Efficient Large Size Transforms for High-Performance Video Coding

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1 Efficient Large Size Transforms for High-Performance Video Coding Rajan Joshi, Yuriy. Reznik *, and Marta Karczewicz Qualcomm Inc, 5775 Morehouse Drive, San Diego, C, US 9 STRCT This paper describes design of transforms for extended block sizes for video coding. The proposed transforms are orthogonal integer transforms, based on a simple recursive factorization structure, and allow very compact and efficient implementations. We discuss techniques used for finding integer and scale factors in these transforms, and describe our final design. We evaluate efficiency of our proposed transforms in VCE's H.65/JMKT framework, and show that they achieve nearly identical performance compared to much more complex transforms in the current test model. Keywords: discrete ine transform, DCT, factorization, multiplicative complexity, scaled transform, video coding, H.64, H.65, HEVC.. ITRODUCTIO Discrete Cosine Transform of type (DCT-) - is a fundamental operation performed by the majority of today s image and video compression algorithms. It was first suggested by. hmed, T. atarajan, and K. R. Rao, and subsequent research provided a number of theoretical arguments for its use, such as energy compaction property, asymptotic equivalence of DCT to the Karhunen-Loève transform for signals produced by Markov- process with high correlation coefficient, etc. (see, e.g. [, Chapter ] for a survey of the related results). DCT- of size 8 has served as the transform of choice in H.6, JPE, MPE-, MPE-, H.6, and MPE-4 visual standards.,4-9 More recent standards, such as MPE-4 VC H.64, VC-, and VS have adopted integer approximations of DCT- with transform sizes: 4, 8, and 6. n emerging JPE-R image compression standard uses overlapping transforms, which are also based on 4-point DCT- kernels. new emerging standard, High Efficiency Video Coding (HEVC), currently under development by Joint Collaborative Team of video experts from MPE and ITU-T S6 (JCT-VC), 4 includes a number of integer transforms of sizes ranging from 4 to 64. s video resolutions keep increasing, it is possible that even larger transforms will be considered in the future. In this paper we describe design of scaled integer transforms, which are numerically stable, fully recursive in structure, and remain orthogonal with perfect scaling (in the absence of quantization). s such, they are well suitable for use in future video coding applications. Described transforms have been proposed to ITU-T S6 Q6 (VCE) standardization committee, and were also included in Qualcomm s response to JCT-VC call for proposals. This paper is organized as follows. In Section, we describe design of underlying factorization that we use in the transform. Section discusses conversion to integer arithmetic and other implementation aspects. Section 4 provides experimental results obtained using this transform in ITU-T S6 Q6 JMKT video coding model. Conclusions are drawn in Section 5.. FCTORIZTIO Let { xn}, n =,..., be a sequence of input samples (i.e. line of pixel values). DCT- and its inverse transform over this sequence are defined as follows: (n+ ) k k = λ( k) xn, k =,...,, n= * Corresponding author. yreznik@ieee.org, Phone:

2 (n+ ) k xn = k λ( k), k = n =,...,, where λ ( k ) = /, if k =, and otherwise. DCT-IV transform and its inverse are defined as follows: IV k = xn (n )(k ), + + n 4 = k =,...,, IV xk = k (n )(k ), + + n= 4 n =,...,. We note that in video coding, we usually work with x matrices of input data, and so the above transforms need to be applied to all rows and columns in a separable fashion, to produce corresponding matrices of transform coefficients. Hereafter we will adopt such separable model in our design of integer D transforms, and will focus mainly on speeding up computations of the component D transforms. For further convenience, we omit normalization factors ( / and λ ( k ) ) and define matrices: ( n+ ) k C ( n, k) =, n, k =,...,, ( n+ )( k + IV ) C ( n, k) =, n, k =,...,, 4 representing coefficients of DCT-, and DCT-IV transforms correspondingly. It is well known, that even-sized DCT- matrix can be factored into a product containing direct sum of smaller DCT- and DCT-IV matrices as follows, : C I J C = P IV C J J, () I where P is a permutation matrix producing reordering: x = x, x = x, i =,,..., /, () i i + i i+ and where I and J denote identity and order reversal matrices respectively. Chen-Smith-Fralick 5, Wang, and many other well-known DCT- factorizations, rely on factorization () as a basic step in their decimation process. We next apply the following decomposition to the DCT-IV block in (): IV T I I I C I C = P R, () I I E J C E where: P is a reordering matrix (), E is the diagonal sign-alteration matrix k {( ) } E = diag -, k =,,... /, (4) While faster non-separable D designs can possibly be created, their large expanded structure, and the need for support of multiple (including hybrid, e.g. 8x6) block sizes makes this approach much less appealing in practice.

