DENOISING OF MAGNETIC RESONANCE AND X-RAY IMAGES USING VARIANCE STABILIZATION AND PATCH BASED ALGORITHMS

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1 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December DENOISING OF MAGNETIC RESONANCE AND X-RAY IMAGES USING VARIANCE STABILIZATION AND PATCH BASED ALGORITHMS V N Prudhv Raj and Dr T Venkateswarlu Assocate Professor, VR Sddhartha Engneerng College, Vjayawada, 57, Inda nagaprudhv@yahoo.com Professor, Unversty College of Engneerng, SV Unversty, Trupat, Inda ABSTRACT Developments n Medcal magng systems whch are provdng the anatomcal and physologcal detals of the patents made the dagnoss smple day by day. But every medcal magng modalty suffers from some sort of nose. Nose n medcal mages wll decrease the contrast n the mage, due to ths effect low contrast lesons may not be detected n the dagnostc phase. So the removal of nose from medcal mages s very mportant task. In ths paper we are presentng the Denosng technques developed for removng the poson nose from X-ray mages due to low photon count and Rcan nose from the MRI (magnetc resonance mages). The Posson and Rcan nose are data dependent so they won t follow the Gaussan dstrbuton most of the tmes. In our algorthm we are convertng the Posson and Rcan nose dstrbuton nto Gaussan dstrbuton usng varance stablzaton technque and then we used the patch based algorthms for denosng the mages. The performance of the algorthms was evaluated usng varous mage qualty metrcs such as PSNR (Peak sgnal to nose rato), UQI (Unversal Qualty Index), SSIM (Structural smlarty ndex) etc. The results proved that the Anscombe transform, Freeman & Tukey transform wth block matchng 3D algorthm s gvng a better result. KEYWORDS Varance Stablzaton Transform, Posson Nose, Nonlocal Means, Block matchng.. INTRODUCTION Medcal magng became an ntegral part of medcal dagnoss n present days. Varous medcal magng modaltes are developed for varous applcatons snce last few decades. These modaltes are used to acqure the mages of the anatomcal structures wthn the body to be examned wthout openng the body. X-rays, Computed Tomography, Ultrasound, Magnetc resonance Imagng and Nuclear magng are the popular modaltes at present to dagnose the varous dseases. However these modaltes are sufferng wth a bg problem called nose. Every modalty s sufferng from nose n mage acquston and transmsson stage such as Quantum nose n X-rays and Nuclear magng, speckle nose n ultrasound magng, Rcan nose n Magnetc resonance magng etc. The nose present n the mages wll degrade the contrast of the mage and creates problems n the dagnostc phase. So denosng s very mportant to remove the nose from these mages []. The nose may be addtve or multplcatve dependng on the modalty used for medcal mage acquston. The nose due to electronc components n the acquston hardware wll be modeled wth Gaussan nose whch ndependent of data, the data dependent nose such as quantum nose n X-ray magng s modeled wth Posson dstrbuton, the speckle nose n ultrasound magng s modeled wth Raylegh dstrbuton and the nose n MRI s modeled wth Rcan dstrbuton. DOI :.5/jma

2 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December Here n ths paper we are attemptng to denose the mages corrupted wth quantum nose n X-ray and Nuclear magng and Rcan nose n Magnetc Resonance Imagng [, ]. The mathematcal modelng of degradaton and restoraton process s gven as (, ) = (, ) (, ) +η (, ) (, ) = (, ) (, ) + (, ) g x y f x y h x y x y G u v F u v H u v N u v () Where g ( x, y) s the nosy and blurred observaton, H s the blurrng kernel and f (, ) x y s the sgnal we are recoverng. In the case of denosng problem the blurrng kernel wll be dropped and the degradaton model wll be gven as (, ) = (, ) +η (, ) (, ) = (, ) + (, ) g x y f x y x y G u v F u v N u v () In the case of multplcatve nose the model s gven as (, ) = (, ) η (, ) g x y f x y x y (3). MATHEMATICAL PROPERTIES OF NOISE. Posson Nose The nose n X-ray magng and Nuclear Imagng (PET, SPECT) s modeled wth Posson nose. X-ray photons ncdent on a receptor surface n a random pattern. We cannot force them to be evenly dstrbuted over the receptor surface. One area of the receptor surface may receve more photons than another area, even when both the areas are exposed to the same average x-ray ntensty. In all medcal magng procedures usng gamma or x-ray photons most of the mage nose s produced by the random behavour of the photons that are dstrbuted wthn the mage. Ths s generally desgnated quantum nose. Each ndvdual photon s a quantum (specfc quantty) of energy. It s the quantum structure of an x-ray beam that creates quantum nose [9]. P(k).5. λ=5 λ= λ=5 λ= k Fgure : Posson dstrbuton for varous values of lambda 54

