JOINT ADAPTIVE WAVELET THRESHOLDING AND BIT ALLOCATION FOR DATA COMPRESSION OF NOISY IMAGES

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1 ONT ADAPTVE WAVELET THRESHOLDNG AND BT ALLOCATON FOR DATA COMPRESSON OF NOS MAGES ur Behtn Department of Automatcs and Contro nformaton Technooges Ryazan State Radoengneerng Unversty Gagarn St. 59/ Ryazan RUSSA 395 ABSTRACT t s proposed an adaptve data-drven aveet soft threshodng and bt aocaton for ont fterng and data compresson of nosy mages. The suggested method uses one common crteron to smutaneousy estmate the threshod vaues and ntervas of quantzaton for each subband and taes nto consderaton the change of dynamc range of threshoded aveet coeffcents. The resuts of modeng sho the effectveness of the desgned coder n comparng th SPHT.. NTRODUCTON Many practca appcatons are deang th mages corrupted by nose hch s ntroduced by devces or regstraton schemes because of an nterna nature of ts appearance. For exampe the rough surfaces of detas are gvng the random phase changes n the bacscatterng refectvty of mcroave umnaton n vson systems SAR magery (spece-nose). The same character of dstortons one can dscover n nfrared mages here nose s condtoned by the foatng of the R-matrx eement transfer coeffcents near unty (fxed pattern nose). n genera the mutpcatve mode s consdered: = X Z () here s the observed mage X the unnon orgna Z mutpcatve nose th unty mean and dfferent probabty densty functons (pdf). The random varabes X and Z are assumed to be ndependent especay hen the nose ntensty s sgnfcant (e.g. fudeveoped spece nose n SAR mages). Dfferent decsons on aveets to compress data of nosy mages have been suggested n the ast decades. One group of agorthms methods s based on expotng any propertes of sgnas coders or ther behavor under some condtons (for exampe see [8 9]). Another group of methods represented by Burrus [4] Vtter [] and other authors s amed for gettng the best threshod vaues for soft or hard threshodng under some crtera or decdng the bt aocaton probem. n other ords denosng by threshodng quantzaton and bt aocaton ere anayzed separatey. n the resut so-caed suboptma estmators of threshod vaues and quantzaton ntervas ere obtaned. The detaed dscusson on ho to appy the adaptve soft threshodng and the vast bbography on de-nosng durng compresson one can fnd n []. Here e frsty do an attempt to combne a search for the optma aveet threshod and the nterva of quantzaton n the sense of one common crteron.. PROBLEM DEFNTON N TERMS OF THE RD-THEOR mage mutresouton anayss ntroduced by Maat (the fast aveet transformaton [3]) gves the Q-eve mage aveet decomposton usng one-dmensona fter ban n the to dmensona case. Equatons to obtan the eve q+ from the eve q are the foong n the pxe poston (a : q+ ( a = h h q( a + b+ ) = [ H q] a b h W q+ ( a = g h q( a + b + ) = [ Gh q] a b v W q+ ( a = h g q ( a + b + ) = [ Gv q ] a b W = here () d q+ ( a = g g q( a + b + ) [ Gd q] a b A q s the approxmaton of the observed mage q at the scae (the o-frequency content n the subband [ π / ]); W q + W q + and W q + the q h v d aveet coeffcents correspondng to horzonta vertca and dagona orentatons respectvey (so-caed detas ); h g the coeffcents hch defne the fter ban. [q] We can determne an operator W permttng to obtan the q eve hgh-frequency mage (aveet coeffcents) for any orentaton as the resut of q successve convoutons: q (3) W = G H and W ( a = [ W ] a b ~ ~ here W ( a = H G( a + b + ) and defne the sze of a ndo centered n (a and ~ q ~ [ q ] q H = h ; G = g h. Then the aveet decomposton of nosy mage () can be rtten as:

2 W = W = W XZ = W X + W X ( Z ) (4) = W X + W XZc = WX + WΞ here W ( ) centered and uncorreated random processes X = W X WΞ = W XZ c E [ W ] = E[ W ] = X Ξ E[ WXWΞ ] = W W E[ ] E[ Zc ] =. Therefore n aveet doman the mutpcatve mode () becomes the addtve one (4). Usng the reaton (4) e can rte for any aveet coeffcent at the subband contanng () coeffcents: = + =... =... (5) ξ here s the number of subbands and =3Q+ for the fast aveet transformaton. Let the nose n aveet coeffcents be represented as: ξ = σ (6) here e e ξ σ ξ > e ξ and here e consder condtonay that s an addtve Gaussan nose th zero mean and σ stands for ts varance =.... There are some ors here t s suggested dfferent hypotheses about nosy aveet coeffcents dstrbuton. n partcuar the generazed Gaussan dstrbuton [] as very e adopted n magery on aveets. n spte nvestgatons on ths topc are st conducted. Here e consder that the hgh-pass fterng eads to an approxmatey symmetrca form of the aveet coeffcents dstrbuton aong th the centered moments. Hence the propertes of nose n spata doman are changed n such manner that e can use the addtve hte nose mode n aveet doman. Ths s our oney presumpton not havng the strong mathematca bacground. Therefore the estmator of any aveet coeffcent ˆ can be represented through a regresson mode X aong th quantzaton: [ ˆ = round ( + ϕ( τ )) / Δ Δ (7) { } ] here Δ s an nterva of quantzaton at the subband ; ϕ ( τ ) a functon of threshodng th threshod vaue τ =.... t s uncomfortabe to drecty use the reaton (7) so e be usng a more tradtona form of the expresson: ˆ = + ϕ ( τ ) + σ =... (8) here σ uq uq s the error of unform quantzaton of the aveet coeffcent th number from the array (subband) th number =.... n order to obtan the optma estmators of the aveet coeffcents e shoud rte the quaty crteron hch concdes outsde but not nsde the Sten s unbased rs estmaton (SURE) crteron [6]: ( ) ˆ (9) D ( τ Δ ) = ( ) mn X X τ Δ here to st unnon parameters τ and Δ determne the quaty of mage enhancement after compresson. Let N be the tota number of bts n the observed mage; be the number of bts correspondng the N subband ; α = N / N be the reatve subband sze; B = ( b... b ) contans the bt engths aocated to the subbands; σ be the subband varance cacuated for a of aveet coeffcents and = = σ = for detas; () ( ) ( ) for approxmaton; ~ σ be the subband varance of the threshoded aveet coeffcents; the dynamc range of the aveet coeffcents s changed after soft threshodng; and M ( +ϕ( τ )) for detas; σ~ = M () σ for approxmaton f t eeps up. Here M s the number of the sgnfcant coeffcents. The quantzaton error at the subband depends on bt aocaton on subbands and the varance of threshoded aveet coeffcents but aso depends on bt rate [3 5] as γ ( σ ~ K ) b for o bt rates (ess than bbp); σuq = ( σ ~ () b K ) for hgh bt rates. Beo e constran our nvestgaton by unform quantzaton. Because the quantzaton nterva Δ under unform quantzaton comprses the quantzaton error as Δ = σ [3] then the threshod vaue τ can be uq put equa to dead zone of a coder τ = θδ and t yeds: τ = θ σuq =... (3) here θ s the coeffcent controng the reaton beteen the dead zone and the nterva of quantzaton (many coders have θ=). After substtuton the expresson (8) n the crteron (9) and tang nto consderaton the resuts obtaned by Sten for SURE [6] and that = for detas e get for dstortons:

3 D ( τ Δ ) =σw + ϕ( τ ) ξ (4) σ W ξ + ϕ( τ ) +σ. uq e Let the estmator of nose varance n the aveet coeffcents fnd out anyay (e.g. by averagng of oca σˆ Wξ varances cacuated thn some tte-szed ndos) then the expresson (4) usng reatons () (3) yeds to: ~ ( D ( τ Δ ) =σˆ W + ϕ ) ( ) b ξ (5) ˆ σw ξ ~ ( + ) ( ) ( ) ϕ b +σ uq b = D b e ~ =... and here ϕ ( b ) s the modfed threshodng functon after some agebra. After threshodng and quantzaton under the gven parameter θ the rest dstortons at the subband depend on the number of bts b of the gven subband. f the bt rate (or the quota of avaabe bts) R ( B ) = R c s ponted by the user then the probem defnton of an optma bt aocaton aong th de-nosng s represented as foos: D( = D ( b ) mn (6) under constrants R( = = = α. b The tas of condtona optmzaton (6) s beng resoved through Lagrange functon as a tas of uncondtona optmzaton [3 5]: L( = [ D( + λr( ] mn (7) here λ s an undetermned Lagrange mutper. Therefore the tas of ont de-nosng and quantzaton of aveet coeffcents s descrbed by the expresson (7) n terms of the rate dstorton theory. Hoever here the mnmzaton of the functon (7) depends on a nd of the threshod functon ϕ( τ ) =... = CALCULATON OF THE THRESHOLDED WAVELET COEFFCENTS VARANCE Before mnmzaton of the functon (7) t s necessary to