3 R is the matrix of ivens rotations: sin 4 4 sin 4 4 O ( ) ( ) sin, (5) 4 4 R = ( ) ( ) sin 4 4 O sin 4 4 sin 4 4 and where C denotes matrices of the remaining half-sized DCT- transforms. This decomposition of DCT-IV is very similar to the one derived by. Plonka and M.Tache 7. In our formulation () all normalization factors of / are outside the transform. Finally, by nesting () and () we can now close the recursion. For example, an =6 DCT- will be decomposed into 8-point DCT- and DCT-IV. Then 8-point DCT-IV will be split into two 4-point DCT- transforms. The 8-point DCT- will be split into 4-point DCT- and DCT-IV. This is continued until we reach -point blocks (i.e. simple butterflies) for both DCT- and DCT-IV. We show full flow-graph of this factorization in Figure. This factorization is numerically stable: only planar rotations are used throughout, it is fully recursive, and in terms of instruction count it is equivalent to best known practical algorithms, such as C. Loeffler, et.al. 6. COVERSIO TO ITEER RITHMETICS In order to convert transform to a fixed-point arithmetic we introduce common scale factors for each interconnected group of butterflies in the transform. For example, for a 6-point transform shown in Figure, this will be: - factor associated with factors and in the left-most butterflies in the transform, - factor ζ associated with 4 factors C -- E in the -rd stage of the transform, and - factor associated with 8 factors -- in the -nd stage of the transform. s long as the following set of conditions are met: C C E F = ( 8 ); = sin ( 8 ); ( 6 ); sin ( 6 ); ( 6 ); sin ( 6 ); ς ς ς ς 7 H 7 I 5 J 5 K L M = ( ); = sin ( ); = ( ); = sin ( ); ( ) ( ) ( ) ( ) = = = = (6) = ; = sin ; = ; = sin ;. the resulting scaled transform is the DCT-. If this set of fractions only approximates the associated ine and sine values, we will say that the resulting transform is an approximation of DCT-. ote that (6) allow tradeoffs between values of factors inside the transform - and scale factors, ς,. Such flexibility was already exploited to gain extra precision in the design of fixed-point algorithms in references 5,8-. In our case, we are also concerned with retaining orthogonality of scaled transform. This creates additional conditions: = + ; ς = C + D = E + F ; = + H = I + J = K + L = M + ; (7)

4 x 4 x x x x 4 x 5 x 6 x 7 F D F E C C E ζ 4 ζ ζ 4 ζ 6 x 8 x 9 x x x x x 4 x 5 L J H J L M K I I K M Figure.. =6-point DCT- factorization produced by recursive use of decompositions () and (). ote that it uses identical 4-point DCT- blocks. Scale factors, ς, are introduced to simplify conversion of this transform to integer arithmetic. Ideally, the following equations must be achieved to make this transform a DCT-: C C E F 7 H 7 = ( 8 ); = sin ( 8 ); = ( 6 ); = sin ( 6 ); = ( 6 ); = sin ( 6 ); = ς ς ς ς ( ) = ( ) 5 J 5 K L M = ( ); = sin ( ); = ( ); = sin ( ); = ( ); = sin ( );. I ; sin ; Hence, considering all above, the task of design of an integer approximation of DCT- now boils down to finding some small integers -, such that conditions (7) are met, and such that the resulting fractions produce good approximations of sine and ine factors in the transform (6). One can easily solve this task by straightforward enumeration. s an example, we show several example solutions found for factors and (and associated scale factor ) in Table.

