SELECTION OF VARYING SPATIALLY ADAPTIVE REGULARIZATION PARAMETER FOR IMAGE DECONVOLUTION

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1 SELECTION OF VARYING SPATIALLY ADAPTIVE REGULARIZATION PARAMETER FOR IMAGE DECONVOLUTION Dmitiy Paliy a, Vladimi Katkovnik a, Sakai Alenius b, Kaen Egiazaian a a Instutite of Signal Pocessing, Tampee Univesity of Technology, P.O. Box 553, Tampee, Finland, stname.lastname@tut., b Nokia Reseach Cente, Tampee, Finland, stname.lastname@tut.. ABSTRACT The deconvolution in image pocessing is an invese illposed poblem which necessitates a tade-off between - delity to data and smoothness of a solution adjusted by a egulaization paamete. In this pape we popose two techniques fo selection of a vaying egulaization paamete minimizing the mean squaed eo fo evey pixel of the image. The st algoithm uses the estimate of the squaed point-wise bias of the egulaized invese. The second algoithm is based on diect multiple statistical hypothesis testing fo the estimates calculated with diffeent egulaization paametes. The simulation esults on images illustate the efciency of the poposed technique. Intoduction A vaiety of image captuing pinciples can be modeled by the integal z(x) = R X v(x; t)y(t)dt; x; t X R ; whee v is a point-spead function (PSF) of the system, y is an image intensity function, and z is an obseved image. A natual simplication is that the PSF v is shift-invaiant which leads to the convolution opeation in the obsevation model. We conside a discete noisy appoximation of the poblem so that the obsevation z is given in the following fom: z(x) = (v ~ y)(x) + "(x); (1) whee " ~ " denotes the discete convolution, the agument x is dened on the egula n 1 n lattice, x X = f(x 1 ; x ) : x i = 0; 1; :::; n i 1; i = 1; g, and " is a andom noise. It is assumed that the noise is white Gaussian with zeo-mean and vaiance ; "(x) s N(0; ): In the D fequency domain fo the cicula convolution the model (1) takes a fom: Z(!) = V (!)Y (!) + "(!); () whee the notation Ffg is used fo the discete Fouie tansfom (DFT) Z(!) = Ffz(x)g, V (!) = Ffv(x)g; Y (!) = Ffy(x)g; "(!) = Ff"(x)g, and! W; W = f(! 1 ;! ) :! i = k i =n i ; k i = 0; 1; :::; n i 1; i = 1; g, is the nomalized D discete fequency. The estimation of y fom the obsevation z is a emoval of the degadation caused by a PSF. It is an invese poblem. Usually this poblem is ill-posed which esults in instability of the solution which, in paticula, is vey sensitive with espect to the additive noise. Stabilizing effects can be intoduced by constaints imposed on the solution. A geneal appoach to this kind of constained estimation efes to the methods of Lagange multiplies and the Tikhonov egulaization [1]. The egulaized (constained) invese (RI) lte can be obtained as a solution of the least squae poblem with a penalty tem: J = kz V Y k + ky k ; (3) whee 0 is a egulaization paamete and kk denotes Euclidean nom. Hee, the st tem kz V Y k evaluates the delity of the model V Y to the available data Z and the second tem ky k bounds the powe of this estimate. The egulaization paamete balances these two tems in the citeion J. In (3), and futhe, we omit the agument! in the Fouie tansfoms. We obtain the estimate of the image by minimizing (3): V by (x) = F 1 fy b g; Y b = jv j Z; (4) + whee the sta ( ) means the complex-conjugate vaiable. A pope selection of the egulaization paamete in (4) is a key point of the egulaization technique oveall. Thee ae numeous publications concening this poblem. Roughly speaking thee ae two types of methods: with a pio knowledge and without a pio knowledge about the noise vaiance. The L-cuve method, sometimes also called the Tikhonov cuve method, belongs to the goup of methods with no infomation on the value of (e.g. Mille [], and Tikhonov and Asenin [1]). The tech- nique uses a log log plot with the log Z V Y b as an abscise and log Y b as an odinate, with as a paamete along this cuve. The tansition between unde- and oveegulaization coesponds to the cone of the L-cuve and the coesponding value of is poposed as an optimal value of the egulaization paamete. Futhe, Hansen has developed this idea in [3] whee he has stipulated conditions when the cone exists. The cone is dened as the maximal cuvatue point of the log log plot. Methods fo detection of this point can be seen in [4], [5].

