Reconstructions that Combine Cell Average Interpolation with Least Squares Fitting

Transcription

1 App Math Inf Sci, No, ) 7 Appied Mathematics & Information Sciences An Internationa Journa Reconstructions that Combine Ce Average Interpoation with Least Squares Fitting Francesc Aràndiga and José Jaime Noguera Departament de Matemàtica Apicada Universitat de Vaència, Spain Received: 9 Jun 5, Revised: 7 Aug 5, Accepted: 8 Aug 5 ubished onine: Jan 6 Abstract: In this paper we present two reconstructions in the ce average framewor of mutiresoution a a Harten The first one combines interpoation and east squares fitting and the second one is based on east squares fitting We study some of their properties as we as its approximation order We aso anayze how different adaptive techniques ENO and SR) can be used within these reconstructions We appy them to noise remova and compare the resuts that we obtain with other existing techniques Keywords: Ce average, Least squares, Interpoation, Noninear mutiresoution, Denoising Introduction A common probem in approximation theory is the reconstruction of a function from a discrete set of data which gives information on the function itsef This information usuay comes either as point-vaues or ce-averages of the function over a finite set of points or ces, respectivey The function is then approximated by an interpoant, that is, another function whose vaues or ce-averages at the given set of points or, respectivey, ces, coincide with those of the origina one The interpoation is a inear procedure of the vaues on the given set of points, but in this case the accuracy of the approximation in the presence of a singuarity is imited by its order, so that any stenci crossing the singuarity wi resut in an unsatisfactory approximation This means that increasing the degree of the poynomia wi produce arger regions of poor accuracy around singuarities The choice of stencis that avoid crossing singuarities, whenever this is possibe, is crucia for the improvement of the accuracy of the approximation This is the ey underying the ENO essentiay non-osciatory) technique, introduced by Harten et a [9] in the context of high resoution shoc capturing HRSC) schemes for conservation aws With the ENO interpoant the region affected by each singuarity is reduced to the interva that contains it, assuming the singuarities are sufficienty we separated It is possibe to improve this resut using the Harten s subce resoution technique SR) [,,7]) If the ocation of the singuarity within the ce or a sufficient good approximation of it) is nown, then the oss of accuracy can be avoided In [] we study the appication of these reconstructions to the case where the data are contaminated with noise On one hand we use interpoatory reconstructions and on the other hand we rey on east squares We do not get good interpoants in a cases This motivates us to introduce new reconstructions using both ideas The first one combines interpoation with approximation in the east squares sense caed in this paper Interp-L S ) and the second reconstruction is based, foowing [], on approximate the data in the east square sense caed in this paper L S ) Here we present these reconstructions in the ce average framewor which is more appropriate in the presence of noise In [] we can see these reconstructions in the point vaue context We aso anayze how can be used combined with different non inear techniques ENO and SR see [7] and [9] resp) The paper is organized as foows: We reca in Section the discrete framewor for mutiresoution introduced by Harten focusing on ce average discretizations We introduce the Interp-LS reconstruction for ce averages in Section There, we aso present how this reconstruction can be combined with non-inear techniques In Section 4 we present the east squares Corresponding author e-mai: arandiga@uves

