Image Denoising using Fast Bilateral Filter

Transcription

1 Iteratioal Joural of Curret Egieerig ad Techology E-ISS , P-ISS IPRESSCO, All Rights Reserved Available at Research Article Riku A Joseph * ad Maju James Departmet of Electroics ad Commuicatio Egieerig, M.G Uiversity, Kottayam, Idia Accepted 0 ov 015, Available olie 6 ov 015, Vol.5, o.6 (Dec 015) Abstract Image de-oisig is the process of mappig a oisy image to a oise free image. The major problem of image deoisig algorithms is that edges are vaished after the de-oisig process. The bilateral filter is a o liear filter that does spatial averagig without smoothig edges. It cosists of a spatial kerel as well as a rage kerel to limit averagig to the eighbourhood pixels havig similar itesity. The rage kerel operates by actig o the pixel itesities. This makes the averagig process o liear ad computatioally itesive, maily whe the spatial filter is large. Those algorithms whose executio time is idepedet of size ad shape of filter are commoly kow as costat time or O(1)algorithm. Here we will see how averagig algorithms ca be leveraged for realizig the bilateral filter i costat time, by usig trigoometric rage kerels. Keywords: Bilateral Filter, Image Deoisig, Costat time filter, Spatial domai filters 1. Itroductio 1 Digital images play a very sigificat part, both i applicatios such as magetic resoace imagig, computer tomography as well as i the field of sciece ad techology such as geographical iformatio system ad astroomy. The data collected by image sesors ad other devices are geerally cotamiated by oise. oise ca also be itroduced due to trasmissio errors ad compressio. Hece deoisig is ofte a ecessary ad first step to be performed before image data is aalysed ad processed. A efficiet de-oisig techique must be applied to compesate for such data corruptio. Image de-oisig remais as a sigificat challege for researchers because de-oisig process removes oise but itroduces artefacts ad also causes blurrig.. Bilateral Filter A bilateral filter is a o-liear, edge-preservig ad oise-reducig smoothig filter for images. The itesity value at each pixel i a image is replaced by a weighted average of itesity values from earby pixels. This weight ca be based o a Gaussia distributio. Crucially, the weights deped ot oly o Euclidea distace of pixels, but also o radiometric differeces (e.g. rage differeces, such as color itesity, depth distace, etc.). This preserves sharp edges by systematically loopig through each pixel ad adjustig weights to the adjacet pixels accordigly. Its *Correspodig author: Riku A Joseph ability to decompose a image ito differet scales without causig haloes after alteratio has made it global i computatioal photography applicatios such as toe mappig, style trasfer, relightig, ad deoisig. Bilateral filterig algorithm is a o-liear ad oiterative image de-oisig method i spatial domai which makes use of the spatial iformatio ad the itesity iformatio betwee a poit ad its eighbours to smooth the oisy images while preservig edges well. The bilateral filter is preferred for oe uique reaso: It reduces oise while preservig details. The bilateral filter embodies the idea of a groupig of domai ad rage filterig. The domai filter averages the earby pixel values ad thus acts as a low-pass filter. The rage filter stads for the o liear compoet ad plays a sigificat part i edge preservig. This compoet allows averagig of similar pixel values oly, despite their positio i the filter widow. If the value of a pixel i the filter widow diverges from the value of the pixel beig filtered by a certai amout, the pixel is skipped. The bilateral filterig of a image f(x) is give by, ~ 1 x,, f ( ) f ( where w ( x,, f ( ) 3654 Iteratioal Joural of Curret Egieerig ad Techology, Vol.5, o.6 (Dec 015) (1) () Here x deotes the pixel of iterest ad y deotes the eighborig pixel. The term w(x, deotes the

