, Nirav G
• Application Number:
  US11/767,452
• Publication Number:
  US 20070297504A1
• Time in Patent Office: 2,363 Days
• Field of Search: 382/232, 375/240, 708/203
• US Class Current: 382/232
• CPC Class Codes: G06F 17/147 Discrete orthonormal transforms, e.g. discrete cosine transform, discrete sine transform, and variations therefrom, e.g. modified discrete cosine transform, integer transforms approximating the discrete cosine transform G06F17/145 takes precedence
• CPC Class Codes: H04N 19/45 performing compensation of the inverse transform mismatch, e.g. Inverse Discrete Cosine Transform [IDCT] mismatch
• CPC Class Codes: H04N 19/60 using transform coding
• CPC Class Codes: H04N 19/61 in combination with predictive coding

Abstract

Techniques are described to reduce rounding errors during computation of discrete cosine transform using fixed-point calculations. According to these techniques, a discrete cosine transform a matrix of scaled coefficients is calculated by multiplying coefficients in a matrix of coefficients by scale factors. Next, a midpoint bias value and a supplemental bias value are added to a DC coefficient of the matrix of scaled coefficients. Next, an inverse discrete cosine transform is applied to the resulting matrix of scaled coefficients. Values in the resulting matrix are then right-shifted in order to derive a matrix of pixel component values. As described herein, the addition of the supplemental bias value to the DC coefficient reduces rounding errors attributable to this right-shifting. As a result, a final version of a digital media file decompressed using these techniques may more closely resemble an original version of a digital media file.

Claims

1. A method comprising:

generating a matrix of biased coefficients by adding one or more bias values to a coefficient of a matrix of source coefficients;

generating a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of biased coefficients;

generating a matrix of output coefficients by right-shifting coefficients in the matrix of transformed coefficients; and

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal inverse discrete cosine transform;

causing a media presentation unit to output audible or visible signals based on the matrix of output values;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of transformed coefficients by a first magnitude and results from dividing the coefficients in the matrix of transformed coefficients by 2 raised to the power of the first magnitude, without regard to precision.

2. The method of claim 1,

wherein coefficients in the matrix of output coefficients are pixel component values;

and

wherein the method further comprises constructing a block of pixels that include the pixel component values.

3. The method of claim 2, wherein causing a media presentation unit to output audible or visible signals based on the matrix of output values comprises causing the media presentation unit to display the block of pixels.

4. The method of claim 2, wherein the method further comprises:

generating a first video frame by combining the block of pixels with other blocks of pixels; and

using temporal decoding to generate one or more subsequent video frames that use the first video frame as a reference video frame.

5. The method of claim 1, wherein the coefficient is a DC coefficient of the matrix of source coefficients.

6. The method of claim 1,

wherein the method further comprises adding a midpoint bias term to the coefficient;

and

wherein midpoint bias value is equal to 2<sup>P−

1</sup>, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

7. The method of claim 1, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

8. The method of claim 1,

wherein generating the matrix of biased coefficients comprises adding a supplemental bias value to the coefficient;

and

wherein the supplemental bias value is equal to −

1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

9. The method of claim 1,

wherein the method further comprises selecting on a pseudo-random basis a value that is equal to either −

1 or 0; and

wherein generating the matrix of biased coefficients comprises adding the selected value to the coefficient.

10. The method of claim 1, wherein the method further comprises:

generating a matrix of de-quantized coefficients by de-quantizing a matrix of quantized coefficients; and

generating the matrix of source coefficients by left-shifting coefficients in the matrix of de-quantized coefficients by a number of mantissa bits of fixed-point numbers used while generating the matrix of transformed coefficients.

11. The method of claim 10,

wherein the number of mantissa bits is three;

and

wherein, given the matrix of source coefficients, the matrix of output coefficients satisfies precision requirements of the Institute of Electrical and Electronics Engineers (IEEE) 1180 standard.

12. The method of claim 1, wherein generating the matrix of transformed coefficients comprises using fixed-point arithmetic to repeatedly apply the transform to coefficients in the matrix of biased coefficients without using multiplication or division operations.