2 Figue 1. Bief scheme of the egilaization paamete selection using the poposed techniques. Figue. MSE, bias, and vaiance as a function of the egulaization paamete. Galatsanos and Katsaggelos [6] popose a technique fo selection of the asymptotically optimal egulaization paamete povided that the vaiance of noise in (1) is known. This appoach is based on calculation of the deivative of the mean squaed eo (MSE) functional. Simila idea is exploited by Nielamani et al. in [7] whee the optimal invaiant egulaization paamete is found by minimizing an uppe bound of MSE calculated in the Fouie- Wavelet domain. Wufan Chen et al. use in [8] the iteative constained total least-squae adaptive pocedue, and show its stability and convegence. The eview of the methods fo invaiant egulaization paamete selection can be found in [6] and [9]. Bege et al. in [10] popose to use a spatially vaying egulaization paamete and descibe a method based on the local weighted standad deviation analyzing the diffeence signal of the estimate. Wu et al. in [11] choose both the spatially adaptive egulaization paamete and egulaization opeato by estimation of the local noise vaiance and detecting edges in the image. Also, eview on the egulaization paamete selection methods in invese poblems can be found in [1]. In this pape we develop two methods based on minimization of the point-wise mean squaed eo. The st algoithm is based on estimation of the squaed point-wise bias. The second algoithm is based on diect multiple statistical hypothesis testing fo the estimates calculated with diffeent egulaization paametes. The intesection of the condence inteval (ICI) ule is used in this testing [13], [14]. The stuctue of the pape is as follows. In Section 1 we popose two techniques fo egulaization paamete selection. Results of simulations ae given in Section, and nally conclusions ae done. 1. SPATIALLY ADAPTIVE REGULARIZATION TECHNIQUES The MSE of the estimato ^y (x) ; 8x X; can be decomposed as a sum of the vaiance of the andom component and the squaed bias (eo of the deteministic component): MSE (; x) = E e (x) = bias (; x) + va () ; (5) whee e(x) = y(x) ^y (x): Accoding to the Paseval theoem the vaiance tem is computed as va() = n 1 n kh V k, (6) whee H V = V jv j + : It is a monotonically deceasing function of with va ()! 0 as! 1. The bias tem bias (; x) depends on the unknown tue signal y(x) and calculated as bias (; x) = = ((y ~ h ) (x)) ; whee h = F 1 fh g; H = jv j +. Unde quite mild assumptions on v and y fo small, bias (; x)! 0 as! 0, and fo lage, bias (; x)! y (x) as! 1. Assuming a monotonic inceasing of the bias as a function of we aive to the cuves fo the vaiance and the bias shown in Figue. In this case always thee is a unique minimum value of the MSE achieved at + (x) = ag min (MSE (; x)). Ou goal is selection of the values of the egulaization paamete close to the optimal + (x). In geneal this optimal value is diffeent fo diffeent x as the bias depends on the estimated signal y(x). Thus, the vaying point-wise adaptive selection of the egulaization paamete has the point-wise optimization of the accuacy of the image econstuction as a main motivation. In Section 1.1. Bias estimation appoach In this appoach we evaluate the citeion (5) and use its minimization fo estimation of the optimal egulaization paamete + (x). Povided a given value of the vaiance va () in the citeion (5) can be easily calculated fo any. The situation is much moe difcult fo the bias as it is signal-dependent. n It can be veied that E F 1 Z o H V H n 1n V = ((y ~ h ) (x)). Inseting this expession fo the squaed bias in (5) we epesent the citeion

3 Oacle Bias ICI Oacle Test images invaiant estimation ule vaying Cheese Cameaman Lena Babaa Boat Table 1. ISNR values in db fo RI technique. in the fom MSE(; x) = E ( ) F 1 Z H + V + n 1 n kh V k H n 1 n V : Then a natual andom and unbiased estimate of the citeion is MSE(; x) ' F 1 Z H + (7) V + kh V k H n 1 n n 1 n V ; whee the ight hand side can be calculated as it depends only on the obseved signal z. Howeve, it is not so simple as the invese of V used in (7) is quite simila to the oiginal invese poblem in (1)-() and we have hee the same calculating difculties. It is natual to eplace this invese by the egulaized invese (4) and ewite (7) as follows MSE (; x) ' F 1 V Z H jv j + + n 1 n kh V k n 1 n H V jv j + : (8) Hee is the egulaization paamete inuencing the accuacy of the appoximation of MSE (; x) by the ighthand side of (8). The new egulaization actually means that the bias eo calculated oiginally as bias (; x) = F 1 fh Y g is eplaced by F 1 jv j fh jv j +Y g. Fo small this change is not signicant fo the bias and fo the following calculations as it has been veied by simulation. Replacing the citeion MSE (; x) by the estimate we calculate the estimate of + (x) as follows " F 1 ^ + (x) = ag min n 1 n kh V k V Z H jv j + + # H V n 1 n jv j +. (9) In pactice the estimates by (x) ae calculated fo the set of egulaization paametes R = f 1 ; ; :::; N g, whee 1 < < ::: < N. Using these N estimates the minimization of citeion function (9) yields the value ^ + fo evey pixel x. In pactice, it is implemented as follows. The estimates by (x) ae calculated fo a set of egulaization paametes R = f 1 ; ; :::; N g accoding to (4) (Fig.1). Using these N estimates, the minimization of citeion function (9) yields the value + fo evey point x (block Citeion fo selection in Fig.1). The nal estimate by + (x)(x) is built using coesponding estimates. 