Application of the pseudoinverse computation in reconstruction of blurred images

Transcription

1 Fiomat 26:3 (22), DOI 2298/FIL23453M Pubished by Facuty of Sciences and Mathematics, University of Niš, Serbia Avaiabe at: Appication of the pseudoinverse computation in reconstruction of burred images Sadjana Mijković a, Marko Miadinović a, Predrag Stanimirović a, Igor Stojanović b a University of Niš, Department of Mathematics and Informatics, Facuty of Science, Višegradska 33, 8 Niš, Serbia b University of Goce Decev, Facuty of Computer Science, Macedonia Abstract We present a direct method for removing uniform inear motion bur from images The method is based on a straightforward construction of the Moore-Penrose inverse of the burring matrix for a given mathematica mode The computationa oad of the method is decreased significanty with respect to other competitive methods, whie the resoution of the restored images remains at a very high eve The method is impemented in the programming package MATLAB and respective numerica exampes are presented Introduction Recording images is a very frequent event in everyday human ife However, the recorded images are degraded with respect to the origina images The question of removing these imperfections, in many scientific areas such as sateite imaging, medica imaging, astronomica imaging etc, is of crucia importance Some of the reasons for having different types of degradations in images are the phenomena described as: bur, noise, geometrica degradations, iumination or coor imperfections A number of outstanding overview artices, journa papers and textbooks dea with the probem of image restoration [, 3, 3, 6, 7] In the present artice, we investigate the probem of removing the bur from images, caused by a uniform inear motion Our assumptions are that the inear motion corresponds to an integra number of pixes, and it is aigned with the horionta (or vertica) samping We are concentrated on the usage of the Moore- Penrose inverse soution of a given matrix equation which represents a mathematica mode of the uniform inear motion bur The methods of image restoration, based on the usage of the Moore-Penrose inverse, have been expoited in many recent papers [6 8] Severa methods for computing the Moore-Penrose inverse have been introduced in [2] One of the most commony used methods, is the method of Singuar Vaue Decomposition (SVD) This method is very accurate but aso time-intensive since it requires a arge amount of computationa resources, especiay in case of arge matrices An agorithm for fast computation of the Moore-Penrose inverse is aso presented in the recent work of P Courrieu [9] Courrieu s agorithm is 2 Mathematics Subject Cassification Primary 68U; Secondary 94A8 Keywords Image restoration, X-ray, motion bur, matrix equation, Moore-Penrose inverse Received: 29 September 2; Accepted: November 2 Communicated by Dragan S Djordjević Research supported by the Serbian Ministry of Science, research project No 743 Emai addresses: sagana256@yahoocom (Sadjana Mijković), markomiadinovic@gmaicom (Marko Miadinović), pecko@pmfniacrs (Predrag Stanimirović), igorstojanovik@ugdedumk (Igor Stojanović)

