EE 5359: MULTIMEDIA PROCESSING PROJECT PERFORMANCE ANALYSIS OF INTEGER DCT OF DIFFERENT BLOCK SIZES.Introduction to IntDCTDefinition:Transforms used in some standardsI. DCTIDCTDCT IITaking the example of the standard H.264:II. ORDER-4 INTEGER TRANSFORM III. ORDER-8 INTEGER TRANSFORMIV. ORDER-16 INTEGER TRANSFORM(1) Order-8 transform matrix(2) Order-16 transform matrix derived from order-8 transform matrixPerformance Evaluation:References:Slide 16Slide 17EE 5359: MULTIMEDIA PROCESSING PROJECTPERFORMANCE ANALYSIS OF INTEGER DCT OF DIFFERENT BLOCK SIZES.Guided by Dr. K.R. RaoPresented by:Suvinda Mudigere SrikantaiahUTA ID: 1000646539Introduction to IntDCTDiscrete 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 matrixK 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 standardsStandard Transform1. MPEG-4 part 10/H.264 8 X 8, 4 X 4 integer DCT, 4 X 4, 2 X 2 Hadamard2. WMV-9 8 X 8, 8 X 4, 4 X 8, 4 X 4 integer DCT3. AVS China Asymmetric 8 X 8 integer DCTTable no.1: Transforms used in standards H.264, WMV-9 and AVS china [4].I. DCTThe 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 – 1First of all we apply this transformation to the rows, then to the columns of image data matrixIDCTThe Inverse Discrete Cosine Transform (IDCT) of N samples is formulated by:for x = 0, 1, . . . , N – 1, whereThe 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 – 1For image decompression we use this DCT.:DCT IIThe 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 bywhere l = u,vTaking 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 TRANSFORMConsider 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 byIV. ORDER-16 INTEGER TRANSFORMOrder-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
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