13. The method of claim 1, wherein fixed-point numbers used while generating the matrix of transformed coefficients are 16-bit fixed-point numbers.

14. The method of claim 1, wherein generating the matrix of transformed coefficients comprises:

generating a matrix of intermediate coefficients by using fixed-point arithmetic to apply a series of butterfly structure operations to each row vector of the matrix of biased coefficients; and

generating the matrix of transformed coefficients by using fixed-point arithmetic to apply the series of butterfly structure operations to each column vector of the matrix of intermediate coefficients.

15. The method of claim 14,

wherein performing any of the butterfly structure operations comprises performing butterfly structure operations of the form:

u=(((x*C′

)+(y*S′

)+(1<

<

(k−

1)))>

>

k;

v=((x*S′

)+(y*−

C′

)+(1<

<

(k−

1)))>

>

k,

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

16. The method of claim 14, wherein differences between results generated by one of the butterfly structure operations and results that would be generated by an equivalent butterfly structure operation using unlimited precision arithmetic are centered around zero and have positive or negative magnitude of 1 or less.

17. The method of claim 16,

wherein performing any of the butterfly structure operations comprises performing butterfly structure operations of the form:

u=((x*C)>

>

k)−

((y*−

S)>

>

k);

v=((x*S)>

>

k)−

((y*C)>

>

k),

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

18. A wireless communication device comprising:

a coefficient biasing module that generates a matrix of biased coefficients by adding one or more bias values to a coefficient of the matrix of source coefficients;

an inverse transform module that generates a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of biased coefficients;

a right-shift module that generates a matrix of output coefficients by right-shift coefficients in the matrix of transformed coefficients; and

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal inverse discrete cosine transform; and

a presentation driver that causes a media presentation unit to output audible or visible signals based on the matrix of output coefficients;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of transformed coefficients by a first magnitude and results from dividing the coefficients in the matrix of transformed coefficients by 2 raised to the power of the first magnitude, without regard to precision.

19. A device comprising:

a coefficient biasing module that generates a matrix of biased coefficients by adding one or more bias values to a coefficient of the matrix of source coefficients;

an inverse transform module that generates a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of biased coefficients;

a right-shift module that generates a matrix of output coefficients by right-shift coefficients in the matrix of transformed coefficients; and

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal inverse discrete cosine transform; and

a presentation driver that causes a media presentation unit to output audible or visible signals based on the matrix of output coefficients;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of transformed coefficients by a first magnitude and results from dividing the coefficients in the matrix of transformed coefficients by 2 raised to the power of the first magnitude, without regard to precision.

20. The device of claim 19,

wherein coefficients in the matrix of output coefficients sare pixel component values;

and

wherein the device further comprises a pixel reconstruction module that constructs a block of pixels that include the pixel component values.

21. The device of claim 20,

wherein the presentation driver causes the media presentation unit to display the block of pixels.

22. The device of claim 20, wherein the device further comprises:

a block combiner module that generates a first video frame by combining the block of pixels with other blocks of pixels; and

a predictive frame generation module that generates one or more subsequent video frames that use the first video frame as a reference video frame.

23. The device of claim 19, wherein the coefficient is a DC coefficient of the matrix of coefficients.

24. The device of claim 19,

wherein the coefficient biasing module generates the matrix of biased coefficients by adding a midpoint bias value to the coefficient;

and

wherein the midpoint bias value is equal to 2<sup>P−

1</sup>, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

25. The device of claim 19, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

26. The device of claim 19, wherein the coefficient biasing module generates the matrix of biased coefficients by adding a supplemental bias value that is equal to −

1 when the coefficient is negative and that is equal to 0 when the coefficient is non-negative.

27. The device of claim 19,

wherein the coefficient biasing module generates the matrix of biased coefficients by selecting, on a pseudo-random basis, a value that is equal to either −

1 or 0, and adding the selected value to the coefficient as a supplemental bias values.

28. The device of claim 19, wherein the device further comprises:

a de-quantization module that generates a matrix of de-quantized coefficients by de-quantizing a matrix of de-quantized coefficients; and

a scaling module that generates the matrix of source coefficients by left-shifting coefficients in the matrix of de-quantized coefficients by a number of mantissa bits of fixed-point number used while generating the matrix of transformed coefficients.

