COMPARISON OF 8 × 8 INTEGER DCTs USED IN H.264, AVS-CHINA AND VC-1 VIDEO CODECS

Submitted by,

Ashwini Urs and Sharath Patil

Under guidance of

Dr.K.R.Rao

Introduction

Integer DCT

• KLT is the statistically optimal transform.• The performance of DCT is close to the

performance of KLT [1].• DCT is a well-known transform and is widely

used by majority of coding standards.• Though integer DCT contains only integers, it

has similar energy-packing ability as that of DCT [1].

Integer DCT (Continued)

• Integer cosine transform does not involve floating point computations and hence is used in video coding standards such as H.264 [2], VC-1 [3] and AVS [4].

• Integer cosine transform has been implemented with transform sizes of 4, 8 and 16 [1].

• Even larger size transforms (up to 64) have been used for high resolution videos to achieve higher coding gain [1].

Integer DCTs compared

Integer DCT matrix for AVS-China, H.264 and VC-1

AVS-China [2]

2691010962

410104410104

6102992106

88888888

9210661029

104410104410

1096226910

88888888

361012121063

48844884

612310103126

88888888

103126612310

84488448

121063361012

88888888

H.264 [3]

491516161594

616166616166

916415154169

1212121212121212

154169916415

166616166616

161594491516

1212121212121212

VC-1 [4]

Integer DCT matrix for AVS-China, H.264 and VC-1

• The orthogonality of the 3 matrices was checked by evaluating [INTDCTi] x [INTDCTi]*T

.

• The orthogonalised matrices are:1. AVS-China = diag (512, 442, 464, 442, 512, 442,

464, 442)2. H.264 = diag (512, 578, 320, 578, 512, 578, 320,

578)3. VC-1 = diag (1152, 1156, 1168, 1156, 1152, 1156,

1168, 1156)

Order-16 Integer DCT matrix used in AVS-China [26]

8888888888888888

10109966222266991010

101044441010101044441010

99221010666610102299

8888888888888888

441010101044441010101066

441010101044441010101044

22669910101010996622

22669910101010996622

441010101044441010101044

66101022999922101066

8888888888888888

99221010666610102299

101044441010101044441010

10109966222266991010

8888888888888888

16T

Comparison of the properties of integer DCTs

Comparison of interger DCT matrices

• The properties of the 3 integer DCT matrices were compared by considering a covariance matrix R for a Markov-I process with ρ = 0.95 and N=8.

• Rjk = [ρ|j-k|] for j, k = 0, 1,…, N-1, where ρ is the adjacent correlation coefficient.

• Covariance matrix in transform domain is given by

where DOT is discrete orthogonal transform and [Σ] is the covariance matrix in spatial

*~

TDOTDOT

Properties used for comparison of integer DCTs

1. Variance distribution: The diagonal elements of correspond to the variances in the transform domain [7].

2. Rate versus distortion: RD is the minimum average rate (bits/sample) for coding a signal at a specified distortion D [7]. For fixed average distortion D, rate distortion function RD is computed as

Choose values of θ betweent 0.1 and 1. For the same values of θ, D and RD are calculated [7].

Properties used for comparison of integer DCTs

3. Normalized basis restriction error, Jm: The compaction of energy in a few transform coefficients can be represented by the normalized basis restriction error defined as [7]:

where are arranged in decreasing order [7].2~

kk

Properties used for comparison of integer DCTs

4. Residual correlation: An indication of the extent of decorrelation in transform domain can be gauged by correlation left undone by the discrete transform, which is measured by the absolute sum of cross-covariance (off diagonal elements) in the transform domain i.e.,

for N = 8 as a function of ρ [7].

Properties used for comparison of integer DCTs

5. Transform coding gain GTC: Transform coding gain is defined as the ratio of arithmetic mean to geometric mean of variances

where is the variance of the ith co-efficient in the transform domain.

• As sum of all the variances is in invariant under orthogonal transformation, by minimizing geometric mean GTC can be maximized [7].

2~

ii

2~

ii

Results and Conclusion

Variance distribution versus N

Rate versus distortion

Normalized basis restriction error versus samples retained m

Residual correlation versus correlation co-efficient

Conclusion

• Variance distribution, normalized basis restriction error and transform coding gain of these 3 codecs are almost comparable.

• Transform coding gain, GTC for AVS, H.264 and VC-1 are 8.2916, 8.0155 and 7.5477 respectively. From this, we observe that AVS achieves maximum GTC.

• For a fixed average distortion D, the rate distortion function characteristics of H.264 and AVS are indistinguishable.

• The residual correlation for ρ > 0.5 is indistinguishable for these 3 codecs.

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