5 Table. Example factors, and = + found as solutions for transform approximation problem. Factors pproximation errors Complexity of multiplications = + ( 8 ) sin ( ) 8 by factors and shift addition + shifts additions + shift additions + shifts additions + shifts Table. Complete set of factors selected for the design of 6-point transform. C D E F L J H I K M Table. ormalized set of factors selected for the design of 6-point transform. C D E F L J H I K M /4 5/ /64 /64 /64 7/64 4/64 8/64 4/64 4/64 ased on Table, it can be observed that larges values and produce better approximations, but this also increases complexity (in terms of bit-width and gates count) of the transform. ut we also note that even small numbers (such = and =5) seem to achieve pretty good precision (around %) in this case. We repeat similar exhaustive search for other groups of factors as well. full set of factors that we have selected for =6 point transform design is shown in Table. In order to reduce the dynamic range of factors within the transform we further normalize them by nearest dyadic numbers. This optimization is prompted by the fact that divisions by dyadic numbers can be implemented as binary shifts, which are gratis on many platforms. The resulting normalized quantities are now shown in Table. Finally, we now need to compute exact values of scale factors moved outside the transform. In the original form, as shown in Figure, such factors are: diag,,,,,,,,,,,,,,, S = ; 4 4ζ ζ ζ 4ζ y incorporating scale factor values from our integer solutions (Tables ), and their subsequent normalization, we obtain: fter multiplication by 4 and conversion to floating point representation, these scale factors are: which is reasonably tight, considering 4 stages of the transform. In our final design these factors are cross-multiplied to produce a scale matrix for the entire D transform, which is subsequently merged with the quantization matrix of video encoder.

6 4. EPERIMETL RESULTS The proposed scheme is tested in the JMKT video encoder under the VCE common testing conditions 5. The reference transforms is a much more complex, direct approximation of matrices of DCT- transforms by using 8-bit fixed-point factors. Full matrix multiplication is used. The configuration parameters UseRDO_Q, UseHPFilter and UseewOffset are set to in both the reference and our proposed solution. Extended block sizes, MDDT 4 (for intra coded macroblocks), and CC are also enabled in both the reference as well as our proposed solution. Table 4 shows the percentages of D-Rate reduction for all test sequences (up to 8P resolution) under the VCE common testing conditions. On average, the use of proposed transforms for extended block sizes, results in a.59% and.59% D- Rate increase for IPPP and Hier configurations. Table 4: Simulation results for proposed transforms for extended block sizes. Sequence Conditions: iglocks, MDDT, CC, UseRDO_Q=, UseewOffsert =, UseHPFilter =, IPPP Hier CIF Foreman Mobile Paris Tempete verage WQV Flower Keiba uts verage WV Flower Keiba uts verage SV Janine verage P igships City Crew ight Jets Raven verage P CrowdRun ParkJoy Traffic Toysandcalendar Subnflower verage.8 -. Overall verage