cacuate the varance vaue ~ σ of the threshoded aveet coeffcents () hch changes aong th [ changng of the threshod vaue τ ]. t s needed for smpcty of the searchng procedures and estmaton of dstortons under o bt rates. f e can sort the aveet coeffcents by ther absoute magntudes then each coeffcent s gettng a ne ndex caed rang : B B [... ]. (8) + t as shon by S.Maat and F.Fazon n [5] that the sorted aveet coeffcents have the fast decreasng. t as proved for mages that have dscrete reguar structures (e.g. havng arge segments th homogenety textures). n ths case after aveet transformaton there are appeared Besov s spaces here each aveet coeffcent γ of rang s mted up to C f the constants C > and γ >/ exst. Usuay γ. Hoever there s a arge dfference beteen the rea curve of the sorted aveet coeffcents and the curve of the approxmaton under codng th o bt rate or hen the mage s strongy textured. Ths phenomenon has been nvestgated by the S.Maat and F.Fazon n [5] here they suggested that the second dervate s not great than some constant ε > : d og ( ) ε ( ] (9) ( d og ) here = / [; ] s the normazed rang γ( ) vaues ( ) = s the approxmated aveet coeffcents. Hence the exponent γ() s soy changng and depends on og [3 6]: d og ( ). () γ( ) = d og Hoever e can not use the reatons (9) () due to the foong reasons.. Because the nosed mage s undergone to be encoded then the nose contrbutes addtona hghfrequency components n the mage content destroyng the o-frequency reguar structures. The dfference beteen the rea curve of the sorted aveet coeffcents and the curve of the theoretca approxmaton ( ) = s ncreasng hen the nose var- γ( ) ance s becomng arger. t s ustrated by Fgure here the modeed data of the e-non mage Lena are represented by the dfferent curves obtaned under nose th exponenta pdf th varance aveet CDF 9.7 Q=3.. The dfference beteen the rea curve of the sorted aveet coeffcents and the curve of the theoretca γ( ) approxmaton ( ) = s becomng much orse hen the aveet coeffcents of any subband are consdered ony. 3. The resuts n [5] determne a ot of condtons; n partcuar the constant ε > s strongy mtng the number M = ε of the aveet coeffcents n approxmaton. Thus t s necessary to precsey as possbe determne the functon of approxmaton because t gves the approxmaton of the varance vaue ~ σ. From the reaton () e have for the normazed sorted aveet coeffcents:

4 ( ) / max { } = [ / max { }] og ( ) og ( / ). () No e can study the behavor of the degree factor og / max{ } n (). The curve of the degree factor behavor for one of the subband s shon on Fgure under constrant M = ε th ε =.9 (.e. for 9 % of a subband coeffcents). From Fgure one can mar that the ogarthmca curve s softy changng thout umps ups and dons (n the nterva of ε from.8-.9 to the curve s suddeny gong don). t aos us to approxmate ths part of the curve by the poynoma mode of the thrd degree and to get the reatvey o error of the approxmaton: og ( / max{ }) 3 = () a3 + a + a + a = a ( ). The coeffcents of the mode () a can 3 a a a be fnd out usng any numerca method. Fgure aso contans the curve of the approxmaton hch has been cacuated by usng the Vandermonde matrx hose eements are 3 s ( ) here υ ; ps = / p p = K s =3 and sovng the system of near equatons. One can see from Fgure 3 that ths approxmaton method gves the reatvey good approxmaton of the curve of the sorted aveet coeffcents thn the subband (the mean square error (MSE) s equa 5 for 3 ths exampe). t eads to the sma MSE n the case of approxmaton of the correspondng cumuatve sum of squares S of the sorted aveet coeffcents. Fgure 4 shos the curves of ncreasng the cumuatve sum of squares of the aveet coeffcents and ts approxmaton for the subband of the decomposed nosed mage Lena. t s seen from Fgure 4 that the curves concde for the arge part of the sgnfcant aveet coeffcents. Therefore usng the suggested method of approxmaton the varance of the threshoded aveet coeffcents can be represented through the approxmated cumuatve sum of squares as foos: ( max{ }) M σ~ r ( ) [ M ] = (3) M here ( ) r ( ) = a ( ) / og and M < and the threshod vaue can be found as τ M =. M n order to compute the dstortons under o bt rate a hat s needed s to fnd the vaue ~ σ εσ ( M ). max { } x 5 Fgure. The curves of the sorted aveet coeffcents under σ =.85 (); 3.5 (); 39.5 (3) and the n theoretca approxmaton (4) { } og max Fgure. The normazed ogarthmca curve of the sorted aveet coeffcents for detas h (the sod max W ne) and ts approxmaton (the dash ne) under σ =39.5 { } [ ] n Fgure 3. The curves of the sorted aveet coeffcents for detas h and ts approxmaton (the dash ne) S W W Fgure 4. The curves of the cumuatve sums of squares of the aveet coeffcents for detas h and ts approxmaton (the dash ne) A