2 74 F Aràndiga, J J Noguera: Reconstructions that combine ce average reconstruction for ce averages L S C ) Section 5 is devoted to show some numerica experiments where we can see the resuts that we obtain with the techniques presented in this paper Finay, in Section 6 we present the concusions Harten s framewor for mutiresoution anaysis The buiding bocs of a mutiresoution anaysis as described by Harten see [8] and references therein) are a sequence of inear decimation operators {D + }L, a nested sequence of inear vector spaces V +, with D + : V + V which, from a vector of V +, computes a coarser representation of it), and a sequence of prediction operators { + } L,+ : V V + which, from a vector of V obtains an approximation to its finer) representation in V + In most appications, V + is a space of sequences of rea numbers Our description impies that the resoution increases with The decimation and prediction operators have to satisfy the compatibiity condition: D + + I V ) Given v + V +, the prediction error e + Q + v + : I V + + D + )v+ obtains the information in v + V + that cannot be predicted from v D + v+ V by the operator + It is easiy seen that the error vector, e + : Q + v +, beongs to the nu space of D +, denoted by N D + ) []) Let G + : V + N D + ) be the operator which assigns to each vector e + V + the coefficients d + of its representation in terms of a given basis, {µ + j }, of N D + ) V +, and et E + be the canonica injection N D + ) V + Ceary G + E + I N D + ) and E + G + I V + The non-redundant information in the error vector is contained in the set of coefficients {di + }, caed the scae coefficients at eve We note that v + and{v,d + } have the same cardinaity These ingredients aow to construct an aternative representation of a vector v + V + Given v + we evauate: { v D + v+, d + G + I V + + D ) + )v+, and given v and d +, computed by ), the vector v + is recovered by the inverse formua v + + v + E + d + This gives the equivaence between v + and {v,d + } By repeating step ) for v one obtains its corresponding decomposition {v,d } Iterating this process from L to we find that a mutiresoution setting {{V } L,{D + }L } and a sequence of corresponding prediction operators { + } L satisfying ) define an invertibe mutiresoution transform In Harten s framewor, the decimation and prediction operators are buit from a sequence of discretization operators and a sequence of compatibe reconstruction operators Given a function beonging to a certain functiona space F the discretization operator D + : F V + obtains a discrete representation of it at the resoution eve defined by V + Conversey, the reconstruction operator R + : V + F obtains a function in F from discrete vaues in V + The reconstruction operators R + are compatibe with D + provided that D + R + I V + ) is satisfied for each The decimation and prediction operators corresponding to these sequences are obtained as D + D R + and + D + R The compatibiity condition ) is a consequence of ) In Harten s framewor, the discretization process specifies the setting, then the choice of a reconstruction operator defines a mutiresoution transformation whose properties are cosey reated to those of the reconstruction From the point of view of data-compression appications, accuracy of the reconstruction is an important feature Ce average mutiresoution anaysis Let f F L [,]) Considering the set of nested dyadic grids defined by X {x i }N i, N N, x i ih, h N,,,L where N N The ce-average discretization operator D + : F V + is defined in [8] as foows: f i + :D + f) i h + + i x + i The decimation can then be computed by: f i i h fx)dx x i + h + x + fx)dx, i N + 4) fx)dx f + + f + ) 5) Let p i x) the poynomia of degree r nr+ n, nr,n N, such that h i+s x i+s p i x)dx f i+s, s n,,nr, 6) then, the reconstruction is defined as foows: I C r x; f ) p i x), x [x i,x i], i,,n

3 App Math Inf Sci, No, ) / wwwnaturaspubishingcom/journasasp 75 The prediction operator is in this case: + f ) ) D + I C r x; f )) h + + x + + f ) D + I C r h + + x + I C r x; f )dx, ) x; f )) As a consequence of 6) we have I C r x; f )dx + f ) + + f ) ) f i 7) and, from 5), e + + e+ Then, we can define di + e + The agorithms for the direct 8) and inverse 9) transforms in the ce average framewor are: f L M f L { f,d,,d L } Direct) for L,, f i f + + f + ), i N d i + f + + f ), i N end M f L M M f L Inverse) for,,l f + + f ) + di +, i N f + f i f + + f ) di +, i N end 9) 8) Interp-L S reconstruction for ce averages Interp-L S reconstruction for point vaues I L S ) As in [] the Interp-L S reconstruction is defined as foows: Given and the interva Ii [x i,x i ] we construct the poynomia of degree r such that interpoates x i, x i and approximates in the sense of east squares at the n + nr > r nodes x i+ j, j n,,,,,nr If we denote q I L S r x; f ) this poynomia, then the reconstruction is I L S r x; f )q I L S r x; f ), x [x i,x i ] and the prediction is + f ) I L S r x + ; f ) Interp-L S reconstruction for ce averages I L S C ) Let p I L S C r x; f ) r +)a x the poynomia of degree r <n + n r, Π r ) such that and h i+s x i+s h i x i p I L S C r x; f )dx f i ) p I L S C r x)dx f i+s, s n,,nr, s, ) nr,n N, in the east square sense That is, that minimizes ) min x i+s p i x)dx f i+s p i x) Π r h x i+s s n,,nr, s If we define q i x) r x pi L S C y; f )dy r a x + and we assume, without oss of generaity, that i and x i+s s, then ) is equivaent to q ) q ) r Since a f r a, then a f q x) f r r a )x+ a x + x f r + a x + x) Conditions ) are equivaent to q s+) q s) f + r in the east square sense and ) a s+) + s + f s ) r a s+) + s + ) f s f, s n,,nr, s Then we have to obtain â R r such that minimizes where: A I L S C r r A I L S C min â R r â ) ˆf n+nr f, n+ ) n) n+ ) r n) r ) ) r r r nr+ ) nr) nr+ ) r nr) r, )

4 76 F Aràndiga, J J Noguera: Reconstructions that combine ce average â [a,,a r ] T, ˆf [ f n,, f, f,, f nr ]T and n,,) }{{} T n Then, using the norma equations, ) T â A I L S C r A I L S C r T ˆf n+nr f ), A I L S C r a f r T â Once we have {a } r define the reconstruction as r and pi L S C x; f ) we I L S C r x; f ) p I L S C r x; f ), x [x i,x i ], 4) and the prediction + f ) h + nr n + f ) h + f i + x + γ r ) f + x + + I L S C r x; f )dx i+, I L S C r x; f )dx f ) Tabe : Fiters, γ r ), of the approximations to f + obtained with the I L S C r reconstruction + f ) nr n γr ) f i+ 4 4 γ, ) γ, ) γ 4, ) γ, ) γ, ) γ,4 ) γ, ) γ, ) γ 4,4 ) In Tabe, we show the fiters, γ r ), for different degrees of the poynomia, r, and different sizes of stenci As in [] and [] we can prove the foowing resuts: roposition The matrix A I L S C r, r>, n+nr>r, defined in ), has fu ran roofwe wi prove that A I L S C r has fu ran, ie rana I L S C r ) r, by induction on the degree of the poynomia p I L S C r When r we have a singe vector, so the ran is Let r N, r >, and assume that rana I L S C r ) r We want to see whether rana I L S C r ) r, n+ n+a,nr+b a+n+b>r, a, b, a N, b N Since rana I L S C r ) r, there is a submatrix M r of A I L S C r such that M r Without oss of generaity, we suppose that n, a,botherwise we coud use the ast row and appy the same reasoning) Then A I L S C r n+,nr n) n ) n) r+ n ) r+ A I L S C r nr+ ) r+ nr r+ Given that n and r : n) r+ n ) r+ ) r+ r+ r+ )n r+ ) 5) Then, n) r+ n ) r can be used as the pivot eement to nuify the eements of its coumn If we denote N r as the resut of appying to M r the inear transformations that enabe nuify such eements, we can define the r r matrix: a a r n)r+ n ) r+ A r N r, where its first row eements a, j A r are a, j a, j A I L S C r n+,nr, such that a i, j M r Therefore, by 5) and the induction hypothesis, A r ) r+ n) r+ n ) r ) N r ) r+ n) r+ n ) r ) M r, and then, rana I L S C r n+,nr )r Finay, by the principe of induction we have the resut roposition The Interp-L S reconstruction for ce averages has a unique soution roofsince the matrix A I L S C r, defined in ), has fu ran by roposition, foowing section, the Interp-L S reconstruction for ce averages has a unique soution roposition If p I L S C r x; f ) is the poynomia I L S C of degree r, the Interp-L S reconstruction for ce averages recovers exacty poynomias of degree s, s r