7 5. COCLUSIOS We have proposed design of fast orthogonal integer transforms for video coding. The design is based on a simple recursive factorization structure, allowing very compact and efficient implementations. We have implemented and tested our transforms in VCE's H.65/JMKT framework. We have showed that under common VCE test conditions our proposed transforms achieve nearly identical performance compared to much more complex transforms in the current test model. REFERECES []. hmed, T. atarajan, and K. R. Rao, Discrete Cosine Transform, IEEE Trans. Computers,, 9-9 (974). [] K. R. Rao and P.Yip, Discrete Cosine Transform: lgorithms, dvantages, pplications, cademic Press (99). [] V.ritanak, P.Yip, K.R.Rao, Discrete Cosine and Sine Transforms: eneral Properties, Fast lgorithms and Integer pproximations, cademic Press (6). [4].J. Sullivan, Standardization of IDCT approximation behavior for video compression: the history and the new MPE-C parts and standards, Proc. SPIE 6696, pp (7). [5] Y.. Reznik,.T.Hinds, C.Zhang, L.Yu, and Z.i, Efficient fixed-point approximations of the 8x8 inverse discrete ine transform, Proc. SPIE 6696, pp (7). [6] ISO/IEC 98- ITU-T Rec. T.8, Information Technology Digital Compression and Coding of Continuous- Tone Still Images Requirements and uidelines (99). [7] ISO/IEC 88- ITU-T Rec. H.6, Information Technology eneric Coding of Moving Pictures and ssociated udio Information: Video (997) [8] ITU-T Rec. H.6, Video Coding for Low it Rate Communication (5). [9] ISO/IEC 4496-, Information Technology Coding of udio-visual Objects Part : Visual (). [] ISO/IEC ITU-T Rec. H.64, dvanced Video Coding for eneric udiovisual Services (7). [] SMPTE Standard 4M-6, VC- Compressed Video itstream Format and Decoding Process (6). [] PRC ational Standard (VS Working roup) /T 9.-6, Information Technology dvanced Coding of udio and Video, Part : Video (6). [] S. Srinivasan, C. Tu, S. L. Regunathan,. J. Sullivan, and R.. Rossi, HD Photo: ew Image Coding Technology for Digital Photography, Proc. SPIE, Vol (7). [4] ISO/IEC JTC SC9 W ITU-T S6 Q6 (JCT-VC), Test Model under Consideration (TMuC), JTC-VC document JCTVC-5 (). [5] W. Chen, C. H. Smith, and S. C. Fralick, Fast Computational lgorithm for the Discrete Cosine Transform, IEEE Trans. Comm., 5 (9) 4-9 (977). [6] C. Loeffler,. Ligtenberg, and. S. Moschytz. "lgorithm-architecture mapping for custom DCT chips," Proc. International Symposium on Circuits and Systems, (988). [7].Plonka and M.Tashe, Fast and numerically stable algorithms for discrete ine transforms, Linear lgebra and pplications, 94 () 9-45 (5). [8] Y.. Reznik,. T. Hinds, and J. L. Mitchell, Improved precision of fixed-point algorithms by means of common factors, Proc. International Conference on Image Processing (ICIP), (8). [9] R. K. Chivukula and Y.. Reznik, Efficient implementation of a class of MDCT/IMDCT filterbanks for speech and audio coding applications, Proc. Int. Conf. coustics Speech and Signal Processing, -6 (8). [] Y.. Reznik and R. K. Chivukula, Design of Fast Transforms for High-Resolution and Video Coding, Proc. SPIE 744, (9). [] R. Joshi, Y. Reznik, and M. Karczewicz, Simplified Transforms for Extended lock Sizes, Input contribution to ITU-T S6 Q6 (VCE), document VCE-L (9). [] M. Karczewicz, P. Chen, R. Joshi,. Wang, W.-J. Chien, R. Panchal, Video coding technology proposal by Qualcomm, Input contribution to JCT-VC, document JCTVC- (). [] P. Chen, Y. Ye, and M. Karczewicz, Video coding using extended block sizes, input contribution to ITU-T S6 Q6 (VCE), document COM 6 C E (9). [4] Y. Ye and M. Karczewicz, Improved intra coding, input contribution to ITU-T S6 Q6 (VCE), document VCE- (7). [5] TK Tan,. Sullivan, and T. Wedi, Recommended Simulation Common Conditions for Coding Efficiency Experiments Revision 4, ITU-T S6 Q6 (VCE) document VCE-Jr (8).

LOSSLESS INTRA CODING IN HEVC WITH INTEGER-TO-INTEGER DST. Fatih Kamisli. Middle East Technical University Ankara, Turkey

LOSSLESS INTRA CODING IN HEVC WITH INTEGER-TO-INTEGER DST Fatih Kamisli Middle East Technical University Ankara, Turkey ABSTRACT It is desirable to support efficient lossless coding within video coding