5 4. WAVELET THRESHOLDNG FOR MAGE DE-NOSNG AND COMPRESSON There are some threshod functons n the terature that are used for de-nosng many. Hoever the hard threshodng s an nterna essence of the modern aveet-coders (EZW PEG and SPHT [3]) because ones use the dead zone an enarged nterva of quantzaton near zero: W > τ f W W ) = (4) f W τ. The aveet coeffcents caught by the dead zone are consdered to be nsgnfcant and put to be equa to zero. Hence e can expect some de-nosng durng data compresson of nosy mages. Hoever eepng up of the sgnfcant deta coeffcents does mean that the nose s aso preserved n t. f e use the dead zone ony then the enhanced mage contans artfacts e pseudo-gbbs oscatons especay under o bt rates. The man reason of one s appearance s n the destroyed reguar mage content by nose rather than n dscontnutes n the reconstructed sgna. n order to avod much destroyed mage content after reconstructon t seems sutabe to use the soft threshodng here the aveet coeffcents are reduced by the threshod vaue τ and the dynamc range of the aveet coeffcents s decreasng: sgn( W )( τ ) > τ W f W W ) = (5) f W τ. t as shon n some ors [ 4] that the soft threshodng provdes some de-nosng n the case of data compresson of nosy mages. Besdes t s orth to remnd that there are other schemes of aveet threshodng. The nd of the soft threshodng s the functon of Vdaovc [7]: sgn( W ) ( ) ( τ ) > τ W f W W ) (6) = f W τ. The another nd of the soft threshodng s the fucton of Bruce Gao [] but t preserves the aveet coeffcents aong th some modfcaton n the zero zone hence t s not sutabe for compresson. Here the man probem s ho to fnd an optma threshod vaue because then t s necessary to determne an nterva of quantzaton. The threshod vaue and the vaue of the quantzaton nterva are computed separatey as t has been aready done n many prevous ors cted above. n ths case e can get the stuaton hen the error of quantzaton be much more than the error from the resdua nose. The quaty of ont de-nosng and data compresson of nosy mages can be estmated by usng such crtera as MSE and PSNR. The vaues of MSE and PSNR depend on both the threshod vaue and the vaue of the quantzaton nterva. Therefore choosng the vaues for the threshod and the nterva of quantzaton e need to get the mnmum of the expresson (6) under constrants R ( B ) = R c tang nto consderaton the reaton (5). 5. BT ALLOCATON UNDER SOFT THRESHOLDNG Let the functon of the soft threshodng be represented as: ϕ( τ ) = sgn( ) τ Thr( τ ) (7) Thr( < τ ) here Thr (...) s the symbo of comparng to the threshod τ. Usng the resuts obtaned by Sten n [6] the expresson (5) for dstortons can be rertten n the foong form: D ( b ) = σˆ W + τ Thr ( τ ) ξ (8) ( ) + ( ) ( ) Thr < τ = σˆ W Thr ( < τ ) + σ uq Thr ( τ ). ξ Then ettng a symmetry of pdf for aveet coeffcents Thr( < τ ) = Thr( τ e have D( b) =σ σˆ Wξ (9) + ( ˆ ) ( ). ( ) τ + σ +σ τ Wξ uq Thr The threshodng Thr( τ ) determnes the mted quantty M = ε of the aveet coeffcents formng the sum and then ~ D ( ) = ( σ εσ ) + σˆ (ε ) + ε( τ + σ ). (3) b Wξ After soft threshodng the error of unform quantzaton be proportona the varance of the threshoded (sgnfcant) aveet coeffcents σ : ~ b ~ uq = α σ σ and the functon of Lagrange (7) be L ( = D( + λr( (3) = ( ~ σ εσ + (ε ) σˆ + ετ ) + ε = b ~ α σ + λ = = α Wξ b mn. Because the to ast members of the expresson (3) depend on b then after dervaton of them e obtan: εσ = (n ) ~ og. (3) b λ After substtuton (3) n constrants R ( e can fnd the unnon Lagrange mutper: b uq )