2 S Mijković et a / Fiomat 26:3 (22), based on the reverse order aw for matrix pseudoinverse (eq 32 from [9]), and on the fu-rank Choesky factoriation of possiby singuar symmetric positive matrices Another very fast and reiabe method for estimation of the Moore-Penrose inverse of fu rank rectanguar matrices is given by V Katsikis and D Pappas [4] The method uses a specia type of tensor product of two vectors A methods for computing the Moore-Penrose inverse, mentioned above, are either iterative or use some kind of matrix factoriation The method we propose, expore the structure of the degradation matrix of the mode and generates the Moore-Penrose inverse anayticay, by means of a set of rues The motivation behind, is the very proper structure of the matrix which participates as a degradation system in the image formation process The introduced method is very fast, which is its main advantage On the other hand, the main disadvantage of the proposed method is its imitation to uniform inear motion bur degradations The presented numerica resuts caim the expected decrease in CPU time The paper is organied as foows In the second section we present the ideas behind the processes of image restoration which are based on the usage of the Moore-Penrose inverse In the third section we present a new method for reconstruction of burred images, by generating the Moore-Penrose inverse of the mode matrix anayticay Finay in the ast section, by reporting numerica resuts, we observe certain enhancement in the computationa time and the Improvement in Signa to Noise Ration (ISNR), compared to other standard methods for image restoration 2 Preiminaries The matrix A which satisfies the foowing properties AA A = A, A AA = A, (AA ) T = AA, (A A) T = A A, is caed the Moore-Penrose inverse of the matrix A This section refers to the formation of the mathematica mode that refects the process of removing the bur in images, which is caused by a uniform inear motion as we as the matrix pseudoinverse soution of the probem Suppose that the matrix F R r m corresponds to the origina image with picture eements f i,j, i =,, r, j =,, m and G R r m with pixes i,j, i =,, r, j =,, m, is the matrix corresponding to the degraded image Let be an integer indicating the ength of the inear motion bur in pixes and n = m + In practice the degradation (index ) is rarey known exacty, so that it must be identified from the burred image itsef To estimate the index, two different cepstra methods can be used: one dimensiona or two dimensiona cepstra method [5] To avoid the probem when the information from the exact image spis over the edges of the recorded image, we suppement the origina image with boundary pixes that best refect the origina scene Without any confusion we are using the same symbo F for the enarged origina image (matrix) with remark that F now becomes a matrix of dimensions r n First, we suppose that the burring is a horionta phenomenon Let us denote the degradation matrix by H R m n For each row f i of the matrix F and the respective row i of the matrix G we consider an equation of the form T i = H f T i, T i R m, f T i R n, H R m n () The objective is to estimate the origina image F, row by row, using the corresponding rows of the known burred image G and a priori knowedge of the degradation phenomenon H Equation () can be written in the matrix form as G = ( HF T) T = FH T, G R r m, H R m n, F R r n (2) There is an infinite number of exact soutions for f that satisfy the equation () But, ony the Moore- Penrose inverse soution soves uniquey the next minimiation probem (see, for exampe [2]): min f 2, subject to min H f 2 (3)

3 S Mijković et a / Fiomat 26:3 (22), The unique vector f satisfying (3) represents a row of the restored image [5 8], and it is defined by f = H (4) The matrix form of the equation (4) ie, the restored image F is given by F = G(H ) T (5) The matrix F defined in (5) is the minimum-norm east-squares soution of the matrix equation (2) The matrix equation which characteries the vertica motion burring process is given by G = HF, G R r m, H R r n, F R n m, n = r + (6) The corresponding restored image can be computed using the Moore-Penrose inverse by the foowing formua F = H G (7) We assume that the burring is a oca phenomenon, spatiay invariant as we that the imaging process captures a ight and no additiona noise is incuded Taking into consideration the given requirements, the degradation matrix of the burring process reduces to a matrix H = toepit(h, h ) The matrix H is non-symmetric Toepti matrix consisting of m rows and n = m + coumns, determined by its first coumn h = (h i, ) m i= and its first row h = (h,j ) n as foows: j= h i, = { /, i =,, i = 2,, m, and h,j = { /, j =,,,, j = +,, n (8) An arbitrary ith row of the burred image can be expressed using the ith row of the origina image extended with the boundary pixes as i, i,2 i,3 i,m = f i, f i,2 f i,3 f i,n, (9) where eements of the vector f i, are not actuay the pixes from the origina scene; rather they are boundary pixes How many boundary pixes wi be added above the vector f depends of the nature and direction of the movement However, the rest of them, ie, minus the number of pixes added above the vector f, woud present the boundary pixes right of the horionta ine, and are added beow the vector f [] The process of non uniform burring assumes that the burring of the coumns in the image is independent with respect to the burring of the rows In this case two matrices participate in the formation of the process and the reation between the origina and the burred image can be dispayed with the foowing reation G = H c FH T r, G R m m 2, H c R m r, F R r n, H r R m 2 n, () where n = m 2 + r, r = m + c, r is the ength of the horionta burring in pixes and c is the ength of the vertica burring in pixes In this case, the Moore-Penrose soution of the system () is given by F = H c G(H T r ) ()

