EE 5359: MULTIMEDIA PROCESSING PROJECT

PERFORMANCE ANALYSIS OF INTEGER DCT OF DIFFERENT

BLOCK SIZES.

Guided by Dr. K.R. Rao

Presented by:

Suvinda Mudigere Srikantaiah

UTA ID: 1000646539

Introduction to IntDCT

Discrete cosine transform has been serving as the basic elements of video coding systems.

The integer discrete cosine transform is an integer approximation of the discrete cosine transform.

It can be implemented exclusively with integer arithmetic.

It proves to be highly advantageous in cost and speed for hardware implementations [1].

Definition:

ICT matrix is in the form [2,3]: I = KJ where I is the orthogonal ICT matrix

K is a diagonal matrix whose elements take on values that serve to scale the rows of the matrix J so that the relative magnitudes of elements of the ICT matrix I are similar to those in the DCT

matrix. The matrix J is orthogonal with

elements that are all integers.

Transforms used in some standards

Standard Transform

1. MPEG-4 part 10/H.264 8 X 8, 4 X 4 integer DCT, 4 X 4, 2 X 2 Hadamard

2. WMV-9 8 X 8, 8 X 4, 4 X 8, 4 X 4 integer DCT

3. AVS China Asymmetric 8 X 8 integer DCT

Table no.1: Transforms used in standards H.264, WMV-9 and AVS china [4].

I. DCT

The forward Discrete Cosine Transform (DCT) of N samples is formulated by [11]

for u = 0, 1, . . . , N - 1, where

The function f(x) represents the value of the xth sample of the input signal.

F(u) represents a Discrete Cosine Transformed coefficient for u = 0, 1, … , N – 1

First of all we apply this transformation to the rows, then to the columns of image data matrix

IDCT

The Inverse Discrete Cosine Transform (IDCT) of N samples is formulated by:

for x = 0, 1, . . . , N – 1, where

The function f(x) represents the value of the xth sample of the input signal.F(u) represents a Discrete Cosine Transformed coefficient for u = 0, 1, … , N – 1

For image decompression we use this DCT. 

DCT II

The DCT-II is probably the most commonly used form, and is often simply referred to as "the DCT" [6].

Given an input function f(i,j) over two integer variables i and j (a piece of an image), the 2D DCT transforms it into a new function F(u,v), with integer u and v running over the same range as i and j. The general definition of the transform is:

where i,u = 0,1,…,M − 1; j,v = 0,1,…, N − 1; and the constants C(u) (or C(v)) are determined by

where l = u,v

Taking the example of the standard H.264:

H.264 [4] uses an adaptive transform block size, 4 X 4 and 8 X 8 (high profiles only)

For improved compression efficiency, H.264 also employs a hierarchical transform structure.

The DC coefficients of neighboring 4 X 4 transforms for the luma signals are grouped into 4 X 4 blocks and transformed again by the Hadamard transform.

For blocks with mostly flat pel values, there is significant correlation among transform DC coefficients of neighboring blocks.

Therefore, the standard specifies the 4 X 4 Hadamard transform for luma DC coefficients for 16 X 16 Intra-mode only, and 2 X 2 Hadamard transform for chroma DC coefficients.

II. ORDER-4 INTEGER TRANSFORMThe 4X4 IntDCT matrix is obtained using matrix H,

The varables a,b,c are as follows:

Thus Order-4 Integer transform,

  III. ORDER-8 INTEGER TRANSFORM

Consider the ICT of a one-dimensional data vector X of size 8 [7]. This ICT is implemented by pre-multiplying X by the orthogonal matrix C, given by

 

IV. ORDER-16 INTEGER TRANSFORM Order-16 ICT has been shown to be very close to

order-16 DCT [8]. A simple integer transform for video coding is

considered and its test is based on a set of CIF sequences.

The order-16 transform considered is an extended version of the order-8 ICT adopted in AVS.

