STATISTICAL DISTRIBUTIONS OF DISCRETE WALSH HADAMARD TRANSFORM COEFFICIENTS OF NATURAL IMAGES

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1 STATISTICAL DISTRIBUTIOS OF DISCRETE WALSH HADAMARD TRASFORM COEFFICIETS OF ATURAL IMAGES Vjay Kumar ath and Deepka Hazarka Department o Electroncs and Communcaton Engneerng, School o Engneerng, Tezpur Unversty (A Central Unversty), apaam, Tezpur, Assam, Inda. vknath@tezu.ernet.n and deepka@tezu.ernet.n ABSTRACT For low bt rate applcatons, the dscrete Walsh Hadamard transorm (WHT) shows almost comparable results when compared to the popular dscrete Cosne transorm (DCT) n terms o compresson ecency, peak sgnal to nose rato (PSR) and vsual results. The dscrete WHT s a best choce whch compromses between the computatonal complexty and compresson ecency. The great advantage o the dscrete WHT s ts relatvely very low computatonal complexty when compared to DCT. However there s no dentve study reported n lterature regardng the statstcal dstrbutons o dscrete WHT coecents o natural mages. Ths study perorms a χ goodness o t test to determne the dstrbuton that best ts the dscrete WHT coecents. The smulaton results show that the dstrbuton o a majorty o the sgncant AC coecents can be modelled by the Generalzed Gaussan dstrbuton. The knowledge o the approprate dstrbuton helps n desgn o optmal quantzers that may lead to mnmum dstorton and hence acheve optmal codng ecency. KEYWORDS Image Compresson, Dscrete WHT, χ test, Statstcal analyss. ITRODUCTIO DCT [] has been very popularly used n many mage and vdeo codng standards lke JPEG, MPEG, MPEG, H.6 [,3]. But the computatonal complexty o the DCT s relatvely very hgh whch may not be preerred n several real tme applcatons. The dscrete WHT [4,5,6,8] s a orthogonal non snusodal transorm whch shows almost comparable perormance n mage compresson applcatons when compared to DCT. The dscrete WHT s popular or ts very low computatonal complexty. The dscrete WHT s consdered as the best opton whenever there s a compromse between the energy compacton capablty and computatonal complexty. However tll now, no study has been reported n the lterature, on the dstrbuton o dscrete WHT coecents o the natural mages. In contrast, there s a huge amount o studes reported n the lterature dealng wth the dstrbutons o D DCT coecents o natural mages [7,8,9,0]. In [7], consderng Gaussan, Laplacan, Gamma and Raylegh dstrbutons as probable models, Rennger and Gbson used Kolomogrov-Smrnov (KS) test and shown that DC coecents can be well approxmated by Gaussan dstrbuton and AC coecents can be well approxmated by Laplacan dstrbuton. In [8], the authors concluded that no partcular densty uncton can be used or each o the coecents but Laplacan ts majorty o the coecents and when all DOI : 0.5/spj