29. The device of claim 28,

wherein the number of mantissa bits is three;

and

wherein, given the matrix of source coefficients, the matrix of output coefficients satisfies precision requirements of the Institute of Electrical and Electronics Engineers (IEEE) 1180 standard.

30. The device of claim 19, wherein the inverse transform module uses 16-bit fixed-point numbers when using fixed-point arithmetic to repeatedly apply the one-dimensional transform to coefficients in the matrix of biased coefficients.

31. The device of claim 19, wherein the inverse transform module uses fixed-point arithmetic to apply a series of butterfly structure operations to each row vector of the matrix of scaled coefficients to produce a matrix of intermediate coefficients and uses fixed-point arithmetic to apply the series of butterfly structure operations to each column vector of the matrix of intermediate coefficients to produce the matrix of transformed coefficients.

32. The device of claim 31,

wherein the butterfly structure operations are of the form:

u=(((x*C′

)+(y*S′

)+(1<

<

(k−

1)))>

>

k;

v=((x*S′

)+(y*−

C′

)+(1<

<

(k−

1)))>

>

k,

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

33. The device of claim 31, wherein differences between results generated by one of the butterfly structure operations and results that would be generated by an equivalent butterfly structure operation using unlimited precision arithmetic are centered around zero and have positive or negative magnitude of 1 or less.

34. The device of claim 33,

wherein the butterfly structure operations are of the form:

u=((x*C)>

>

k)−

((y*−

S)>

>

k);

v=((x*S)>

>

k)−

((y*C)>

>

k),

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

35. A device comprising,

means for generating a matrix of biased coefficients by adding one or more bias values to a coefficient of a matrix of source coefficients;

means for generating a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of biased coefficients;

means for generating a matrix of output coefficients by right-shifting coefficients in the matrix of transformed coefficients;

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal inverse discrete cosine transform; and

means for causing a media presentation unit to output audible or visible signals based on the matrix of output values;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of transformed coefficients by a first magnitude and results from dividing the coefficients in the matrix of transformed coefficients by 2 raised to the power of the first magnitude, without regard to precision.

36. The device of claim 35,

wherein coefficients in the matrix of output coefficients are pixel component values;

and

wherein the device further comprises means for constructing a block of pixels that include the pixel component values.

37. The device of claim 36, wherein the means for causing a media presentation unit to output audible or visible signals based on the matrix of output values comprises means for causing the media presentation unit to display the block of pixels.

38. The device of claim 36, wherein the method further comprises:

means for generating a first video frame by combining the block of pixels with other blocks of pixels; and

means for using temporal decoding to generate one or more subsequent video frames that use the first video frame as a reference video frame.

39. The device of claim 35, wherein the coefficient is a DC coefficient of the matrix of source coefficients.

40. The device of claim 35,

wherein the means for generating the matrix of biased coefficients comprises means for adding a midpoint bias value to the coefficient;

and

wherein the midpoint bias value is equal to 2<sup>P−

1</sup>, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

41. The device of claim 40, wherein first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

42. The device of claim 35,

wherein the means for generating the matrix of biased coefficients comprises means for adding a supplemental bias value to the coefficient;

wherein the supplemental bias value is equal to 31 1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

43. The device of claim 35,

wherein the method further comprises selecting on a pseudo-random basis a value that is equal to either −

1 or 0; and

wherein the means for generating the matrix of biased coefficients comprises means for adding the selected value to the coefficient.

44. The device of claim 35, wherein the device further comprises:

means for generating a matrix of de-quantized coefficients by de-quantizing a matrix of quantized coefficients; and

means for generating the matrix of source coefficients by left-shifting coefficients in the matrix of de-quantized coefficients by a number of mantissa bits of fixed-point numbers used while generating the matrix of transformed coefficients.

45. The device of claim 44,

wherein the number of mantissa bits is three;

and

wherein, given the matrix of source coefficients, the matrix of output coefficients satisfies precision requirements of the Institute of Electrical and Electronics Engineers (IEEE) 1180 standard.