4 3 New image restoration method S Mijković et a / Fiomat 26:3 (22), First, we define a method of image restoration in the case when the number of coumns of the image, enarged by boundary pixes, can be divided by an appropriate number of burring pixes, ie, when the equaity n = p hods We show that in this case the Moore-Penrose inverse H can be generated anayticay, without any iterations Later, we generaie the method to the case when the dimension n is arbitrary Let us suppose that n = m + = p, where the number of burring pixes is a positive integer In the rest of the section, we construct the matrix H = [ h ij ], i =,, n, j =,, m and show that it is actuay the Moore-Penrose inverse of the degradation matrix H A eements of the matrix H, excuding ero eements, can be represented by the foowing two sequences: x k = (k ) (m (k ) ) = m, n p k =, 2,, p, y k = m (k ) (m (k )) =, n p k =, 2,, p Additionay, we put = y p = p The genera ayout of the matrix H C n n in the case n = m + = p is given in Figure 3 B B 2 - B 3 upper ine B p- m m C p- ower ine C 3 C 2 - C Figure 3 Genera ayout of the matrix H in the case n = m + = p The paraeogram between the bocks B k and C k is caed ero ayer The ine denoted by upper ine refers to the eements above the ero ayer of the matrix H which actuay constitute the diagona of the square m m matrix formed from the first m rows of the matrix H Simiary, ower ine refers to the eements beow the ero ayer of the matrix H which constitute the diagona of the square m m matrix formed from the ast m rows of the matrix H The upper ine, the ower ine, the first eements of the first row of H and the ast eements of the ast coumn of H wi be denominated as sides of the ero ayer

5 S Mijković et a / Fiomat 26:3 (22), Further, we preview the structure of the bocks B k, k =,, p Each bock B k can be represented via appropriate bock of the foowing form: = x k x k x k x k x k x k x k x k + x k y k+ Zero paraeogram bocks are of dimensions ( 2) ( ) The bock B k is obtained by pasting up k times the bock, from the bottom upwards, and then from the resuting bock we cut by horionta ine the most upper triange which has a vertica side containing the upper 2 eements of the bock The missing eement of the resuting bock B k is fied by the vaue of y k+ The three steps in the formation process of the bocks B k are presented in Figure 32 Cutting ine y k+ (k th) (k th) (k th) (2nd) (2nd) (2nd) = B k ( st) ( st) ( st) st step 2nd step 3rd step Figure 32 Formation of the bock B i from the bock P i In order to write the anaytica form of the matrix H we wi use the foowing notation Let q i, q j, r i and r j be integers such that for a given ith row and jth coumn hod i = q i + r i and j = q j + r j From the previous considerations it is cear that: i j if and ony if h ij B k, k =, 2,, p Simiary, i and i > j if and ony if h ij C k, k =, 2,, p Taking this into account, we present the anaytica form of the matrix H as foows

6 S Mijković et a / Fiomat 26:3 (22), y qj +, i j, r j =, r i =, + x qj, i j, r j =, r i =, ( ) d+ x qj +, i j, r j, q j q i, (r i j = or r i j = ) hij = + x p qj, i, i > j, r j =, r i =, y p qj, i, i > j, r j =, r i =, ( ) d x p qj, i, i > j, r j, q j q i, (r i j = or r i j = ) (2), r j =, r i, r i, otherwise, where d is if r i j = and d = otherwise The first case in (2) gives the eements that are not equa to ero or of the bocks B k, k =,, p pus y from the first coumn The second case produces the eements that are not equa to ero or of the bocks C k, k =,, p pus y from the ast coumn In order to store the whoe matrix we need ony to store 2p eements which is ower than n in the case > 2 To iustrate our description we give the fu form of the matrix H observing the case p = 4 y x y 2 x 2 y 3 x 3 y 4 x x x 2 x 2 x 3 x 3 x x x 2 x 2 x 3 x 3 x x x 2 x 2 x 3 x 3 y 4 x + x x 2 + x 2 x 3 + x 3 + x 3 x 3 y 2 x 2 y 3 x 3 y 4 x 3 x 3 x 2 x 2 x 3 x 3 x2 x 2 x 3 x 3 x 3 x 3 x 3 x 3 x 2 x 2 x 3 x 3 y 4 x 3 y 3 x 2 + x 2 x 3 + x 3 + x 3 x 3 + x 2 x 2 y 3 x 3 y 4 x 3 x 3 x 2 x 2 x 3 x 3 x3 x 3 x 3 x 3 x 2 x 2 x 3 x 3 x 2 x 2 x 3 x 3 y 4 x 3 y 3 x 2 y 2 x 3 + x 3 + x 3 x 3 + x 2 x 2 + x x y 4 x 3 x 3 x 2 x 2 x x x 3 x 3 x 2 x 2 x x x 3 x 3 x 2 x 2 x x y 4 x 3 y 3 x 2 y 2 x y