6 λ = Fnay e have for ~ α og ((n ) εσ ) RC = b. b = og(nεσ ~ ) α og(nεσ~ ) + R. (33) C = After substtuton (33) n (3) e obtan that the tota dstortons D( = D ( b ) depend on the varances of = the threshoded aveet coeffcents σ~ because the square of the threshod vaue τ s the ast eement of the cumuatve sum of the threshoded aveet coeffcents. 6. THE CODER ALGORTHM Summarzng the moments descrbed above e have the next agorthm of a coder for data compresson of nosy mage.. Under the gven quota of avaabe bts R C and the chosen constant ε (the recommended vaue s.9) t s performed the Q-eve fast aveet transform and gettng arrays of aveet coeffcents. For each subband t s cacuated the estmator of the nose varance. σˆ Wξ. n order to use the approxmaton (3) nstead of recacuaton the cumuatve sum of the aveet coeffcents durng each teraton the aveet coeffcents of each subband are sorted by descend and the poynoma coeffcents of r() are computed. 3. The functon (3) s beng formed n accordance th ~ (33). Because the varabes σ are ndependent then the mnmum of the functon (3) can be found by any numerca method. As t as obtaned by modeng the functon (3) s the mut-extrems functon havng a goba mnmum (see Fgure 9). There are some cases hen the functon (3) not have the goba mnmum. t corresponds the negatve vaue of b. When R C s too sma the resutng bt ength b can be negatve for subbands th very sma sgna varances. Hence the bt aocaton steps are repeated th these subbands removed ( bts). Aso t s necessary to reduce the number of subbands appropratey and adust the quota of bts R C. Searchng of the optma threshod vaue starts th the argest rang M = ε then M = M δ here δ s the reasonabe step vaue and so on. The constant ε s beng recacuated each tme hen the quantty of the aveet coeffcents s changed. 4. After fndng the optma vaues of σ~ t s computed the vaue of the quantzaton nterva through the reaton Δ = σ. uq r M ( 5. Then the threshod vaue max{ } ) / τ = M s cacuated here M s the rang of the ast aveet coeffcent n the approxmaton of the cumuatve sum of the threshoded aveet coeffcents. 6. A ne aveet threshodng functon shoud be set (see Fgure 7). 7. Usng the ne threshodng aveet functon the estmators ˆ of the aveet coeffcents are com- puted. 8. The aveet coeffcents are to be encoded and sent through a channe or zpped. The enhanced (ftered) mage s obtaned by appyng the nverse aveet transformaton to the estmators =... =.... ˆ 7. THE RESULTS OF MODELNG Beo e paced the resuts of processng for the Lena mage. For comparng e used the SPHT agorthm and an dea coder the threshod vaue of hch s cacuated on the prncpe of Orace (.e. hen the orgna (non-nosed) aveet coeffcents are non): τ Orace = argmn τ ( ) + ϕ( τ ). () The mutpcatve nose n () as modeed by usng the random number generator th the exponenta pdf th unty mean and the varance changng n the experments. We used three eves of the fast aveet transform on the base of the borthogona fter ban CDF 9.7. Ths fter ban as t as found n the prevous ors has aso good de-nosng propertes. Fgure 5 shos the orgna nosed mage Lena th =5; the mages processed by SPHT and the suggested method are on Fgure 6 and Fgure 7 Fgure 8 respectvey. As expected the suggested agorthm provdes more effectve de-nosng because of the optma threshod vaues thn each subband and the errors of quantzaton are very sma. Vsua dfferences are practcay nvsbe for human sghtng n Fgure 7 and Fgure 8 obtaned under dfferent bt rates. t s proved by the dependences for PSNR on Fgure. Tabe contans the numerca resuts for the suggested method and the SPHT-based and the dea coders for comparng. The bt aocaton and the threshod vaues for horzonta (H) vertca (V) dagona (D) orentatons and approxmaton (A) obtaned by the experment are paced n Tabe. t s seen from Tabe that the most part of the quota of bts beongs to the frst eve of aveet decomposton. The typca dependence shong the changng of the functon (3) by decrementng the ranges of the subband aveet coeffcents s on Fgure 9. One can mar that ths dependence has a mut-extrems behavor. The threshod functons for each subband are shon on Fgure.