46. The device of claim 35, wherein the means for generating the matrix of transformed coefficients comprise means for using fixed-point arithmetic to repeatedly apply the transform to coefficients in the matrix of biased coefficients without using multiplication or division operations.

47. The device of claim 35, wherein the means for generating the matrix of transformed coefficients uses 16-bit fixed-point numbers when using fixed-point arithmetic to repeatedly apply the one-dimensional transform to coefficients in the matrix of biased coefficients.

48. The device of claim 35, wherein generating the matrix of transformed coefficients comprises:

means for generating a matrix of intermediate coefficients by using fixed-point arithmetic to apply a series of butterfly structure operations to each row vector of the matrix of biased coefficients; and

means for generating the matrix of transformed coefficients by using fixed-point arithmetic to apply the series of butterfly structure operations to each column vector of the matrix of intermediate coefficients.

49. The device of claim 48,

wherein the means for performing any of the butterfly structure operations comprises means for performing butterfly structure operations of the form:

u=(((x*C′

)+(y*S′

)+(1<

<

(k−

1)))>

>

k;

v=((x*S′

)+(y*−

C′

)+(1<

<

(k−

1)))>

>

k,

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

50. The device of claim 48, wherein differences between results generated by one of the butterfly structure operations and results that would be generated by an equivalent butterfly structure operation using unlimited precision arithmetic are centered around zero and have positive or negative magnitude of 1 or less.

51. The device of claim 50,

wherein the means for performing any of the butterfly structure operations comprises means for performing butterfly structure operations of the form:

u=((x*C)>

>

k)−

((y*−

S)>

>

k);

v=((x*S)>

>

k)−

((y*C)>

>

k),

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

52. A non-transitory computer-readable medium comprising instructions, when executed the instructions cause a processor to:

generate a matrix of biased coefficients by adding one or more bias values to a coefficient of a matrix of source coefficients;

generate a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of biased coefficients;

generate a matrix of output coefficients by right-shifting coefficients in the matrix of transformed coefficients;

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal inverse discrete cosine transform; and

cause a media presentation unit to output audible or visible signals based on the matrix of output values;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of transformed coefficients by a first magnitude and results from dividing the coefficients in the matrix of transformed coefficients by 2 raised to the power of the first magnitude, without regard to precision.

53. The computer-readable medium of claim 52,

wherein coefficients in the matrix of output coefficients are pixel component values;

and

wherein the instructions further cause the processor to construct a block of pixels that include the pixel component values.

54. The computer-readable medium of claim 53, wherein the instructions cause the processor to cause the media presentation unit to output audible or visible signals based on the matrix of output values by causing the processor to cause the media presentation unit of display the block of pixels.

55. The computer-readable medium of claim 53, wherein the instructions further cause the processor to:

generate a first video frame by combining the block of pixels with other blocks of pixels; and

use temporal decoding to generate one or more subsequent video frames that use the first video frame as a reference video frame.

56. The computer-readable medium of claim 52, wherein the coefficient is a DC coefficient of the matrix of coefficients.

57. The computer-readable medium of claim 52,

wherein the instructions further cause the processor to add a midpoint bias value to the coefficient;

and

wherein the midpoint bias value is equal to 2<sup>P−

1</sup>, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

58. The computer-readable medium of claim 52, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

59. The computer-readable medium of claim 52,

wherein the instructions further cause the processor to generate the matrix of biased coefficients cause the processor to add a supplemental bias value to the coefficient;

and

wherein the supplemental bias value is equal to −

1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

60. The computer-readable medium of claim 52,

wherein the instructions further cause the processor to select on a pseudo-random basis a value that is equal to either −

1 or 0; and

wherein the instructions that cause the processor to generate the matrix of biased coefficients cause the processor to add the selected value to the coefficient.

61. The computer-readable medium of claim 52, wherein the instructions further cause the processor to:

generate a matrix of de-quantized coefficients by de-quantizing a matrix of quantized coefficients; and

generating the matrix of source coefficients by left-shifting coefficients in the matrix of de-quantized coefficients by a number of mantissa bits of fixed-point numbers used while generating the matrix of transformed coefficients.

