Impact of bilateral filter parameters on medical image noise reduction and edge preservation

Transcription

1 The Univesity of Toledo The Univesity of Toledo Digital Repositoy Theses and Dissetations 2009 Impact of bilateal filte paametes on medical image noise eduction and edge pesevation Michael D. Lekan Medical Univesity of Ohio Follow this and additional woks at: Recommended Citation Lekan, Michael D., "Impact of bilateal filte paametes on medical image noise eduction and edge pesevation" (2009). Theses and Dissetations This Thesis is bought to you fo fee and open access by The Univesity of Toledo Digital Repositoy. It has been accepted fo inclusion in Theses and Dissetations by an authoized administato of The Univesity of Toledo Digital Repositoy. Fo moe infomation, please see the epositoy's About page.

2 Health Science Campus FINAL APPROVAL OF THESIS Maste of Science in Biomedical Sciences (Medical Physics - Diagnostic) Impact of Bilateal Filte Paametes on Medical Image Noise Reduction and Edge Pesevation Submitted by: Michael Lekan In patial fulfillment of the equiements fo the degee of Maste of Science in Biomedical Sciences Examination Committee Signatue/Date Majo Adviso: Michael Dennis, Ph.D. Academic Advisoy Committee: E. Ishmael Pasai, Ph.D. Diana Shvydka, Ph.D. Associate Dean College of Gaduate Studies Doothea Sawicki, Ph.D. Date of Defense: Novembe 11, 2009

3 A Thesis Entitled: Impact of Bilateal Filte Paametes on Medical Image Noise Reduction and Edge Pesevation Michael D. Lekan Univesity of Toledo Health Sciences Campus Decembe 2009

4 Abstact The objective of this pape is to examine the bilateal filte and its application to digital medical imaging. The bilateal filte is a non-linea adaptive smoothing filte which is capable of peseving the edges of image stuctues. Selection of the bilateal filte s paametes geatly influence how well an image is smoothed and how well edges ae peseved. The paametes of the bilateal filte wee analyzed individually. The noise eduction fo each paamete is elated to the much simple box (aveaging) filte. Edges wee also analyzed fo each paamete. Techniques fo pope paamete selection ae studied and geneal guidelines ae suggested. Afte the popeties of each paamete wee examined, they wee ecombined and applied to CT images. Solutions addessing the limitations of this filte fo high noise CT images ae also consideed. It was found that the bilateal filte is capable of poviding effective smoothing and at the same time peseves edges which ae othewise blued by othe filtes. The viability of this filte geatly depends on the selection of its paametes. ii

5 Dedication I dedicate this wok to my daughte Elena. iii

6 Acknowledgements This study would not have been possible without the suppot of the following individuals. I would fist like to thank D. Michael Dennis fo his constant suppot and advice. Without his encouagement to exploe this topic and insight in the aea of image pocessing, this wok would not have been completed. I also thank D. Ishmael Pasai and D. Diana Shyvdka fo thei suppot, advice and seving on my committee. This wok would not have been complete had it not been fo the Univesity of Toledo Medical Cente Radiology Depatment. I especially thank the adiology technologists fo taking the time out of thei busy day to assist in the scanning of the CT phantom and acquiing the clinical images. I eceived unmeasued pesonal suppot fom my wife Rebecca, bothes Mak and Todd and my paents. Finally, I d like to thank my gandpaents who played such impotant ole in defining who I am today. iv

7 Table of Contents Intoduction..1 Liteatue Mateials. 26 Methods...29 Results. 33 Summay of Results & Discussion Conclusions Bibliogaphy v

8 Intoduction The numbe of photons used fo adiogaphic images is elatively small. Photogaphic images utilize appoximately photons/mm 2. In geneal adiogaphic images make use of about 10 5 photons/mm 2, (Huda and Slone 2003). Photon statistics follow a Poisson distibution which has a standad deviation given by N, whee N is the numbe of photons pe esolution element o pixel. The image matix fomed by those photons will have pixels which may deviate fom the mean intensity, i.e. some pixels will have gey levels which deviate fom the mean gey level. This is the oigin of image noise. As the numbe of photons incident on the detecto inceases, the elative noise level deceases. Image noise can significantly educe the quality of adiogaphic images. When noise is pesent, objects have educed visibility and infomation egading thei size and shape is compomised. Thus it is often advantageous to apply a smoothing filte to emove noise. Image smoothing educes the vaying photon intensities pe pixel by aveaging them togethe. This modifies the image so that adjacent pixels ae moe unifom in intensity. Howeve, smoothing can significantly blu stuctues in an image. Like noise, pixels which epesent stuctues will have a diffeent intensity fom thei non-stuctue neighbos. In medical images significant anatomy and pathology can be blued. Thus, 1

9 thee is a need to develop filtes which can accomplish smoothing yet peseve stuctual detail. In geneal the methods of image pocessing can be divided between linea and non-linea opeations. Linea filteing is commonly used fo smoothing to educe noise. Linea pocessing is well undestood and has pedictable outcomes. The outcomes of non-linea pocessing ae moe ambiguous and ae not evesible, and theefoe have limited application, (Gonzalez & Woods 2008). Howeve, non-linea filtes can pocess images in a way which allows them to adapt to the distinctive attibutes of a neighbohood within an image. Consequently non-linea pocessing can poduce bette esults than linea filtes. A discussion of linea and non-linea opeations and filtes can be found in Gonzalez and Woods (2008). Filteing can take place in the spatial o fequency domain. Pocessing in the spatial domain is descibed mathematically by the sum of poducts o a convolution: a b wxy (, ) f( xy, ) wst (, ) f xsy, t (1) satb Whee wxy (, ) is the filte mask and f ( xy, ) is the input image. The mask is of size mn and a ( m1) 2and b ( n 1) 2 (Gonzalez et al. 2008). The filte pefoms some opeation, such as aveaging the pixel values in the neighbohood. A new pixel is fomed fom that opeation with the same coodinates as 2

10 the cente pixel of the neighbohood. The filte is passed ove the entie image pixel by pixel and a new image is fomed. Some common spatial linea filtes used to smooth noise fom an image ae the box (aveaging) filte and the spatial Gaussian filte. These filtes take the weighted aveage of the pixels in a neighbohood. An aveaging filte, which has filte coefficients that ae equal, is called a box function. Equation 2 shows how f ( xy,, ) the input image, is pocessed with a box filte. The pocessed image is gxyand (, ) M N is the numbe of pixels being aveaged: m n i j (2) gxy (, ) 1 ( M N f( xy) i1 j1 In this and subsequent equations the summation fom 1 to m o fom 1 to n epesents stepping though all of the pixel values within the filte kenel. Filtes can also pefom a weighted aveage with vaying coefficients. A common aveaging filte with this quality is the spatial Gaussian filte. The spatial Gaussian weights the cente value highe than any othe pixel in the neighbohood. The othe pixels ae weighted based on the Euclidean distance fom the cente. To calculate the coefficients of the filte: wxy (, ) 1 k m n i1 j i x j y (x -cente ) +(y -cente ) domain e (3) 2 3

11 Whee d is the spatial standad deviation o domain value, i and j ae the coodinates of x and y, m and n ae the dimensions of the kenel and 1 k is the nomalization facto. The lage the standad deviation the boade the Gaussian distibution. A lage domain value will allow inclusion of moe distant pixels and geate smoothing will occu. A key popety of non-linea filtes is they ae not mathematically evesible. A common non-linea filte is the median filte. It is a type of odeed statistic filte. The median filte woks by soting the pixels in the neighbohood. The median intensity pixel is found and that pixel is used fo the new pixel in the filteed image. Median filtes ae paticulaly effective at educing speckle o spike noise. Othe ode statistic filtes ae the maximum and minimum filtes. These filtes ae commonly used to find bight and dak pixels in an image. Additional non-linea filtes ae descibed in the Liteatue section. Fo completeness, fequency filteing is biefly discussed. Fequency domain filteing is sometimes a moe suitable way to filte images. Fo example sinusoidal intefeence can be filteed in the fequency domain by designing an appopiate filte to emove that intefeence. To filte an image in the fequency domain it must be tansfomed into the fequency domain and then back to the spatial domain though the following pocess, (Gonzalez et al. 2008): 1.) The image is tansfomed to the fequency domain though the following Fouie tansfomation: 4

12 M1N1 j2 ux Mvy N f xye (4) Tuv (, ) (, ) x0 y0 Whee Tuvis (, ) the Fouie tansfom, f ( xy, ) is the input image, M and N ae the matix dimensions and u and v ae the fequency domain vaiables. 2.) An appopiate filte is applied that will enhance o depess the intensity at cetain fequencies. That is: T( u, v) H( u, v) T( u, v) 3.) The image is tansfomed back to the spatial domain though the invese Fouie tansfom: 1 f x, y T ( u, v) e MN M1N1 j2 ux Mvy N (5) x0 y0 Whee f, xy is the filteed image. The tansfomation given in these equations ae discete Fouie tansfoms since the data is given as a seies of discete elements o pixels, athe than as a continuous mathematical function. Smoothing pefomed in by Fouie filteing is equivalent to the coesponding convolution filteing in the spatial domain. Some common fequency domain filtes ae the Buttewoth low-pass filte and the Gaussian low-pass filte. The Buttewoth filte is mathematically chaacteized by: 5

13 1 Huv (, ) 1 [ D u, v ] (6) n 2 D 0 D u v is the distance of point uv, fom the oigin in the fequency domain and whee, D 0 is the limiting adius o cutoff fequency of the filte. Using a smalle D 0 esults in educing the high fequency content of the filteed data. The ode n detemines how smooth the tansition is fom highe and lowe fequencies. A smoothe cutoff will esult in less Gibb s inging atifact in the image. All fequencies past D 0 ae filteed out of the image. Hence, the highe fequencies on the outskits of the Fouie space ae emoved fom the image. The Gaussian low-pass filte is mathematically chaacteized by: 2 2 D ( uv, ) 2D0 Huv (, ) e (7) A lage value of D 0 tanslates into a smoothe tansition between the low and high fequencies. The effect of the Gaussian filte applied in the fequency domain is simila to the spatial Gaussian convolution filteing except a lage σ (domain) maximizes smoothing and a lage D o minimizes smoothing. Using local statistics fo image pocessing was suggested by Ketcham (1976) and Wallis (1976). It was extended to contast and edge enhancements by Lee (1981). In his oiginal pape Lee used this method fo additive and multiplicative noise. His technique pocessed pixels independently. The assumption is that a sample mean and vaiance of a 6

14 pixel is equal to the mean and vaiance of all pixels in a neighbohood suounding it (Lee 1980). The advantage of this method is that it is non-iteative and image modeling is not equied as with the othe techniques such as Wiene filteing. Lee s papes ae discussed futhe in the Liteatue chapte. Wiene filteing equies that an estimate of the spectal popeties of the uncoupted image is known o assumed. Moe specifically: gxy (, ) hxy (, ) f( xy, ) ( xy, ) (8) Whee gxy (, ) is the coupted image, hxy (, ) the degadation function and ( x, y) the additive noise. The input image is convolved with the degadation function. The point of the Weine filte is to minimize the mean squae eo between the estimate of the uncoupted image (coupted image degadation function) and the oiginal uncoupted image. The bilateal filte was fist intoduced by Tomasi and Manduchi (1998). Simila to Lee s technique, the bilateal filte does not equie iteation o image modeling and employs the use of local statistics. The motivation fo ceating this filte was to emove image noise and peseve edges. The filte opeates by examining both spatial and photometic chaacteistics (intensity diffeences) of the pixels in a neighbohood, (Tomasi et al. 1998). 7

15 As descibed in Tomasi et al. (1998), domain filteing is based on spatial location o closeness of neighbohood pixels. Range filteing is based on photometic similaity o similaity in gay levels. Domain filteing alone filtes an image based only on the distance, of a given pixel, to the cente pixel of the neighbohood. The assumption is that pixels close to the cente ae likely to be of the same stuctue. Domain filteing tends to blu edges because no consideation is given to pixel intensity diffeences o whethe they ae tuly of the same stuctue. Range filteing examines the pixel intensities of a neighbohood with espect to the intensity of the cente pixel. The following deivation of the bilateal filte is fom Tomasi et al. (1998): (9) k 1 ( x) f( ) c(, x) s( f( ), f( x)) d The tem c(, x) standad spatial Gaussian filte: is the weighting facto fo the spatial component. This tem is a 1 d(, x) 2 (10) d c(, x) e 2 The tem d(, x) is the distance between the cente value of the neighbohood and one of the neighboing pixel,. The tem s( f( ), f( x)) is the weighting facto fo the 8

16 photometic similaity. It is also Gaussian in fom and measues the intensities between the cente pixel and a neighboing pixel: 1 2 f (, fx 2 s( f( ), f( x)) e (11) The tem k 1 ( x) is the nomalization and given by: kx ( ) c(, xsf ) ( ( ), f( x)) d (12) The esult is a new pixel at location x found by aveaging the neighbohood values centeed on x with pixels that ae spatially close and photometically close. The final equation is given by: hx ( ) f ( ) c(, x) s( f( ), f( x)) d c(, x) s( f( ), f( x)) d (13) The theoetical undepinnings of the bilateal filte and its elationship to well known iteative algoithms, such as anisotopic diffusion ae discussed in Elad (2002) and Baash (2002). 9

17 The domain standad deviation, of the bilateal filte, will detemine the amount of bluing. If lage, moe distant pixels will be used and edges will become blued. The domain filte may be limited by specifying a limited kenel size, m n, which effectively cuts the tails off o tuncates the spatial Gaussian function. If the kenel is small and d is lage, the filte appoaches a box filte. On the othe hand if is elatively small thee will be little smoothing. The bilateal filte weighting factos, in the discete case, ae calculated by the following: w(, ) ange domain f (x i,y j)-f (cente x, y) (xi-cente x) +(y j-cente y) m n 1 2 ange domain e k i1 j1 (14) 2 Whee m and n ae the dimensions of the kenel and 1 k is the nomalization tem. The two tems of the exponential ae the ange and domain components espectively. This pape will analyze how the bilateal filte pefoms on CT images. Thus a discussion of CT image econstuction is intoduced. As an example, a small ound attenuating object is placed in the CT scanne. An x-ay is tansmitted though the object and the attenuation acoss the x-ay path is measued by a detecto opposite the souce of that ay. Each element along the ay s path contibutes equally to the total attenuation and the ay is backpojected onto an empty image matix. 10

18 Moe tansmissions ae geneated at multiple angles and the esult is a summed attenuation coefficient fo each element. Each backpojection fom each ay is added and aeas whee the ays intesect poduce a geate intensity, due to the attenuation of the object. The point spead function of the object diminishes with 1/. Whee is the distance fom the point whee the ays ovelap. Round Attenuating Object Figue 1 As the image above indicates, thee is bluing aound the object whee all of the ays convege. In ode to emove the blu a shapening filte is applied befoe the ays ae backpojected onto the matix. In the spatial domain the pojection data can be convolved with the debluing kenel as follows: f ( x) f( x) w( x) (15) 11

19 f ( x) is the filteed pojection data which is backpojected to fom the CT image. The filteing in Equation 15 can also be pefomed in the fequency domain as was shown by Equations 4 an 5. Once in the fequency domain the image data can be multiplied by the fequency domain kenel (de-bluing kenel) and tansfomed back to the spatial domain though the invese Fouie tansfom. The Fouie tansfom pai of Equation 15 is given by: f( x) w( x) H( ) F( ) (16) Equation 16 illustates the Convolution Theoem that the convolution of two functions in the spatial domain can be expessed by the poduct of the Fouie tansfom of these two functions in the fequency domain. The Lak filte, amp filte, is given by L( f) (Ramachandan and Lakshminaayanan, 1971). f and depicted gaphically in Figue 2 Lak Filte Amplitude Fequency Figue 2 The 1/ bluing due to the backpojection in the spatial domain coesponds to a 1/f degadation in the fequency domain. 12

20 When multiplying the image in the fequency domain by the Lak filte, the bluing is emoved,1 f f 1, (Bushbeg et al. 2002). The poduct of the image tansfom, H ( ), and L( f ) is then tansfeed back to the spatial domain though the invese Fouie tansfom. When noise is pesent it is beneficial to have some oll off at high fequencies. The Shepp-Logan filte (Shepp and Logan, 1974) has oll off at high fequencies and is depicted in Figue 3. Shepp-Logan Filte Amplitude Fequency Figue 3 The Hamming filte can also be applied to give even moe high fequency ole off and shown in Figue 4. 13

21 Hamming Window Amplitude Fequency Figue 4 The poduct of the Lak filte and the Hamming window is depicted in Figue 5. Lak Filte Hamming Window Spatial Domain Figue 5 In CT, filteing a high pojection value causes negative contibutions to the adjacent filteed pojection values. This appopiately econstucts the image fom the tue signal data, when combined with othe filteed data in the backpojection pocess. Noise, howeve, is andom and the negative contibution to adjacent pojection values causes coelation of the noise in the econstucted images. Refeing to the spatial filte in Figue 5, the neighbos of the cental pixel ae lowe in intensity. If the cental pixel is a noise spike its neighbos will be lowe in value. Coelation of noise in CT images will influence the pefomance of spatial smoothing filtes. This will be shown in the Results chapte. 14

22 As discussed the bilateal filte has both a domain and ange component, both of which ae epesented by a Gaussian distibution in this pape. Analyzing them individually povided a bette undestanding of each component s contibution to the filte s oveall pefomance. The domain potion of the bilateal filte was analyzed by compaing its noise eduction and edge pesevation chaacteistics to the box filte. The box filte woks well fo smoothing ove noise and its popeties ae staightfowad. Howeve, its limitations include poo edge pesevation and smoothing ove fine detail. Because the ange component does not yield meaningful esults by itself (Tomasi et al. 1998), it was combined with the box filte which defines the spatial neighbohood o kenel. How the ange component influenced the box filte s noise eduction pefomance povided insight into how the ange pefomed in aeas of unifomity and edges. Afte the domain and ange components wee analyzed individually they wee combined in the fom of the bilateal filte. This study was pefomed in two pats. Fist the spatial box filte, spatial Gaussian filte and the bilateal filte wee applied to images of unifom intensity and images with defined edges. Gaussian noise is chosen because of its mathematical tactability. It can povide a good appoximation of Poisson noise and often used, (Wenick and Aasvold 2004). 15

23 The second pat is the application of the bilateal filteing to CT images. Both phantom and clinical CT images will be studied. The CT phantom images will include low contast and esolution slices with vaying mas o noise levels. The slices will be econstucted with a soft tissue and bone econstuction kenels. The aim of the analysis is to poduce highe quality images by educing noise, peseve edges and stuctue detail. The objectives of this pape ae to investigate the bilateal filte and assess its utility fo medical images, specifically CT bain images. This filte claims to poduce images with deceased noise while peseving edges. These ae highly sought goals of medical imaging. Thus, it is wothy to futhe examine this filte s pefomance on medical images. Due to the geneality of this filte its paametes will be examined with espect to noise eduction and edge pesevation. As will be demonstated, pope selection of the bilateal s paametes can significantly change the esponse of the filte. This pape will specifically study the paametes of the bilateal filte. The following popositions will be examined: 1. The bilateal filte can sepaated into ange and domain components and studied individually. 2. Each paamete, with espect to noise eduction, can be linked to the box filte. 16