2 the coecents are lumped nto one densty uncton, the Cauchy dstrbuton provdes the best t. In [9], Muller ound that the Generalzed Gaussan dstrbuton best approxmates the statstcs o the D DCT coecents. Smoot and Reeve [9] study the statstcs o the DCT coecents o the derental sgnal obtaned ater moton estmaton. They observe that the statstcs are best approxmated by the Laplacan dstrbuton. In ths paper, consderng Gaussan, Laplacan, Gamma, Generalzed Gaussan and Cauchy dstrbutons as probable models [,], we study the statstcal dstrbuton usng χ Goodness o Ft test [3]. The model dstrbuton that gves the mnmum χ statstc s chosen as the best t. The χ test o t showed that Generalzed Gaussan dstrbuton best models the statstcs o dscrete WHT coecents o natural mages. These results can be used to desgn optmal quantzers or dscrete WHT coecents. The organzaton o the paper s as ollows. Secton provdes the ntroducton to dscrete WHT. χ Goodness o Ft test s explaned n Secton 3. Secton 4 descrbes the model statstcal dstrbutons. Expermental results are presented n Secton 5 and the paper s concluded n Secton 6.. DISCRETE WALSH HADAMARD TRASFORM The bass unctons o dscrete WHT [4] are not snusods unlke Fourer and Cosne transorms. The bass unctons are based on rectangular waveorms wth peaks o ±. The -pont - D dscrete Walsh Hadamard transorm o a sgnal x( n) s gven as V ( z) = x( m) yz ( m), z = 0,,..., () m= 0 where y ( m) s the Walsh uncton o order z and s dened recursvely as: z or z = 0, χ GOODESS OF FIT TEST yz ( m ) = 0 or m < 0 and m > y ( ) 0 m = or m = 0,..., z y ( m) = y ( m) + ( ) y (m ) z z z z z z z y + ( m) = y ( m) ( ) y (m ) χ and KS test [3] are the wdely used goodness o t test. The KS goodness o t test statstc s a dstance measure between the emprcal cumulatve dstrbuton uncton (CDF) or a gven data set and the gven model cumulatve dstrbuton uncton. In contrast to the KS test, the χ goodness o t test compares the model probablty densty unctons wth the emprcal data and nds out the dstorton by the ollowng equaton k e ( O E ) χ = () E where the range o data s parttoned nto k e dsjont and exhaustve bns B, =,,3,..., ke. E = nc p s the expected requency n bn B where p = P( x B ) and O s the observed requency n bn B. nc s the total number o data samples. The model probablty densty uncton whch gves the mnmum χ value can be consdered as the best t. The χ test s preerred over the KS test because or source codng the devaton rom an assumed probablty densty uncton s more nterestng than devaton rom a dstrbuton uncton [9]. 76

3 4. PROBABILITY DISTRIBUTIOS We use χ goodness o t tests consderng Gaussan, Laplacan, Gammaauchy and Generalzed Gaussan dstrbutons [,] as probable models as these dstrbutons are commonly used or statstcal modelng o DCT coecents. The parameters o the dstrbutons were ound usng the maxmum lkelhood (ML) method. 4.. Gaussan Probablty Densty Functon The Gaussan probablty densty uncton s gven by ( x µ ) ( x) = exp πσ σ where µ s the mean and σ s the varance. The ML estmates o µ and σ are gven by n (3) µ = x (4) n n σ = ( x µ ) (5) n 4.. Laplacan Probablty Densty Functon The Laplacan probablty densty uncton s gven by x µ ( x) = exp a a where µ s the mean and a s the scale parameter. The varance s gven by estmate o the parameter a s gven by where µ s estmated usng (4). (6) a. The ML a$ = x µ (7) 4.3. Gamma Probablty Densty Functon The Gamma probablty densty uncton s gven by [4] x µ ( x) = exp 8πσ x µ σ The parameters µ and σ are estmated usng (4) and (5) respectvely Generalzed Gaussan Probablty Densty Functon The Generalzed Gaussan probablty densty uncton [,,5] s gven by where (.) x ( x) = exp α αγ Γ s the Gamma uncton gven by t z ( ) 0 Γ z = e t dt, z > 0 and α, respectvely are known as the scale parameter and the shape parameter. For the specal cases = or =, the generalzed Gaussan pd becomes a Gaussan or a Laplacan pd respectvely. The ML estmaton o the parameters α and can be obtaned as ollows: (9) (8) 77

