EEE TRANSACTIONS ON IMAGE PROCESSING. VOL 1. NO. 2. APRIL 192 Fig. 5 represents one stage in a multiscale pyramidal In Fig. 8(a)we can see the normalized detail subimages decomposition of an image: wavelet coefficients of the at different resolution levels m=1, m=2, and m =3 Image are computed, as in the one-dimensional case(Sec- (wavelet coefficients)and in Fig. 8(b)the low resolution tions I-A and II-B. 1), using a subband coding algorithm. level subimages The filters h and g are one-dimensional filters. This de- composition provides subimages corresponding to differ- III. IMAGE CODING APPLICATI ent resolution levels and orientations(see Fig. 6). The A. Statistical Properties of Wavelet Coefficients reconstruction scheme of the image is presented Fig. 7 To compare the three different filters presented in this and direction can be determined by the statistics of the each of these filters. The results are presented in Fig. 8. function (PDF/ abimage, i.e., its probability density have decomposed the image Lena(Fig. 16)with corresponding
ANTONINI er al.: IMAGE CODING USING WAVELET TRANSFORM ow resolution Horiranto Resolution m-2 Resolution m-2orientation sub-image sub amage Resolution m=l Vertical Diagonal orientation sub-imageorientation sub-image Fig. 6. Image decomposition COLUMNS 2} [ 21 Console with alter x Dnt column e A typical PDF and different approximations are .Im.d 2 leads to the well-known Gaussian PDF in Fig. 9, where we plot the true pDF for resolution I leads to a Laplacian PDF m=I and direction d vertical together with three model functions: a Gaussian, a Laplacian, and an intermediate The variance of this approximation model is set equal function, the so-called generalized Gaussian [2] to the variance of the corresponding subimage. Thus the This generalized Gaussian law is given explicitly by parameter rm, d is computed in order to match the real PDF Pm. d(r)=am. d exp(-bm,dx rm.d) using the well-known chi-squared test. In this case the optimum parameter was 0.7. Other experiments for other resolutions (except the lowest resolution) lead to very 3 Fig 9 that the real PDF(scale dm. d and vertical orientation) is closely eralized Gaussian law with parame B. Encoding of Wavelet Coefficients Using Vector where om, d is the standard deviation of the subimage Different techniques involving vector or scalar quanti- (m, d), and r( )is the usual Gamma function zation can be used to encode wavelet coefficients The general formula(13)contains the other two ex- According to Shannons rate distortion theory, better amples as particular cases: sults are al ways obtained when vectors rather than sca