24 3. The noise dependency of the bilateal filte can be epesented by the atio. This atio can then be linked to a box equivalent size fo a bilateal ange noise filte with a box domain. 4. The ange value can be effectively detemined by object contast. 5. The techniques of using a pe-filte and iteating the bilateal filte will incease noise eduction and peseve edges. They will allow using a low ange value even when ange noise is small. To accomplish the objectives of this pape the bilateal filte will be tested by the following methods: 1.) A compute geneated phantom with vaious objects and vaying gay levels will be ceated and noise will be added. The bilateal s geneal pefomance will be tested in a Gaussian distibuted noise envionment. Gaussian noise offes the most geneal and mathematically tactable types of noise. 2.) A CT phantom will be used to obtain images that ae then post pocessed with the bilateal filte. Noise eduction, edge pesevation and low contast will be examined. 3.) Clinical CT bain images will be pocessed with the bilateal filte. 17

25 The tools that will be used to achieve the goals of this pape ae MatLab and MatLab s image pocessing tool box. An M-file, a use defined MatLab function, will be implemented to ceate the bilateal filte. MatLab s pedefined function nlfilte will be used fo the convolutions. All medical images will be in the dicom fomat and a standad dicom viewe will be used to access the pocessed images. 18

26 Liteatue Seach Adaptive filteing can be implemented in a vaiety of ways. Howeve all adaptive techniques shae the same geneal paadigm. That paadigm is to utilize the unique chaacteistics of a pixel neighbohood to ceate a new (filteed) image. Linea filteing does not take into account the unique chaacteistics of pixel neighbohoods. Theefoe edges and fine detail can become lost with linea smoothing filtes. The fist uses of local statistics in image pocessing can be found in Ketcham (1976) and Wallis (1976). The theoetical advantages and a desciption of geneal adaptive neighbohood pocessing ae discussed in Debayle and Pinoli (2006). This pape elates the adaptive appoach to mathematical mophology and demonstates that a pioi infomation about image stuctues can be deived fom the local mean and vaiance. This diveged fom pevious methods which elied on a coelation model to find the mean and vaiance. The advantage is that contast can be bette peseved. Utilizing the local vaiance and mean fo image filteing was discussed by Lee (1980). In this pape Lee demonstated that that the local mean and vaiance can be successfully applied to images with additive o multiplicative noise. The novelty of this pape is extending the use of local statistics to contast enhancement. It is also demonstated that the computational efficiency is supeio to such techniques as fequency domain o 19

27 ecusive pocessing. Thus Lee s technique is bette fo eal time pocessing because each pixel can be individually pocessed. In a following pape by Lee a technique was developed to educe noise nea edges. Depending on the oientation of an edge, the local mean and vaiance ae calculated fom a educed numbe of pixels (Lee 1981). Using local gadient infomation allows the mean and local vaiance to be calculated on one side of the edge. These ealy woks by Lee demonstated that impoved image contast can be obtained by use of local pixel statistics. Rangayyan et al. (1998) poposed an adaptive neighbohood filte which gows fom a single pixel fo images with signal dependent noise. This filte woks by estimating the noise and signal of a neighbohood. The neighbohoods ae gown fom a single pixel based on the type of noise and powe of the noise. Only pixels that belong to the same object ae included in the neighbohood. Simila to Rangayyan s appoach Guis et al. (2003) suggested an adaptive neighbohood appoach fo mammogaphic phantom images. Local statistics aound a given pixel ae calculated. Based on those local statistics, the neighbohood will gow to include only those pixels which ae simila to the efeence pixel. This method uses a maximum gowth size fo the neighbohood. Gowth stops if the maximum size is eached o the pecentage of pixels in the neighbohood ae ove the pe-detemined theshold by 60%. 20

28 Anisotopic diffusion as intoduced by Peona and Malik (1990), uses the idea of scalespace filteing. Scale space filteing is based on hieachical oganization Peona et al. (1990). The pocess of scale-space filteing involves embedding the oiginal image with a family of deived images with a one vaiable paamete, esolution. This can be accomplished though a diffusion pocess o a convolution with a Gaussian kenel. Koendeink (1984). The scale space design involves a convolution of an image with a Gaussian filte that has a scale space paamete t, that is: I( xyt,, ) I( xy, ) Gxyt (,, ) (17) 0 Whee I 0 is the input image and Gxyt (,, ) is a Gaussian kenel. Lage values of t coespond to coase esolutions. It is show in by Koendeink (1984) that the family of deived images can be viewed as the solution to the diffusion equation. Peona et al. (1990) extended scale based filteing though the use of a modified diffusion equation. The diffusion equation is given by: I div((, c x y,) t I) (18) t whee cxyt (,, ) is the diffusion coefficient. The esult I t is a scala field. The modification of the diffusion equation includes an edge stopping function: cxyt (,, ) g Ixyt (,, ) (19) 21

29 Whee the edge stopping function is zeo in the inteio of a egion. This pevents smoothing outside of the egion. Black et al. (1998) develops a elationship between anisotopic diffusion and obust statistics. The edge peseving popety of this method consides edges between constant egions to be outlies, Black et al. (1998). The pupose of this method is to use obust statistics to analyze and design isotopic diffusion. A connection between the bilateal filte and the anisotopic diffusion (non-linea diffusion equation) was established by Baash (2002). In his pape the bilateal filte and the diffusion equation ae linked though adaptive smoothing. The diffusion coefficient and the kenel, in the bilateal filte, pefom the same pupose. In addition thee is global dependence on intensity fo the kenel of the bilateal filte. The gadient (the diffusion coefficient) shows only local pixel dependence. Thus thee is a need to pefom seveal iteations with the diffusion equation. The bilateal filte was fist intoduced by Tomasi et al. (1998). The filte was fomally defined in the Intoduction chapte. The bilateal filte utilizes both a spatial and a photometic (intensity) component. The spatial component is efeed to as the domain and the photometic component is efeed to as the ange. The oiginal pape uses a Gaussian distibution fo both components. Howeve, othe kenels could be used. Combining the domain and ange components is essential to the filte. The ange component alone simply modifies the gay map of the image. Howeve when combined 22

30 with spatial constaints the emapping is local and dissimila at diffeent locations in the image (Tomasi et al. 1998). In the pesence of edges the bilateal filte woks especially well due to the ange component. Fo example, the kenel is at the bode between a light and dak egion and centeed on the light egion. The pixels on the light side of the kenel will be photometically simila. Thus the ange weighting factos fo those pixels will be appoximately one. The dak side of the kenel will have low ange weighting factos. Those pixels will have little influence on the calculation of the new pixel. A pape by Elad (2002) bidges the bilateal filte to moe classical filteing appoaches such as anisotopic diffusion. This pape demonstates that bilateal filte emeges fom a Bayesian appoach. Speeding up the bilateal filte and inceasing its smoothing is also poposed. A theoetical backgound fo the oigin of the bilateal filte is poposed in this pape. Anothe pape by Duand and Dosey (2002) uses the bilateal filte fo the display of high dynamic ange images on low dynamic ange media. The image is decomposed into a base laye and a detail laye. The base laye is obtained with the bilateal filte and it is only this laye which has its contast educed. That is lage scale featues have thei contast educed while detail is peseved. This pape demonstates the vesatility of the bilateal filte to othe image pocessing demands. 23

31 Xie et al. (2008) discusses a technique to ovecome the poblem of choosing a kenel size fo the bilateal filte. The kenel size is impotant when balancing edge pesevation and noise eduction. Constant kenels ae eplaced by a function that deceases with bounday saliency. The idea is that unifom aeas will have a boad kenel and at edges a shape kenel. A pixel in an image plane has a saliency measue which is the saliency of the geatest edge acoss it. Fo example point p is located on a salient edge. If anothe point, q, is not located on the edge the similaity is computed using only a sub-egion containing p and q. If q is located on the edge the entie filteing window is used fo the similaity calculation. This technique has a stong dependence on pope calculation of the salient featues. Walke et al. (2006) used the bilateal to minimize the smoothing, typically pefomed in low signal fmri images, in the pesence of bain lesions. The pupose of this pape was to delineate boundaies between a bain lesion and nomal bain tissue. Obtaining a good bounday allows minimal esection of nomal functional bain tissue and maximal emoval of the pathological bain tissue. Zhou et al. (2007) applied the bilateal filte to PET imaging. Eibenbege et al. (2008) evaluated the bilateal filte, anisotopic diffusion and the wavelet appoach fo CT images. This pape analyzed the feasibility of each filteing technique fo the segmentation of ogans. It was concluded that the bilateal filte and anisotopic diffusion had to have thei paametes adapted to each ogan and fo diffeent noise levels fo pope segmentation. The paametes fo the wavelet appoach had to 24