7 S Mijković et a / Fiomat 26:3 (22), In order to verify that the matrix H is actuay the Moore-Penrose inverse of the matrix H, ie, that H = H we wi use the foowing two emmas: Lemma 3 The equaity H H = I hods for the matrix H given by (2) Proof Each row of the matrix H contains non ero constant eements equa to /, so that it is obvious that the eements of the matrix H H are (H H) ij = i+ hsj, i =,, m and j =,, m s=i Therefore, we need to expore the properties of jth coumn of the matrix H, j =,, m, ie, the sums of its consecutive eements For this reason, from the representation of the matrix H given by (2) we distinguish two different cases: case : r j = Each set of consecutive eements contains ony two eements different from If both of them are above the ero ayer their sum is y qj + + ( ) + x qj = If both of them are beow the ero ayer their sum is x p qj + ( ) + y p qj = Otherwise, if one of them is above the ero ayer and the other one is beow the ero ayer, then their sum is y qj + + y p qj = Unfortunatey, as we mentioned before, the ast is the case ony when i = j 2 case : r j Each set of consecutive eements contains ony two non ero eements If those two eements are above the ero ayer or both of them are beow the ero ayer then it is obvious that their sum is If one of them is above the ero ayer and the other one is beow the ero ayer, thus i = j, their sum is x qj + x p qj, which by easy cacuations can be shown that equas So finay for the both cases we have (H H) ij = i+ {, i = j hsj =, i j s=i, i =, m, j =, m ie H H = I Lemma 32 The equaity ( HH) T = HH hods for the matrix H given by (2) Proof For a given i =,, n and j =,, n, we shoud show that ( HH) ij = ( HH) ji The eements of the matrix HH can be presented as ( HH) ij = min {j,m} s=max {,j +} Let us denote by s = min {j, m} max {, j + } his, i =,, n and j =,, n (3) Consequenty, we shoud show that the sum of s consecutive eements in the ith row of the matrix HH where the ast eement is in the jth coumn; equas the sum s consecutive eements in the jth row of the matrix HH, where the ast eement is in the ith coumn So the case when i = j is cear, actuay, for a given i these eements actuay present the sum of the s consecutive eements that beong to the ero ayer as we as his sides We continue with the opposite case when i j Let us expore the properties of each row i of the matrix H, i =,, n First we reca that if i j that means that h ij is either y or beongs in a bock B k, k =,, p And, if i > and i > j that means that h ij is either y or beongs in a bock C k

8 S Mijković et a / Fiomat 26:3 (22), case : i < j, r i = and r j (i =,, n and j =,, n) From (3) and definition of the matrix H we have ( HH) ij = (y q j + + x qj +) = That means that the sum of each s consecutive eements equas, if the eements are in the (k + )st row, k =,, p, and the ast eement is not in the (k + )st coumn, k =,, p Aso since i < j the ast eement is above the diagona of the square matrix constituted of the first m rows of H We need to compare these vaues with the vaues of ( HH) ji For this situation we anaye the opposite case ie we interchange the conditions for i and j Suppose, i > j, r i and r j = (i =,, n and j =,, n ) From here we continue with two different possibiities, denoted by a and a2 a: if r i = then a2: if r i then ( HH) ij = {, j = + x p qj x p qj =, j > ( HH) ij = ( x q j + + x qj ) = Thus the first case is competed ie if i < j, r i = and r j (i =,, n and j =,, n) then ( HH) ij = ( HH) ji 2 case : i < j, r i = and r j = (i =,, m and j =,, m) then ( HH) ij = y qj + + x qj = ( ) Note: Since m = (p ) + and n = m +, if j > m it foows that r j If the conditions i > j, r i = and r j = (i =,, m and j =,, m ) are satisfied we obtain { + xp = ( ), j = ( HH) ij = + x p qj x p qj = ( ), j, which competes the proof in the case 2 3 case : i < j, r i and r j = (i =,, m and j =,, m) then ( HH) ij = ( x q j + + x qj ) = Under the assumptions i > j, r i = and r j (i =,, m and j =,, m ) one can verify ( HH) ij = ( + x p q j x p qj ) =, so that the verification of the statement in case 3 is competed 4 case : i < j, r i and r j (i =,, n and j =,, n) Simiary as in the previous cases, after considering severa possibiities, one can derive the foowing ( HH) ij = { ( ), ri = r j, i m, otherwise Under the opposite assumptions i > j, r i and r j (i =,, n and j =,, n ) we get ( HH) ij = { ( ), ri = r j, j m, otherwise, and the proof is competed

9 S Mijković et a / Fiomat 26:3 (22), Theorem 3 The matrix H given by (2) is the Moore-Penrose inverse of the matrix H Proof Since the matrix H is fu row rank matrix its Moore-Penrose inverse is its right inverse From this fact and from the previous two emmas foows the proof of the theorem Agorithm 3 Image deburring method (MP method) Require: The burred image G of dimensions r m defined in the burring process (9) : If m + mod then add quotient(m +, ) + m boundary pixes, ese add boundary pixes 2: Compute the matrix H according to the formua (2) 3: Appy formua (5) 4: Return F 4 Experimenta resuts In this section we present numerica resuts which are obtained by testing the method proposed in Agorithm 3 (MP method) on X-ray images In order to confirm the efficiency, we compared our method with three recenty announced methods for computing the Moore-Penrose inverse of the matrix H Summariing, the foowing four methods are compared: The MP method, 2 Pappas method, defined by the MATLAB function ginvm from [4], 3 Pappas2 method, defined by the MATLAB function qrginvm from [8], 4 Courrieu method from [9] The experiments are done using Matab programming anguage [2] on an Inte(R) CPU 86 GH 87 GH 32-bit system with 2 GB of RAM memory Tests are made for severa images of dimensions r m The index that takes vaues between and is the varying parameter for a given image Aso we compared the efficiency of four different strategies of image restoration: the approach based on the Moore-Penrose inverse, the Wiener fiter (WF), the constrained east-squares (CLS) fiter, and the Lucy-Richardson (LR) agorithm For the impementation of the Wiener fiter, the constrained east-squares fiter, and Lucy-Richardson agorithm we used buit in functions from the Matab package [] In image restoration the quaity enhancement of the restored image over the recorded burred one is evauated by the signa-to-noise ratio improvement (ISNR) The ISNR of the recorded (burred) image is defined in decibes as foows: ( n,n2(g(n, n 2 ) F(n, n 2 )) 2 ) ISNR = og (4) n,n2( F(n, n 2 ) F(n, n 2 )) 2 Figure 4 presents one practica exampe for restoring burred X-ray image The image is taken from the resuts obtained from Googe Image search with the keywords X-ray image The picture in Figure 4 caed Origina image shows the origina X-ray image The image is divided into r = 843 rows and m = 5 coumns To prevent oosing information from the boundaries of the image, we assumed ero boundary conditions, which impies that vaues of the pixes of the origina image F outside the domain of consideration are ero (back) This choice is natura for X-ray images since the background of these images is back The picture named as Degraded image presents the degraded X-ray image with a uniform horionta motion of ength = 8 The pictures named as Moore-Penrose Inverse, Wiener Restored Image, Constrained LS Restored Image and Lucy-Richardson Restored Image denote the images obtained after the appication of the corresponding restoration agorithms From Figure 4, it is cear that the MP method produces the best resut

