See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2423281 Video Compression using the Three Dimensional Discrete Cosine Transform (3D-DCT) Article · August 1997 DOI: 10.1109/COMSIG.1997.629976 · Source: CiteSeer CITATIONS READS 33 784 2 authors, including: G. de Jager University of Cape Town 110 PUBLICATIONS 1,128 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: UCT Digital Image Processing Masters Studies View project All content following this page was uploaded by G. de Jager on 08 September 2015. The user has requested enhancement of the downloaded file. Video Compression using the Three Dimensional Discrete Cosine TransformDDCT Marc Servais Gerhard De Jager Member IEEE Abstract Sequences of digital video images typically re timation p erformed on each frame These blo cks are quire vast amounts of electronic memory for storage and generated from noninterlaced High Denition Television o ccupy much bandwidth during transmission Widely used HDTV frames image and video compression standards such as JPEG and The three dimensional DCT describ ed in this pap er MPEG use the twodimensional Discrete Cosine Transform DCT to achieve nearoptimal compression of individual utilises the high degree of temp oral correlation b etween frames This is done by decomp osing the the frames into successive frames in a video sequence In contrast to mo comp onents of dierent spatial frequencies This pap er tion vector implementations of interframe compression presents an extension of the DCT to the temporal dimen sion It describ es the implementation of a software based p erforming a D DCT involves using the same technique compression scheme which involves the computation of a in all three dimensions horizontal vertical and temp oral three dimensional DCT of successive groups of eight frames This compression scheme transforms the eight image frames I I Definition into eight DCT frames with comp onents of b oth spatial and temp oral frequencies The compression scheme was im The twodimensional Discrete Cosine Transform DCT plemented and tested with standard MPEG video test se of an N xN blo ck of pixels and the Inverse Discrete Co R C quences It showed compression p erformance comparable to sine Transform IDCT are dened by Rao and Yip as various MPEG implementations Keywords Discrete Cosine Transform DCT Three di Forward D DCT mensional D Video compression Transform co ding I Introduction N 1 N 1 R C X X S v u v u sy x cos t cos t 2D 1 2 RANSFORMS and in particular integral transforms y =0 x=0 are used primarily for the reduction of complexity in T mathematical problems The Fourier Transform and the Inverse D DCT KarhunenLo eve Transform KLT which decorrelates a signal sequence are wellknown examples in the digital sig N 1 N 1 R C nal pro cessing area X X sy x v uS v u cos t cos t 2D 1 2 The KLT is a series representation of a given random v =0 u=0 function This transform is optimal in that it completely decorrelates the random function ie the signal sequence where in the transform domain and pro duces uncorrelated co e y v x u cients Decorrelation of the co ecients is very imp ortant t t 2 1 N N C R for compression b ecause each co ecient can b e treated in dep endently without loss of compression eciency r The Discrete Cosine Transform DCT was rst applied v u C v C u 2D to image compression by Ahmed Natara jan and Rao in N N R C They showed that this particular transform was very and 1 close to the KLT for natural images The two dimen p k 2 C k sional DCT has b een applied to b oth image compression otherwise and intraframe compression of video frames sy x is a pixel value within the N xN image blo ck R C Rao and Yip rep ort several implementations of the three and S v u a DCT co ecient in the corresp onding N xN R C dimensional DCT This compression technique has b een DCT blo ck applied to multisp ectral scanner data based on xx If in addition to the two spatial dimensions one consid cub es comp osed of x blo cks from each of the four sp ectral ers the dimension of time then the N xN blo ck referred R C bands A compression ratio of was achieved 1 to ab ove can b e extended to an N xN xN cub e There F R C Probably the most wellknown application is the D DCT of threedimensional blo cks displaced by motion es 1 Technically cub e would only b e correct for the case N N F R N A more general threedimensional term such as blo ck may C G De Jager is with the Digital Image Pro cessing Group De b e considered preferable However blo ck is commonly used to partment of Electrical Engineering University of Cap e Town South refer to part of a frame as in macroblo ck Consequently the