Sequence Mirroring Properties of Orthogonal Transforms Having Even and Odd Symmetric VectorsResearch PurposePrevious Works on Transform Domain Image/Video Processing Transforms having even and odd symmetric row vectorsProposed AlgorithmProposed AlgorithmExtension to the 2-D Case Two-Dimensional ExtensionVideo Editing Application Mirroring or Rotation of portion of Lena in spatial domain by manipulating the IntDCT coefficients of the original block Mirroring or Rotation Conclusions ReferencesReferencesSequence Mirroring Properties of Orthogonal Transforms Having Even and Odd Symmetric VectorsK. R. RaoDept of Electrical Engineering Univ. of Texas at Arlington, TX, USAResearch Purpose Image/video coding/compression utilizes discrete orthogonal transforms. Compress images to save storage space or transmit and then edit in the compressed-domain.  Transform-domain image mirroring and rotation scheme in this presentation.Previous Works on Transform Domain Image/Video Processing Smith and Rowe: Video dissolve and caption insert. Linearity of the DCT. Chang and Messerschmitt: Reconstruct DCT block using neighboring4 DCT blocks. DCT is distributive to matrix multiplication.  Merhav and Bhaskaran: Fast DCT-domain bilinear interpolation and motion compensation.  Shen, Sethi and Bhaskaran: logotype insertion. DCT-domain convolution which corresponds to multiplication in the spatial domain is more efficient because of the orthogonality of the DCT.  Shen and Sethi [7]: DCT-domain image flipping schemeTransforms having even and odd symmetric row vectors Let one of these orthogonal matrices be As is orthogonal,[]oddevenoddevenghhgeffecddcabbaS⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−−−=4[]4S[][ ][]ISST=44Proposed Algorithm Let be transform coefficients of mirroring a sequence .[][]44JSXM=x[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=00010010010010004J[][]44SKI=x[]()1,1,1,14−−= diagKIMXx[]XKI4=Proposed Algorithm Type 1 orthogonal transforms: DCT, DST, slant and their integer revisions like IntDCT Type 2 orthogonal transforms: Hadamard No such properties: MDCT, Haar, Hartley[]()11,1,1,1,1,1,18−−−−= diagKI[]()11,1,1,1,1,1,18−−−−= diagKIIExtension to the 2-D Case  Image can be represented as nonoverlapping blocks of size (NN) (for example N = 8).  Each block: Transform: 2-D transform coeff: 2D-transform of horizontally/vertically mirrored sequence[]70,2121)},({==nnnnxx[]70,1111)},({==nmnmsS×[][][][ ]SxSXT=[][][]IHKXX8=[][][]XKXIV 8=[]⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛−−−=010000000010000000010000000010000000010000000010000000018IK⎟⎠⎜⎝−10000000Two-Dimensional Extension Let represent element-by-element multiplication. The 2D-transforms of the 90°, 180° and 270° rotated blocks[][][][][][][][][]ITTTTKSxSSJxSXo8890==[][][]TVITXKX ==8⊗[][][][][][]XWKXKXIIIo⊗==88180[][][ ][][][][][][ ]TTITTSxSKSxJSXo88270==[][][]THTIXXK ==8[]⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−=1111111111111111111111111111111111111111111111111111111111111111IW[]⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=OOJ1118$Video Editing Application  Select a group of consecutive 2D-transform blocks of an image  Flip horizontally according to the following steps:  Apply the (8 x 8) 2D-transform to the nonoverlapping blocks of size (8 x 8) of the image. Set the size of a rectangular block to be horizontally flipped. Horizontal and vertical sizes should be multiples of eight according to the transform size. Compute transform -domain image flipping for each transform block by using  Rotate horizontally transform blocks within the rectangular block. The most left transform block goes to the most right, and vice versa. [][][]IHKXX8=Mirroring or Rotation of portion of Lena in spatial domain by manipulating the IntDCT coefficients of the original block(a) horizontal mirroring(b) vertical mirroring (c) rotation by 90°(d) rotation by 180°(e) rotation by 270°(a)(b)(c)(d)(e)Mirroring or Rotation (a) horizontal mirroring(b) vertical mirroring (c) rotation by 90°(d) rotation by 180°(e) rotation by 270°(a)(b)(c)(d)(e)Slant TransformDST-IHadamard TransformDCT-IIConclusions  We showed sequence mirroring properties of orthogonal transformshaving even and odd symmetric vectors As applications, we flip images horizontally and/or vertically in the spatial domain by appropriately changing the signs of the transform coefficients.  Similarly rotation of the images are accomplished. This technique does not need any multiplications and only needs to change signs of the DST coefficients.References[1] K. R. Rao and P. Yip, The transform and data compression handbook.CRC Press, 2001.[2] W. K. Cham, “Development of integer cosine transforms by the principle of dyadic symmetry,” IEE Proc. I, Commu., Speech & Vision, vol. 136, pp. 276-282, Aug. 1989.[3] G. J. Sullivan, P. Topiwala and A. Luthra, “The H.264/AVC advanced video coding standard: overview and introduction to the fidelity range extensions”, in Proc. SPIE Conf. on Applications of Digital Image Processing XXVII, pp. 53-74, Aug. 2004.[4] S. Srinivasan et al., “Windows media video 9: overview and applications,” Signal Processing: Image Communication, vol. 19, pp. 851-875, Oct. 2004. [5] Y.-J. Chen and S. Oraintara, “Video compression using integer DCT,”in Proc. IEEE ICIP, pp. 844-845, Sept. 2000.References[6] W. Gao et al., “AVS – The Chinese next-generation video coding standard”, NAB, Las Vegas, Nevada, April 2004.[7] B. Shen and I. K. Sethi, “Inner-block operations on compressed images,” in Proc. of the Third ACM International Conference on Multimedia, pp. 489-498, San Francisco CA, 1995. [8] V. Britanak, P. Yip and K. R. Rao, Discrete cosine and sine transforms. Orlando, FL: Academic Press (Elsevier), 2007.[9] W. K. Pratt, W.-H. Chen and L. R. Welch, “Slant transform image coding,” IEEE Trans. Commun., vol. 22, pp. 1075-1093, Aug. 1974. [10] A. K. Jain, “Fundamentals of digital image processing,” Englewood Cliffs, NJ: Prentice Hall, 1989.[11] S.-K. Kwon, A. Tamhankar and K. R. Rao, “Overview of H.264/MPEG-4 part 10,” J. Vis. Commun. Image R., vol. 17, pp.