Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March -4, 7 67 Sesitivity Aalysis of Daubechies 4 Wavelet Coefficiets for Reductio of Recostructed Image Error DEVINDER KAUR EECS, Uiversity of Toledo OH 4366 USA PAT MARSHALL 4 Avioics Circle WPAFB, OH 45433-7334 USA Abstract: Quatizatio of compressed image file reduces the requiremets for memory storage ad the trasmissio badwidth. However, the cost of quatizatio is loss of iformatio. O the receiver ed whe image is recostructed from lossy de-quatized file usig the wavelet recostructio coefficiets, it is impossible to create a perfect image exactly like the origial image. This paper records the sesitivity aalysis of wavelet coefficiets to determie which coefficiets cotribute to the image recostructio ad i what way ad how they should be chaged to recostruct a perfect image. Some iterestig fidigs have bee recorded i the coclusio sectio. Key Words: - Daubechies 4 Wavelet Coefficiets, Image Recostructio, Mea Squared Error, Sesitivity Aalysis. Itroductio Raw image data requires large amout of storage space ad i order to reduce the memory requiremets may compressio algorithms have bee developed. Historically the discrete cosie trasform (DCT) used i the JPEG compressio stadard was used. However, with the adoptio of Joit photographic Experts Group s JPEG stadard for still image compressio, which is based o discrete wavelet trasforms, wavelets have become the most popular trasforms for image compressio. Fig. illustrates the image compressio ad recostructio with wavelet coefficiets. C Cq DWT Q Ecoder Recostructed Image Cq DWT - Q - Decoder C: Compressed Image Cq: Quatized Image Cq: de-quatized image Fig. : Image Compressio with Discrete Wavelet Trasforms
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March -4, 7 68 Wavelets trasform cotiuous or discrete time domai sigals ito frequecy domai. DWT covolves the sigal agaist specific wavelet istaces at various time scales ad positios resultig i a compressed represetatio of the origial image. The compressio is reversed by applyig the Iverse discrete wavelet trasform DWT -, which covolves the sigal agaist a iverted order of the origial wavelet istaces to produce a approximatio of the origial sigal. Wavelets coserve eergy ad redistribute most of the eergy to the first tred sub- sigal. The eergy outside of the first tred sigal is isigificat ad ca be elimiated without much sigificat loss of iformatio providig a favorable compressio rate at the expese of perfect recostructio. Fig. illustrates the eergy distributio i a compressed sigal obtaied by first level of discrete wavelet trasform of the AF museum. Two mai compoets of DWT are the scalig fuctio φ(t), ad the wavelet fuctio ψ(t), which are defied as follows: h g ψ (t) = g ( t ) ψ () Where is the impulse respose of the scalig filter ad is the impulse respose of the wavelet filters. Moreover, h cotais the set of filter coefficiets correspodig to the projectio of the basis fuctios for low pass filterig sectio of the DWT, ad g cotais filter coefficiets correspodig to the projectio of the basis fuctios for high filterig sectio of the DWT. Oce trasformed, the aalysis of sigal x(t) results i discrete sets of data i the wavelet domai. The iverse DWT - is used to trasform coefficiets from the wavelet domai back ito the origial sigal domai. Thus the iverse trasform produces the origial sigal x(t) from the wavelet ad scalig coefficiets. ( t) = k, k= = x d k ( t) Where, ψ (3) ( ) φ (t) = h φ t () d + k, ψ k, = ( t) x ()dt t x(t) (4) 6 Compressed sigal c for AF Museum 5 4 3 -.5.5 3 x 5 Fig. a: Origial Satellite AF Museum Fig. b: Compressed Sigal of AF Museum after oe level of DWT
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March -4, 7 69 For Daubehies 4 wavelet coefficiets have the followig values: (Lo_D) h = {-.94,.4,.8365,.489} Lo_D (:4) i Matlab (Hi_D) g = {-.483,.8365, -.4, -.94} Hi_D(:4) i Matlab (Lo_R) h = {.483,.8365,.4, -.94} Lo_R(:4) i Matlab (Hi_R) g = {-.94, -.4,.8365,.483} Hi_ R(:4) i Matlab h is the set of wavelet umbers for the forward discrete wavelet trasforms (DWT). g is the set of scalig umbers for the DWT. h is the set of wavelet umbers for the iverse DWT (DWT - ). g is the set of scalig umbers for the (DWT - ). A two-dimesioal D DWT of a discrete iput image f with M rows ad N colums (M ad N beig eve) is computed by first applyig the