32 have its paametes adjusted only once fo pope segmentation of diffeent ogans with diffeent noise levels. This can be addessed by selecting the ange value based on the lowest contast object in the image and the image noise level. This will be discussed futhe in the Results chapte. The tilateal filte was poposed by Wong et al. (2004). Like the bilateal filte the tilateal uses the spatial and ange components. Howeve it adds a thid component, which examines the local stuctual oientations between neighboing pixels. Oientation tensos fom eigen decomposition ae used to identify the local stuctual infomation. Fo medical imaging smalle kenel sizes ae necessay so that detail is not smoothed fom the image. Howeve in ode to obtain good noise eduction moe iteations ae equied. Fewe iteations ae equied with the tilateal filte. As eviewed in this chapte, the bilateal filte has been successfully used in medical image pocessing and othe aeas. Impovements have also been poposed. Compaed to othe filteing techniques, like anisotopic diffusion, the bilateal filte is faily staightfowad in design. Though this filte is simple in concept its paametes need thoough analysis as they geatly influence its total esponse. In addition those paametes have many dependencies. As discussed in the Intoduction, this pape will examine the paametes of the bilateal filte. In doing so bette pedictions can be made of how the bilateal will espond fo diffeent noise levels and types of noise. 25

33 Mateials A windows based compute was used in this study. MatLab Image Pocessing Toolbox was used fo all image pocessing. A use defined M-File function was implemented fo the bilateal filte. MatLab s nlfilte was used fo convolving the bilateal filte. Sante Dicom Viewe was used fo viewing the CT clinical and phantom images. The image captue pogam, Snagit was used to captue images fo this pape. Open Office 3.1 speadsheet pogam was used fo the data analysis, gaphs and tables. A compute geneated phantom was developed to evaluate the bilateal filte and is depicted in Figue 6. The phantom contains the following stuctues: Intensity Pyamid: The top of the pyamid begins at intensity 0 (on the left side) and 1 (on the ight side). Descending the pyamid, the intensities ae incemented ove 128 pixels gadually blending to 0.5 at the base of the pyamid. The pupose of the pyamid is to asses how each filte pefomed on edges with vaious levels of contast. The intensity of the phantom s backgound is 0.5. Low Contast Boxes: Thee ae five ows and five boxes in each ow. The ows incease by 5%, above backgound in intensity fom the top ow to the bottom. The sizes of the 26

34 boxes ae: 33, 55, 77, 99 and The low contast boxes wee used to assess how well each filte was able to maintain the visibility of small stuctues. Resolution Bas: Thee ae five sets with thee bas fo each set. The fist set of bas ae 15 pixels wide and 15 pixels apat. The subsequent bas ae 10, 5, 3 and 2 pixels wide and sepaated by thei width. Compute Geneated Phantom Figue 6 The phantom in Figue 6 was used to evaluate the bilateal filte unde Gaussian noise conditions. The noise was geneated using MatLab s function imnoise. The CT phantom used was the Phantom Laboatoy s Catphan 600. The low contast, esolution and unifomity slices wee used. The mas levels wee 900, 400, 200 and 100mAs. All, except the 900mAs, wee 4mm slices. The 900mAs 16mm slice was used fo the measuing the aveage CT numbes fo the thee Supa-Slice taget goups. The nominal taget contast levels fo the thee Supa-Slice taget goups wee, 0.3%, 0.5% and 1.0%. The taget goup diametes anged fom 2.0mm to 15mm. The sub-slice 27

35 vaying length ods wee not examined since thei contast vaied with the slice thickness. The econstuction filtes used fo this study wee the bone and soft tissue filtes. CT data was obtained fom a Toshiba Aquillion 16 including scans of a 20 cm diamete Ameican College of Radiology CT Acceditation Phantom. An anonymized axial head scan was acquied unde an exempt IRB potocol. CT phantom and head scans wee econstucted utilizing diffeent econstuction filtes. 28

36 Methods As discussed in the Intoduction chapte, the bilateal filte is non-linea adaptive smoothing filte which can peseve edges. It combines both domain and ange filteing. The amount of noise eduction and edge pesevation is dependent on the kenel size, domain value, ange value, and noise level. This pape will analyze the impact the bilateal filte paametes have on smoothing and edge pesevation. In analyzing the paametes a moe concete pediction can be made, fo a given paamete combination, egading noise eduction. It will demonstate how fo diffeent ange values, some image stuctues ae bette peseved. A link, with espect to noise eduction, will be made to the simple box (aveaging) filte. Finally a method fo the selection of the ange value will be evaluated. The amount of noise eduction fo the filtes examined in this pape can be found though the standad deviation of a weighted mean. The standad deviation of a weighted mean is: 2 2 x w n (20) i1 i i In the case of unifom noise, i is the same fo all samples. Fo a box filte all of the weighting values equal 1 N and the standad deviation of the mean is: 29

37 n 2 2 x 1 N (21) i1 i 1 ( x) (22) N whee N is the numbe of pixels in the neighbohood and ( x) is the noise of the pocessed image. Fo example if the box filte kenel size was 33 then: 1 x / (23) 9 The eduction in noise is 1 3. Fo the spatial Gaussian filte the noise eduction is given by: ( x) (x -cente ) +(y -cente ) i x j y m n 1 2 d e k i1 j1 (24) 2 Whee m and n ae the dimensions of the kenel and is the domain value and ( x) is the noise of the pocessed image. Noise eduction fo the spatial Gaussian filte is dependent on the kenel size and the domain value. The noise eduction of the bilateal filte is given by: d 30

38 ( x) f (x i,y j)-f (cente x, y) (xi-cente x) +(y j-cente y) m n 1 2 ange domain e k i1 j1 (25) Noise eduction fo the bilateal filte depends on the ange value and the same dependencies of the spatial Gaussian filte. Because the weighting factos of the bilateal ae geneated by compaing intensities (a neighbohood pixel is compaed to the cente intensity) the bilateal is also dependent on noise. Fo example should the kenel be centeed on a noise spike the neighbos of that spike will eceive low weighting values and the noise spike will be peseved. To examine how the domain and ange values affect the esponse of the filte, they wee sepaated and studied individually. The spatial Gaussian filte was used to evaluate the domain component. Noise eduction fo a given kenel size and domain value, of the spatial Gaussian filte, was linked to the noise eduction of the box filte. In doing so the dependence of noise eduction is effectively educed to only the kenel size. It will also be shown that the spatial Gaussian filte peseves edges bette than the box filte fo equivalent noise eduction. The ange component was combined with a spatial box filte and the amount of noise eduction was linked to the box filte. By linking the noise eduction of the bilateal (Gaussian ange/box domain) filte to the box filte, a bette pediction could be made of the bilateal s noise eduction in a unifom aea. In addition genealizations can made 31

39 egading the maximum amount of noise eduction fo a given ange value. Fo object contast the ange value will be selected based on what objects ae to be delineated. The bilateal filte was evaluated fist with Gaussian noise. The compute geneated phantom fom Figue 6 was used fo this pat of the analysis. Edge pesevation was evaluated by an examination of the low contast boxes and intensity pyamid. The bilateal filte was then applied to CT phantom images. Some geneal conclusions ae dawn fo using the bilateal filte fo bone and soft tissue econstucted images. The techniques of iteating the bilateal and applying a pe-filte pio to the bilateal ae also studied. These techniques ae useful when the noise value is high compaed to the desied ange value. The teminology used thoughout this pape follows the oiginal pape by Tomasi et al. (1998) and summaized below: 1. Domain Value o d : Refes to the spatial spead of the bilateal filte. It also efeences the standad deviation fo the Gaussian convolution kenel. 2. Range Value o : Refes to the photometic spead of the bilateal filte. 3. Noise level o noise : Refes to the spead of the noise in an image scaled [0,1]. 32

40 Results As discussed in the methods section, the paametes of the bilateal (domain and ange vaiables) filte ae sepaated and studied individually. This pape uses a Gaussian distibution to detemine the weighting factos fo both the ange and domain components. If the ange component of the bilateal filte is emoved the filte effectively becomes a Gaussian spatial filte. Domain values can be analyzed, independent of the ange component, via the spatial Gaussian filte. Since the box filte has vey staightfowad noise eduction popeties, its noise eduction will be elated to the Gaussian spatial filte. The ange component will be analyzed by combining it with a spatial box domain. The noise eduction popeties of the bilateal filte (Gaussian ange/box domain) will also be elated to the box filte. The dependence on noise will be genealized by using a ange/noise atio. A method fo selecting the ange value based on object contast will also be examined. The Results chapte will conclude with an examination of CT phantom and CT clinical images fo bone and soft tissue econstuction kenels. The pe-filte and iteative techniques will be analyzed. These techniques will be essential when the ange/noise atio is low. 33

41 The Results section is outlined as follows: 1) Compute Geneated Phantom with Gaussian Noise a) Domain Component Analysis b) Range Component Analysis i) Method fo Selecting the Range Value 2) CT Images a) Soft Tissue Kenel b) Bone Kenel c) High Resolution Slice d) Clinical CT Images 1) Compute Geneated Phantom with Gaussian Noise a) Domain Component Analysis Figue 7 displays the obseved and theoetical noise eduction vs. kenel size fo the box filte. The theoetical matches vey closely to what was obseved fo a image of unifom intensity. 34

42 Box Filte Factional Theoy vs. Obseved Noise Standad Deviation: Factional Noise Kenel Size Obseved Powe Regession fo Obseved Theoy Powe Regession fo Theoy Figue 7 Noise eduction fo the Gaussian spatial filte is dependent on the geometic spead, d, and the kenel size. The value of d influences the effective size of the kenel. Figue 8.A-C displays the un-nomalized weighting values fo 33, 55, 77 and 99 kenel fo vaious domain values. Neighbohood Sizes 9x9 7x7 5x5 3x3 35

43 d = A d = B d = C Figue 8.A-C Figue 8.A-C shows that an inceasing domain value boadens the Gaussian weighting value which will be tuncated by the limits of the kenel size. Fo example when d =2.0 the 33 kenel is effectively a 33 box filte. The figues also show that when 36