62. The computer-readable medium of claim 61,

wherein the number of mantissa bits is three;

and

wherein, given the matrix of source coefficients, the matrix of output coefficients satisfies precision requirements of the Institute of Electrical and Electronics Engineers (IEEE) 1180 standard.

63. The computer-readable medium of claim 52, wherein the instructions cause the processor to generate the matrix of transformed coefficients by causing the processor to repeatedly apply the transform to coefficients in the matrix of biased coefficients without using multiplication or division operations.

64. The computer-readable medium of claim 52, wherein the instructions cause the processor to use 16-bit fixed-point numbers when the instructions cause the processor to use fixed-point arithmetic to repeatedly apply the one-dimensional transform to coefficients in the matrix of biased coefficients.

65. The computer-readable medium of claim 52, wherein the instructions cause the processor to generate the matrix of transformed coefficients by causing the processor to:

generate a matrix of intermediate coefficients by using fixed-point arithmetic to apply a series of butterfly structure operations to each row vector of the matrix of biased coefficients; and

generate the matrix of transformed coefficients by using fixed-point arithmetic to apply the series of butterfly structure operations to each column vector of the matrix of intermediate coefficients.

66. The computer-readable medium of claim 65,

wherein the instructions cause the processor to perform any of the butterfly structure operations by causing the processor to perform butterfly structure operations of the form:

u=(((x*C′

)+(y*S′

)+(1<

<

(k−

1)))>

>

k;

v=((x*S′

)+(y*−

C′

)+(1<

<

(k−

1)))>

>

k,

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

67. The computer-readable medium of claim 65, wherein differences between results generated by one of the butterfly structure operations and results that would be generated by an equivalent butterfly structure operation using unlimited precision arithmetic are centered around zero and have positive or negative magnitude of 1 or less.

68. The computer-readable medium of claim 66,

wherein the instruction cause the processor to perform any of the butterfly structure operations by causing the processor to perform butterfly structure operations of the form:

u=((x*C)>

>

k)−

((y*−

S)>

>

k);

v=((x*S)>

>

k)−

((y*C)>

>

k),

wherein u, v, x, and y are fixed-point numbers,

wherein x and y are input values and u and v are output values; and

wherein C, S, and k are integers.

69. A method comprising:

generating a matrix of adjusted coefficients by left-shifting each coefficient in a matrix of source coefficients;

generating a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of adjusted coefficients;

generating a matrix of scaled coefficients by scaling each coefficient in the matrix of transformed coefficients;

generating a matrix of biased coefficients by adding one or more bias values to one or more coefficients in the matrix of scaled coefficients; and

generating a matrix of output coefficients by right-shifting coefficients in the matrix of biased coefficients by a first magnitude;

wherein the matrix of source coefficients represents media data;

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal forward discrete cosine transform;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of biased coefficients by the first magnitude and results from dividing the coefficients in the matrix of biased coefficients by 2 raised to the power of the first magnitude, without regard to precision.

70. The method of claim 69, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

71. The method of claim 69,

wherein generating the matrix of biased coefficients comprises adding a supplemental bias value to one or more coefficients in the matrix of scaled coefficients;

and

wherein the supplemental bias value is equal to −

1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

72. The method of claim 69,

wherein the method further comprises selecting, on a pseudo-random basis, a value that is equal to either −

1 or 0; and

wherein generating the matrix of biased coefficients comprises adding the selected value to one or more coefficients in the matrix of scaled coefficients.

73. A device comprising:

a left-shift module that generates a matrix of adjusted coefficients by left-shifting each coefficient in a matrix of source coefficients;

a forward transform module that generates a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of adjusted coefficients;

a scaling module that generates a matrix of scaled coefficients by scaling each coefficient in the matrix of transformed coefficients;

a coefficient biasing module that generates a matrix of biased coefficients by adding one or more bias values to one or more coefficients in the matrix of scaled coefficients; and

a right-shift module that generates a matrix of output coefficients by right-shifting coefficient in the matrix of biased coefficients by a first magnitude;

wherein the matrix of source coefficients represents media data;

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal forward discrete cosine transform;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of biased coefficients by the first magnitude and results from dividing the coefficients in the matrix of biased coefficients by 2 raised to the power of the first magnitude, without regard to precision.