4 The shape parameter s the soluton o the equaton where (.) ψ x log log x x = + + = 0 x ψ s the dgamma uncton gven by ψ ( z) Γ = Γ ( z) ( z) The ML estmate oα (or known ML estmate ) s gven by where s number o observatons. (0) α = x () The s determned usng the ewton Raphson teratve procedure wth the ntal guess rom the moment based method descrbed n [6] Cauchy Probablty Densty Functon The Cauchy probablty densty uncton [,,7] s gven by γ ( x) = () ( ) π x x0 + γ where x0 s the locaton parameter and γ s the scale parameter. For the Cauchy dstrbuton wth L x,... x ; x, γ s gven by sample sze, the lkelhood uncton ( ) 0 L( x,... x ) ; x, γ = 0 πγ { + ( x ) x0 / γ } The logarthm o the lkelhood s gven by log L = logπ logγ log + Maxmzng the log lkelhood uncton wth respect to y0 and γ gves: (3) x x0 (4) γ x x0 x x0 + = 0 γ γ (5) x x0 + = 0 γ (6) We use ewton Raphson teratve procedure to solve (5) and (6) or x0 and γ. The medan s used as an ntal value or x0 as descrbed n [7]. 5. EPERIMETAL RESULTS The χ goodness o t perormance was evaluated usng dscrete WHT coecents (8x8 block sze) or teen natural test mages havng derent characterstcs but here we report only or popular Lena, Barbara, Aeral, House, Boats and Mandrll mage. The parameters o all dstrbutons were ound usng maxmum lkelhood method. The model dstrbuton whch 78

5 provdes the mnmum χ statstc s consdered to be the best t under the χ crteron. The nne AC coecents C and C used n the experments were chosen because they usually have the most eect on the mage qualty. Table. χ statstcs or a ew dscrete WHT coecents o Lena (5x5) mage Coecent Gaussan Laplacan Gamma Gen. Gaussan Cauchy C 5.94x C 4.36x C 4.6x C 4.3x C.0x C.5x C.4x C 6.38x C.6x Table. χ statstcs or a ew dscrete WHT coecents o Barbara (5x5) mage Coecent Gaussan Laplacan Gamma Gen. Gaussan Cauchy C.00x C.x C.47x C 5.48x C.53x C.67x C.04x C.00x C.59x Table 3. χ statstcs or a ew dscrete WHT coecents o Aeral (56x56) mage Coecent Gaussan Laplacan Gamma Gen. Gaussan Cauchy C C C C C C C C C

6 Table 4. χ statstcs or a ew dscrete WHT coecents o House (56x56) mage Coecent Gaussan Laplacan Gamma Gen. Gaussan Cauchy C C 5.69x C.4x C C 8.8x C 9.8x C.34x C.8x0 3.33x C Table 5. χ statstcs or a ew dscrete WHT coecents o Boats (5x5) mage Coecent Gaussan Laplacan Gamma Gen. Gaussan Cauchy C.9x C 8.60x C 7.48x C.68x C.39x C.5x C 6.8x C.9x x C Table 6. χ statstcs or a ew dscrete WHT coecents o Mandrll (5x5) mage Coecent Gaussan Laplacan Gamma Gen. Gaussan Cauchy C C C C C C.3x C C C I x M s the sze o the mage ater computaton o D dscrete WHT, then each coecent M wll have x values or the mage to be used n KS and χ test. Table,, 3, 4, 5 and show the χ statstcs o dscrete WHT coecents or Lena, Barbara, Aeral, House, Boats and Mandrll mages respectvely. The result ndcates that the χ test statstc s smallest or the 80

