Modulation Transfer Function Compensation through a modified Wiener Filter for spatial image quality improvement

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1 Modulation Tanse Function Compensation though a modiied Wiene Filte o spatial image quality impovement JEICA WONG OO MEE Malaysia pace Cente National pace Agency o Malaysia (ANGKAA) Lot 33, Jalan Tui, g. Lang, 4700, Banting, elango MALAYIA jessica@angkasa.gov.my Abstact: - The knowledge o Modulation Tanse Function (MTF) o a degaded image povides a mean to compensate o the image degadation, which impoves the image quality in tems o shapness. MTF compensating is in the image estoation; hence it is undamentally an ill-posed poblem. This pape poposes a stable and lexible ilteing technique that executes an optimal tadeo between shapness and noise to waant an acceptable esult o image estoation. The MTF compensating is peomed though a modiied Wiene ilte, and can be educed to a well-posed poblem by incopoating the egulaization method. The modiied Wiene ilte employed the L-cuve method o selection o optimal egulaization paamete, and intoduces an invese contol and smooth contol paamete that allows eedom o tuning when the impotance o image shapness vesus noise is being compomised. The data sets used in the analysis wee synthetically blued emotely-sensed images simulated om Level-A poduct o IKONO. The esults by modiied Wiene ilte wee analyzed and pesented, they wee ound to be able to educe the Mean quae Eo by moe than 50% with espect to the oiginal image, and it esults in a signiicant impovement o spatial image quality. Key-Wods: - Modulation Tanse Function, patial image quality, Wiene ilte, Image estoation, Ill-posed poblem, L-cuve 1 Intoduction patial image quality is one o the key paametes o chaacteizing and validating image data. The technical chaacteistics o image data must be well undestood beoe image analysis can be done, since the quality o the analysis depends on the quality o the data. Image quality can be measued by a vast numbe o actos, such as contast, bightness, noise vaiance, shapness, adiometic esolution, ganulaity, modulation, contast tanse unction and many moe [1]. Among othe measues, image shapness is vital o chaacteizing images, o much o the inomation o an image esides in its edges. Image bluiness, which limits the visibility o details, can be objectively measued by the point spead unction (PF), o its amplitude spectum, which is the modulation tanse unction (MTF). patial image degadation can happen in many ways. Fo satellite imaging, image acquisition occus while obiting the eath, and due to the satellite s altitude detemination contol o maneuveing, the instantaneous ield o view (IFOV) o the imaging system can be geate o lesse than the nominal esolution at any point in time duing image captuing. This esults in MTF degadation popotional to the atio o IFOV and Gound ample Distance (GD) []. Moeove, egadless o how well an image system is abicated, it will inevitably sue om some degee o blu. These sots o degadations need to be compensated, and they can be compensated though the knowledge o the degadation unction which is the MTF o that image estoation. MTF compensating is in the image estoation; hence it is undamentally an illposed poblem. This pape poposes a stable and lexible ilteing technique that executes an optimal tadeo between shapness and noise to waant an acceptable esult o image estoation. MTF Compensation MTF Compensation (MTFC) is a estoation technique; Restoation techniques ind iltes that apply invese pocess in ode to ecove its oiginal image. In image estoation, the main challenge is to pevent noise o input data om being ampliied to unacceptable atiacts in the estoed image. Image estoation is ill-posed, which means the MTFC is also an ill-posed poblem, meaning that it does not have a unique solution, even with the absence o noise [3]. Howeve, scientists and enginees ae usually less concened with existence and uniqueness o the solution to the ill-posed poblem and woy moe about the stability o the solution. With the concept o egulaization, the ill-posed poblem o the IN: IBN:

2 existing MTFC can be educed to a well-posed poblem by intoducing a egulaization paamete as a pioi constaints to poduce a stable and lexible deconvolution ilte o image estoation. In this case, it is stable in the sense that it will always poduce an acceptable image estoation esult, while also being lexible by allowing eedom o tuning when the impotance o image shapness vesus noise is compomised..1 Deconvolution Filte As a deconvolution ilte, Wiene ilteing is an invese poblem, theeoe it is likely to be ill-posed. Regulaization is necessay while solving invese poblem because o the naïve least squaes solution by Wiene ilte. By using egulaization, the eo contibution can be damped and the esidual nom can be kept in easonable size. In othe wods, by incopoating egulaization paamete into the Wiene ilte, it is possible to contol the powe o invese ilteing, and subsequently minimizes the ampliication o noise in the image. The Wiene ilte has been used by othe eseaches [4] [5] as an MTFC technique to compensate o the degadation o emotely-sensed images. And it also has been used by hacham et al.[6] as blind estoation technique to compensate o the degadation o astonomical data. Howeve, none o them applied the egulaization method in thei iltes, thus a modiied Wiene ilte based on the idea in the egulaization method is employed in this wok..1.1 The Modiied Wiene Filte In this pape, the poposed deconvolution ilte consides images and noise to be a andom pocess, and is ooted in one o the most widely used deconvolution iltes, which is the Wiene ilte. In ode to egulaize the invese poblem o the Wiene ilte, the concept o the Tikhonov egulaization was adopted. Accoding to Wiene s theoy [7], given a degaded image g ( x,, the degaded image can be estoed by W in a Fouie sense, as shown by the ollowing equations: H W = (1) H n H = () n 1 = (3) H n whee epesents the degadation unction; n epesents powe spectum o the noise; epesents powe spectum o the undistoted image and the pai ( epesents the location in the spatial equency domain. Fom equation (3), the Wiene ilte can be peceived to have two sepaate pats: an invese ilteing pat and a noise smoothing pat. It not only peoms the deconvolution by invese ilteing but also emoves the noise with a compession opeation. These popeties wee advanced to solve the eseach poblem. In the absence o noise (i.e. n = 0 ), the Wiene ilte educes to the invese ilte. On the othe hand, in the pesence o noise (i.e. n > 0), the v ) denominato o () is neve zeo. In the limiting case 0, then n 1 H 0 H W = (4) 0 H = 0 thus no ill-posed poblems aise in Wiene ilteing. The tem n in (3) can be egaded as a tem that v ) smooths the invese ilte 1. Howeve, o H ( u, v ) equencies at which n, estimation o W( becomes nealy equal to 1, which means the H ( u, v ) Wiene ilte behaves as an invese at the equencies (. At high spatial equencies, is oten much lage than n, thus condition holds tue. Fo degadation such as blu, H at high spatial equencies is elatively small. When ( appoaches a egion with a lage value o u v, H is nealy zeo. Theeoe, when n and H 0, the Wiene estimate w = 1 becomes ill-posed. Consequently, this H ( u, v ) estimate will lead to a poo estoation esult. Fo the ill-posed invese poblem to be solved, it is necessay to egulaize the solution by intoducing a pioi constaints [8]. This can be done by using a numeical egulaization paamete to contol the powe o the invese ilte, theeby penalizing those ( with vey high spectal equencies. The n in equation v ) (3) is assumed to be constantly popotional to the invese o the signal-to-noise atio (NR) o the image. This leads to the ollowing modiication o W( in equation (3): 1 H W = (5) H λ H NR n IN: IBN:

3 Thus, the equency domain solution to optimize the degadation poblem is expessed as 1 Fˆ( = G( (6) λ NR whee F ˆ epesents the estoed image, G epesents the degaded image, and λ epesents the egulaization paamete. The egulaization paamete was detemined by using the L-cuve method. The L-cuve is a paametic plot o the size o the egulaized solution and its coesponding esidual [9]. The undelying idea is that a good method o choosing the egulaization paamete o discete ill-posed poblems must incopoate inomation about the solution size, in addition to using inomation about the esidual size. The L-cuve method equies both the egulaized solution and its coesponding esidual to ind the cone o the L-shape cuve o the optimal selection o egulaization. With the undelying degadation is the linea, position-invaiant degadation that is descibed in a spatial domain by g = H η (7) whee g epesents the degaded