oe-dimesioal (D) trasform defied by the coefficiets from set h ad g to the colums of f, ad the applyig the same trasform to the rows of the resultig sigal []. Similarly, D DWT - is performed by applyig the D DWT - defied by sets h ad g first to the rows ad the to the colums of a previously compressed sigal. A oe-level DWT decomposes f ito M/ by N/ sub-images h, d, a ad v, where a is the tred sub mage where most of the eergy of the sigal is cocetrated. h, d, ad v are its first horizotal, diagoal, ad vertical fluctuatio subimages, respectively. Oe-level DWT may be repeated k log (mi (M, N)) times. The size of the tred sigal a i at level i of decompositio is /4 i times the size of the origial image f (e.g., a three-level trasform produces a tred sub image a 3 that is /64 th the size of f). Nevertheless, the tred sub image will typically be much larger tha ay of the fluctuatio sub images; for this reaso, the MRA scheme computes a k-level DWT by recursively applyig a oe-level DWT to the rows ad colums of the discrete tred sigal a k-. Similarly, a oe-level DWT - is applied k times to recostruct a approximatio of the origial M-by-N sigal f. [] : Impact of Recostructio Coefficiets o the Recostructed Image The wavelet recostructio coefficiets based o Daubechies 4 wavelets (db i Matlab) are used for the recostructio of compressed images. The image recostructed with these wavelets has certai amout of error expressed i mse (mea squared error). However, there is o kowledge base available which tells i what way each of the eight recostructio wavelet coefficiets Lo_R (:4) ad Hi_R (:4) impact the recostructio of the image. Each time the coefficiets are evolved they evolve to some umbers which deped o the image. For each image the evolved coefficiets are differet but we do ot uderstad the relatioship betwee the properties of the image ad the evolved coefficiets. This study is a step i that directio. Approach to the Problem: I order to see the impact of each coefficiet oly oe parameter was varied ad all the other coefficiets were fixed to the values obtaied as the wavelet coefficiets. The followig wfilters fuctio o the wavelet toolbox gives the coefficiets: [Lo_D,Hi_D,Lo_R,Hi_R] =filters (wave_type); Each of the recostructio coefficiets is varied from - to i steps of. ad image is recostructed usig each of these values. The mea squared error of the recostructed image was computed for each step value of the parameter. The value which gives the miimum mse is chose for each coefficiet ad compared with stadard wavelet coefficiet. The first ru of the study was doe o the.bmp image. Later it was exteded to other images. Fig. 3a illustrates the best value (the value for which the error is miimum) for the Lo_R (Low Frequecy Wavelet Recostructio Coefficiets). Fig. 3b illustrates the best value for Hi_R (High Frequecy Wavelet Recostructio Coefficiets).
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March -4, 7 7.5 x 4.5 x 4.5.5 Wavelet recostructio coefficiet.5 x 4 Figure: Lo_R() Best Value Aalysis - - -.5.5 Wavelet recostructio coefficiet Figure: Lo_R() Best Value Aalysis.5 x 4.5.5 Wavelet recostructio coefficiet 3 Wavelet recostructio coefficiet 4 Figure: Lo_R(3) Best Value Aalysis Figure: Lo_R(4) Best Value Aalysis Fig. 3a: The Best Value Aalysis of Iverse Low Frequecy Wavelet Coefficiet The experimets were repeated with more images to get reasoable data to arrive at some coclusio. Images i two differet categories were aalyzed. Some images fall i the category of satellite ad others are called o-satellite. It was observed from these experimets that the low frequecy coefficiets were very close to the stadard wavelet coefficiets which have the fixed value of: Lo_R(:4):.483.8365.4 -.94 The last two digits are always as i.xx. These discrepacies are because the step size was chose to be.. If the step size chose was. the we ca get a exact value up to the four decimal places, however, the computatio time would icrease times. The best values obtaied for the Hi_R coefficiets were differet from the fixed oe level wavelet coefficiets. The stadard Hi_R coefficiets are: Hi_R(:4) -.94 -.4.8365 -.483 Sice these recostructio coefficiets work o compressed ad de-quatized image which has lost iformatio because of quatizatio, these Hi_R wavelet coefficiets have to adapt i order to recostruct image which is as close to the origial image.
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March -4, 7 7 5 7 45 4 35 3 6 5 4 5 3 5 Wavelet recostructio coefficiet 5 7 6 5 4 3 Hi_R() Best Value Aalysis Wavelet recostructio coefficiet 7 Hi_R(3) Best Value Aalysis Wavelet recostructio coefficiet 7 55 5 45 4 35 3 5 5 Hi_R() Best Value Aalysis Wavelet recostructio coefficiet 8 Hi_R(4) Best Value Aalysis Fig. 3b: The Best Value Aalysis of Iverse High Frequecy Wavelet Coefficiets If the best value for each recostructio wavelet coefficiet Lo_R (:4) ad Hi_R (:4) is picked for recostructig the image, the overall quality of the image worses as is show i Fig. 4a of Fruits where the mse of the best value coefficiet became 4.453 i compariso to the mse of 3.39 for the image costructed with wavelet coefficiets. However, if oly those coefficiets for the best value are picked for which the error was less tha the wavelet coefficiets the there is small improvemet i the image over the image costructed with the wavelet coefficiets. The followig Fig. 4b shows that the mse of the recostructed image reduced to 8.66 whe the best values for Hi_R (:4) were picked ad the rest were left with the stadard wavelet coefficiets. The above experimets show that the recostructio wavelet coefficiets are ot idepedet. The best value for each recostructio parameter is best oly whe the other seve were fixed. Whe the other seve parameters are chaged best value for a particular parameter does ot remai best. This shows that all the parameters should be take i uiso for fidig the best value which will miimize the error of the recostructed image.
Proceedigs of the 6th WSEAS Iteratioal Coferece o SIGNAL PROCESSING, Dallas, Texas, USA, March -4, 7 7 Fig. 4a: Image Recostructio with best value for Lo_R(:4) ad Hi_R (:4) Parameters Fig. 4b: Image Recostructio with best values from Hi_R(:4) 3. Coclusio The experimets i this study coclude that wavelet coefficiets are ot idepedet of each other. The best value for Recostructio Coefficiets Lo_R ad Hi_R deviate from the origial discrete wavelet coefficiets substatially. This is because the recostructio coefficiets work o the quatized compressed image to recostruct the image. There is loss of iformatio i the quatized compressed image. Therefore i order to miimize the error i the recostructed image the best value recostructio parameters have to deviate from the stadard costat values of discrete wavelet coefficiets. Hi_R coefficiets deviate the most ad that explais why the error i the recostructed image is mostly cofied aroud the edges. The techiques to reduce the error i the recostructed image should focus o fidig the best value of all the recostructio coefficiets take together ad ot take idividually. Refereces: [] Taubma, D. ad M. Marcelli, JPEG: Image Compressio Fudametals, Stadards, ad Practice, Kluwer Academic Publishers,. [] I. Daubechies, Te Lectures o Wavelets, SAIM, 99. [3] B. E. Usevitch. A tutorial o moder lossy wavelet image compressio: foudatios of jpeg. IEEE Sigal Processig Magazie, pages 35, September. [4] C. Christopoulos, A. Skodras, ad T. Ebrahimi, The JPEG still image codig system: a overview, IEEE Trasactios o Cosume Electroics, Vol. 46, No. 4, pp. 3 7, Nov.. [5] G. Davis ad A. Nosratiia, Wavelet-based Image Codig: A Overview, Applied ad Computatioal Cotrol, Sigals, ad Circuits, Vol., No., 998. [6] MATLAB Geetic Algorithm ad Direct SearchToolbox.http://www.mathworks.com/ access/helpdesk/help/toolbox/gads/, 5. [7] J. Walker, A Primer o Wavelets ad Their Scietific Applicatios, CRC Press, 999. [8] C. Christopoulos, A. Skodras, ad T. Ebrahimi, The JPEG still image codig system: a overview, IEEE Trasactios o Cosume Electroics, Vol. 46, No. 4, pp. 3 7, Nov.. Ackowledgemet: This work was supported by Visitig Faculty Research Grat from Air Force Research Laboratory at Wright Patterso Base i Dayto, Ohio, USA, durig summer of 6.