Direct Conversions Between DV Format DCT and Ordinary DCT

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1 Direct Conversions Between DV Format DCT Ordinary DCT Neri Merhav HP Laboratories Israel* HPL-9-4 August, 99 digital recording, DVC format, - DCT, 2-4- DCT A fast algorithm is developed for direct conversions between the 2-4- DCT mode associated with the digital video cassette (DVC) stard the ordinary - DCT associated with other formats of compressed video. The algorithm avoids explicit transformations to from the original pixel domain thus saves a considerable amount of computations. Internal Accession Date Only *Hewlett-Packard Laboratories Israel, Technion City, Haifa 32, Israel Copyright Hewlett-Packard Company 99

2 Introduction DV (formerly DVC) is a new digital recording format being backed by the main manufacturers such as Sony, Philips, Thomson, Hitachi, Matsushita (Panasonic). It was the rst digital recording format in the reach of consumer markets. Its main advantage is that it lends itself easily to video editing operations. DV uses a 5: DCT-based compression scheme at a xed rate of 25Mbps the signal-to-noise ratio (SNR) is typically around 54dB. Depending on whether the dierence between two elds is small or large, the encoder adaptively decides whether to compress picture elds separately (small dierence) or combine two elds into a single compression block (large dierence) [, p. 27]. Therefore, in a certain sense, DV coding can be thought of as a stard that lies in the midway between Motion JPEG MPEG. Unfortunately, however, the DV format is not related to any other compressed/uncompressed video format. As a rst step towards the goal of building an interface between DV other compressed video formats, such as MPEG-I, MPEG-II, H.26, H.263, etc. (for applications like transcoding), we develop propose, in this document, a fast algorithm that converts the 2-4- DCT of the DV format (which is used for largely dierent elds) to the ordinary - DCT which is used by all the above-mentioned video compression stards. Since the proposed algorithm operates directly in the DCT domain, without explicit transformation to from the pixel domain, it saves a great deal of the computations without additional memory requirements. 2 Background Problem Formulation The DV stard contains two DCT modes, called - DCT mode 2-4- DCT mode to improve the picture quality after the bit rate reduction. The - DCT mode should be selected when the dierence between two elds is small, whereas the 2-4- DCT mode should be selected when the dierence between two elds is large. The two DCT modes are dened as follows. - DCT: The - DCT is the ordinary, type-ii two-dimensional DCT [2] that transforms a block fx(n; m)g 7 n;m= in the spatial domain into a matrix of frequency components fx(k; l)g 7 k;l= according to the following equation X (k; l) =c(k)c(l) n= m= 2n + 2m + x(n; m) cos( k) cos( l) () 6 6 where c() = =(2 p 2) c(i) ==2 for i>. The inverse transform is given by x(n; m) = k= l= 2n + 2m + c(k)c(l)x (k; l) cos( k) cos( l): (2) 6 6 In a matrix form, let x = fx(n; m)g 7 n;m= X = fx (k; l)g 7 k;l=. Dene the -point DCT operator matrix C = fc (k; n)g 7 k;n=, where 2n + c (k; n) =c(k) cos( k): (3) 6

3 Then, X = C xc t (4) where the superscript t denotes matrix transposition. Similarly, let the superscript,t denote transposition of the inverse. Then, x = C, X C,t = C t XC (5) where the second equality follows from the orthonormality of C DCT: The 2-4- DCT combines two elds in one block in the following way. Let us assume that the spatial domain block x = fx(n; m)g 7 n;m= consists of two interlaced elds, where the odd rows (starting to count from zero) are from the rst eld the even rows are from the second eld. Then, the 2-4- DCT is dened as follows. For k =;;2;3 l =;; :::; 7 dene X 24 (k; l) =c(k)c(l) 3X X 24 (k+4;l)=c(k)c(l) n= m= 3X n= m= 2n + [x(2n; m)+x(2n +;m)] cos( The inverse transform is given by x(2n; m) = x(2n +;m)= 2 [x(2n; m),x(2n+;m)] cos( 3X k= l= cos( 2 2n + 3X k= l= cos( c(k)c(l)[x 24 (k; l)+x 24 (k +4;l)] 2n + 2m + k) cos( l) (6) 6 2n + 2m + k) cos( l): (7) 6 2m + k) cos( l) () 6 c(k)c(l)[x 24 (k; l), X 24 (k +4;l)] where n =;;2;3 m =;; :::; 7. In a matrix form, let F = 2m + k) cos( l); (9) 6,,,, 2 ()

4 C 24 = C 4 C 4! where C 4 = fc 4 (k; n)g 3 k;n= is the 4-point DCT operator matrix whose entries are given by We can now rewrite eqs. () (9) as () 2n + c 4 (k; n) =c(k) cos( k); k; n =;;2;3: (2) X 24 = C 24 F xc t (3) x = F, C, 24 X 24C ; (4) respectively, where X 24 = fx 24 (k; l)g 7 k;l=. Our goal is to devise fast algorithms that convert from X 24 to X vice versa. Our algorithms will be based on sparse matrix factorizations of C 4 C. The best known factorization of C to sparse matrices corresponds to the Winograd -point DCT which was derived from the 6-point Winograd FFT [3] (see also [4]). According to this factorization, C is represented as a product: where D is a diagonal matrix given by C = DPB B 2 MA A 2 A 3 (5) D = diagf:3536; :2549; :276; :37; :3536; :45; :6533; :24g; (6) P is a permutation matrix given by P = the remaining matrices are dened as follows: B =,, 3 (7) ()