As shown in (1), T8 is the order-8 transform matrix. Without significant increase in complexity, T8 can be extended to order-16 transform T16 as shown in (2).

The normalized basis vectors of T16 have the waveforms similar to that of DCT.

(1) Order-8 transform matrix

(1) T8: Order 8 transform matrix [5].

(2) Order-16 transform matrix derived from order-8 transform matrix

(2) T16: Order 16 transform matrix [5].

(2)

Performance Evaluation:

The efficiency of a transform is generally defined as its ability to decorrelate a vector or random elements. In finding efficiency of integer DCT, standard images are applied as an input signal. Transforms considered will be DCT, Integer DCT of different block sizes.

 The following operations are performed:a) Variance distribution for I order Markov process , ρ = 0.9 ( Plot and Tabulate ) b) Normalized basis restriction error vs. # of basis function ( Plot and Tabulate ) c) Obtain transform coding gains d) Plot fractional correlation ( 0<ρ<1)

References:

1. N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete Cosine Transform", IEEE Trans. Computers, vol. C-32, pp. 90-93, Jan 1974.

2. W. K. Cham and Y. T. Chan” An Order-16 Integer Cosine Transform”, IEEE Trans. Signal proc. vol. 39, issue no. 5, pp. 1205 – 1208, May 1991.

3. W. K. Cham, “Development of integer cosine transforms by the principle of dyadic symmetry,” in Proc. Inst. Electr. Eng. I: Commun. Speech Vis., vol. 136. no. 4, pp. 276–282, Aug. 1989.

4. S. Kwon, A. Tamhankar, K.R. Rao, “Overview of H.264/MPEG-4 part 10”, Special issue on “ Emerging H.264/AVC video coding standard”, J. Visual Communication and Image Representation, vol. 17, pp.183-552, Apr. 2006.

5. W. Cham and C. Fong “Simple order-16 integer transform for video coding” IEEE ICIP 2010, Hong Kong, Sept.2010.

6. R. Joshi, Y.A. Reznik and M. Karczewicz, “ Efficient large size transforms for high-performance video coding”, SPIE 0ptics + Photonics, vol. 7798, paper 7798-31, San Diego, CA, Aug. 2010.

7. M. Costa and K. Tong, “A simplified integer cosine transform and its application in image compression”, Communications Systems Research Section, TDA Progress Report pp. 42-119, Nov 1994.

8. A.T. Hinds, “Design of high-performance fixed-point transforms using the common factor method”, SPIE 0ptics + Photonics, vol. 7798, paper 7798-29, San Diego, CA, Aug. 2010.

9. S. Chokchaitam, M. Iwahashi and N. Kambayashi, “Optimum word length allocation of integer DCT and its error analysis”, Elsevier, Signal Processing: Image Communication vol. 19, pp. 465–478, July 2004.

10. C Wei, P. Hao Q. Shi, “Integer DCT-based Image Coding”, National Lab on Machine Perception, Peking University Beijing, 100871, China.

11. P.C. Yip and K.R. Rao, “ The transform and data compression handbook,” Boca Raton, FL: CRC Press, 2001

12. Y. Zeng, et al “Integer DCTs and Fast Algorithms”, IEEE Trans. Signal proc. vol. 49, No. 11, Nov 2001.

13. P. Chen, Y. Ye and M. Karczewicz, “Video Coding Using Extended Block Sizes,” ITU-T Q.6/SG16, T09-SG16- C-0123, Geneva, Jan 2009.

14. B. Lee, et al “A 16×16 Transform Kernel with Quantization for (Ultra) High Definition Video Coding,” ITU-T Q.6/SG16 VCEG, VCEG-AK13, Yokohoma, Japan, April 2009.

15. G. Mandyam, N. Ahmed, and N. Magotra, “Lossless image compression using the discrete cosine transform”, Journal of Visual Communication and Image Representation, Vol.8, No.1, pp. 21-26, March, 1997.

THANK YOU!!!