74. The device of claim 73, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

75. The device of claim 73,

wherein the coefficient biasing module generates the matrix of biased coefficients by adding a supplemental bias value to one or more coefficients in the matrix of scaled coefficients;

and

wherein the supplemental bias value is equal to −

1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

76. The device of claim 73,

wherein the coefficient biasing module selects, on a pseudo-random basis, a value that is equal to either −

1 or 0 and adds the selected value to one or more coefficients in the matrix of scaled coefficients.

77. A device comprising:

means for generating a matrix of adjusted coefficients by left-shifting each coefficient in a matrix of source coefficients;

means for generating a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of adjusted coefficients;

means for generating a matrix of scaled coefficients by scaling each coefficient in the matrix of transformed coefficients;

means for generating a matrix of biased coefficients by adding one or more bias values to one or more coefficients in the matrix of scaled coefficients; and

means for generating a matrix of output coefficients by right-shifting coefficient in the matrix of biased coefficients by a first magnitude;

wherein the matrix of source coefficients represents media data;

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal forward discrete cosine transform;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of biased coefficients by the first magnitude and results from dividing the coefficients in the matrix of biased coefficients by 2 raised to the power of the first magnitude, without regard to precision.

78. The device of claim 77, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

79. The device of claim 77,

wherein the means for generating the matrix of biased coefficients generates the matrix of biased coefficients by adding a supplemental bias value to one or more coefficients in the matrix of scaled coefficients;

and

wherein the supplemental bias value is equal to −

1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

80. The device of claim 77,

wherein the means for generating the matrix of biased coefficients selects, on a pseudo-random basis, a value that is equal to either −

1 or 0 and adds the selected value to one or more coefficients in the matrix of scaled coefficients.

81. A non-transitory computer-readable medium comprising instructions, wherein the instructions cause a programmable processor to:

generate a matrix of adjusted coefficients by left-shifting each coefficient in a matrix of source coefficients;

generate a matrix of transformed coefficients by using fixed-point arithmetic to repeatedly apply a one-dimensional transform to coefficients in the matrix of adjusted coefficients;

generate a matrix of scaled coefficients by scaling each coefficient in the matrix of transformed coefficients;

generate a matrix of biased coefficients by adding one or more bias values to one or more coefficients in the matrix of scaled coefficients; and

generate a matrix of output coefficients by right-shifting coefficients in the matrix of biased coefficients by a first magnitude;

wherein the matrix of source coefficients represents media data;

wherein coefficients in the matrix of output coefficients are approximations of values that would be produced by transforming the matrix of source coefficients using an ideal forward discrete cosine transform;

wherein the addition of the bias values to the coefficient causes positive errors and negative errors to be, on average, equal in magnitude and to be, on average, symmetric about zero, and

wherein the errors represent differences between values that result from right-shifting limited precision fixed-point representations of coefficients in the matrix of biased coefficients by the first magnitude and results from dividing the coefficients in the matrix of biased coefficients by 2 raised to the power of the first magnitude, without regard to precision.

82. The computer-readable medium of claim 81, wherein the first magnitude is equal to 2^P, wherein P is equal to the number of mantissa bits in fixed-point numbers used in the transform plus a number of bits that are added to coefficients in the matrix of transformed coefficients by applying the transform.

83. The computer-readable medium of claim 81,

wherein the instructions cause the processor to generate the matrix of biased coefficients by causing the processor to add a supplemental bias value to one or more coefficients in the matrix of scaled coefficients;

and

wherein the supplemental bias value is equal to −

1 when the coefficient is negative and equal to 0 when the coefficient is non-negative.

84. The computer-readable medium of claim 81,

wherein the instructions cause the processor to generate the matrix of biased coefficients by causing the processor to:

select, on a pseudo-random basis, a value that is equal to either −

1 or 0; and

to add the selected value to one or more coefficients in the matrix of scaled coefficients.