5 M = B 2 =,, :77,:9239,:327 :77,:327 :9239 A 2 = A 3 = A =,,,,,,,,, (9) (2) (2) (22) (23) 4

6 In the next section, we will present a parallel sparse matrix factorization of C 4 that corresponds to the -point Winograd FFT. We will then show how to use these factorizations to derive a fast conversion algorithm from - DCT to 2-4- DCT vice versa. 3 A 4-Point Winograd DCT Let us concentrate on D transforms, where it should be understood that 2D transforms are obtained in a row-column separable fashhion. The 4-point Winograd DCT will be based on the -point Winograd FFT, which works as follows (see also [5, p. 63]). Given an input vector x = (x ;x ; :::; x 7 ), the -point DFT X =(X ;X ; :::; X 7 ), is computed in the following way:. Compute t =(t ;t 2 ; :::; t ) according to t = x + x 4 t 2 = x 2 + x 6 t 3 = x + x 5 t 4 = x, x 5 t 5 = x 3 + x 7 t 6 = x 3, x 7 t 7 = t + t 2 t = t 3 + t 5 2. Dene u = =4, i = p, compute m =(m ;m ; :::; m 7 ) according to 3. Compute s =(s ; :::; s 4 ) according to m = t 7 + t m = t 7, t m 2 = t, t 2 m 3 = x, x 4 m 4 = (t 4, t 6 ) cos(u) m 5 = i(t 3, t 5 ) m 6 = i(x 2, x 6 ) m 7 = i(t 4 + t 6 ) sin(u) s = m 3 + m 4 s 2 = m 3, m 4 s 3 = m 6 + m 7 s 4 = m 6, m 7 4. Finally, compute the output DFT according to X = m X = s + s 3 X 2 = m 2 + m 5 X 3 = s 2, s 4 X 4 = m X 5 = s 2 + s 4 X 6 = m 2, m 5 X 7 = s, s 3 It is well-known (see, e.g., [4]) that an N-point DCT can be obtained from the real part of a(2n)-point DFT if the input vector is dened by the symmetric extension (x ;x ; :::; x N,;x N,;x N,2; :::; x ): A nice property of the Winograd FFT is that there is complete separation between the real part the imaginary part. Since the DCT is a real transform, the imaginary part can be therefore ignored altogether. In our case, N = 4. If we use the above algorithm, taking advantage of the facts that (i) x i = x 7,i, i =;;2;3, (ii) the imaginary part can be ignored, we end up with a further simplied algorithm for computing the 4-point DCT. 5

7 The resulting Winograd DCT algorithm corresponds to the following factorization of C 4 : where C 4 = D GHL (24) ) c(i) D = diag( 2 cos(i=) ;i=;;2;3 (25) H = For later use, we will also dene G = L =, 2 2,, :77 :77,,! D D 2 = D G 2 = G G H 2 = H H L 2 = L L Clearly, since C 4 = D GHL, wealso have C 24 = D 2 G 2 H 2 L 2. 4 Derivation of the Algorithm (26) (27) : (2) ; (29)! ; (3)! ; (3)! : (32) We are now ready to develop the conversion algorithm. First, observe that according to eqs. (3), (4), the subsequent derivations, we have X = C xc t = C (F, C, 24 X 24C )C t = C F, C, 24 X 24 = DPB B 2 MA A 2 A 3 F, L, 2 H, 2 G, 2 D, 2 X 24: (33) 6

8 Precomputation of R = MA A 2 A 3 F, L, 2 H, 2 (34) results in a fairly sparse matrix given by R = :5 2a a :5 a,b b :5,a=2 c c :5 :25 :25 ; (35) where a = :77, b = :366, c = :542. The proposed conversion algorithm calculates X from X 24 according to where G, 2 = X = DPB B 2 RG, 2 D, 2 X 24: (36) :5 :5 :5,:5 :5 :5 :5,:5 : (37) This means that the input 2-4- DCT is rst premultiplied by D 2,, then the result is premultiplied by G, 2, so on. Assuming that the multiplication by D, 2 can be absorbed in the dequantization of the 2-4- DCT coecients (as well as compensation for the weight factors [, p. 2, subsection 7.5.2]), that multiplication by D can be absrobed in the re-quantization of the - DCT coecients, the only phase, in this algorithm, that contains nontrivial multiplications is the pre-multiplication by the matrix R. This is because multiplications by powers of two are implementable by simple shifts. Given a column vector u =(u ; :::; u ) t, the vector v =(v ; :::; v ) t dened as v = Ru can be eciently computed by the following steps: v = u =2 w = (2a)u v 2 = 2u 7, w v 3 = au 2 v 4 = (u 2 + w)=2 7