7 General Gaussan dstrbuton or most o the tested coecents o all the mages. Fg. shows the emprcal pd o the dscrete WHT coecents along wth the tted Gaussan, Laplacan, Gamma, Generalzed Gaussan and Cauchy pds. From Fg. as well as rom the values o the χ statstc, t s clear that the Generalzed Gaussan dstrbuton provdes a better t to the emprcal dstrbuton then that acheved by the Gaussan, Laplacan, Gamma and Cauchy pds. 6. COCLUSIO In ths paper, we perorm the χ goodness o t tests to determne an approprate statstcal dstrbuton that best models the dscrete WHT coecents o mages. The smulaton results ndcate that no sngle dstrbuton can be used to model the dstrbutons o all AC coecents or all natural mages. However the dstrbuton o a majorty o the sgncant AC coecents can be approxmated by the Generalzed Gaussan dstrbuton. The knowledge o the statstcal dstrbuton o transorm coecents s very mportant n the desgn o optmal quantzer that may lead to mnmum dstorton and hence acheve optmal codng ecency. REFERECES []. Ahmad, T. atarajan, and K. Rao, Dscrete Cosne Transorm, IEEE transactons on Computers, vol. C-3, no., pp , 974. [] V. Bhaskaran and K. Konstantndes, Image and Vdeo compresson standards - Algorthms and archtectures, Kluwer Academc Publshers, 997. [3] W. B. Pennebaker and J. L. Mtchell, JPEG stll mage data compresson standard, ew York: Van ostrand Renhold, 99. [4] A. K. Jan, Fundamental o Dgtal Image Processng. Englewoods Cls, J : Prentce Hall, 989. [5] R. Costantn, J. Bracamonte, G. Rampon, J. agel, M. Ansorge and F. Pellandn, A low complexty vdeo coder based on dscrete Walsh Hadamard transorm, In EUPSICO 000 : European sgnal processng conerence, pages 7 0, 000. [6] A. Aung, Boon Poh g. T. Shwe, A new transorm or document mage compresson, n Proc. o 7 th Internatonal Conerence on Inormatonommuncatons and Sgnal Processng, 009. [7] R. C. Rennger and J. D. Gbson, Dstrbutons o the two dmensonal dct coecents or mages, IEEE Transactons on Communcatons, Vol. 3, o. 6, pp , 983. [8] J. D. Eggerton and M. D. Srnath, Statstcal dstrbutons o mage DCT coecents," Computer Electrcal Engneerng, vol., pp , 986. [9] F. Muller, Dstrbuton shape o two dmensonal DCT coecents o natural mages, Electroncs Letters, vol. 9, no., pp , 993. [0] R. L. Josh and T. R. Fscher, Comparson o Generalzed Gaussan and Laplacan modelng n DCT mage codng, IEEE Sgnal Processng Letters, vol., no. 5, pp. 8-8, 995. []. L. Johnson, S. Kotz,. Balakrshnan: Contnuous Unvarate Dstrbutons John Wley and Sons,994, Vol., st edton. []. L. Johnson, S. Kotz,. Balakrshnan: Contnuous Unvarate Dstrbutons John Wley and Sons,994, Vol., nd edton. [3] V. K. Rohatg and A. K. E. Saleh, An Introducton to Probablty and Statstcs, John Wley and Sons, 00. [4]. S. Jayant and P. oll, Dgtal Codng o waveorms. Prentce Hall, 984. [5] S. Mallat, A theory or Multresoluton sgnal decomposton: The wavelet representaton, IEEE Transactons on Pattern Recognton Machne Intellgence, vol., pp , 989. [6] K. Shar and A. Leon-Garca, Estmaton o shape parameter or Generalzed Gaussan dstrbutons n subband decompostons o vdeo," IEEE Transactons on Crcuts and Systems or Vdeo Technology, vol. 5, no., p. 556, 995. [7] G. Haas, L. Ban and C. Antle, Inerences or the Cauchy dstrbuton based on maxmum lkelhood estmators. Bometrka, Vol. 57, o., pp , 970. [8] A.. Akansu and M. S. Kadur, Dscrete Walsh Hadamard transorm vector quantzaton or moton compensated rame derence sgnal codng Internatonal Conerence on Accoustcs, Speech and Sgnal Processng (ICASSP), UK,

8 [9] S. R. Smoot and L. A. Rowe, Study o dct coecent dstrbutons, SPIE, vol. 657, pp. 4 4, 996. (a) C 0 (b) C (c) C (d) C 0 (e) C 0 () C 0 Fgure. Logarthmc Hstograms o the dscrete WHT coecents or (a) Lena (b) Barbara (c) Aeral (d) House (e) Boats () Mandrll mages and the Gaussan, Laplacan, Gamma, Generalzed Gaussan and Cauchy pds tted to ths hstogram n log doman 8