Africa Email gdjelecenguctacza term cub e although not entirely general will b e used to refer to a threedimensional array of either pixels or DCT co ecients with M Servais is a Masters student at the University of Cap e Town N frames N rows and N columns Email marcsdipeeuctacza F R C are then N successive frames of an N xN blo ck forming F R C a cub e in DDCT space The three dimensional DCT and IDCT are dened as Forward D DCT N 1 N 1 N 1 F R C X X X S w v u w v u f 3D z =0 y =0 x=0 sz y x cos t cos t cos t g 1 2 3 Inverse D DCT N 1 N 1 N 1 F R C X X X sz y x f w v u 3D w =0 v =0 u=0 S w v u cos t cos t cos t g 1 2 3 where x u y v z w t t t 1 2 3 N N N C R F r w v u C w C v C u 3D N N N F R C and 1 p k 2 C k otherwise sz y x is a pixel value in one of the N image frames F z N and S w v u is a DCT co ecient in the F corresp onding N xN xN DCT cub e The N xN xN F R C F R C DCT cub e then contains information regarding each of the N image frames An example D DCT is illustrated b e F low Eight successive frames taken from a video clip are transformed pro ducing a corresp onding eight frames in the DCT domain as depicted in Figures and resp ectively Note that in this example altogether cub es are trans 2 formed into the DCT domain In the ab ove example it is apparent that DCT frame contains much of the information in the DCT cub e while the opp osite is true for DCT frame Most of the infor mation in DCT frames to is contained in the area of motion ie the b ottom right hand corner It will b e shown that DCT frame can b e thought of as the DC Frame conveying the information common to each of the image frames The other DCT Frames all convey AC information which corresp onds to motion in the original image sequence I I I Properties A Separability Fig A sequence of eight images Note the motion of the arm in the b ottom right The most straightforward metho d of implementing the D DCT or IDCT is to follow the theoretical equations Equations and For an xx cub e this corresp onds 2 The image size is x to multiplication and addition op erations p er co ecient However the DCT is a separable transform This im plies that a multidimensional DCT may b e implemented as a series of onedimensional discrete cosine transforms Numerous fast algorithms for implementing the DCT and IDCT in b oth one and two dimensions exist Thus for example a fast DDCT could b e implemented on the rows and columns of each of the N frames This could b e F followed by a fast DDCT along the time axis ie corre sp onding pixels in each of the frames B Relationship to the DDCT The Deltamodication process In this section the relationship b etween the three dimen sional DCT of an image sequence and the two dimensional DCT of the rst frame of the same sequence is examined Consider N frames of a N xN image blo ck F R C Let frame b e DDiscreteCosineTransformed to pro duce one frame with DDCT co ecients S v u F r ame0 Next let all N frames ie frame to frame N F F b e DDiscreteCosineTransformed to pro duce a cub e of N frames with DDCT co ecients S w v u The co F ecients S v u can b e calculated from Equation by setting w This gives N 1 N 1 N 1 F R C X X X p v u f S v u 2D N F z =0 y =0 x=0 sz y x cos t cos t g 1 2 where sz x y corresp onds to a pixel value within the original image sequence Then letting sz y x s y x z y x the op erations on s y x and z y x can b e considered separately Thus S v u can b e decomp osed into two comp onents a term prop ortional to S v u and F r ame0 a term prop ortional to the amount of motion present in 3 the N image frames relative to the rst frame F Accordingly this gives p N S v u S v u S v u F F r ame0 where N 1 N 1 N 1 F R C X X X p S v u v u f 2D N F z =0 y =0 x=0 z y x cos t cos t g 1 2 Fig The D DCT of the image frames in Figure Grey rep The ab ove equations are illustrated with an example resents zerovalued co ecients white represents p ositivevalued co ecients and black represents negativevalued co ecients Using the eight image frames shown in Figure as the in put sequence S v u and S v u were calculated The 3 Note that z x y indicates a pixel value relative to the corre sp onding pixel in the rst frame of the N frame sequence F resulting DCT co ecients are represented graphically in Figure demonstrates an interesting phenomenon the 4 Figures and resp ectively entropy of DCT frame has b een further reduced Notice that no information has b een lost since the original D For regions of low movement within the N frames com F DCT co ecients of frame can b e determined as shown in prising the image sequence it is evident that the mo di Equation ed co ecients S v u are clustered more