2 geometric proximity betwee x ad y. The fuctio ϕ(f(x), f() measures itesity values of the pixels x ad its eighbor y. Here η idicates the ormalizig factor used to preserve local mea. We have w(x,=w(x- sice it is traslatio ivariat. The rage filter depeds o itesity differece ϕ(f(x), f()=ϕ(f(x) - f() Thus the filter is give by ~ 1 f ( x f ( x Where w ( f ( x ) Here w(x) represets the spatial kerel ad ϕ(s) deotes the rage kerel. The local support of spatial kerel that cosists of the eighborhood over which the averagig takes place is idicated usig the symbol Ω. The spatial ad rage kerel are usually gaussia. For the bilateral filter i (1), the differece f(x-- f(x) is close to zero i homogeous regios, ad hece ϕ(f(x--f(x)) 1. I this case, (1) simply results i the averagig of pixels i the eighborhood of the pixel of iterest. O the other had, if the pixel of iterest is i the surroudig area of a edge, ϕ(f(x--f(x)) is large whe x-y belogs to the same side of the edge as x ad is small whe x-y is o the other side of the edge. As a result, the averagig is limited to eighborhood pixels that are o the same side of the edge as the pixel of iterest. This is the basic idea that allows oe to carry out smoothig while preservig edges at the same time. Sice its begiig, the bilateral filter has foud extesive use i several image processig, computer graphics, ad computer visio applicatios. This icludes deoisig, video abstractio, demosaicig, optical-flow estimatio, ad stereo matchig, to ame a few. It is the presece of term ϕ(f(x--f(x) i (1) that makes the filter oliear. I the absece of this term, that is, whe ϕ(s) is costat, the filter is simply give by averagig f ( x (3) Our preset idea is to leverage these fast averagig algorithms by expressig (1) i terms of (3), where the averagig is performed o the image ad its simple poit wise trasforms. Our observatio is that we ca do so if the rage kerel is of the form By pluggig (4) ito (1), we ca write the itegral as cos( ) cos( f ( x ) f ( x si( ) si( f ( x ) f ( x This is clearly show to be the liear combiatio of two spatial averages, performed o images cos(ϒf(x))f(x)) ad si(ϒf(x))f(x)). Similarly, we ca write the itegral i () as cos( ) cos( f ( x ) si( ) si( f ( x ) I this case, the averagig is o images cos(ϒf(x)) ad si(ϒf(x)). This is the trick that allows us to express (1) i terms of liear covolutio filters applied to poit wise trasforms of the image. 3. Proposed Work Algorithm Iput: Image f (x) - Variace of spatial kerel - Variace of Rage kerel The rage kerel is defied as, S ( s) cos( s) cos T Dyamic Rage [-T, T] Set / ad Calculate the maximum Dyamic Rage T ( r) =.405 T r For 0,set up images h exp( j ( ) ) / g h ad coefficiets d j ( ) ( C )exp Filter ad with a gaussia of variace to get (x) ad (x) ~ d 0 d 0 g h r ( s) cos( s),t s T (4) Retur: Filtered Image (x) 3655 Iteratioal Joural of Curret Egieerig ad Techology, Vol.5, o.6 (Dec 015)

3 4. Methodology ad Desig Here our aim is to brig the bilateral filter i oliear form to the liear form. This is achievable if the rage kerel is of the form S s) cos( s) cos T, -T s T 4.1 Geeral Trigoometric kerels costat whe is eve, ad we have oe less auxiliary image to process i this case. 4.3 Gaussia Kerel Approximatio Figure below shows raised cosies of order =1 to =5. As icreases ϕ(s) becomes more gaussia like, over the half period [-Π, Π]. The idea metioed above is exteded to more geeral trigoometric fuctios havig the form s) a a cos( s)... a cos( ) ( 0 1 s Rewritig i terms of complex expoetials, we have s) c exp( js) The coefficiets must be real ad symmetric. 4. Raised Cosies I order to qualify as a valid rage kerel, the rage kerel should be oegative ad mootoic, aside from beig symmetric. Fig. Raised cosie approximatio of gaussia kerel s s lim [cos( )] exp( ) (5) It is clear that raised cosie offers better approximatio tha its polyomial couterpart. 4.4 Cotrol of width of rage kerel Fig. 1 Family of raised cosies, over the amic rage -T S T,as goes from 1 to 5 (outer to ier curves) The properties of symmetry, oegativity, ad mootoicity are cocurretly ejoyed by the family of raised cosies of the form s) cos( s),t s T We have cos ( e jx e jx ) / Ad applyig the biomial theorem we see that The approximatio i (5) also suggests a meas of cotrollig the variace of the raised cosie, amely, by cotrollig the variace of the target Gaussia. The target Gaussia (with ormalizatio) has a fixed variace of. This ca be icreased by simply rescalig the argumet of the cosie i (5) by some ρ>1. I particular, for sufficietly large s [(cos )] s exp( 5. Experimetal Results The Lea image of size 56 x 56 i pg format was chose as the test image, iitially. Additive white gaussia oise with sigma=0 was added i to the origial image. The oisy image was filtered usig the proposed algorithm ad the output obtaied was plotted. ) s) 0 C exp( j( ) s) Sice ϕ(s)has a total of (+1) terms, this gives a total of (+1) auxiliary images. The cetral term =/ is 3656 Iteratioal Joural of Curret Egieerig ad Techology, Vol.5, o.6 (Dec 015)

4 (a) (c) (d) (b) Table 1 shows the maximum amic rage obtaied for six stadard test images, lea, barbara, boats, checker ad house. The maximum value of amic rage of a grayscale image is 55. From the table, it is clear that the maximum amic rage obtaied is much lower tha the worst case estimate (T=55). The de-oisig results obtaied usig the proposed algorithm for various stadard test images were compared both visually ad i terms of psr values. Table Compariso of psr values Sl.o Test oisy Proposed SBF FBF Image Image Method 1 Checker Lea Barbara Boats House Peppers Table shows the compariso of the peak sigal to oise ratio (psr), obtaied for six stadard images. Colums 3, 4 ad 5 represets the psr obtaied for stadard bilateral filter (SBF), Fast bilateral filter (FBF) ad the proposed method respectively for the six stadard images. The psr value obtaied for checker image usig stadard bilateral filterig is 6.70 db ad with fast bilateral filterig is 8.85 db. The proposed method resulted i the rise of psr value to db, a rise of about 6 db. Thus from the above table, it is clear that the quality of de-oised image ca be improved by usig this algorithm. Coclusio (e) Fig. 3 Compariso of outputs of SBF, FBF ad proposed method usig lea image Figure (a) represets the clea Lea image. The image obtaied after addig additive white gaussia oise with σ= 0 is show i figure (b), with a psr value of.14 db. Figure (c) shows the output of stadard bilateral filter (SBF), havig a psr value of value of 6.77 db. Figure (d) shows the output of fast bilateral filterig (8.85 db) ad the output of proposed method is show i figure (e), with a psr value of db. Figure (f) shows the residual image obtaied by subtractig the filtered image from the origial. (f) The ehacemet of oisy image is a essetial task i image processig. The objective of ay de-oisig algorithm is to de-oise the image i miimum time, preservig the edge iformatio as well as the image details. Thus edge preservig bilateral filter was realized with the raised cosie approximatio of rage kerel of the bilateral filter. It is clear that raised cosies of large order provide close approximatios of the Gaussia kerel. The de-oisig results obtaied usig this algorithm was compared both visually ad i terms of psr values. Alog with the ehacemet i quality, the reductio i computatioal complexity is aother advatage of this method. This algorithm ca also be exteded for removig oise from color images. Refereces Table 1 Maximum Dyamic Rage Sl,o Image Maximum Dyamic Rage (T) 1 Lea 44 Barbara 6 3 Boats Checker 44 5 House 09 Kual araya Chaudhury, Daiel Sage, ad Michael User(011), Fast O(1) Bilateral filterig usig Trigoometric Rage kerels, IEEE trasactios o image processig, Vol.0, o. 1 C.Tomasi, R.Maduchi (1998), Bilateral fillterig for gray ad color images, IEEE iteratioal Coferece o Computer visio, Bombay, Idia Fredo Durad,Julie Dorsey (011),Addedum to Fast O(1) bilateral filterig usig trigoometric rage kerels, IEEE trasactios o image processig, Vol. 0, o Iteratioal Joural of Curret Egieerig ad Techology, Vol.5, o.6 (Dec 015)

5 K.. Chaudhary (01), Acceleratio of the shiftable O(1) algorithm for bilateral filterig ad o-local meas, IEEE Trasactios,Image Processig Sweety Deswal, Shaileder Gupta ad Bharat Bhusha (015),A Survey of Various Bilateral Filterig Techiques, iteratioal Joural of Sigal Processig, Image Processig ad Patter Recogitio Vol.8, o.3, pp Michael M.Brostei (014), oises ad Image De-oisig Techiques : A Brief Survey, Iteratioal Joural of Emergig Techology ad Advaced Egieerig, ISS , ISO 9001:008 Certified Joural, Volume 4, Issue 3, Kollipara Rithwik ad Kual. Chaudhury (015) :A Simple Yet Effective Improvemet to the Bilateral Filter for Image De-oisig, IEEE Trasactios, image processig D.Vishu Vardha, K. Jayachadra Red (013):Acceleratio of Shiftable O (1) Algorithm for Bilateral Filterig ad olocal meas, Iteratioal Joural of Sciece ad Research (IJSR), ISS (Olie): Iteratioal Joural of Curret Egieerig ad Techology, Vol.5, o.6 (Dec 015)