QUANTIZATION by Joseph RONSIN Outlnes Scalar quantzaton Dstorton Non unform quantzaton Vector quantzaton
Quantzaton 3 : Dscretsaton of color space One value for a set of values on an nterval Defnng number of ntervals Depends on dsplay (physcal factors) Depends on human vsual propertes (SVH) x Contnuous ampltude Quantzaton Y = Q(x) Dscrete values Quantzaton Topcs: Acquston Processng: reducton of number of grey levels or colors nsde orgnal mage Mnmal Dstorson 4 Interest Reducton of number of bts Dsplayng a pcture wth N bts on a dsplay wth M<N bts Quantzaton types Scalar (lnear or not) Vectoral
Type of mages Dfferent Quantzatons and correspondng types of mages bnary I(x,y) { 0,1 } black whte Monochrome or grey I(x,y) [ a, b ] often a = 0 et b = 55 black whte 5 color RVB I(x,y) = I r (x,y) I v (x,y) I b (x,y) Outlnes Scalar quantzaton 6 Dstorton Non unform quantzaton Vector quantzaton
Scalar Quantzaton : 7 Quantzaton levels q -1 q q +1 Output Qx d -1 d d +1 decson thresholds Input X Q ( x ) q d x 0 mn d x L max f d x d 1 wth 0,..., L 1 x x, x mn max Scalar Quantzaton Other representaton: Quantzaton Characterstc 8 Unform Quantzaton N reconstructon levels Q(x) Output q N/ q d d +1 d N/-1 X : Input
Impossble d affcher l mage. Dstorson Measure of dstorson: objectve crteron 9 For an orgnal mage MxN represented wth B bts dynamc of symbols : B -1 = L x,j and y, pxels of orgnal mage and quantzed mage MSE (Mean Square Error) 1 MSE MN M N 1 j1 x j y j PSNR (Peak Sgnal-to-Nose Rato) PSNR 0log 10 L MSE Non unform leads to ntervals of dfferent szes Adaptaton to dstrbuton of values to quantze Optmal Quantzer Objectve: Fnd best d and q hypothess: optmsaton crteron probablty densty p(x) Crteron of Mean Square Error Non unform Quantzaton Mnmzaton of MSE non unform probablty densty non lnear quantzaton 10
Impossble d affcher l mage. Non unform Quantzaton MSE Mnmzaton MAX Quantzer reconstructon levels: centroïds of areas defned by p(x) and decson regons d 1 d decson thresholds d : n the mddle of lmt values of ntervals Symetry wth 0 ( x q ) p( x) dx 0 0 q q d d d 1 et L 1,,..., 0 L 1,,..., 1 L q q 11 Outlnes Scalar quantzaton 1 Dstorton Non unform quantzaton Vector quantzaton
Vector Quantzaton Prncpe / example 4 color Image Symbols 00 / 01 / 11 / 10 4 color Image Image: Symbols 0/1 13 00 00 00 01 01 00 01 10 11 01 1 1 00 01 10 11 11 01 10 11 11 VECTOR QUANTIZATION Bnary Dctonnary Q = 1 0 0 1 Index choce for the nearest one of current regon Resultng number of bts Let: Image : matrx MxN I(x,y) [ L mn, L max ] Necessary number of bts for representaton of grey levels n L s K 14 So for scalar quantzaton: L = K Total number of bts: b = M x N x K Then for vector quantzaton: blocks m x n p blocks to code M : dctonary sze b = p x log (M)
Bblographe [1] Véronque Coat, Cours et supports de cours, INSA Rennes [] Max Mgnotte, "Tratement d'mages Introducton", support de cours, Unversté de Montréal [3] Grégory Bzarr, "Etude des mécansmes de dégradaton du lumnophore", thèse de doctorat, décembre 003 [4] Sofane Laran, "Percepton et nterprétaton de sectons et blocs ssmques: oculométre et analyse d'mages", Thèse de l'ujf, Mathématques Applquées, Grenoble, 4 Octobre 000. [5] http://www.chusa.jusseu.fr/pedagoge/pcem1/bophysque/opt_phys_a_005.pdf [6] Adelson, «E.H. Lghtness Percepton and Lghtness Illusons». In The New Cogntve Neuroscences, nd ed., M. Gazzanga, ed. Cambrdge, MA: MIT Press, pp. 339-351, (000). [7] Perre Kornprobst, Cours et supports de cours, INRIA [8] M. Burel, C. Obert, «DICOM Quantfcaton vectorelle», 005 [9] Perre MATHIEU, Cours, DEA ARAVIS, Polytech Nce-Sopha 15 MAX Quantzer Decson thresholds and reconstructon Levels for MAX s quantzer Unforme Gaussen Laplacen Raylegh 16 bts d r d r d r d r 1-1.0000-0.5000 - -0.7979 - -0.7071 0.0000 1.657 0.0000 0.5000 0.0000 0.7979 0.0000 0.7071.0985.9313 1.0000 - - - -1.0000-0.7500 - -1.5104 - -1.8340 0.0000 0.8079-0.5000-0.500-0.9816-0.458-1.169-0.4198 1.545 1.7010-0.0000 0.500 0.0000 0.458 0.0000 0.4198.1667.635 0.5000 0.7500 0.9816 1.5104 1.169 1.8340 3.465 3.8604 1.0000 3-1.0000-0.8750 - -.1519 - -3.0867 0.0000 0.5016-0.7500-0.650-1.7479-1.3439 -.3796-1.675 0.7619 1.0-0.5000-0.3750-1.0500-0.7560-1.57-0.8330 1.594 1.4966-0.500-0.150-0.5005-0.451-0.533-0.334 1.737 1.9688 0.0000 0.150 0.0000 0.451 0.0000 0.334.18.4675 0.500 0.3750 0.5005 0.7560 0.533 0.8330.7476 3.077 0.5000 0.650 1.0500 1.3439 1.57 1.675 3.3707 3.7137 0.7500 0.8750 1.7479.1519.3796 3.0867 4.14 4.7111 1.0000
MAX Quantzer 17 Unforme Gaussen Laplacen Raylegh bts d r d r d r d r 4-1.0000-0.9375 - -.736 - -4.4311 0.0000 0.3057-0.8750-0.815 -.4008 -.0690-3.740-3.0169 0.4606 0.6156-0.7500-0.6875-1.8435-1.6180 -.5971 -.1773 0.7509 0.8863-0.650-0.565-1.4371-1.56-1.8776-1.5778 1.0130 1.1397-0.3750-0.315-0.7995-0.6568-0.9198-0.787 1.5064 1.677-0.500-0.1875-0.54-0.3880-0.5667-0.4048 1.7499 1.871-0.150-0.065-0.58-0.184-0.664-0.140 1.9970.10 0.0000 0.065 0.0000 0.184 0.0000 0.140.517.3814 0.150 0.1875 0.58 0.3880 0.644 0.4048.518.6550 0.500 0.315 0.54 0.6568 0.5667 0.757.801.949 0.3750 0.4375 0.7995 0.943 0.9198 1.1110 3.1110 3.779 0.5000 0.565 1.0993 1.56 1.3444 1.5778 3.4566 3.6403 0.650 0.6875 1.4371 1.6180 1.8776.1773 3.8588 4.077 0.7500 0.815 1.8435.0690.971 3.0169 4.3579 4.6385 0.8750 0.9375.4008.736 3.740 4.4311 5.0649 5.4913 1.0000 ² 18 END