44 the domain value is small elative to the kenel size, the weighting values fall off to zeo with distance fom the cente. Fo example, when d =1.0 and the kenel size is 77 the esult is an un-tuncated Gaussian distibution, see Figue 2.B. Noise eduction of the spatial Gaussian filte and the box filte can be elated by solving Equations 22 and 24 fo neighbohood size N. The esult is: N( d ) m n i1 j1 e i x j y (x -cente ) +(y -cente ) d 2 2 (26) Figue 9 plots effective box kenel sizes vs. domain values of the spatial Gaussian filte. Effective Box Kenel Size Box Kenel Size vs Domain Value Noise Reduction Equivalent Domain Value 3x3 5x5 7x7 9x9 Figue 9 37

45 Recall that noise eduction fo the box filte is simply 1 n. By elating the spatial Gaussian filte to the box filte, though Equations 22 and 24, the influence of the domain value and kenel size on noise eduction can be simplified. When woking with the bilateal filte, selection of the domain value and kenel size can be thought of in tems of the box filte though this elationship. The advantage of the spatial Gaussian ove the box filte is that consideation can be given to pixel distance. The box filte weights all of the pixels in a neighbohood with the same value. Though the selection of the domain value, the Gaussian weights pixels based on thei distance fom the cente pixel. Thus, it is expected that the spatial Gaussian s pefoms bette fo edge pesevation. A compute geneated phantom was ceated to test how well a filte peseves edges. The phantom descibed in the Mateials chapte is displayed in Figue 10. Compute Geneated Phantom Figue 10 38

46 A 5 5 spatial Gaussian filte with d =0.86, poduces the same noise eduction (0.34) as a 3 3 box filte. As an example, a 5 5 spatial Gaussian filte with d =0.86 will be compaed to a 3 3 box filte. The filtes ae applied to the phantom in Figue 10. Figue 11 A -D shows the esult. Highest Contast Row, Fist Thee Boxes 33 Box Filteed, Low Contast Boxes (A) Highest Contast Row, Fist Thee Boxes 55 Spatial Gaussian Filteed, =0.86 d (B) 33 Lowest Intensity Contast Box 33 Box Filteed 55 Spatial Gaussian, d =0.86 (C) Figue 11 (D) 39

47 As can be seen in Figue 11 the spatial Gaussian filte peseved moe of the low contast boxes than the box filte. Figue 12 shows esults fo the left side of the intensity pyamid afte being filteed. As descibed in the mateials section, the two sides of the intensity pyamid gadually blend to an intensity of 0.5 ove 128 pixels. The 55 Gaussian kenel with d =0.86 peseved the edge bette than the 33 box filte. Intensity Intensity Pyamid Descending Edge Pixel Location Unfilteed 3x3 Box 5x5 Gaussian, D=0.86 Figue 12: Intensity Measuements fo the left side of intensity pyamid. The point whee the plots mege is when the two filtes begin to poduce simila esults with espect to edge pesevation. The Gaussian s use of weighting factos educes the bluing of edges. This is because pixels on the peiphey of the kenel ae weighted less than the cente. The pevious 40

48 example demonstated that fo equivalent smoothing, the spatial Gaussian filte peseves edges bette than the box filte. Thus, the spatial Gaussian filte is easonable choice fo the domain component of the bilateal filte. The point of thooughly examining the Gaussian spatial filte, is that the ange filte has no notion of space, Tomasi et al. (1998). Theefoe, the ange filte must combined with a domain filte. When using a spatial Gaussian fo the domain, the kenel size and d will detemine how lage of a spatial aea the photometic spead,, will extend. b) Range Component Analysis The ange component of the bilateal filte is epesented gaphically in Figue 13. The gaph displays how weighting factos ae assigned when the kenel is centeed on intensity 0.5. The y-axis is the assigned elative weighting facto fo a given intensity. The intensities within the cuve ae weighted based on thei similaity fom the cente. The width of the cuve Gaussian cuve of Figue 13 is contolled by, the photometic spead. If the is boade than the intensity spead, in a paticula neighbohood, the esponse of the filte will be simila to a domain filte only. That is the domain component dominates the filte. 41

49 Bilateal Filte Range Component Only Intensity Figue 13 The bilateal filte is dependent on the noise level of an image. This is because the weighting factos fo the ange component ae found by examining the intensities of pixels in a neighbohood in efeence to the cente pixel. The ange value with espect to noise will influence the noise eduction of the filte. Figue 14 is a plot of the effective box kenel size vs. noise fo a unifom aea. Effective Box Size Gaussian Range/Spatial Box Domain Effective Box Kenel Size Range Value/Noise 3x3 5x5 7x7 9x9 Figue 14 42

50 In Figue 14, a box filte was chosen as the domain component so an equivalent box size could be found. Recall, fo noise eduction in a unifom aea, the spatial Gaussian, fo a given domain value and kenel size, could be elated to an equivalent box size. Theefoe if a spatial Gaussian wee used as the domain component, the domain value and kenel size could be deived fom Figue 9 fo a given noise. When the plots on the gaph of Figue 14 each dn ( d ) 0, the bilateal filte effectively becomes a box filte, that is it aveages all values within the kenel. The only limiting facto of the bilateal filte, when using a box as the spatial component, is and the kenel size. The noise eduction of the bilateal filte with a Gaussian spatial component is epesented by Equation 27. noise ( x) f (x i,y j) - f (cente x, y) (xi-cente x) + (y j-cente y) m n 1 2 ange domain e k i1 j1 (27) Note fom the equation that if d o is lage, the espective tem appoaches zeo. When this happens the esponse of the filte becomes eithe a domain o ange dominated filte. Fo noise eduction analysis a Gaussian ange and a box domain component was used. In doing so the bilateal could be linked to an effective box kenel, fo a given noise. If 43

51 a spatial Gaussian was chosen fo the domain component the kenel size and domain value, of the spatial Gaussian, could be deived fom the effective box kenel size fo a given noise. To compae edge pesevation, the spatial Gaussian filte and bilateal filte wee matched with espect to noise eduction. A 55 spatial Gaussian filte with a d value of 0.65 poduced factional smoothing (unifom aea) of The bilateal with a d value of 1.2 and of poduced factional smoothing (unifom aea) of Figue 15 A & B display the esult of filteing the phantom with the above mentioned paametes. 44

52 55 Bilateal Filteed d =0.65 A 55 Bilateal Filteed d =1.2, =0.195 B Figue 15 45

53 Note fom Figue 15 that the spatial Gaussian filteed image blued moe than the bilateal filteed image. This is especially noticeable fo the high contast esolution bas as shown in Figue 16. High Contast Bas 55 Bilateal Filteed d =1.2, =0.195 A 55 Spatial Gaussian Filteed d =0.65 B Figue 16 With egad to the low contast boxes, Table I shows how the ange weighting facto changed fo each ow of the low contast boxes. Low Contast Box Row Low Contast Boxes Intensity Range Weighting Facto Table I Table I shows that the low contast boxes in ow five will have the best edge pesevation. The impotance of choosing an appopiate value of is citical fo edge pesevation. The weighting factos in Table I ae lage enough that when the kenel is 46

54 centeed on a low contast box and pat of the kenel includes backgound pixels, those backgound pixels will be assigned faily lage ange weighting factos. This will cause bluing of the low contast boxes. Figue 17 ae plots of the cente of the left side of the intensity pyamid. Intensity Bilateal vs. Gaussian Spatial Filte Intensity Pyamid Pixel Location Unfilteed 5x5 D=1.2, R= x5 D=0.65 Figue 17 What is obseved in Figue 17 is that not until appoximately pixel seventy does the bilateal filte begin blu the edge. At pixel seventy the left side of the pyamid has an intensity of 0.24 and the ight side Thus, fo the given ange and domain values, the bilateal would peseve edges that have an intensity pecent diffeence of up to 68% to thei neighbos. Afte which the bilateal gadually esponds moe like a spatial Gaussian filte. The domain and ange values of the pevious example wee chosen to compae the bilateal to the spatial Gaussian filte fo equivalent noise eduction. A moe pecise method fo choosing the ange value will be discussed late. 47

55 The pefomance of the bilateal was next analyzed with noise of =0.1 added to the phantom image. As befoe, the bilateal paametes wee: a kenel size of 55, domain of 1.2 and ange of Fo the spatial Gaussian filte the kenel size was 55 and the domain value Recall those paametes poduced the same noise eduction in a unifom aea. The esults fo the low contast boxes ae displayed in Figue 18 A-D. 48

56 Spatial Gaussian, d =0.65, noise =0.47 Bilateal, d =1.2, =0.195, noise =0.48 A B Median Filteed 55 Filte, noise = Filte, noise =0.56 C Figue 18 D The bilateal filte peseved the low contast box edges bette than the othe filtes. Also note that some noise outlies wee peseved by the bilateal filte. This occus when the kenel is centeed on an outlie. Low weighting values will be assigned to the est of the pixels in the neighbohood. This will cause some outlies to be peseved. 49

57 Figue 19 shows the lagest (1111) contast box with the highest intensity (0.67) fo the unfilteed and filteed images Highest Contast Box Figue 19 The median filteed image, especially the 55, is moe unifom than the othes. This is because the median filte selects the median of the neighbohood. Thus few outlies will be peseved. 50