1.. ICI ule appoach In this appoach selection of the egulaization paamete is based on compaison of the signal estimates calculated with diffeent values of the egulaization paamete. The ICI concept developed oiginally fo the window size selection in signal/image denoising is descibed in a numbe of publications [13], [14], [15]. It gives the window size close to its ideal value minimizing the coesponding MSE. Fist, we wish to note that the ICI ule is applicable fo selection of the egulaization paamete. Indeed, the output by (x) of the egulaized invese lte is the estimate of the signal y. As it is illustated in Fig. the bias and the vaiance of this estimate ae inceasing and deceasing functions of the egulaization paamete, espectively. This behavio of the bias and the vaiance ae the basic assumptions used in deivation and justication of the ICI ule fo the window size selection [15], [16]. It follows that application of the ICI ule to the estimate by (x) with vaying theshold paamete (x) allows to nd the value of the egulaization paamete close to the one minimizing the citeion (5). The idea of the ICI ule is as follows. The estimates by (x) ae calculated fo R. The adaptive paamete ^ + is dened as the lagest of those in R which estimate does not diffe signicantly fom the estimates coesponding to the smalle values of. This idea is implemented as follows. We conside a sequence of condence intevals D s = by s (x) s ^y s (x); by s (x) + s ^ys (x) ; whee s = 1; ::; N; s > 0 is a paamete and ^ys (x) is the standad deviation of the estimate by s, calculated accoding to (6) as ^ys (x) = p va(). The ICI ule is stated as follows: conside the intesection of the condence intevals I s = T s i=1 D i and let s + be the lagest of

4 the indices s fo which I s is non-empty. Then the adaptive egulaization paamete + is dened as ^ + = s + and the coesponding adaptive estimate is by + (x). This pocedue is epeated fo each x and, in this way, the adaptive + (x) is spatially vaying. In geneal, the ICI allows the maximum degee of smoothing the estimate stopping befoe ovesmoothing begins [17]. The paamete s is an impotant element of the algoithm as it says when a diffeence between estimate deviations is lage o small. Too lage value of this paamete leads to signal ovesmoothing and too small value leads to undesmoothing. Usually the ICI ule is used with a constant s fo all s intevals. Optimization of s can be poduced fom some heuistic and theoetical consideations (e.g. [13], [14], [15], [18]). In this pape we use a vaying paametes s with diffeent values fo the intevals D s ; s = 1; ::; N: It is woth to note that this set of the paametes s is xed fo all ou expeimental tests.. SIMULATIONS The esults ae obtained by the Monte-Calo simulation (100 uns) fo the following test images: Cameaman of the size 5656 (Fig.3a), Lena 5656, Babaa 5151, Boat 5151, and Cheese Cheese is a black-and-white image with two egions of constant values of the image intensity. The blu is dened as the 99 boxca point-spead function (mean lte). The level of the Gaussian noise is calculated as the blued signal-to-noise atio (BSNR) 0 1 BSNR=10 log 1 1 X v ~ y (v ~ y)(x) A n 1 n xx equal to 40 db in ou expeiments. A selection of the set R = f 1 ; ; :::; N g of the egulaization paamete values is a special poblem as it should cove an inteval including the optimal value. In geneal, these optimal value depends on the image, the point-spead function and the noise level. The following constuction denes the set R in the manne univesally applicable fo vaiety of scenaios: i = i n 1 n kv(x)k 1 Pf jz(f)j n 1 n ; i = 1; :::; 5; (10) whee f1:3; :3; 6:3; 9:6; 30g and kv(x)k 1 = = ( P x jv(x)j). It is employed fo all test images used in ou simulation expeiments and simila to implementation poposed in [7]. The citeion used to evaluate the pefomance of the poposed techniques is impoved signal-to-noise atio (ISNR): ISNR = 10 log 10 ky zk ky byk db: The esults of statistical Monte Calo expeiments ae pesented in Table 1. The st column gives the names of the test images. The thid and the foth columns gives ISNR values obtained by the bias estimation (9) and by the