10 S Mijković et a / Fiomat 26:3 (22), Figure 4 Remova of bur, caused by a uniform horionta motion The difference in quaity of the restored images among the three methods (WF, CLS and LR) is insignificant, and can hardy be seen by human eye For this reason, the ISNR is appied in order to compare the quaity of the restored images Figure 42 (eft) shows the corresponding ISNR vaue of the restored images as a function of for the MP method and the other mentioned cassica methods This figure iustrates that the MP method for image restoration has the best quaity of the restored image with respect to the other methods Moore Penrose inverse Wiener fiters Constrained east squares fiter Lucy Richardson Agorithm ISNR (db) 5 t(sec) Proposed Method Pappas Method Pappas2 Method Courrieu Method (pixes) (pixes) Figure 42 (eft) Improvement in signa-to-noise-ratio vs ength of the burring process (right) Computationa time vs ength of the burring process As we have aready mentioned, the main advantage of the proposed method for computing the Moore- Penrose inverse, is the time required to obtain the restored image compared to other methods for computing the Moore-Penrose inverse Figure 42 (right) shows the corresponding computationa time, denoted by t(sec), as a function of < pixes for the MP method and the other three considered methods for computing the Moore Penrose inverse Obviousy, the minima computationa time is reached by the MP method Additionay, we compare the method based on the usage of the Moore-Penrose inverse with another image restoration method, which uses a east-squares soution of the foowing system, arising from (9), i,j = k= f i,j+k, i =,, n, j =,, m, (5) where n = m + The corresponding soution is derived by using the standard Matab function mrdivide(), and it wi be caed LS soution in the test exampes The function mrdivide(b,a) (or equivaenty

11 S Mijkovic et a / Fiomat 26:3 (22), B/A) performs matrix right division (forward sash) [8] Matrices B and A must have the same number of coumns If A is a square matrix, B/A is roughy the same as B*inv(A) If A is an n n matrix and B is a row vector with n eements, or a matrix with severa such rows, then X = B/A is the soution of the equation XA = B computed by using Gaussian eimination with partia pivoting If B is an m n matrix with m = n and A is a coumn vector with m components, or a matrix with severa such coumns, then X = B/A is a soution in the east-squares sense to the under- or overdetermined system of equations XA = B In other words, X minimies norm(a*x - B) The rank k of A is determined from the QR decomposition with coumn pivoting The computed soution X has at most k nonero eements per coumn If k < n, this is usuay not the same soution as X = B*pinv(A), which returns a east-squares soution with the smaest norm X The presented resuts show that using the pseudo inverse approach eads to better improvements than soving the system directy The respective comparison is iustrated in the next figure The eft image in Figure 43 shows the restoration determined by using the soution of the system (5), whie the right image shows the restoration based on the usage of the Moore-Penrose inverse Figure 43 a) Restoration arising from the LS soution; b) Restoration based on the Moore-Penrose inverse The next two exampes refer to the case when, in the inception, the image is degraded by incuding an image noise and ater it is foowed by a uniform inear bur The corresponding mathematica mode generaies the horionta burring process presented with (2) and it becomes GN = (F + N)HT = FN HT, (6) where N is an additive noise and GN is the burred noisy image To obtain approximation of the origina image, we appy two steps: Cacuate the restored matrix FN by using (5), to produce F N = GN (H )T ; 2 Obtain the restored image F by appying fitering process on the image F N obtained in Step Simiary, we can formuate a process in the case of having two ways degraded image (with noise and vertica bur) The noisy image, the burred noisy image and the restored images obtained by using different methods are presented in Figure 44 Figure 44 Remova of bur, caused by noise and uniform horionta motion with r = 843, n = 5 and = 4

12 S Mijković et a / Fiomat 26:3 (22), The resuts for ISNR and the peak signa-to-noise ratio (PSNR) [4] for the origina image given in Figure 4, are presented in Figure 45 The origina image is degraded in two ways: by sat and pepper (white and back) noise with noise density of 3 and after that it is burred by a uniform inear motion with pixes The 2-D median fitering is used for the image restoration The image restoration procedures based on the Moore-Penrose inverse, appied to images degraded by both the motion bur and noise, are more reiabe and accurate compared to other image restoration methods Moore Penrose inverse Wiener fiters Constrained east squares fiter Lucy Richardson Agorithm ISNR (db) Moore Penrose inverse Wiener fiters Constrained east squares fiter Lucy Richardson Agorithm PSNR (db) (pixes) (pixes) Figure 45 (eft) Improvement in signa-to-noise-ratio vs ength of the burring process (right) Peak signa-to-noise ratio vs ength of the burring process The ISNR and PSNR vaues presented in Figure 45 indicate that MP method is the best among the other methods Simiar numerica resuts are generated when the image is burred by a non-uniform motion detrmined with the reations () and () The case when the image is burred by a non-uniform burring with parameters c = 35 and r = 25, is presented in Figure 46 Figure 46 Remova of bur, caused by the non-uniform burring mode with c = 35 and r = 25 A confirmation that the proposed MP method is faster than the other methods for computing the Moore-Penrose inverse is iustrated in Figure 47