58 b.i) Method fo Selecting the Range Value The ange value in the pevious example was chosen only to compae the bilateal filte to the spatial Gaussian filte fo equivalent noise eduction. A method fo selecting the ange value based on object contast is now examined. When the kenel is centeed on an object, a linea filte will use the pixels in the neighbohood without egad to thei intensity similaity to the cente. If those pixels suounding the cente ae dissimila to the cente they will destoy the cente pixel s uniqueness. A linea filte may extend past the object and pull in pixels that can be vey diffeent in intensity. If a small object is filteed with a filte lage than the object, that object can be blued fom the image. An adaptive filte, such as the bilateal, excludes o includes pixels based on the unique intensity chaacteistics of a neighbohood. The ange value will evaluate pixels in the neighbohood based on thei intensity similaity to the cente of the kenel. If a pixel is spatially distant fom the cente but simila in intensity it will eceive a highe ange weighting facto. Examining the equation fo the ange component, of the bilateal filte, povides insight into the selection of the ange value. 2 1 w( ) k 1f (x i,y j) - f (cente x, y) m n 2 e (28) i1 j1 51

59 If f (x,y) object - f (cente x, y), then the ange weighting facto is appoximately 0.606, 12 ( w( ) e 0.606). Fo example, when the cente of the kenel is placed outside one of the phantom low contast boxes, pixels of the neighbohood which happen to be inside the low contast box should be weighted the least. When (1 4) f (x,y) object - f (cente x, y ) the weighting facto fo pixels inside a low contast box appoach zeo. Thus, to eliminate the backgound fom being blued into the lowest contast box (box intensity = 0.526, backgound intensity=0.5) a = (¼)( ), would sepaate this low intensity contast box fom backgound. Figue 20 shows appoximate weighting facto fo diffeent values of. Range Weighting Facto Bilateal Filte Range Weighting Factos Diffeence R=1Delta R=Delta/4 R=4Delta Figue 20 52

60 The abscissa, of Figue 20 is the photometic diffeence ( k (f (x,y) object - f (cente x, y)) ). Fo example when = f (x,y) object - f (cente x, y), a pixel whose diffeence with the cente pixel of the neighbohood is, that pixel will eceive a ange weighting value of appoximately 0.6. If the diffeence with the cente pixel was 2, that pixel would eceive a ange weighing facto of appoximately Figue 21 illustates how using the geneal ange values in Figue 20 will effect the esponse of the filte fo noiseless data. The 1111 low contast box with intensity of 0.76 and fo 0.55 on a backgound of 0.5 is used as an example. Response of the Bilateal Filte Figue 21 53

61 When the ange value is high ( 4 ( I, I ) ) the smoothing esult is simila to the Box Backgound spatial Gaussian filte. When the ange value is lowe ( ( I, I ) 4) the 1111 Box Backgound low contast box is peseved completely. Fo efeence, the 1111 low contast box with an intensity of 0.55 (lowe ight image of Figue 21) will be blued when the ange value is set using the 0.76 intensity box. This demonstates how some objects in an image may be filteed in a diffeent way than othes. The method fo selecting the ange descibed peviously has limitations. It was shown ealie in Figues 9 and 14 that the atio noise should be elatively high fo acceptable noise eduction. In ode to use low ange values with espect to noise, a pe-filte and/o the bilateal should be iteated. The bilateal filte can be used as a pe-filte. When noise esults in little o no smoothing, the atio can be inceased with a highe ange value fo the bilateal pe-filte. The ange value then can the be loweed fo subsequent iteations afte noise level has been educed by the pe-filte. The spatial Gaussian could also be used fo the domain component. The domain value in Figue 9 and noise value in Figue 14 can be combined into the bilateal filte fo a cetain equivalent box size. Using a spatial Gaussian fo the domain component instead of the box filte is advantageous given the bette edge pesevation of the spatial 54

62 Gaussian. Table II shows the change in noise eduction when the domain value fom Figue 9 and noise values fom Figue 14 ae combined. Bilateal: Gaussian Range/Box Domain noise (Fo Column 1) Spatial Gaussian Table II d (Fo Column 2) Box Equivalent Size (Fo columns 1 & 3) Adjusted Box Equivalent Bilateal Filte (Columns 2 & 4) Values in Table II obtained fom a unifom aea. Table II is found by meging Figue 9 and 14. Columns one and two ae the values fo the bilateal filte that poduced the noise eduction in column five. Columns thee and fou ae the values fo the spatial Gaussian filte which poduced the noise eduction also in column five. Column six is the box equivalent size fo the bilateal filte using the values of column two and fou. The ange value fom column two can be extacted by solving fo. The pupose of Table II is to link noise eduction of the spatial Gaussian filte and bilateal filte (with a box domain) though the box filte. Table II shows the 55

63 effect the ange value has on noise eduction with a Gaussian ange and Gaussian domain. As an example, noise with 0.04 is added to the phantom. The lowest contast boxes in the phantom have an intensity of Recall the backgound intensity is 0.5. Thus a ange value of ( and =0.015) would be the optimum ange value to peseve all of the low contast boxes in the phantom. The value of ( ) is Thus the noise eduction is appoximately zeo, see noise Figue 14. In ode to use a ange value of 0.015, the noise must be educed. A box filte will educe the noise by appoximately by 1/3, see Table II column six. With a noise level of 0.04, the noise should be educed to about This is below the desied ange value of The noise fo a box filte is 2.7, see Table II. Thus a ange value of will be used fo the pe-filte ( and 0.108) with a domain value of 1.39, see Table II. The measued noise, afte the bilateal pe-filte, is The unfilteed and filteed images ae shown in Figue 22 A and B. 56

64 Unfilteed 55 Bilateal Filteed, d = 1.39, = A Figue 22 B A ange value of can be used now that the noise has been inceased, =1.25. In summay the ange value can have a global edge pesevation if it is detemined by using the lowest contast object in the image. In doing so all objects highe in contast will be peseved. If howeve, noise is low, a pe-filte may need to be applied. A link fo noise eduction was established between the spatial Gaussian and bilateal (box domain) filte though the box filte. This allowed a bette pediction of the noise eduction fo a given value of d and noise. A method was analyzed fo detemining the ange value based on object contast. It was shown that finding the diffeence between two intensities and then dividing that diffeence by fou oughly povides the maximum edge delineation between those to intensities. 57

65 When noise is elatively low it was shown that noise eduction may be minimal and a pe-filte could be used. The bilateal filte can wok as a pe-filte by aising the ange value. Afte an image is pe-filteed and the noise is educed to an acceptable level, the ange value can then be loweed. The median and aveaging filte can also be used as pe-filtes. They will be analyzed as pe-filtes with CT images. 2) CT Images Noise in CT images is coelated (see Intoduction fo a desciption of CT noise). In CT, images ae smoothed as pat of the econstuction pocess. The soft tissue econstuction kenel smoothes an image moe than the bone econstuction kenel. This is because the soft tissue kenel has moe high fequency oll off. The bone kenel will poduce highe esolution images and the soft tissue will have less image noise. Table III lists the noise fo each mas level and econstuction kenel type. mas Table III Soft Tissue (HU) noise Noise Bone (HU) noise

66 Table IV lists the data fo the lagest taget of the low contast slice fo each mas and econstuction kenel. The pupose of the table is to ensue that the data is consistent fo all the CT goups. mas Lage Taget Avg. CT # Table IV Soft Tissue Kenel Backgound Avg. CT # % Contast mas Lage Taget Avg. CT # Bone Kenel Backgound Avg. CT # % Contast The CT phantom used fo the following analysis is descibed in the mateials and methods section. Noise in CT is multiplicative and due to the econstuction pocess, coelated. The goal of this section is to study the bilateal s pefomance with CT noise. The CT phantom was scanned using the soft tissue and bone kenels. The slices wee 4mm and the mas levels wee: 400, 200 and

67 a) Soft Tissue Kenel Ealie it was found that noise eduction fo the spatial Gaussian filte could be elated to the spatial box filte, fo a given kenel size and domain value, though the following equation: Aea( d ) m n i1 j1 e i x j y (x -cente ) +(y -cente ) d 2 2 (29) Noise eduction fo the box filte, fo the soft tissue econstuction kenel ove seveal mas levels was found to be appoximately: 1 x 2 (30) N This elationship was veified within 12% fo images econstucted with the soft tissue kenel. The adjustment of the noise eduction equation fo the box filte is because images ae smoothed fom the soft tissue econstuction kenel. Figue 23 ae plots of the effective box kenel size vs. noise of bilateal box domain filte. 60

68 Bilateal Filte: Gaussian Range/Spatial Box Soft Tissue Kenel Effective Kenel Size Range Value/Noise 3x3 Kenel 5x5 Kenel 7x7 Kenel 9x9 Kenel Figue 23 In Figue 23 the least amount of noise eduction occus when the ange value is low with espect to the noise. Also note that the equivalent box size values in Figue 23 ae about half of those in Figue 14. In CT a high pojection value causes negative contibutions to the adjacent filteed pojection values. This appopiately econstucts the image fom the tue signal data, when combined with othe filteed data in the backpojection pocess. Howeve, noise is andom and the negative contibution to adjacent pojection values causes coelation of the noise in the econstucted images. Due to the coelation, when the high noise pojection value occupies the cente of a neighbohood, the neighbos will eceive a low ange weighting facto. This may tend to peseve the noise with bilateal filteing. A 61

69 potential solution is to use a median filte pio to using the bilateal filte. In doing so the median of the neighbohood, athe than the noise spike will be used. Noise will ultimately limit what ange value can be used. If the ange value is low with espect to the noise thee will be little noise eduction, see Figue 23. A solution is to emove some noise fom the image pio to applying the bilateal filte. This can be done with a median filte. The median filte has good edge peseving popeties and is moe likely to be epesentative of the neighbohood. The bilateal filte can also be iteated seveal times afte the median filte is applied. This will educe the noise futhe and peseve edges. As descibed, if the ange value is ( I, I ), and the kenel is centeed on a Taget backgound taget pixel, the backgound pixels will eceive a weighting facto of appoximately 0.6. The appoximate point at which the bilateal filte will espond as a spatial filte only (the ange component effectively becomes one) is when the ange value is lage i.e. ange = 4 ( I, I ) t aget backgound. When the ange = t aget backgound ( I, I ) 4the ange weighting facto will assign oughly zeo to the backgound pixels. The stategy listed above is a ule of thumb and infomation about the contast levels of the image must be known. Distinguishing between CT numbes can be done though this method. Figue 24 is the 900mAs 16mm soft tissue econstucted low contast slice used to measue the taget CT numbes. 62