5 App Math Inf Sci, No, ) / wwwnaturaspubishingcom/journasasp 77 rooflet px) s b x be a poynomia of degree s, with s r We define{ f j } m j, m>r, as: and j+ j px) f j, j,,m, 6) qx) px) s a x +, 7) where a b +,,,s Thans to 6) we have that A I L S C s,m â ˆf m f, where â[a,,a s ] T, ˆf [ f,, f m ]T and A I L S C s,m is defined in ) Then A I L S C s,m â ˆf m f ) 8) On the other hand, we now that the Interp-L S probem for ce averages has a unique soution by roposition In particuar, whether p I L S C r r i,,m + )a x is the soution of the Interp-L S probem defined by ) and ), with { f j } m j, m > r, defined by 6), and since qx) is the soution of this probem by 8), we have that p I L S C qx), and therefore the Interp-L S reconstruction recovers exacty qx) Coroary The order of the Interp-L S reconstruction for ce averages is r Noninear Reconstructions for ce averages When using the Interp-L S reconstruction for ce averages we have that if r increases, r mnr+n+, the interpoation process has higher order accuracy, ie the detais di + wi be smaer when f is smooth on [x i n,x i+nr ] On the other hand, the interva [x i n,x i+nr ] gets arger with nr,n so that a singuarity in x i,x i ) wi affect more detai coefficients Non-inear essentiay non-osciatory ENO) interpoation techniques, which were firsty introduced in [9], circumvent this drawbac ENO Interp-L S reconstruction for ce averages EI L S C ) The idea of ENO interpoation technique is to repace in 4) the poynomia p I L S C r x; f ) by p I L S C r i,n i,m n i x; f ) seected among the m+ poynomias {p I L S C r i,s,m s x; f )} m s, where m nr+n+, in order to avoid the infuence of the singuarity In [,8,9] the seection process is made by picing the east osciatory poynomia using numerica information on the divided differences of f at the points x That is, using the stenci that minimizes f[x i,,x i+m ], m Here, inspired by [], we use a seection which gives us better resuts in the presence of noise First, we define E i,s,m) m i s+ h x i s+ and now ) p I L S C r i,s,m s x, f )dx f i s+ 9) E i,n i,m)min s,,m {E i,s,m)} ) Once this seection is made, we thus define EI L S C r x; f ) p I L S C r i,n i,m n i x; f ), x [x i,x i] and + f ) h + + f ) f i + x + EI L S C r x; f )dx, + f ) SR Interp-L S reconstruction for ce averages SI L S C ) The ENO interpoatory technique sti produces arge detais di + when a singuarity is contained in the interva [x i,x i ] In order to reduce further the interpoation error, subce resoution methods SR) were introduced in [7] Now et s see how we can appy the SR technique [,7]) in the Interp-L S reconstruction for ce averages: Assume that fx) has a jump in [x i,x i ] Then the primitive function of fx), Fx) fy)dy C[,]), has a corner a discontinuity in the derivative) there Note that the sets { f i }N i and F {Fi }N i are equivaents due to the reations F i Fx i ) i fy)dyh i j f j and f j F j F j h Then, as in [] or [], we can use the function ) to detect where the singuarity is The steps to appy the SR technique in the Interp-L S reconstruction for ce averages are: - Taing stencis with m n + nr+ nodes, we cacuate the stenci ENO as in the previous section - If n i m and n i+ the stencis for the ces Ii and I i+ are disjoint We abe the ce as suspect of containing a discontinuity