13 S Mijković et a / Fiomat 26:3 (22), t(sec) Proposed Method Pappas Method Pappas2 Method Courrieu Method r (pixes) Figure 47 Computationa time versus variabe ength r and c = 25 of the burring process The resuts presented in Figure 47 are made on an Inte(R) Core(TM) i5 CPU 227 GH 64/32-bit system with 4 GB RAM memory 5 Concusions We introduce a new method for restoration of images which are burred by a uniform inear motion The method is based on the usage of the Moore-Penrose inverse soution of the matrix equation which presents a mode of the motion bur Our method expoits the structure of the burring matrix and generates its pseudoinverse directy, without any iterations The main advantage of the method is found in the decrease of the CPU time with respect to other methods for paseudoinverse computation We iustrate the theoretica findings by comparing the Moore-Penrose inverse method against the Wiener fiter, Constrained eastsquares fiter and Lucy-Richardson agorithm Aso, we present numerica resuts in which we compare our method (caed MP method) with we-known Pappas, Pappas2 and Courrieu methods References [] M R Banham and A K Katsaggeos, Digita image restoration, IEEE Signa Processing Magain 4(2) (997) 24 4 [2] A Ben-Israe and TNE Grevie, Generaied inverses: theory and appications, (2nd edition), Springer, 23 [3] J Biemond, R L Lagendijk, R M Mersereau, Iterative methods for image deburring, Proceedings of the IEEE 78(5) (99) [4] A Bovik, The essentia guide to the image processing, Academic Press, 29 [5] A Bovik, Handbook of image and video processing, Academic Press, San Diego, San Francisko, New York, Boston, London, Sydney, Tokyo, 2 [6] S Chountasis, V N Katsikis, D Pappas, Appications of the Moore-Penrose inverse in digita image restoration, Mathematica Probems in Engineering, Voume 29, Artice ID 7724, 2 pages doi:55/29/7724 [7] S Chountasis, V N Katsikis, D Pappas, Digita Image Reconstruction in the Spectra Domain Utiiing the Moore-Penrose Inverse, Mathematica Probems in Engineering, Voume 2, Artice ID 75352, 4 pages doi:55/2/75352 [8] S Chountasis, V N Katsikis, D Pappas, Image restoration via fast computing of the Moore-Penrose inverse matrix, Systems, Signas and Image Processing, (29), 4 pages [9] P Courrieu, Fast Computation of Moore-Penrose Inverse matrices, Neura Information Processing-Letters and Reviews (25) 8:25 29 [] R C Gonae, R E Woods, S L Eddins, Digita Image Processing Using MATLAB, Prentice-Ha, 23 [] PC Hansen, JG Nagy, DP O Leary, Deburring images: matrices, spectra, and fitering, SIAM, Phiadephia, 26 [2] Image Processing Toobox User s Guide, The Math Works, Inc, Natick, MA, 29 [3] A K Katsaggeos, editor, Digita Image Restoration, Springer Verag, New York, 99 [4] V Katsikis, D Pappas, Fast computing of the Moore- Penrose Inverse matrix, Eectronic Journa of Linear Agebra 7 (28) [5] F Krahmer, Y Lin, B Mcadoo, K Ott, J Wang, D Widemann, and BWohberg, Bind image deconvoution: Motion bur estimation, In IMA Preprints Series (26) [6] D Kundur and D Hatinakos, Bind image deconvoution: an agorithmic approach to practica image restoration, IEEE Signa Processing Magain 3(3) (996) [7] R L Lagendijk and J Biemond, Iterative Identification and Restoration of Images, Kuwer Academic Pubishers, Boston, MA, 99 [8] MATLAB 7 Mathematics, The Math Works, Inc, Natick, MA, 2 [9] M A Rakha, On the Moore-Penrose generaied inverse matrix, Appied Mathematics and Computation 58 (24) 85 2