70 900mAs 16mm Soft Tissue Reconstucted Slice Figue 24 The lagest taget in the goup of tagets was measued to obtain the aveage CT value. The goups ae labeled one though six. Taget goups fou though six ae ods of constant attenuation and vaiable lengths. Thei aveage CT numbe is simila to taget goup one. Table V displays the aveage CT# fo each taget and backgound. Table V Taget Goup Aveage (HU) Backgound (HU) Diffeence (HU)

71 Example 1: 400mAs/4mm Soft Tissue Reconstucted Low Contast Slice If the diffeence of taget goup thee and backgound wee selected as the ange value and the kenel is centeed on a taget in goup thee, a typical backgound pixel within the neighbohood will eceive a weighting value of appoximately 0.6. Table VI displays the weighting factos fo the backgound when the kenel is centeed on a taget goup when the ange value is 4.23HU. Table VI Taget Goup Weighting Facto HU The values in Table VI ae dependent on how close the measued aveage intensity of each taget is to the actual intensity of the taget. It also depends on the how close the backgound is to the actual backgound. Thus, these weighting factos ae appoximate. A ange value of 4.23HU assigns a elatively high ange value to backgound pixels. Theefoe, the ange value is deceased to 2.5HU which is ( I, I ) 4. This GoupI Backgound ange value will assign appoximate ange weighting factos listed in Table VII. 64

72 Taget Goup Table VII Backgound Weighting Facto, = 2.5HU HU Although these ae ough values, they povide a geneal idea on how the bilateal filte will ange weight backgound pixels o taget pixels when the kenel is centeed on a taget pixel o backgound pixel. The measued noise fo the 400mAs 4mm soft tissue econstucted slice is 4.42HU. If a ange value of 2.5HU wee to be used noise = Refeing to Figue 23, the noise eduction would be appoximately zeo fo the bilateal box domain filte. A pe-filte will be necessay given the lowe ange value. A 55 median filte is chosen. It was found that noise eduction fo the box filte and median filte wee simila, see Figue

73 0.8 Unifom Aea, 400mAs/4mm: Soft Tissue Kenel Median and Aveaging Filte CT Noise Reduction Median Filte Avg Filte Noise Reduction Kenel Size Figue 25 Howeve, edge pesevation should be bette with the median filte. The noise will be educed to appoximately 2.21HU with a 55 median pe-filte. The images in Figue 26 A and B demonstate that applying a median pe-filte followed by the bilateal does not negatively impact edge pesevation. 66

74 Taget I Goup I Median/Bilateal Compaison A 5x5 Median Noise: 1.94HU Unfilteed Noise: 4.42HU Figue 26 B 55 Median Pe-Filte d, = 2, 2.5HU Noise: 1.49HU Thee was little change in edge pesevation between A and B. Thee was a noticeable decease in noise fo Figue 26.B. This demonstates that applying a median pe-filte pio to the bilateal does not degade edges any moe than the median filte alone. The iteative appoach was next applied to the 400mAs soft tissue econstucted image. The image was fist pocessed with a 55 median filte and then with thee iteations of the bilateal filte. The esult is displayed in Figue 27 A-E. 67

75 Iteative vs. Non-Iteative A 5x5 Median 9x9 Bilateal D/R Values: 2, 2.5HU Noise: 1.54HU B 5x5 Median 9x9 Bilateal 3 Iteations D/R Values: 2, 2.5HU Noise: 0.98HU Taget I Goup I Non-Iteated Bilateal Iteated C D 68

76 Taget I Goup II Non-Iteated Bilateal Iteated E Taget I Goup III Non-Iteated Bilateal F Iteated G H Taget 1 Goup 1 Spatial Gaussian Filteed, 1111, d =3 Noise Level: 1.1HU I Figue 27 69

77 Fom Figue 27 the iteated bilateal had significant noise eduction ove the non-iteated bilateal. Figue 28 shows the subtacted image (iteated bilateal non-iteated) fo the fist taget of goup one. Taget Goup I Tagets I & II (5x5 Median 9x9 Bilateal 3 Iteations d & Values: 2, 2.5HU) (5x5 Median 9x9 Bilateal d & Values: 2, 2.5HU) Figue 28 The fist and second tagets of goup one showed some diffeence between the two images. The slightly highe intensity inside the fist two tagets of goup one means that the non-iteated bilateal was lowe in intensity inside the taget. The lowe intensity ing on the outside of the taget means the non-iteated bilateal was highe in intensity. This tanslates into moe blu inside and outside the taget fo the non-iteated bilateal. Some atifacts ae pesent in Figue 29.B (55 Median, 99 Bilateal 3 Iteations, d =2, = 2.5HU). These can be educed by using a lage pe-filte. The esults ae displayed in Figue 29 A and B. 70

78 Atifact Suppession A 5x5 Median 9x9 Bilateal 3 Iteations D/R Values: 2, 2.5HU Noise: 0.98HU B 9x9 Median 9x9 Bilateal D/R Values: 2, 2.5HU Noise Unifom: 0.68 Figue 29 Using lage pe-filte educed the noise moe than the iteated smalle pe-filte, Figue 29.A. As a final example the bilateal filte is compaed to the median filte. The esults ae displayed in Table VIII. The spatial Gaussian filte is also displayed fo efeence. Table VIII Filte Pe-Filte Range Kenel Size Goup 1 Taget 1 Iteations Gaussian, D=3 None N/A 15x Median None N/A 15x Med/Bilat, D=2 Median, 7x x Bilateal, D=5 None x

79 Fom Table VIII, the bilateal without a pe-filte o iteations must have a ange value of 100 befoe it can match the noise eduction of the othe filtes. With a ange value of 100 the esponse of the bilateal would effectively be a spatial Gaussian filte. To compae edges, Figue 30A-D shows selected images fom Table VIII. Median Filte vs. Bilateal A 15x15 Median Noise: 0.74 B 7x7 Median 9x9 Bilateal D/R Values: 2, 2.5HU Noise: 0.83 Taget 1 Goup 1 Median Filteed Bilateal Filteed C Figue 30 D 72

80 Fo compaable noise eduction the bilateal filte peseved moe edges than the median filte. Figue 31 is the esult of subtacting the pevious two images, the 1515 median was subtacted fom the iteative bilateal Figue 30 A and B. (7x7 Median 9x9 Bilateal D/R Values: 2, 2.5HU 3Iteations)- (15x15 Median) Figue 31 The tagets in goup two ae the second lowest in contast fom the backgound and ae displayed in Figue

81 Taget Goup II Taget I A 15x15 Median Noise: 0.74HU B 7x7 Median 9x9 Bilateal D/R Values: 2/2.5HU 3 Iteations Noise Level: 0.83HU Figue 32 C 13x13 Gaussian, D =3 Noise: 1HU The iteative technique is useful fo educing noise and at the same time maintaining edges. Fo the median filte o spatial Gaussian filte to poduce the same amount of noise eduction as the iteative bilateal, thei kenel sizes had to be lage and it is advantageous to use smalle kenel sizes. Smalle objects can be effectively smoothed out of an image with lage kenel sizes. Thus, the iteative bilateal is a sound technique fo filteing low contast images. When selecting a ange value consideation should be given to the noise level and what stuctues in an image ae desied to be peseved. When the ange is low with espect to the noise level the noise eduction will be minimal. If set too high the filte s esponse is simila to a spatial Gaussian filte. In geneal, fo maximum edge pesevation, the ange value should be set as low as possible. The techniques demonstated ae ways of loweing the noise level so that a lowe ange value can be used. In geneal knowledge of how the ange value effects noise eduction 74

82 should be chaacteized. It was also shown that the bilateal filte can be compaed to the spatial box filte fo equivalent smoothing. The noise eduction fo the box filte is appoximately 2 1 N fo the soft tissue econstucted images. Being able to elate the bilateal filte to the spatial box filte though the ange value allows one to undestand, with a given noise level, how well a paticula ange value will fai. If the ange must be kept low, to peseve low contast stuctues, the median pe-filte and/o iteative techniques can help achieve inceased noise eduction and at the same time maintain edges. Using the lowest contast object fo the calculation of the ange is advantageous in that all othe stuctues will be peseved. Example 2: 100mAs/4mm Soft Tissue Reconstucted Low Contast Slice When filteing in high noise envionments, such as the 100mAs low contast slice, the iteative bilateal filte and o a pe-filte will be necessay. The noise level fo the 100mAs slice is 9.06HU. The value of noise is 0.46, with a value of 4.23HU. The ange value was inceased due the high noise. A 99 median pe-filte will educe the noise level to oughly 2.5HU. The bilateal with a ange value of 4.23HU can then be applied with iteations. The esults ae displayed in Figue 33 A and B. 75