3 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December A Posson model assume that each pxel ' x' of an mage ' f ( x) ' s drawn from a Posson dstrbuton of parameter ' f ( x) densty s gven as λ = ' where ' f ' s the orgnal mage to recover. The Posson λ k e λ ( ( ) k ) k! P f x = = (4) The above fgure gves the Posson dstrbuton for varous values of ' λ ' as the ' λ ' ncreases the Posson dstrbuton turns towards the Gaussan dstrbuton.. Rcan Nose Magnetc Resonance Imagng (MRI) s a non-nvasve wdely used modalty n medcal dagnoss such as cardac related dseases and neurologcal dsorders. The MRI magng wll suffer from low sgnal to nose rato (SNR) or contrast to nose rato (CNR) because of whch the mage analyss tasks such as segmentaton, reconstructon and regstraton wll become complcated. So the nose reducton n MR mages s very mportant as a pre-processng task before gong to mage analyss and to mprove the dagnostc qualty of the mages [9,]. Thermal nose s the major source of nose n MR magng. The MR mages are reconstructed from the raw data by applyng the nverse Fourer transform to t. The sgnal component s present n both real and magnary channels whch are orthogonal to each other and are affected by addtve whte Gaussan nose. Hence the nose n the reconstructed date s complex whte Gaussan nose. Normally the magntude mage of the reconstructed complex data s used for vsual nspecton. So the magntude of the MR sgnal s the square root of the sum of the squares of the data present n real and magnary channels, the nose s the square root of the two ndependent Gaussan varables. Hence the nose n MR mages s no longer Gaussan. Let ' A' be the pxel ntensty n the absence of nose and ' M ' be the observed or measured pxel ntensty. In the presence of nose the probablty dstrbuton for ' M ' s gven as M ( M + A ) e A M PM ( M ) σ = I σ σ (5) Where ' σ ' denotes the standard devaton of the Gaussan nose n the real and magnary mages whch s consdered as equal here and ' I ' s the modfed zeroth order of Bessel functon of the o frst knd. Ths s called as Rce densty. For small values of SNR ( A / σ ) the rce dstrbuton s far from beng Gaussan and from ratos as small as A / σ = 3t starts to move towards the Gaussan dstrbuton. In the mage regons where sgnal content s much less (approxmately zero.e. A = ) only nose s present then the above equaton s reduces to M ( M ) e PM ( M ) = σ (6) σ Ths s well known as Raylegh dstrbuton. Ths dstrbuton governs the nose n mage regons where no NMR sgnal and only nose s present. The mean and varance of ths dstrbuton s gven as 55

4 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December M π π = σ and σ = σ M (7) These relatons are useful n the estmaton of the true nose power..4. Low SNR Rcan Raylegh Hgh SNR Rcan Gaussan, nave mean Gaussan, corrected mean Fgure : Rcan nose (Low SNR, Hgh SNR) dstrbutons When the SNR s large then PM ( M ) e πσ ( M A σ ) + σ (8) From the above equatons we can say that for mage regons where large sgnal ntenstes are present the nose dstrbuton wll be consdered as a Gaussan dstrbuton wth mean and varanceσ. 3. PROPOSED METHOD A + σ A numerous algorthms are developed by the scentsts and engneers to remove the nose n the natural and medcal mages snce 97. These denosng methods are orgnatng from varous dscplnes such as algebra, probablty & statstcs, lnear and nonlnear flterng, partal dfferental equatons and multresoluton analyss etc. At frst the denosng methods are based on pxel by pxel transformaton later the neghborhood flterng was developed to remove the nose. However these methods wll ntroduce some artfacts and blurrng whle removng the nose from the mages. The mportant features n the mage such as edges and texture are gong to be damaged n the process of nose removal. To preserve the edges and textural nformaton n the mage many denosng methods were developed such as ansotropc dffuson, Laplacan pyramds, steerable pyramds, wavelet transforms, prncple component analyss based methods, dctonary based approaches etc. The ansotropc dffuson wll preserve the edges but loose the texture nformaton. The multresoluton based approaches are senstve to the hgh frequency dscontnutes but fals to detect the spatal changes n the low frequences properly. The dctonary based approaches are very tme consumng and computatonal cost s very hgh to mplement n real tme. 56

5 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December In ths paper we developed the algorthms combnng the advantages of varance stablzaton transforms and the famous patch based algorthms (Nonlocal means and BM3D) to recover the medcal mages corrupted wth Quantum nose (X-ray, CT, and Nuclear Imagng) and the Rcan nose (Magnetc resonance magng). The block dagram of the proposed archtecture s gven below. Frst the nosy mage s processed by the varance stablzaton transform to have the dstrbuton wth constant varance (Posson and Rcan dstrbutons are converted nto Gaussan dstrbuton). Then one of the patch based technque s appled on the stablzed data to remove the nose and then the processed data s converted back by the nverse varance stablzaton transform to get the denosed mage. Each block of the archtecture s explaned n the followng sectons [, 7]. 3. Varance Stablzaton Transforms Fgure 3: Proposed Denosng System Archtecture In statstcs the varance stablzaton transforms are used to convert the dstrbutons wth varable varance nto dstrbuton wth constant varance. These transforms are useful to convert the Posson dstrbuton and Rcan dstrbutons nto Gaussan dstrbuton wth constant varance. The nose n the X-ray magng and nuclear medcal magng such as PET and SPECT s modeled wth Posson dstrbuton because ts varance depends on the mean value and ths makes dffcult the denosng task. Here n ths work we want to use the varance stablzaton transforms as a preprocessng stage for the denosng flter to make the Posson and Rcan dstrbutons nto Gaussan dstrbuton wth known varance and then we perform the denosng operaton. After the preprocessng we wll apply the nverse varance stablzaton transform to get back the orgnal dstrbutons. In ths work we are usng three varance stablzaton transforms and ther nverses they are square root, Freeman & Tukey and Anscombe transforms. The mathematcal representaton of the transforms s gven below. 57

6 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December Table : Varance Stablzaton transforms and ther Inverses The drect nverse and Adjusted nverse wll gve unbased estmates f photon count s sgnfcant. However they wll gve based estmates for the low ntensty data that s photon count s mnmum, the unbased nverse s used n such stuatons. 3. Gaussan Flters After transformng the data nto dstrbutons wth constant varance we wll apply the Gaussan flterng on ths data. In ths work we are usng two algorthms one s nonlocal means algorthm and the second one s BM3D (block matchng 3D). Here n ths secton we wll gve the bref descrpton of these two algorthms. In the earler works of mage denosng there are two assumptons consdered about the nosy mage. The frst assumpton s that the nosy mage conssts of both low and hgh frequences. The nose s treated as non-smooth because of the hgh frequences contaned n t. The orgnal mage only contans low frequences s the second assumpton..e. mages do not contan fne detal. Both the assumptons lead to the damage of fne detals such as edges and lnes n the denosed mages. The Wener and the Gaussan flterng approaches also make these assumptons. The flters attempt to denose the nosy mage by removng the hgher frequences from the mage keepng the lower frequences assumng that nose s present n hgh frequences and most of the mage nformaton s n low frequences. The assumpton wll not sutable for the mages whch may not be smooth. They can contan fne detals and structures whch have hgh frequences. Because these flterng methods unable to dfferentate between the hgher frequences of the orgnal mage and the nose, the hgh frequences of the orgnal mage wll be lost. Ths results n blurrng. In addton, the low frequences of the nose wll stll reman n the denosed mage. Here n ths work we are not consderng the above assumptons and our methods depends on the self-smlarty of the pxels n the mage. 58

7 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December 3.. Nonlocal Means algorthm The nonlocal means flterng s also consdered as the extenson of the fundamental neghborhood flter (Yaroslavsky flter) where the pxel ntensty s decded by averagng the ntenstes of ts neghborhood. In ths method the pxels whch are havng smlar neghborhood are found and averaged n decdng the ntensty of a pxel. Normally smlarty between the pxels s found by observng ther ntenstes, f the two pxels havng the same ntensty we can say that they are smlar otherwse they are dfferent. Here n the nonlocal means algorthm the smlarty s found by consderng the neghborhoods of the pxels. Two pxels are smlar when the neghborhoods of the two pxels are havng same ntenstes. So we can defne the neghborhood of pxel ' ' s a set of pxels ' j ' whose neghborhoods are smlar to the neghborhood of ' '. The followng fgure shows the smlarty wndows n the mage [9]. Fgure 4: Self smlarty n an Image (Fundamental to Patch based Algorthms) Let ' Ω ' s the area of an mage ' I ', ' x ' s the locaton nsde ' Ω ', u( x) and v( x) are the clean and observed nosy mage value at the locaton ' x ' respectvely. Then the nonlocal algorthm s gven as follows Where Ga v( x+.) v( y+.) ( ) NL[ u]( x) = e h v ( y) dy C( x) (9) Ω Ga a Gaussan s functon wth standard devaton ' a ' and ' h' s a flterng parameter Ga v( x+.) v( y+.) ( ) C( x) = e h dz Ω In the dscrete case the non-local means algorthm s gven as NL[ u] = w(, j) v( j) () ( ) j I 59

8 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December Where the weght w(, j) depends on the dstance between observed gray level vectors at ponts ' ' and ' j '. The dstance can be calculated as j, a d = v( N ) v( N ) () N s the neghborhood of the pxel ' ' whch s the squared wndow around the pxel ' '. Every pxel n the mage wll have ther own neghborhood. The sze of the neghborhood can be vared dependng on the applcaton. Now we can defne ' NS ' the neghborhood system of a pxel ' '. It s a set of smlar neghborhoods to the neghborhoods of ' ' ( N ). The smlarty between the neghborhoods s calculated by calculatng the dfference n the ntensty gray levels n the neghborhoods as shown n the above equaton. The pxels ' ' and ' j ' are sad to be smlar f v( N ) s smlar to v( N ). The weght can be defned as 3.. BM3D j w(, j) Where z( ) e z( ) v( N ) v( N j ), a h = () = j e v( N ) v( N j ), a h The BM3D algorthm s based on non-local mage modellng and frequency doman flterng. The algorthm s developed based on enhanced sparse representaton of the mage. Ths can be acheved by the followng steps. Frst the mage s dvded nto number of D fragments. Then the smlar D fragments are grouped together to form 3D stack. Now the collaboratve flterng s performed to acheve the sparsty. And fnally the fltered fragments are returned back to ther orgnal postons. The followng fgure gves the block dagram of the system [3]. Fgure 5: Block dagram of BM3D system used n our algorthm The algorthm s dvded nto two stages. Basc estmaton and Fnal estmaton. In the basc estmaton stage the groups are pre-fltered and ths basc estmate s jontly used wth nosy mage n stage to gve the fnal estmate. The algorthm wll proceed as follows. In both the 6

9 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December stages the estmaton use smlar steps except that n the frst step flterng s carred by hard thresholdng and n the second step flterng s done by wener flterng. Basc Estmaton. Groupng.. 3D Transform Applcaton. 3. Hard Thresholdng. 4. Inverse 3D Transform. 5. Returned estmated blocks to ther poston 6. Aggregaton Fnal Estmaton. Groupng (based on BE result). 3D Transform Applcaton. 3. Wener flterng (usng BE result). 4. Inverse 3D Transform. 5. Returned estmated blocks to ther poston. 6. Aggregaton Groupng The frst step n algorthm s to dvde the mage nto number of fragments by defnng a square neghborhood around each pxel. The groups are formed by performng block matchng wth reference to a reference frame to fnd the smlar fragments at varous spatal locatons n the mage. These D fragments are stacked as a 3D array. The block matchng method measures the dfference between the fragments wth reference one and f the dfference s less than certan threshold the fragment s kept n the group. Let ' N ' be the neghborhood around ' '(Reference fragment), ' GN ' be the smlarty group for ths fragment, ' Th ' s a certan threshold then the fragment ' N ' belongs to the above group f t satsfes the followng condton If dfference( N, N ) < Th N GN (3) j j j For groupng the smlar blocks we may use the clusterng algorthms but they wll create the dsjont sets. Where the block matchng algorthm one fragment may become member n more than one group. That s the algorthm produces dsjont sets Smlarty measurement Basc Estmaton: In basc estmaton the smlarty s measured by applyng a D forward transform on both the fragments and hard thresholdng s performed on them. Now the dfference between the spectral coeffcents of both the fragments s measured. If the dstance s smaller than certan threshold then the fragment s smlar otherwse they are dfferent from each other. D( N, N ) j ( ) ' ( D ( )) D ( j ) ( M ) ' Y T N Y T N = (4) j ' Y s the hard thresholdng operator, D n the frst stage. T s the D Lnear Transform and M s the fragment sze Fnal Estmaton: The pre-fltered mage from the basc estmaton s used n formng the groups of second stage. The coordnates of the pre-fltered mage s used to make the groups n the nosy mage. The normalzed Eucldean dstance at ths stage s gven as 6

10 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December D( PN, PN ) j PN PN j = (5) ( M ) Where PN and PN j are the pre-fltered fragments, M s the sze of the fragment n second stage. Based on ths new dstances and new threshold new groups are defned. Collaboratve Flterng: The collaboratve flterng conssts of three steps. They are forward 3D transform, Flterng and Inverse 3D transform. The groups are characterzed by ntra-fragment correlaton (appears between the pxels of each grouped fragment) and nter-fragment correlaton (appears between the correspondng pxels of dfferent fragments). The 3D transform can take advantage of these correlatons and wll produce the sparse representaton of the true sgnal n the group. Ths sparsty wll mprove the effectveness of shrnkage step n removng the nose and preservng the features of the mage. When a group of ' X ' fragments are taken the collaboratve flterng wll produce ' X ' estmates one for each fragment. Every fragment wll partcpate n estmatng the every other fragment so all the fragments are partcpatng n the estmaton process t s called collaboratve flterng. Basc Estmaton: In ths stage the collaboratve flterng s gven as below ( ( ( ))) PNG = T Y T GN (6) 3D 3D Where ' Y ' s the hard thresholdng operator, ' T3 D ' s the 3D lnear transform, ' GN ' s the group of fragments to be processed and ' PNG ' s the output of the collaboratve flterng. The 3D transform s mplemented by performng the D transform on the fragments and D transform along the thrd dmenson of the group. In ths work borthogonal splnes are used n D transformaton and Haar bass s used n D transform. Fnal Estmaton: In ths stage we have two groups one from nosy mage and the other s pre-fltered mage from basc estmaton. Ths stage wll use the wener flterng. The coeffcents of the wener flter are computed from the pre-fltered mage performng 3D transform on t as gven below W C = 3D 3D ( PNG ) ( ) T T PNG + σ (7) Now the collaboratve flterng on the nosy fragments are carred out by multplyng the 3D transformed coeffcents of nosy fragments wth the wener flter coeffcents and takng the nverse 3D transform. Ths process wll gve us the fnal estmate. ( ( )) FNG = T W T NG (8) 3D C 3D Where ' WC ' are the emprcal wener shrnkage coeffcents, T3D s a forward 3D lnear transform, ' NG ' s the fragment group from nosy mage and ' FNG ' s the fnal estmate. Here 6

11 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December n ths stage the D transform s DCT (Dscrete cosne transform) and D transform s Haar Transform Aggregaton: Ths s very mportant step n the algorthm. As the algorthm s not havng dsjont sets n groupng the fragments may present more than one group. Because of ths the fragments wll get more than one estmaton dependng upon ther partcpaton. For example f a fragment s partcpatng n three groups t wll get three estmatons one from each group after performng the collaboratve flterng. The aggregaton n basc and fnal steps are performed by weghted averagng. The weghts are computed based on the varance of the estmaton.e the weghts are nversely related wth the varance. If a fragment n havng hgh varance t s consdered as noser and the weght assocated wth that frame s mnmum. The weghts are calculated as follows Basc estmaton: Where w HTC BE = σ NHTC (9) f N otherwse NHTC s the number of non-zero coeffcents after hard thresholdng Fnal estmaton: w FE = () σ W C Where ' W C ' are the wener flter coeffcents. Fnally the weghted averagng s computed and each fragment s returned ts orgnal poston. 4. EVALUATION CRITERIA FOR DENOISING ALGORITHMS To evaluate the qualty of the mage processng algorthms there are several metrcs proposed n the lterature. The metrcs are classfed as pxel dfference based measures, correlaton based measures, edge based measures, spectral dstance measures, context based measures and Human vsual system based measures. Here we are comparng our denosng algorthms usng a group of metrcs drawn from the above class and performance of the algorthms was observed. 4. Pxel dfference based measures 4.. Mnkowsk metrcs The L γ norm of the dssmlarty of two mages can be calculated by calculatng the mnkowsk average of the pxel dfferences spatally and then chromatcally as gven below K M N γ γ (, ) ˆ ε = fk x y fk ( x, y) K k = MN () x= x, y= Where f ( x, y) s the reference mage, f ˆ( x, y) s the estmated mage of f ( x, y) by our denosng algorthm wth the nput g( x, y) whch s a nosy verson of f ( x, y ). γ 63

12 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December For γ = we obtan the absolute dfference (AD), for γ = we wll obtan the mean square error (MSE). Along wth these two measures we are calculatng mnkowsk measures for γ = 3 and γ = 4 n ths paper to observe the performance of our algorthms. 4.. PSNR (Peak Sgnal to Nose Rato) PSNR s the peak sgnal-to-nose rato n decbels (db). The PSNR s only meanngful for data encoded n terms of bts per sample, or bts per pxel. For example, an mage wth 8 bts per pxel contans ntegers from to 55. B PSNR = log () MSE Where B represents bts per sample and MSE (Mean Squared error) s the mean square error between a sgnal f ( x, y) and an approxmaton f ˆ( x, y ) s the squared norm of the dfference dvded by the number of elements n the sgnal. ˆ MSE = f ( x, y) f ( x, y) = f x, y f x, y MN = = M N ( ) ˆ ( ) (3) x y M N RMSE = f x y fˆ x y MN = = (, ) (, ) (4) x y MSE and RMSE measures the dfference between the orgnal and dstorted sequences. PSNR measures the fdelty.e how close a sequence s smlar to an orgnal one Maxmum Dfference Maxmum dfference s defned as ( ( ) ˆ ( ) ) MD = max f x, y f x, y (5) The large value of maxmum dfference means denosed mage s poor qualty Normalsed Absolute Error (NAE) The large value of normalsed absolute error means that denosed mage s poor qualty and s defned as NAE = M N x= y= M N x= y= (, ) ˆ (, ) f x y f x y f ( x, y) (6) 4..5 Sgnal to Nose Rato (SNR) Sgnal to nose rato n an mage s calculated as SNR = µ (7) σ 64

13 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December Where µ s the average nformaton n the sgnal and σ s the standard devaton of the sgnal whch represents the amount of nose present n the mage. There s one more measure s there smlar to the SNR t s sgnal to background rato. µ SBR = (8) σ BG Subtract background from the mage calculate standard devaton from t and fnally compute the above rato. 4. Correlaton based measures The correlaton between two mages can also be quantfed nterms of correlaton functon. These measures measure the smlarty between the two mages hence n ths sense they are complementary to the dfference based measures. 4.. Structural content For an M N mage the structural content s defned as M N ( x, y) (9) f K k x = y = SC = K M N k = fˆ x, y x= y= k ( ) The large value of structural smlarty means that denosed mage s poor qualty 4.. Normalsed cross correlaton measure (NK) The normalsed cross correlaton measure s defned as 4..3 Czekanowsk dstance M N ( ) ˆ, (, ) (3) fk x y fk x y K x = y = NK = K M N k = f x y x= y= k (, ) A metrc useful to compare vectors wth strctly postve components as n the case of mages s gven as ( f ( ) ˆ k x y fk ( x y) ) K mn,,, M N k = C = MN (3) K x= y= f (, ) ˆ k x y + fk ( x, y) k = Ths coeffcent s also called as percentage smlarty measures the smlarty between dfferent samples, communtes and quadrates. 4.3 Edge Based metrcs 4.3. Laplacan Mean Square Error (LMSE) Ths measure s based on mportance of edges measurement. The large value of Laplacan mean square error means that the mage s poor qualty. LMSE s defned as 65