6 78 F Aràndiga, J J Noguera: Reconstructions that combine ce average - For each suspicious ce we define the function G I i L S C x)q I i+,,m x;f L S r ) q I i,m,x;f L S r ) ) If G I i L S C x i ) GI i L S C x i ) < we abe the ce Ii as singuar 4- If G I i L S C x i ) GI i L S C x + ) >, the node x + ies to the eft of the discontinuity Then + f ) is obtained as foows + f ) qi L S r i,m, x+ ;F ) qi,m, I L S rx+ ;F ) h + ) + γn+nr+, r n nr f i+, ) f ) f + i + f ), ) where some of these coefficients are in Tabe use ) because the reconstruction is consistent due to the interpoatory conditions: + f ) + + f + f h + The same coud be said for 5) f ) F i + Fi f i h Thans to the interpoatory conditions, G I i L S C can be expressed in terms of { f i } see [] for detais) For instance, the fiters of ) for n + nr 4 and r are: G I i L S C x i ) h f i 6 f i f i f i f i f i+5 ), G I i L S C x + ) h 6 89 f i f i f i f i 48 5 f i + f i 5 f 48 i f i f i f i f i+5 ), G I i L S C x i ) h f i f i f i f i 6 f i + f i ) Tabe : Fiters, γ r ), of the approximations to f +, obtained with the SI L S C r n+nr+, reconstruction + f ) n nr γr n+nr+, ) f i ) γ, 9 ) γ5, ) 4 γ5, ) 4 γ7, ) 56 γ5, ) 5 γ7, In the other case G I i L S C ) < is ocated to the right of the discontinuity and ), x + + f ) qi L S r + i+,,m x+ n+nr+ β,n+nr+ r x i ) GI L S C i x + ;F ) qi+,,m I L S rx+ ;F ), h + ) f + i+ 4) f ) f + i + f ), 5) where β r,n+nr+ ) γ r n+nr+, ),,,n+ nr+ + f ) ;F ) q I i,m, L S rx+ ;F )) We can Remar) can be cacuated as: h + q I L S r i+,,m x+ 4 Least squares reconstruction for ce averages L S C ) Let p L S C r x; f ) r + )a x the poynomia of the degree r such that h i+s x i+s p L S C r x; f )dx f i+s, s n,,nr, 6) in the east square sense If we define q i x) r x pl S C y; f )dy r a x + and we assume, without oss of generaity, that i and x i+s s, then 6) is equivaent to q s+) q s) and r r a s+) + s +) f s 7) a s+) + s +) f s, s n,,nr Then we have to obtain â R r that minimizes A r L S C min â R r â ˆf, where: A L S C r n+ ) n) n+ ) r n) r nr+ ) nr) nr+ ) r nr) r 8)

7 App Math Inf Sci, No, ) / wwwnaturaspubishingcom/journasasp 79 with â[a,,a r ] T and ˆf [ f n,, f nr] T Then, using the norma equations, ) T â A L S C r A L S C r T ˆf A L S C r Once we have {a } r the reconstruction r and pl S C x; f ) we define L S C r x x; f i+s ) p L S C r x x; f )dx, x [x i,x i ], i+s and the prediction + f ) h + nr n + f ) h + + x + L S C r x; f )dx γ r ) f i+, 9) + x + L S C r x; f )dx nr β r ) f i+ ) n In Tabe and 4 we can see some of these coefficients Tabe : Fiters, γ r ), of the approximations to f + obtained with the L S C r reconstruction, + f ) nr n γr ) f i+ 4 4 γ, ) γ, ) γ, ) γ 4, ) γ, ) γ, ) γ, ) γ,4 ) γ 4, ) γ 4 4,4 ) Tabe 4: Fiters, β r ), of the approximations to f + f ) obtained with the L S C r reconstruction, + nr n β r ) f i+ 4 4 β, ) β, ) β, ) β 4, ) β, ) β, ) β, ) β,4 ) β 4, ) β 4 4,4 ) Observe that, since i h p x i x)dx f i, this i reconstruction do not satisfy ) and ) Then 7) is not f ) d + i true and f i f + + Then, if we define, in 9), f + the inverse transform is M f L M M f L + for,,l f + + f ) + di +, i N f + + f ) di +, i N end 7 64 f ) d + i, ) and the reconstruction is not consistent We ca it L S C NC To ensure the consistency we can use f + f i f + In this case the inverse transform is: M f L M M f L for,,l f + + f ) + di +, i N f + f i f +, i N ) end and we ca it L S C Is we nown by the east squares probem, that the matrix A L S C r, defined in 8), has fu ran Therefore, simiary to roposition we can prove the resuts: roposition 4If p L S C r x; f) is the poynomia L S C of degree r, the L S reconstruction for ce averages recovers exacty poynomias of degree s, s r Coroary The order of the L S reconstruction for ce averages is r 4 ENO and SR L S reconstruction for ce averages EL S C and SL S C ) Whie ENO can be appied as in section obtaining the EL S C reconstruction), we have a probem appying SR in this case The fact that we do not have the consistency impies that we do not now the point vaues of the primitive function as in Section ) and then we can not ocate where the discontinuity is A way to appy the SR in this context is using the function G I i L S C x) ) to detect the singuarity as in