83 100mAs/4mm Soft Tissue Reconstucted A 9x9 Median, 9x9 Bilateal, Domain Value=2, Range Value=4.23HU 2 Iteations Noise: 1.45HU Figue 33 B 9x9 Median, 9x9 Bilateal, Domain Value=2, Range Value=4.23HU 5 Iteations Noise: 0.97HU The image in Figue 33.A shows some globule atifacts. Because the soft tissue econstuction kenel has a fai amount of smoothing, the bilateal filte will peseve these globules. Theefoe, a lage pe-filte and multiple iteations should be used to smooth away the globules. b) Bone Reconstucted Images Figue 34 displays the esults of the noise eduction fo the theoetical and actual box filte. The theoetical is based on: x 1 N. Whee x pocessed image. is the noise of the 76

84 Box Filte Theoy/Actual and Median Filte Bone Kenel 900,400,200,100mAs Levels Theoy Actual Median Filte 0.6 Noise Reduction Kenel Size Figue 34 Afte a kenel size of 33, the box filte fo bone kenel econstucted images begins to espond close to theoy. This may be due to the fact that with negatively coelated noise as found in the bone econstuction kenel, the high cente spike fom the econstuction pocess aises the aveage fo small kenel sizes. As the kenel sizes become lage thee ae moe pixels in the neighbohood to aveage out the high intensity spike. On the othe hand, fo the median filte, the midpoint between the high cente intensity spike and the low may be highe. Thus the noise becomes peseved with these two filtes. Figue 35 shows the plots of the box equivalent vs. noise. 77

85 Effective Box Kenel Size Gaussian Range/Spatial Box Bone Kenel Range Value/Noise 3x3 Kenel 5x5 Kenel 7x7 Kenel 9x9 Kenel Figue 35 Fo the phantom the ange values of inteest ae fom appoximately 1-5HU. These ange values would allow vey little smoothing fo the bone econstucted images. Thus, the noise should be educed pio to applying the bilateal and iteations of the bilateal will be needed to use those ange values. Contast fo taget goups two and thee in Figue 36 ae quite compised by the high noise. 78

86 Unfilteed Bone Kenel Low Contast Slice 400mAs/4mm Figue 36 A lage box filte will be fist applied and the ange value inceased fom 2.5HU. The diffeence between the tagets in goup thee, the lowest contast tagets, and the backgound is 4.23HU. Using 4.23HU fo the value of would assign an appoximate ange weighting value of 0.6 to any backgound pixels which have a diffeence fom the cente pixel of 4.23HU. The bilateal kenel size will also be inceased to an The esults ae displayed in Figue 37 A-C. The 400mAs soft tissue econstucted image is included in Figue 37 along with a median filteed image fo compaison. 79

87 Bone Reconstuction A 9 9 box pe-filte, Bilateal d =3, =4.23HU, 2 Two Bilateal Iteations Noise: 0.73HU Soft Tissue Reconstuction B 400mAs/4mm 7 7 Median Pe- Filte, 9 9 Bilateal =2, =2.5HU, 3 Bilateal Iteations Noise: 0.76HU d 80

88 Median Filteed C 1919 Median Filteed Noise: 0.94HU Figue 37 A-C The image in Figue 37.A was subtacted fom the image in 37.B. The esult is displayed in Figue

89 Figue 37.B 37.A Figue 38 Note that the soft tissue econstucted image, Figue 37.B blued slightly less than the bone econstucted image. Thee was little change fo taget goups two and thee. Figue 39 shows the fist taget of goup one fo both images. 82

90 400mAs Soft Tissue 400mAs Bone A B Refeence Figue 163.B Refeence Figue 163.A Figue 39 Thee was slightly moe blu with the bone econstucted image. Relatively speaking, the noise eduction was fa moe significant with the bone econstucted image than the soft tissue econstucted image. Recall the soft tissue kenel econstuction positively coelates the noise which means thee is geate smoothing in the oiginal image. The bone kenel negatively coelates the noise and thee is less smoothing in the oiginal econstucted image, but geate noise eduction with the image post-pocessing. In summay, fo high noise envionments like the 100mAs soft tissue kenel and the 400mAs bone econstucted images, the bilateal filte can educe noise significantly and yet peseve edges. Depending on the value of noise a pe-filte should be used. The median filte woks well as pe-filte and showed negligible edge degadation compaed to the median filte alone. Iteating the bilateal filte is also a viable technique which does not degade edges. 83

91 Choosing an appopiate ange value will ensue edge pesevation. The ange value was chosen based on delineating CT phantom tagets fom thei backgound. Fo global edge pesevation the ange value should be set with the lowest contast object in mind. All objects of highe contast will be peseved this way. The type of econstuction kenel will influence how well noise eduction will occu. Fo the soft tissue econstucted images it was obseved that, fo the box filte, ( x) 2 1 N. This is because thee is a consideable amount of smoothing fom the econstuction pocess. On the othe hand the bone econstucted images have less smoothing fom the econstuction pocess. Thus, noise eduction will be geate with the pe-filte fo bone econstucted images. c) High Resolution Slices The high esolution slices wee not pe-filteed as they ae consideably highe in intensity than backgound. The bone econstucted 400mAs/4mm esolution slice was used. The noise level was measued as 20.52HU. The bas ae closely spaced so a small kenel size will be used. A 33 kenel with a domain=0.5 will be used on the bone econstucted 100mAs image. Fo the bone kenel the maximum smoothing fo the 33 kenel occus when noise =2.5, see Figue 35. To find the ange value: ; =51.3HU. The esults ae displayed in Figue 40 A-C. 84

92 400mAs/4mm Bone Reconstucted, Bilateal Filteed A 3 3 Bilateal/Box Domain =12.8HU, Noise: 11.55HU 400mAs/4mm Bone Reconstucted, Spatial Gaussian Filteed B 3 3 Spatial Gaussian d =0.5 Noise: 16.57HU 85

93 Bilateal Filteed Spatial Gaussian Filteed A C B Figue 40 Figue 40.C shows that the bilateal box domain filte blued somewhat less than the spatial Gaussian filte. A bilateal with a box domain was chosen to maximize noise eduction. Figue 40 A-C indicates that the bilateal filte can peseve high esolution stuctues as well as the spatial Gaussian filte. In addition the bilateal filteed image had geate noise eduction. In summay, the bilateal filte was applied to CT phantom images. The noise in CT is coelated, multiplicative noise. It was shown that the bilateal can successfully be deployed in this type of noise envionment. Applying a pe-filte can impove the esults of the bilateal filte. In addition epocessing the image with many passes of the bilateal impoved esults as well. 86

94 d) Clinical CT Images The clinical image used in this section is shown in Figue 41. Head Reconstuction Smooth 10mm noise =3.5HU Figue 41 The image is of a gun shot wound with associated bleeding. Measuements wee made of the aveage CT numbe in the wound aea and the suounding aea. The wound aea was appoximately 65HU and the suounding tissue about 32HU. The wound egion had minimal distance of 2mm. The field of view is 240mm and the matix size is The pixel size was calculated to be: 240mm mm. Thus a 33 filteing kenel was selected fo the pocessing of this image. To sepaate the wound aea fom backgound a ange value of about 5-8HU is used, (65HU 32HU)/4=8.25HU. This will assign weighting factos of appoximately zeo fo 87

95 any pixels that have a diffeence of 8.25HU fom the cente pixel. The value of is 8.25/3.5= Fo the soft tissue econstuction filte a bilateal box domain noise filte would with a ange of 8.25HU would be appoximately equivalent to a box filte, see Figue 24. The CT numbes of the wound aea and suounding aea vay. Thus, to be consevative, ange values below 8.25HU ae used fo less noisy images. The techniques used fo the CT phantom analysis will be applied to the clinical images. Figue 42 A and B shows the 5mm Head Smooth econstucted image. Oiginal Unfilteed Image, noise =4.89HU A Head Smooth 5mm Slice, Bilateal Filteed 88

96 33 Kenel, d =0.5, B =5, 5 Iteations, Figue 42 A B noise =3.27HU C (Unfilteed) (3 3 Bilateal, d =0.5, =5, 5 Iteations) Figue 42 The subtacted image in Figue 42.C shows that thee is negligible edge blu afte the bilateal filte was applied. The benefit of using the bilateal filte in this image was educed noise and insignificant edge blu. 89

97 Next the 5mm bone shap econstucted image is analyzed. The esults ae shown in Figue 43 A-C. Unfilteed, 5mm Bone Shap A Unfilteed, 5mm Bone Shap, noise =38.52HU 5mm Bone Shap Bilateal Filteed B Bone Shap 5mm Slice, noise =8.68HU 33 Box Pe-Filte, 33 Bilateal, d =0.8, =8, 2 Iteations 90

98 Figue 43 A B C Figue 43 The bullet fagments ae moe discenable with the filteed bone shap kenel. A box pefilte was fist passed ove the image to educe the high noise spikes. In addition the ange value was inceased to 8HU to futhe suppess noise spikes. Figue 43.C shows that thee was vey little edge degadation afte filteing the image. Finally the 2mm head smooth and bone shap econstucted images ae filteed. Figue 44 A and B show the head smooth econstucted image. 91

99 Unfilteed 2mm Head Smooth Reconstucted A Unfilteed 2mm Head Smooth Reconstucted, noise =9.18HU 2mm Head Smooth Reconstucted Bilateal Filteed B 2mm Head Smooth Reconstucted 33 Median Pe-Filte, 33 Bilateal, d =0.8, =5, noise =5.73HU Figue 44 The 2mm bone shap econstucted image is displayed in Figue 45 A and B. 92

100 Unfilteed 2mm Bone Shap Reconstucted (A) Unfilteed 2mm Bone Shap Reconstucted, noise =72.56HU Unfilteed 2mm Bone Shap Reconstucted Bilateal Filteed (B) 2mm Bone Shap Reconstucted 33 Box Pe-Filte, 33 Bilateal d =0.8, =10HU, noise =17.19HU Figue 45 93