7 L(.788 x Fgure 5. The nosed mage Lena; 5 =5; 8 bpp [ M Fgure 9. The behavor of the functon (33) n dependency of number M of the aveet coeffcents n the subband H under =5;. bpp PSNR db Fgure 6. The processed mage by SPHT; σ =5;. bpp 5 n bpp Fgure. The dependences PSNR bt rates for the suggested method () and SPHT (); = Fgure 7. The processed mage by the suggested method; =5;. bpp Fgure 8. The processed mage by the suggested method; =5;. bpp PSNR db MSE Coder (for. bpp) Tabe n 5 35 Orace SPHT The suggested method Orace SPHT The suggested method Tabe α σ b Parameter τ H() V() D(3) H(4) V(5) D(6) H3(7) V3(8) D3(9) A3() 7 69

8 H() V() D(3) H(4) V(5) D(6) H3(7) V3(8) CONCLUSON The suggested method for data compresson of nosy mages provdes the reatvey hgh PSNR vaue and stabe de-nosng under dfferent bt rates. Obvousy the man roe pays the optma threshod vaues hch are cose to the vaues of SURE and Orace s threshods. The nfuence of quantzaton s much ess than for e-non coders (e.g. SPHT). Nevertheess the dsadvantage of the suggested method s a search for the optma vaues durng ong tme (n fact a of vaues) because of the mut-extrems character of the crteron. So ths method can be recommended to use for tass of archvng nosy mages n dfferent appcatons. 9. REFERENCES [] A. Bruce and H. -. Gao WaveShrn: Shrnage Functons and Threshods Proc. SPE San Dego CA 995. [] S.G.Chang B.u M.Vetter Adaptve aveet threshodng for mage denosng and compresson EEE Trans. on mage Processng Vo. 9 No. 9 pp [3] Gonzaez R.C. and Woods R.E. Dgta mage Processng. Addson Wesey 99. [4] H.Guo.E.Odegard M.Lang R.A.Gopnath.W.Seesnc and C.S.Burrus Waveet based Spece Reducton th Appcaton to SAR based ATD/R EEE nternatona Conf. on mage Processng Vo. pp [5] S.Maat and F.Fazon Anayss of o bt rate mage transform codng EEE Trans. on Sgna Process. vo.46 no. 4 pp.7-4 Apr [6] C.Sten Estmaton of the mean of a mutvarate norma dstrbuton Ann. Statstcs No. 9 pp [7] B.Vdaovc Statstca Modeng by Waveets. ohn Wey & Sons 999. [8] O.-K. A-Snayh R.M.Mercereau Lossy Compresson of Nosy mages EEE Trans. on mage Processng Vo.7 Nr. 998 pp [9] R.Ötem K.Egazaran Transform Doman Agorthm for Reducng the Effect of Fm-Gran Nose n mage Compresson Eectroncs Letters Vo. 35 No. 999 pp D3(9) - - A3() Fgure. The threshod functons for hgh-frequency subbands and approxmaton; =5;. bpp