8 8 F Aràndiga, J J Noguera: Reconstructions that combine ce average section ) and then, if the jump is in [x + evauate 9) + f ) + h + + x + n nr f ) f i + and in the other case ) + f ) h + + n+nr+ + f ) f i +,x+ ], we L S C r n+nr+, x; f )dx γ r n+nr+, ) f i+, ) + f ), 4) x + L S C r,n+nr+, x; f )dx 5) β r,n+nr+ ) f + i+, + f ) 6) In Tabe 5 we show some of the coefficients β and γ r n+nr+, ) β r,n+nr+ ),,,n+ nr+ We ca this reconstruction SL S C Notice that it is consistent because we force it by 4) and 6) We aso can set a non consistent reconstruction as in ), denoted by SL S C NC and defined as SL S C but introducing the foowing changes: Repacing 4) by: + f ) S r pli,,n+nr+ x+ ;F ) p L i,n+nr+, S r x+ ;F ))/h + ), 7) and repacing 6) by: + f ) S r pli,,n+nr+ x+ ;F ) p L i,n+nr+, S r x+ ;F ))/h + 8) ), where p L S r x;f ) denotes the L S reconstruction for point vaues Its definition is the same as in the I L S reconstruction section ) but here the poynomia approximates in the sense of east squares at the m > r + nodes x i+ j, j n,,nr see [] for detais) 4 Combining SI L S C and SL S C Consistent reconstructions recover propery the jumps but with non consistent reconstructions the exact ocation of them is ost On the other hand, SL S C NC is expected to produce smoother resuts than SL S C, because 4) and 6) wi produce osciations in the soutions In order to tae advantage of both schemes we create a combined agorithm between SI L S C and SL S C NC denoted as SI L S C L S C The idea is through the agorithm of SI L S C introduce a second function that cacuates the SL S C NC reconstruction Tabe 5: Fiters, β,n+nr+ r ), of the approximations to f + obtained with the SL S C r,n+nr+ reconstruction + f ) n+nr+ β,n+nr+ r ) f i ) β, ) 4 β, ) 4 β, ) β, The SR decision wi be done by the SI L S C scheme, avoiding osing the ocation of jumps but enabing a east squares approximation both in f and f To sum up, initiay we set { fi L}N L i {gl i }N L i, and introduce the SL S C reconstruction in the SI L S C SR oop: The function G I i L S C x) ) is used to detect the singuarity as in section ) and then, if the jump is in[x +,x+ ], we cacuate: + f ) as in ), + f ) f i + + f ), + ḡ ) as in ) substituting 9) f by ḡ, + ḡ ) as in 7) repacing f by ḡ In the other case the jump is in [x +,x+ ] and: + f ) as in 4), + f ) f i + + f ), + ḡ ) as in 5) substituting 4) f by ḡ, + ḡ ) as in 8) repacing f by ḡ 5 Numerica experiments The purpose of this section is to show how the reconstructions presented in this paper can be used to remove noise We consider the function gx) 4x 5 sin 4x π 5 ) ) if x< 9 π, 4) sinπ 4x 5 )+ π π if 9 x We discretize gx) using 4) with N L 5+L obtaining ḡ L {ḡ L i }N L i Then we obtain f L ḡ L + n, where n is some white Gaussian noise We consider the Signa-to-Noise Ratio, expressed in db, to measure the noise of the signa: SNRg, f) : og N L i ḡl i ) N L i ḡl i f i L) Cacuated with the function awgn of MATLAB R

9 App Math Inf Sci, No, ) / wwwnaturaspubishingcom/journasasp 8 We fix SNR5dB and appy L eves of the the direct transform 8), with different reconstructions, to { f i L}N L i, obtaining { f i }N L/ L i Subsequenty, we eiminate a the detais {d } L and we appy L eves of an inverse transform 9), ) or )) obtaining { ˆf L } N L i Then, we evauate the Root Mean Squared Error, RMSE ˆf L,ḡ L ) N L ˆf i L ḡ L i N ) L i In Figure -a) we can see the resuts that we obtain with the SI L S C reconstruction Section ), using L 5, m 7 and r The noise and the Gibbs phenomenon have been competey removed Now we consider the SL S C reconstruction Section 4), with L 5, m 7 and r, where the position of the singuarity is detected with the function G I i L S C