Lossy coding. Lossless coding. Organization: 12 h Lectures + 8 h Labworks. Hybrid coding. Introduction: generalities, elements of information theory

Transcription

1 Organization: h Letures + 8 h Labworks Lossless oding RLC and Huffman oding Introdution: generalities, elements of information theory Lossy oding Salar and Vetorial oding Arithmeti oding and ditionary oding Hybrid oding Standards: JPEG, MPEG, H64 Transform-based oding: DCT, WT Image Quality

2 TRASFORM BASED Introdution Input : Output : y - Eah y i represents the similarity between and line A i,. - Reovery apaity: (Matri A the reeiver) Code Deode

3 TRASFORM BASED Basi priniple Coding : Transform Input data 0,,, into oeffiients 0,,.., - : - Most oeffiients are zero. The energy is ontained in a redued number of oeffiients. ) Salar Quantizations of oeffiients (bit alloation) : - The most signifiant oeffiients need an important number of levels Entropi oding of quantized values Deoding : ) Entropi deoding Evaluate oeffiients from quantized values Inverse transform

4 TRASFORM BASED Basi Priniple

5 TRASFORM BASED Linearity Orthonormality : Consequene : energy onservation : 0 0,,0 0, 0,0 a a a a T A A 0,,0 0 0, 0,0... a a a a A T 0 0 ) ( ) )( ( ) ( ) ( i i T T T T T T i T i A A A A A A Basis vetors oeffiients

6 From a statisti point of view TRASFORM BASED Correlated Gaussian Proess After de-orrelation Orthogonality : A T =A - A Compression property: b b b 0

7 Transform based orrelation gain TRASFORM BASED G TC i i Q : How to hoose the transformation matri? R : From the Input R s orrelation matri A A i i /,: vetor higly orrelated with signal s,: vetor higly orrelated with signal s i C i oeffiients variation number of oeffiients and linear independent with respet to A(,:) Et

8 TRASFORM BASED Karhunen-Loève Transform(KLT) optimal provides a better gain - KLT matri formulated by the eigen vetors of the proess s R X autoorrelation matri A v v T T The eigen vetors of proess X are the solutions of the equation system: ivi R Xv where R X is the autoorrelation matri of X and i are the eigen values of matri R X Drawbaks: - Highly dependent of the Input data s statistis - Comple omputation

9 TRASFORM BASED Comparison

10 TRASFORM BASED Comparison

11 TRASFORM BASED Separable transforms of interest for images beause the omputation ompleity is redued from O( 4 ) to O( 3 ) Coeffiients () Orthonormal transformation matri () Input Signal () Input Signal () Coeffiients ()

12 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) DCT oeffiients matri definition a ik (k ) i i os, i, k 0,..., 0 0, i 0

13 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) DCT (88) oeffiients histograms for natural images CC Coeffiient uniform distribution CA Coeffiient Laplae distribution

14 TRASFORM BASED Energy distribution of the DCT oeffiients DISCRETE COSIE TRASFORM (DCT)

15 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) Eample The number of retained oeffiients

16 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) Problem : DCT oeffiients bit alloation Genral solution: hoose the bits/oeffiients so that the error s variane is minimum Find R k minimizing: with k Q Q k k k Q R Q R R k k, i X X k i k R R log

17 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) DCT bit alloation pratial algorithm Rk Q k X k - For eah alloated bit, the variane 4 times lower Calulate the variane for eah oeffiient C i. Make R k =0 for k=.. R k = R k + redue variane 4 times If all bits are alloated, stop. Otherwise, go to step 3 55 bits for 64 piels =.86 bits/piel Eample DCT

18 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) DCT oeffiients quantization for a nn oeffiients zone we onstrut a Quantization matri Q(nn) representing the size of the quantization interval Q ij for eah oeffiient Thee oeffiient ij will be oded as : Deoding s ij ' s ij ij 0. 5 Qij ij Q ij Eamples 54., Q 4, 54. s 0.5, ' , Q 6, 54. s 0.5 9, ' , Q, 54. s 0.5 9, ' 5 60

19 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) Inrease the bits/oeffiient number divide by the interval size Redue the bits/oeffiient multiply by the interval size San in zig-zag ode the oeffiients in desending order of their amplitudes

20 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT) Eample Starting image (9 Kb) Image with FFT (Fq= 3, Size = Kb) Image with DCT(Fq= 3, Size = 5 Kb)

21 TRASFORM BASED DISCRETE COSIE TRASFORM (DCT)