THE USE OF WAVELETS FOR NOISE DETECTION IN THE IMAGES TAKEN BY THE ANALOG AND DIGITAL PHOTOGRAMMETRIC CAMERAS

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1 THE USE OF WAVELETS FO NOISE DETECTION IN THE IMAGES TAKEN BY THE ANALOG AND DIGITAL PHOTOGAMMETIC CAMEAS K. Pyka a, *, J. Siedlik b a Depatment of Geoinfomation, Photogammety and emote Sensing of Envionment AGH Univesity of Science and Technology Kakow- kisfoto@agh.edu.pl b Business Goup MGGP AEO Tanow - jsiedlik@mggpaeo.com Commission I, WG I/ KEY WODS: Noise, Wavelets Tansfom, Decomposition,, Detail Component, Kutosis, Vaiance ABSTACT: In the pape the use of wavelet tansfomation fo valoization of andom noise content in photogammetic images is poposed. Thee wee two wavelets indicatos studied. The fist indicatos based on the analysis of the wavelet detail coefficients distibution shape. As the esults pove, noise conclusions based on the shape ae not always objective. As the second noise indicato the analysis of changing of elative vaiance duing decomposition was eseached. Based on the studies, it has been poven that the analysis of the equation of pesevation of image elative vaiance is a good indication of the noise level. The low noise level is poven by a stable incease of the details vaiance along with the level of decomposition. In case of fine-gained image textue, such incease is undistubed. Fo the eseach a set of aeial images taken by two photogammetic cameas, analogue LMK and digital DMC was compaed. In all examined cases the bette paametes of the noise evaluation wee obtained fo the digital camea. The eseaches confimed the possibility to define the noise content indicatos based on the analysis of the wavelet detail coefficients.. INTODUCTION The adiometic quality is often neglected when looking fo the easons of the unsatisfactoy quality of the automatic images analysis. Meanwhile, the image noise can substantially educe the efficiency of image pocessing, especially in the case of images with low contast and containing many fagments with fine-gained textue, with which we deal often in photogammety and emote sensing. We have witnessed the pocess of eplacing the analogue photogammetic cameas with digital ones. Diect digital image acquisition educes the numbe of stages in which noise can occu, but does not liquidate the poblem of its occuence. Noise should be egaded as an immanent featue of the photogammetic images, similaly to, fo example, lens distotion. Fo some yeas the discete wavelet tansfomation has been used in the image pocessing. The wavelet tansfomation is egaded as the most effective method of lossy compession of multitonal images and is moe and moe often used in the photogammetic woking stations. In this pape the using of wavelets fo evaluation of the image andom noise is poposed.. THE NEED FO IMAGE NOISE INDICATOS Noise is any andom o deteministic distubance of luminance of a hypothetical image that would come into existence in the ideal conditions (Moain, 004). Image noise togethe with adiometic esolution, contast, tonal matching ae elements shaping adiometic quality. Noise aises in the diffeent stages of the image acquisition: duing the image foming, sampling, encoding, compession, tansmission and duing image pocessing. andom noise is pesent pactically in any image, but is not always noticed. In the analogue images, the souce of andom noise is the ganula stuctue of photogaphic emulsion. Noise in the digital images is caused by instability of detectos, including - to some extent detecto s own noise. andom noise, because of its unpedictable chaacte, cannot be emoved completely fom the image. One can only smooth ove the effects of its occuence. We face a dilemma: Is it bette to educe the noise level at the expense of the edges shapness o the othe way ound? The photogammetic and emote sensing multitonal landscape images, which ae taken fom lage distances pedominate, have fequently low local contast and small signal-to-noise atio. This is why thee is a need to look fo the indicatos of noise content THE USEFULLNESS OF WAVELET FO NOISE DETECTION THEOETICAL STUDY Basic featues of wavelet tansfomation Wavelet tansfomation demonstates some featues shaed with Fouie tansfomation. The Fouie tansfomation convets the signal fom spatial domain into fequency domain. The wavelet tansfom is a fequency-spatial epesentation, i.e. it is possible * Coesponding autho. 77

2 The Intenational Achives of the Photogammety, emote Sensing and Spatial Infomation Sciences. Vol. XXXVII. Pat B. Beijing 008 to localize each coefficient spatially, what is impossible in the case of Fouie tansfom. 3. Mallat s Multiesolution Analysis The pocedue fo detemining the image wavelet expansions (two-dimensional signals) with the help of multi-level decomposition utilizing one-dimensional filtes, sepaately applied to the ows and columns of the image, was given by (Mallat, 998). Thee ae fou components in the wavelet expansion of the image: so-called coase component (LL) and thee details, named as vetical- (LH), hoizontal- (HL) and diagonal (HH) detail. The chaacteistic featue of wavelet tansfomation is the possibility to continue applying it to the chosen component. This is the coase detail that is expanded most often. 3.3 The dependence of noise and wavelet coefficients distibution Simonceli noticed that wavelet detail coefficients distibution has a shap maximum in zeo and has a good symmety, wheeas the flattening of histogam is coelated with the pesence of noise in the image (Simonceli, 996, 999). The kutosis, which is the fouth moment divided by the squae of the vaiance, was employed as a paamete descibing the histogam shape. E ( I) = E( LL LH HL HH ) () whee E(I) = enegy of image I E(LL ) = enegy of coase component LL on fist level of decomposition E(LH ), E(HL ), E(HH ) = enegy of details components LH,HL,HH (on fist level of decomposition) Futhe decomposition of coase component allows witing: E ( LL ) = E( LL LH HL HH ) () whee E(LL ) E(LH ), E(HL ), E(HH ) = enegy of components on second level of decomposition The geneal fom of equation () and () is shown below: [ E( LH HL ) E( HH ] E( I) = E( LL ) (3) whee = the numbe of level of decomposition 3.5 The equation of elative vaiance pesevation by wavelet decomposition The equation (3) is also tue when we use vaiance instead of enegy (Pyka, 005): [ V ( LH ) V ( HL ) V ( HH ] V ( I ) = V ( LL ) (4) Figue. The histogams of wavelets detail components It was claimed (Pyka, 005) that the estimation of the shape of coefficients distibution should be made fo all thee detail components, but it is enough to limit eseach to wavelet decomposition on one level of esolution. Futhe decomposition of coase component (LL) does not give moe infomation on noise, because each subsequent coase component is the effect of the smoothing of the peceding one, what deceases the noise content. In Figue the typical histogams of thee wavelets components fo a image without noise and fo the same image with white noise ae shown. 3.4 The ule of image pesevation of enegy though its wavelet tansfom The wavelet decomposition peseves the image enegy (Mallat, 998). In case of decomposition on fist level of esolution we can wite: whee V(I ) = vaiance of image V(LL = vaiance of coase components on level of decomposition, V(LH ), V(HL ), V(HH ) = vaiance of detail components on level of decomposition The image vaiance is dependent of pixels value scaling. If we linealy tansfom the image pixels value (called also bightness o DN) then the wavelet components undego the same tansfomation. It is disadvantage of ule given by equation (4). When we divide eithe side of equation (4) by V(I) we eceive the following equation: V ( LL ) = V ( I ) V ( [ V ( LH ) V ( HL ) V ( HH )] I ) The equation (5) shows that the sum of elative vaiance of wavelet tansfom components equals and that is tue fo any levels of decomposition. It is woth to note that the ule is independent of pixels value scaling. Fo images without noise the following ule should be tue: (5) 78

3 The Intenational Achives of the Photogammety, emote Sensing and Spatial Infomation Sciences. Vol. XXXVII. Pat B. Beijing 008 V ( LH ) V ( HL ) V ( HH ) < (6) V ( LH ) V ( HL ) V ( HH ) whee V(LH ), V(HL ), V(HH ), V(LH ), V(HL ), V(HH ) = vaiance of detail components on level and level of decomposition scanne: DELTA SCAN scanning esolution: μm and 4 μm (espectively fo medium and small scale) GSD: cm (medium) and 36 cm (small scale), photo captuing: Octobe 003 (medium) and June 006 (small scale). Images fom digital camea: camea type: DMC (Z/I Imaging), focal length 0 mm, pixel size: μm, photo scale: : 000 (medium) and :38000 (small), GDS=4 cm (medium) and 46 cm (small scale), photo captuing: Octobe 007 (medium) and Septembe 007 (small scale). The images wee taken and deliveed in TIFF fomat without compession by MGGP AEO company fom Tanow, Poland ( The images wee selected in pai which wee composed of analogue and digital image. An example of pai in medium and small scale is showed in Figues 3 and 4 espectively. Figue. The illustation of equation of elative vaiance pesevation by wavelet decomposition The speed of the elative vaiance gowing is connected with the density and powe of edges. In othe wods the gowing of speed depends of scale and image contents. Fo image with noise the following ule was established: Figue 3. The pai of analogue and digital images in medium scale V ( LH ) V ( HL ) V ( HH ) > (7) V ( LH ) V ( HL ) V ( HH ) Theefoe the equation (5) is useful fo compaing multi-level wavelets tansfom of diffeent aeial images taken fom analogue and digital cameas. 4. THE USEFULLNESS OF WAVELET FO NOISE DETECTION PACTICAL STUDY 4. The eseach data Fo the eseach a set of aeial images taken by two cameas, analog and digital photogammetic, was used. The main chaacteistic of the eseach data ae descibed below: Images fom analogue camea: camea type LMK000 (Zeiss Jena), lens type LAMEGON, focal length 53mm, film: AGFA Aviphot Colo X00, developing pocess: C-4, photo scale: :0 000 (medium) and :6000 (small), Figue 4. The pai of analogue and digital images in small scale 4. The eseach method and tools Fo each pai of photogaphs, coesponding fagments of the same contents wee selected. That task was moe challenging fo photos in a smalle scale, as the analogue photogaphs wee taken in 003, and the digital ones in 007. As fa as possible, fagments of unifom land use wee selected, e.g. buildings, paking lots, fields, foests. That made it possible to obseve how the land use affected the wavelet tansfom. 5 fagments of images altogethe wee selected fo the tests in the medium 79

4 The Intenational Achives of the Photogammety, emote Sensing and Spatial Infomation Sciences. Vol. XXXVII. Pat B. Beijing 008 scale and 9 fagments of images in small scale. Pio to the analysis of noise content, coloued images wee eplaced with a esultant luminance image, using equation: I = 0,99 0,587G 0, 4B (8) whee I = luminance image,g,b = channels of colou image Matix algoithm with the thid ode Coiflet filtes was used fo the wavelet tansfom. The tansfom was caied out with the aid of pocedues witten in the envionment. Selected fagments wee sized 04 * 04 pixels. Additionally, fames in individual pais wee so masked to have the same contents. That was indispensable due to vaious scales of the analogue and digital photogaphs. then inceases slowly o is stable as shown by the zone with signatue maks in figues 5 and 6, which encompass the esults fo all examined fagments of photogaphs In ode to bette mak the changes of vaiance fo futhe levels of decomposition, in figues 5 and 6, the aveage changing tendency was maked with an open polygon. Fom the theoetical point of view, figue is moe appopiate, which shows the values of elative vaiance of discete, not continuous, chaacte. Moeove, figues 5 and 6 do not include the vaiance of the coase component. The value of that vaiance can be easily established based on the fomula (5). The wavelet decomposition was continued until the thid esolution level, in which the components had the size of 56 * 56 pixels. Kutosis and vaiance fo individual components was established. Aftewads, the equation of pesevation of image elative vaiance was fomulated. 4.3 The eseach esults It was found that the kutosis fo analogue photogaphs is always lowe than that of the digital photogaphs. The kutosis fo analogue photogaphs vaies between 3 and 3.5, iespective of the method of use and scale of the photos. The vaiability of the kutosis between detailed components is not visible. Fo the analogue photogaphs examined, the distibution of detailed components may be modelled with the aid of a nomal distibution, the kutosis of which equals 3. Figue 5. The changes of elative vaiance fo images in medium scale Accoding to the studies, the kutosis fo digital photogaphs is geneally lage and, at the same time, exhibits geate divesity. Both fo the medium and small-scale photogaphs, the detail kutosis is above 0, with one exception howeve. The exception efes to foested aeas and paks of dense foest stand. The kutosis in such aeas is smalle: fo the medium scale, the kutosis is aound 6, and 4 fo the small scale accodingly. Such phenomenon may be explained by the fact that the distibution of the wavelet components is sensitive to the natual image stuctue. The image of tees in the photogaphs is of gainy stuctue, the fine the lowe the scale of the photos is. As the esults pove, noise conclusions based on the kutosis ae not always objective. Thee ae cases obseved in which a flat distibution, esembling the Gaussian one, does not esult fom the andom noise, but is the effect of natual fine stuctue of the image. Based on the analysis of the equation of pesevation of image elative vaiance, the following ules wee established: the elative vaiance of details in images fom digital camea inceases along with the decomposition as indicated by the gey zone in figues 5 and 6, which encompass the esults fo all examined fagments of photogaphs, the elative vaiance in images fom analogue camea deceases between and decomposition level and, Figue 6. The changes of elative vaiance fo images in small scale 5. CONCLUSIONS It has been confimed in the pape that, based on the distibution of wavelet components, the shae of andom noises in an image can be established. It has been shown that the flattening of the histogam of wavelet coefficients is also 80

5 The Intenational Achives of the Photogammety, emote Sensing and Spatial Infomation Sciences. Vol. XXXVII. Pat B. Beijing 008 affected, apat fom the noise, by the natual image textue. Hence, the shape of the histogam of wavelet details is not always coelated with the noise level. Based on the studies, it has been poven that the analysis of the equation of pesevation of image elative vaiance is a good indication of the noise level. The low noise level is poven by a stable incease of the details vaiance along with the level of decomposition. In case of fine-gained image textue, such incease is undistubed. ACKNOWLEDGEMENTS The investigation has been made within the scope of eseach poject AGH /08. Geat diffeence in the contents of andom noise has been confimed by the compaison of images fom the digital and analogue cameas. The DMC images contain seveal times less andom noise than those fom an analogue camea. The studies descibed ae not exhaustive as egads the use of wavelets fo valoisation of adiometic quality of photogammetic images. Since highe quality of 8-bit DMC images geneated fom the panchomatic and 3 multi-spectal components has been poven, the quality of individual components would have to be veified. It appeas that the wavelet tansfom can also be used fo the optimisation of tonal mapping which takes place while tansfoming the signal of a boad dynamic ange of a digital camea into the 8-bit ange. In the cuent stage of eseach it is not possible to use the esults of the wavelet-components analysis as an absolute measue of the noise content. Defining such a measue equies caying out a seies of expeiments, in which the images taken in the diffeent conditions and seasons, will be studied. Such eseaches should be caied out in the futue, because unsatisfactoy quality of the adiometic images makes automation of the photogammetic technology difficult and educes the intepetation value of the images, including othophoto-maps, so popula nowadays. EFEENCES Mallat S., 998: A Wavelet Tou of Signal Pocessing. Academic Pess, ISBN Moain S.A.,. Zanoni V.M:, 004: Joint ISPS /CEOS -WGCV Task Foce on adiometic and Geometic Calibation. ISPS Istanbul 004, com. I, p Mulcahy C., 997. Image Compession Using The Haa Wavelet Tansfom. Spelman College Science & Mathematics Jounal, Vol, No, Apil 997, p.-3. Pyka K., 005. The Wavelets Indicatos of adiometic Changes Taking Place in Othophoto-map Geneation Pocess (in polish). UWND AGH Kaków, 005, p Simoncelli E.P., Adelson E.H., 996: Noise emoval via Bayesian Wavelet Coing. Poceedings of 3d IEEE Intenational Confeence on Image Pocessing, vol. I, pp Lausanne, Switzeland. 6-9 Septembe 996. IEEE Signal Pocessing Society, 996. Simoncelli E.P., 999: Modeling the Joint Statistic of Images in Wavelet Domain. Poc. SPIE 44th Annual Meeting, vol. 383 Denve, Coloado. July

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