x), ) The imposition of consistency inverse transform )) causes that noise remova is reduced Figure -b)) In Figure -c) we use the SL S C NC reconstruction Section 4) with L5,m7 and r As expected, the ocation of the jump is ost, but the noise reduction improves with respect SL S C In Figure -d), we use the SI L S C L S C reconstruction Section 4) with L 5, m 7 and r Now, the position of the jump is propery detected and the noise reduction maintained Finay, these experimenta resuts are aso compared with other image fitering agorithms VISU [5] and SURE []) from Waveet Shrinage Denoising see p ej [6], [4]), obtaining Figures e) and f) From this experiment we concude that SI L S C is the best choice considering the RMSE, but SI L S C L S C is aso a good choice since it provides a cosey RMSE and visuay smoother soutions In the second experiment we want to see which parameters size of the stenci, m, and degree of the poynomia, r) are more adequate to remove noise We use SI L S C and SI L S C L S C for different noise eves from to db) and we evauate the RMSE between the origina data ḡ 5 and the reconstruction ˆf 5 The resuts that we obtain with degree r and different size of stencis m 5,7,9,, are shown in Figure -a) and c) for SI L S C and SI L S C L S C respectivey When we fix m 9 and tae different degrees, r,,4,5, we obtain the Figure -b) and d) for SI L S C and SI L S C L S C respectivey We deduce from Figure that we have better resuts with ow degree poynomias for both reconstructions Regarding the size of the stenci, for SI L S C it is not an important feature, but for SI L S C L S C better resuts are obtained with m 7 After testing with some signas, we propose that r, 7 m 5 is a genera good choice a) SI L S C ; RMSE75 5 b) SL S C ; RMSE9 5 c) SL S C NC; RMSE768 5 d)si L S C L S C ; RMSE64 5 e) VISU; RMSE85 5 f) SURE; RMSE57 Fig : We appy 5 eves of decimation and 5 eves of reconstruction without detais We use r and m7 from a) to d) Dash-dot ine represent the origina data{ḡ 5 }, the dots are the data with noise { f 5 } and the soid ine the reconstruction

10 8 F Aràndiga, J J Noguera: Reconstructions that combine ce average RMSE RMSE 4 m5 m7 m9 m m SNR 4 a) SI L S C r r r4 r5 In this third experiment we compare the resuts using the reconstructions SI L S C, SI L S C L S C both with m and r ) and SURE We decimate L eves and we eiminate a the detais {d } L Afterwards, we appy L eves of an inverse transform In Figure -a) we show the resuts that we obtain when L5 and in Figure -b) when L We observe that we obtain better resuts with the SI L S C L S C reconstruction, speciay when L 4 SILSC SILSC LSC SURE RMSE RMSE SNR 4 b) SI L S C m5 m7 m9 m m SNR a) SILSC SILSC LSC SURE RMSE RMSE SNR c) SI L S C L S C r r r4 r SNR b) Fig : Comparison between the methods SI L S C, SI L S C L S C eiminating {di }L ) and SURE a) L5, b) L SNR d) SI L S C L S C Fig : For a fixed reconstruction indicated in each picture), and different magnitude of noise we appy 5 eves of decimation and 5 eves of reconstruction without detais RMSE that we obtain is shown, with degree r and different size of stencis m 5,7,9,, for a) and c); and m 9 and different degrees, r,,4,5 for b) and d) Finay, we show in Figure 4 the imit functions of some reconstructions We appy 6 eves of reconstruction to { f i } 5 i with f i, i 7, and f 7 In a) we use I L S C and we obtain a broen ine Despite it means that it woud be inadequate for using in mutiresoution schemes, we have seen in the previous experiments that the resuts are smooth This is because we appy the reconstruction to a sufficienty smoothed signa thans to the ce average decimation In b) we use L S C ), which is consistent The output is extremey broen, due to the imposition of consistency generates peas that are transmitted and increased to the next eve With L S C NC ), Figure 4-c), the imit function is smooth since these peas are not generated

11 App Math Inf Sci, No, ) / wwwnaturaspubishingcom/journasasp 8 Acnowedgement This research was partiay supported by Spanish MTM -74 and MTM a) I L S C b) L S C c) L S C NC Fig 4: Limit functions L6, m7 and r 6 Concusions In this paper we have buit a reconstruction, in the ce average framewor, based on combining interpoation and east squares approximation Interp-L S ) verifying that it meets the requirements for use in mutiresoution schemes a a Harten We have seen how to incude the non-inear techniques ENO and SR in the Interp-L S reconstruction improving the approximation in the vicinity of discontinuities Then, we have presented a reconstruction based on east squares approximations L S ), aso in the ce average framewor In this case we do not have the consistency which is necessary in the mutiresoution schemes a a Harten We have presented an scheme that circumvents this probem Whie ENO can be appied without probems, a way to appy the SR in this context is using the function G I i L S C x), ), to detect the singuarity as in section ) We have shown how these reconstructions can be used to reduce the noise We have compared the resuts that we obtain with the methods introduced in this paper with other existing techniques We observe that in the presence of jumps the resuts we get with the methods presented in this paper SI L S C and SI L S C L S C ) are more accurate in genera References [] F Aràndiga, A Cohen, R Donat, and N Dyn Interpoation and approximation of piecewise smooth functions SIAM J Numer Ana, 4):4 57, 5 [] F Aràndiga and R Donat Noninear mutiscae decompositions: the approach of A Harten Numer Agorithms, -):75 6, [] F Aràndiga and J J Noguera Signa denoising with Harten s mutiresoution using interpoation and east squares fitting Advances in Differentia Equations and Appications SEMA SIMAI Springer Series, 4:7 45, 4 [4] Taswe C The what, how, and why of waveet shrinage denoising Comput Sci Eng, : 9, [5] D L Donoho Idea spatia adaptation by waveet shrinage Biometria, 8):45 455, 994 [6] D L Donoho Denoising via soft threshoding IEEE Transactions on Information Theory, 4:6 67, 995 [7] A Harten ENO schemes with subce resoution J Comput hys, 8:48 84, 989 [8] A Harten Mutiresoution representation of data: A genera framewor SIAM J Numer Ana, :5 56, 996 [9] A Harten, B Engquist, S Osher, and SR Charavarthy Uniformy high-order accurate essentiay nonosciatory schemes III J Comput hys, 7):, 987 [] I M Johnstone and D L Donoho Adapting to smoothness via waveet shinage Journa of the Statistica Association, 94): 4, 995 [] D Mizrachi Remoiving Noise from Discontinous Data hd thesis, Schoo of Mathematica Sciences Appied Math Department, 99 [] J J Noguera Transformaciones mutiescaa no ineaes hd thesis, Appied Math Department, Univertity of Vaencia, Francesc Arandiga was born in Canas, Spain, in 96 He received the hd degree in mathematics from the University of Vaencia, Vaencia, Spain, in 99 From 994 to he has been associate professor and since fu professor of appied mathematics at the University of Vaencia He has been head and secretary of the Department of Appied Mathematics at the University of Vaencia He has aso been coordinator of the graduate program in Computer Science and Computationa Mathematics His research interest incude image processing, mutiresoution anaysis, and waveets He has pubished more than research artices in reputed internationa journas of mathematica and engineering sciences

12 84 F Aràndiga, J J Noguera: Reconstructions that combine ce average José Jaime Noguera received the hd degree in Mathematics at Universitat de Vaència Spain) in He has pubished research artices in conferences and journas reated to appied mathematics His research interests incude Signa Denoising, Image Compression and Mutiresoution Schemes, among others He currenty wors in the fied of teaching