9 v 5 = b(u 4, u 3 ) v 6 = (u 4, au 6 )=2 v 7 = c(u 3 + u 4 ) v = (u 3 +(u 6 +u 5 =2)=2)=2 Thus, pre-multiplication by R can implemented by 5 multiplications, 3 shift&adds, 4 additions, 4 shifts. Since the matrix P is computationally costless, B B 2 require 4 additions per vector each one, G, 2 is associated with 4 additions 4 shifts per vector, the total complexity associated with this algorithm is 5 multiplications, 3 shift&adds, 6 additions, shifts per each column vector. For the sake of comparison, the direct approach (of explicitely undoing the column 2-4- DCT doing the - DCT instead) would require 7 multiplications, 55 additions/shift&adds. This means that if, for example, each multiplication takes 3 cycles on the average, then about 5% of the computations have been saved. If, on the other h, the processor performs multiplication in one cycle, the saving factor is even higher. For a complete DCT block, all the above numbers should be multiplied by. As a nal remark, it should be pointed out that since both C C 24 are orthonormal, namely, C, = C t C 24, = C24, t there are three more ways to write the equation relating X 24 to X (eq. (33)). These are: X = DPB B 2 MA A 2 A 3 F, L t 2 Ht 2 Gt 2 D 2X 24 ; (3) X = D, P,t B,t 2 M,t A,t A,t 2 A,t 3 F, L t 2 H 2 t Gt 2 D 2X 24 ; (39) X = D, P,t B,t B,t 2 M,t A,t A,t 2 A,t 3 F, L, 2 H, 2 G, 2 D, 2 X 24: (4) Accordingly, the matrix R would then be redened as MA A 2 A 3 F, L t 2 Ht 2 ; or M,t A,t A,t 2 A,t 3 F, L t 2 H 2 t ; or M,t A,t A,t 2 A,t 3 F, L, 2 H, 2 ; respectively. However, the rst representation that wehave used (eq. (33)) yields the sparsest R. 5 The Inverse Conversion In view of the last comment in the previous section, there are also four ways to write the inverse transformation from X to X 24 : X 24 = D 2 G 2 H 2 L 2 FA, 3 A, 2 A, M, B, 2 B, P, D, X ; (4) X 24 = D 2 G 2 H 2 L 2 FA t 3 At 2 At Mt B t 2 Bt Pt DX ; (42) X 24 = D 2, G,t 2 H,t 2 L,t 2 FA, 3 A, 2 A, M, B 2, B, P, D, X ; (43)

10 Correspondingly, we dene ~ R as either or X 24 = D, 2 G,t 2 H,t 2 L,t 2 FAt 3 At 2 At Mt B t 2 Bt Pt DX ; (44) H 2 L 2 FA, 3 A, 2 A, M, ; H 2 L 2 FA t 3 At 2 At Mt ; or H 2,t L,t 2 FA, 3 A, 2 A, M, ; or H,t 2 L,t 2 FAt 3 At 2 At Mt : The latter possbility turns out to be best in terms of the computational complexity of the associated algorithm. The inverse algorithm then computes X 24 from X according to where ~R = X 24 = D 2, G,t 2 ~RB2 t Bt P t DX (45) 2a,2b 2c 2b 2c :25,a :5 4,4a 2a where a, b, c are as in eq. (35). The computational complexity of this algorithm is approximately the same as that of the forward algorithm because the computation of v = ~ Ru can be done in 5 multiplications, 5 additions, 2 shift&adds, 2 shifts, using the following procedure: v = u v 2 = (2a)u 3 + u 4 w = (2b)u 5 w 2 = (2c)u 7 v 3 =,w + w 2 + u v 4 = w + w 2 + u 6 v 5 = u =4 v 6 =,au 6 + u =2 v 7 = 4u 2 v = (2a)(u 4, 2u 2 ): (46) 9

11 6 References [] HD Digital VCR Conference, Specications of Consumer-Use Digital VCRs Using 6.3mm Magnetic Tape, Part 2, December 994. [2] K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press 99. [3] Y. Arai, T. Agui, M. Nakajima, \A Fast DCT-SQ Scheme for Images," Trans. of the IEICE, E7():95, November 9. [4] W. B. Pennebaker J. L. Mitchell, JPEG Still Image Data Compression Stard, Van Nostr Reinhold, 993. [5] H. S. Silverman, \An introduction to programming the Winograd Fourier transform algorithm," IEEE Trans. Acoustics, Speech, Signal Processing, vol. ASSP{25, no. 2, pp. 52{65, April 977.

Neri Merhav. and. Vasudev Bhaskaran. Abstract. A method is developed and proposed to eciently implement spatial domain ltering

Neri Merhav. and. Vasudev Bhaskaran. Abstract. A method is developed and proposed to eciently implement spatial domain ltering A Fast Algorithm for DCT Domain Filtering Neri Merhav HP Israel Science Center and Vasudev Bhaskaran Computer Systems Laboratory y Keywords: DCT domain ltering, data compression. Abstract A method is developed