ICI ule, espectively. The paamete in (9) is set to be 0:001. We compae these esults with the efeence ideal (oacle) esults which coespond to minimization of MSE by selection of the optimal invaiant (single fo all x) and the optimal vaying (optimal fo each x) egulaization paamete assuming that the tue image y is known. The column Oacle invaiant of Table 1 contains the values of ISNR obtained using the optimal invaiant egulaization paamete inv = ag min R Px (y(x) by (x)) : The column Oacle vaying contains the values of ISNR obtained using the vaying egulaization paamete va (x) = ag min R (y(x) by (x)). The set of theshold paametes s used in the ICI ule is f0:86; 0:5; 0:8; 0:9; 1:1g : The esults given in the table allow the following conclusions. Fist of all, the both adaptive estimates show the esults which ae bette than those obtained by the oacle invaiant estimate. i.e. the estimates with the vaying adaptive egulaization paamete wok bette that the ideal estimate with the best possible selection of the invaiant egulaization paamete. Compaison of the citeion minimization method vesus the ICI ule is in favo of the latte fo all test images. Howeve, we can see that the vaying oacle estimates demonstate much bette pefomance in compaison with all othe estimates. A diffeence in pefomance of this appoach vesus the estimate with the invaiant oacle selection conms a geneal motivation of this pape to impove the estimation by using the vaying egulaization paamete. In the same time the esults obtained by the adaptive egulaization paamete estimates ae quite fa fom the vaying oacle ones. It says that thee ae esouces fo futhe impovement of the adaptive vaying egulaization paamete algoithms. Visually the effects of point-wise egulaization ae demonstated in Fig.3c,d, whee the esults of Monte-Calo modeling ae shown fo the blued Cameaman image (Fig.3b). These images show the mean values of the egulaization paametes fo diffeent pixels of the Cameaman image. The vaying oacle values of the egulaization paamete va ae small nea edges while lage values coespond to the smoothe aeas of the image (Fig.3c). The mean values of the vaying adaptive egulaization paamete selected by the ICI ule ae shown in Fig.3d. Compaing the images Fig.3c and Fig.3d we note thei similaity. In paticula, the ICI ule gives smalle values of the egulaization paamete nea the edges and lage fo smoothe aeas as the oacle estimato does. The adaptive values of the egulaization paamete accuately delineate the edges of the image similaly to Fig.3c. 3. CONCLUSIONS Two novel appoaches fo spatially adaptive selection of the egulaization paamete fo deconvolution poblems wee poposed in this pape. The st algoithm is based on the bias estimation and MSE minimization fo evey point of an image. The second one uses the statistical ICI ule. The algoithm based on the ICI ule demonstated

5 a) c) b) d) Figue 3. Results obtained by Monte-Calo (100 uns) silmulations: a) tue Cameaman test image; b) Cameaman test image blued with 9x9 boxca PSF and noisy (white Gaussian noise) with BSNR=40dB. c) values of the vaying oacle egulaization paametes; d) vaying egulaization paametes obtained by the ICI ule. bette pefomance than the algoithm using the bias estimate. Compaison of the poposed adaptive techniques' vesus the optimal (oacle) invaiant selection technique fo vaious images showed that the poposed techniques pefoms bette than the estimate with the best possible invaiant egulaization paamete. 4. ACKNOWLEDGMENTS This wok was suppoted by the Finnish Funding Agency fo Technology and Innovation (Tekes) and by the Academy of Finland, poject No (Finnish Cente of Excellence pogam ( ). 5. REFERENCES [1] Tikhonov, A. N., Asenin, V. A.: Solutions of Ill-posed Poblems. Winston & Sons, Washington (1977). [] Mille. K.: Least squaes methods fo ill-posed poblems with a pediscibed bound. SIAM J. Math.Anal., vol. 1, (1970), [3] Hansen, P.C.: Analysis of discete ill-posed poblems by menas of the L-cuve. SIAM Rev., vol. 34, (199), [4] Hansen, P.C., O'Leay, D.P.: The use of the L-cuve in the egulaization of discete ill-posed poblems. SIAM J. Sci.Comput., vol. 14, no. 6, (1993) [5] Oaintaa. S., Kal, W.C., Castanon, D.A., Nguyen, T.Q.: A method fo choosing the egulaization paamete in genealized Tikhonov egulaized linea invese poblems. Intenational Confeence on Image Pocessing 000, Poceedings, vol. 1, 10-13, (000) [6] Galatsanos, N.P., Katsaggelos, A.K.: Methods fo choosing the egulaization paamete and estimating the noise vaiance in image estoation and thei elation. IEEE Tans. on Image Pocessing, vol. 1, issue 3, (199) [7] Neelamani, R., Choi, H., Baaniuk, R.G.: Fowad: Fouie-wavelet egulaized deconvolution fo illconditioned systems. IEEE Tans. on Signal Pocessing, vol. 5, issue, (003)

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