14 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December LMSE = 4.4 HVS based metrcs ( ) M N x= y= ( (, )) ˆ (, ) L f x y L f x y M N x= y= ( ( x, y) ) L f 4.4. Unversal Image Qualty Index (UQI) It s a measure used to fnd the mage dstorton. It s mathematcally defned by makng the mage dstorton relatve to the reference mage as a combnaton of three factors: Loss of correlaton, Lumnance dstorton and contrast dstorton. If two mages f ( x, y ) and fˆ (, ) contanng pxel values f ( x, y ) and fˆ (, ) may be calculated as a product of three components MN σ f fˆ σ σ Q = Where f f ( x, y) σ ffˆ (3) x y are consdered as a matrces wth M column and N rows ffˆ f fˆ ˆ σ fσ fˆ f + f σ f + σ fˆ M N = and fˆ = ˆ f ( x, y) x= y= ( (, ) ) (, ) x y respectvely the unversal mage qualty ndex Q MN ( ) M N ˆ ˆ = f x y f f x y f M + N x= y= M σ f = M N ( ) M N x= y= f ( x, y) f and σ (, ) f + N ˆ x= y= (33) ( f x y f ) = ˆ ˆ M + N M N x= y= The frst component s the correlaton coeffcent whch measures the degree of lnear correlaton between mages. It vares n the range [-,]. The best value s obtaned when the mages are lnearly related. The second component measures how close the mean lumnance s between mages wth a range [, ]. The thrd component measures the contrasts of the mages the value range for ths component s [, ]. The range of values for Q s [-, ]. The best value s acheved f and only f the mages are dentcal Structural smlarty (SSIM) ndex s a method for measurng the smlarty between two mages [8]. The SSIM ndex s a full reference metrc, n other words, the measurng of mage qualty based on an ntal uncompressed or dstorton-free mage as reference. SSIM s desgned to mprove on tradtonal methods lke peak sgnal-to-nose rato (PSNR) and mean squared error (MSE), whch have proved to be nconsstent wth human eye percepton []. The SSIM metrc s calculated on varous wndows of an mage. The measure between two wndows x and y of common sze N N s [8]: ( µ xµ y + c )( σ xy + c ) SSIM ( x, y) = (34) µ + µ + c σ + σ + c Where ( x y )( x y ) µ x s the average of x and µ y s the average of y 66

15 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December σ x s the varance of x and σ y s the varance of y s the covarance of x and y σ xy c = ( kl), c = ( kl) two varables to stablze the dvson wth weak denomnator #bts per pxel L s the dynamc range of the pxel values (typcally ths s ) k =.and k =.3 by default In order to evaluate the mage qualty ths formula s appled only on luma. The resultant SSIM ndex s a decmal value between - and, and value s only reachable n the case of two dentcal sets of data. Typcally t s calculated on wndow szes of 8 8. The wndow can be dsplaced pxel-by-pxel on the mage. 5. RESULTS The algorthms were appled to denose the X-ray mages contamnated wth Posson nose and the MR mages contamnated wth Rcan nose. Three varance stablzaton transforms square root, Freeman & Tukey and Anscombe are used n combnaton wth patch based algorthms Nonlocal means and BM3D. The qualty metrcs were tabulated as below. The denosed mages of Posson nose (Anscombe+NLM, Anscombe+BM3D) and of Rcan nose (Square Root +NLM, Square root+bm3d, Freeman+NLM and Freeman+BM3D) are presented below. Table : Performance evaluaton of denosng algorthms for Posson nose 67

16 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December Table 3: Performance evaluaton of denosng algorthms for Rcan nose 68

17 The Internatonal Journal of Multmeda & Its Applcatons (IJMA) Vol.4, No.6, December 6. CONCLUSIONS AND FUTURE WORK The development and mplementaton of denosng flters usng varance stablzaton transform and patch based algorthms were carred and evaluated ther performance usng varous mage qualty metrcs. The developed algorthms are appled on the x-ray and MR Images and the vsual appearance after denosng was mproved, the contrast of the mages are also mproved. Further 69

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