image, epesents the oiginal image, matix H epesents the degadation unction, and η epesents the additive white noise. In this case, the knowledge o noise vaiance is not equied. Thus the egulaization poblem can be expessed using the Tikhonov egulaization as g Hˆ ˆ λ λ λ (8) Let be the esidual vecto descibed by The g Hˆλ = (9) can be calculated using the ollowing equation: N 1 M 1 = ( x, y= 0 x= 0 1 whee ( x, = I [ G H Fˆ ] (10) (11) substituting the ight side o equation (11) o Fˆ in equation (6) yields [ ] [ ] = I 1 G( λ NR ( x, (1) H λ NR 1 whee I epesents invese Fouie tansom. Meanwhile, equation: ˆλ can be calculated using the ollowing N 1M 1 ˆ = ˆ λ ( x, (13) y= 0 x= 0 whee ˆ ( x, = I [ ˆ (, )] 1 F u v The pocedues o detemining the egulaization paamete ae as ollows: tep 1: peciy an initial value o λ tep : Calculate and ˆλ tep 3: Gadually incease λ until λ = 1; tep 4: Convet both and ˆλ to logaithmic unctions. tep 5: Plot the L-cuve in log-log scale (see Fig.1). tep 6: Tace the tansition o the shape o the cone o the L-cuve plot, and mak the optimal λ, tep 7: Use the optimal λ to compute the optimal estimate F ˆ Fig.1. L-cuve o Tikhonov egulaization [10] As mentioned ealie, the Wiene ilte can be peceived to have two sepaate pats: an invese ilteing pat which contibutes to the shapness o debluing, and a noise smoothing pat o denoising. Theeoe it is possible to contol the egulaization and invese powe independently. ubsequently, it allows moe degees o eedom o image estoation that incopoate both debluing and denoising techniques in one single ilte. Hence the poposed ilte is as expessed as: γ β 1 W = λ (14) NR whee β and γ epesent the invese contol paamete and the smooth contol paamete espectively. IN: IBN:

4 3 Expeimental Results The oiginal image used in this wok is pesented in Fig. ; it is one pat o Level A poduct o IKONO, containing 7800 x 7800 pixels. Fig.. The oiginal image Fig. 3 pesents one o the blued vesion o the images obtained by using the invese Fouie tansom, by multiplying the oiginal (in Fouie sense) with a Gaussian ilte. The Gaussian ilte unction, H G ( which appoximates the optical point spead unction in the equency domain is expessed as: ( u v ) H (, ) σ G u v = e (15) whee σ is the standad deviation o Gaussian cuve. The standad deviation, σ o Gaussian unction o this blued image was set to 80. The data sets o this wok consist o sub-scenes extacted om this synthetically blued image N M ME = [ ( x, ˆ( x, ] (16) MN y= 0 x= 0 ME PNR = 10 log 10 (17) whee, ˆ and epesent the oiginal image, the estoed images and the maximum pixel value o the image, espectively. Two vesions o blued images wee used to pesent the expeimental esult o MTFC. The standad deviation, σ o Gaussian unction o these images is 50 and 80, espectively (see Fig. 4(a) and (b)). The degadation unction,, which is the MTF o the degaded image, was modeled using the measued MTF by edge input method. The measued (estimated) MTF wee compensated using the modiied Wiene ilte as descibed in Equation (14) to estoe these images. The egulaization paamete, λ was tuned between 10-6 and 10. Expeimental esult shows that egulaization paamete, λ = 10-1 is elatively good to optimize the egulaization o estoation poblem. The L-cuve plot o egulaization pocess o the blued image in Fig. 4(a) and (b) is pesented in Fig. 4(c) and (d), espectively. By visual obsevation, it is clea that the estoed images in Fig. 4(e) and () ae consideably bette than thei espective blued image shown in Fig. 4(a) and (b). It can be obseved that the blued images ae shapened without the side eect o noise ampliication. Typically, blued images ae assumed to have a lowe quality as compaed to the estoed images. Theeoe the MTF o the blued images ae expected to plummet to lowe MTF values as compaed to the estoed images. Unde this cicumstance, MTF aea unde the MTF cuve ate estoation will be lage than those beoe estoation. This supposition was conimed by the two MTF plots depicted in Fig. 4. (a) (b) Fig. 3. A synthetically blued image The poposed MTFC algoithm was applied to data sets and evaluated quantitatively using two metic, namely Mean-quaed Eo (ME) and Peak ignal-to- Noise Ratio (PNR), which ae deined as ollows: IN: IBN:

5 (c) (d) images obtained om both classical and modiied Wiene ilte ae pesented in Figue 5(a) and 5(b), espectively. Qualitative measues based on visual obsevation ound that 5(b) displays a moe enhanced details as compaed to 5(a). Fig. 6 shows the compaison o MTF plot between these two iltes; MTF aea unde the MTF cuve o the modiied Wiene ilte ae obviously lage than the classical Wiene ilte, which means that it poduces a bette image quality. (e) () (a) Image estoation by classical Wiene Filte (g) (h) Fig. 4. Restoation esults; (a) Blued image, σ = 50 with its L-cuve plot (c), estoed image (e) and MTF plot (g); (b) Blued image, σ = 80 with its L-cuve plot (d), estoed image () and MTF plot (h); One citeion o the estoation quality is the atio between the aeas unde the MTF cuve ate and beoe estoation. The MTF aea (MTFA) atio o the estoed image in Figue 4(e) and () ae.08 and 1.84, espectively. Table 1 shows othe oms o quantitative esults that quantiied the image quality impovement. Table 1: Quantitative measues o the estoed images Peomance Measue Fig. 4(a) Fig. 4(e) Fig. 4(b) Fig. 4() PNR (db) (b) Image estoation by the modiied Wiene ilte Fig.5. Compaison o image estoation between the classical Wiene ilte and the modiied Wiene ilte ME Fo compaison pupose, the classical Wiene ilte was implemented using Equation (3). Both classical and modiied Wiene ilte wee applied to the synthetically blued image shown in Figue 4(a). The estoed IN: IBN:

6 modiied Wiene ilte to invet blu and suppess noise ampliication at the same time demonstated the obustness o the poposed MTF compensation technique. Fig. 6. MTF Compaison between classical and the modiied Wiene ilte Table summaizes the peomance assessment o these iltes; the esults demonstated that the modiied Wiene ilte was indeed able to poduce a bette quality o estoed image. Table : Peomance assessment o classical and the modiied Wiene ilte Filte PNR (db) ME Classical Wiene Filte Modiied Wiene Filte Conclusion This pape descibes a stable and lexible MTFC technique though Wiene ilte that aims to execute an optimal tadeo between shapness and noise to waant an acceptable esult o image estoation o spatial image quality impovement. The technique is based on deivation o the spatial equency esponse to edge taget input, which is the MTF measuement. The deived MTF measuement was used to model the degadation unction to compensate o image degadation. The main novelty this ilte is the incopoation o L-cuve egulaization method into the classical Wiene ilte, which then becomes the modiied Wiene ilte to ovecome the ill-posed poblem o image estoation. The algoithm was applied to synthetically blued images. Restoation esults, using the estimated MTF and L-cuve egulaization show signiicantly impoved o spatial quality image, whee the estoed image by the modiied Wiene ilte has the Mean quae Eo (ME) educed by moe than 50% with espect to the oiginal image. The ability o the Reeences: [1] P.M. Ein, Review o tandads Deining Testing Pocedues o Chaacteizing the Colo and patial Quality o Digital Cameas Used to Image Cultual Heitage, Mach, Munsell Colo cience Laboatoy Technical Repot, 004. [] G. Fank, Chaacteistics o pace Imaging's One- Mete Resolution atellite Imagey Poducts. The Intenational Achives o the Photogammety, Remote ensing and patial Inomation ciences, 8, 000, pp [3] D. Kundu & D. Hatzinakos, Blind image deconvolution. IEEE ignal Pocessing Magazine, May, 13, 1996, pp [4] Z.G. Wang, & Z.X. Geng, The R images estoation o CBER- based on atmospheic MTF evaluation using meteoological data. The Intenational Achives o the Photogammety, Remote ensing and patial Inomation ciences. 37, 008, pp [5] D.H. Lee, et al., Image Restoation O Calibation And Validation Fo Kompsat-. The Intenational Achives o the Photogammety, Remote ensing and patial Inomation ciences, 38, 008, pp [6] O.hacham,, O. Haik & Y. Yitzhaky, Blind estoation o atmospheically degaded images by automatic best step-edge detection. Patten Recognition Lettes, Novembe, 8(15), 007, pp [7] R.C. Gonzalez & R.E. Woods, Digital Image Pocessing. nd ed. New Jesey: Pentice Hall, 001. [8] M.M. Lavent ev, V.G. Romanov, & hishat.kiˇi,.p.,. Ill-posed poblems o mathematical physics and analysis. Povidence:Ameican Mathematical ociety, 1986 [9] C.L Lawson, & R.J. Hanson, olving Least quaes Poblems. New Jesey: Pentice-Hall, [10] P.C. Hansen,. Analysis o discete ill-posed poblems by means o the L-cuve, ociety o Industial and Applied Mathematics, Decembe, 34(4), 199, pp IN: IBN: