Statistical Noise Models and Diagnostics

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Denis Lloyd
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1 L. Yaroslavsky: Advaced Image Processig Lab: A Tutorial, EUSIPCO2 LECTURE 2 Statistical oise Models ad Diagostics 2. Statistical models of radom iterfereces: (i) Additive sigal idepedet oise model: r = s +. (2.) Examples: Image sesor thermal white Gaussia oise; arrow bad (moire) oise (ii) Multiplicative oise model r = s. (2.2) Examples: Photographic graiiess oise, speckle oise i imagig systems that use coheret irradiatio (sythetic aperture radar, ultrasoud imagig, holography) (iii) Impulse oise model: r = ( - e )s + e; e =,with certai probabitype. (2.3), otherwise Examples: oise i digital image trasmissio ad storage; aomalous oise i aalogue image trasmissio systems with oliear modulatio. (iv) Composite oise model: r = s +. (2.4) (v) m a Sigal depedet oise. Example: quatizatio oise. 2.2 Basic statistical characteristics of radom iterfereces: Probability distributio ad desity: ( ) = Pr obability( ( = ) < ) P ; (2.5) ( + D / 2) - P ( - D / 2) P p ( ) = lim. (2.6) Dfi D Probability desity momets: Mea value: ( ) = p ( ) = AV d. (2.7) -

2 Variace: s ( - ) ) = ( - ) p ( ) = AV d. (2.8) Higher order momets: ( m) ( - ) - m m ( ) = ( - ) p ( ) = AV d. (2.9) - Autocorrelatio fuctios: CF ( x, x ) { ( x ) ( )} 2 = AV x2. (2.) I image processig, statistical (esemble) averagig frequetly does ot make sese ad should be replaced by averagig over certai limited set of image samples. Accordigly, defiitios of statistical characteristics are modified: Distributio histogram (for discrete ad quatized sigals of samples): k k= ( q) = ( q - ) h d, (2.) where d( x) =, x =, otherwise Distributio histogram momets: Mea: Q i= ( q ) = qi h i = k, (2.2) k= where Q is umber of quatizatio leveles, ad { q i } are quatized sigal values. Variace: Q s = ( - ) ( ) = qi h qi ( k - ). (2.3) i= k= Higher order momets: Q [ m ] m = ( - ) ( ) = m qi h qi ( k - ). (2.4) i= k= Correlatio fuctios ad spectra ( t) CF = k k+ t. (2.5) k= ( ) = tr S r CF ( t) exp i2p. (2.6) t= Ł ł 2

3 2.3 oise visibility i images oise visibility i images depeds o: Type o the oise (additive, multiplicatio, impulse, quatizatio, sigal depedet oise) Itesity of oise (variace, dyamic rage, distributio histogram) oise correlatio fuctio ad spectra. Backgroud image Geerally: Correlated oise has higher visibility tha ucorrelated oise of the same variace. Visio is less sesitive to oise i the viciity of object boudaries. Visibility of oise with heavy tails of the distributio histogram (such as, for istace, impulse oise with uiform distributio) is higher the that of oise with Gaussia distributio. Sigal depedet oise (such as quatizatio oise) may cofuse visio by producig image-alike artefacts ad hidig image details. Examples: false cotours ad disappearig of low cotrast image details as cosequeces of quatizatio oise. Figure 2. provides illustratios of these properties of visio. 2.4 Diagostics of radom iterfereces i images 2.4. Basic priciple. The type ad statistical properties of oise i images may be kow from certificatio of imagig systems. However, i may applicatios these data may be uavailable ad should be obtaied directly for available images. Such a empirical oise diagostics i images assumes: Determiatio of oise model Measurig appropriate statistical parameters of oise. Determiatio of oise model is usually doe o the base of image visual aalysis ad a priori kowledge regardig the type of imagig system that geerated images uder questio. I measurig oise statistical parameters requires breakig a vicious circle: i order to measure oise parameters oe eeds to separate oise from the useful 3

4 sigal that ca be doe oly if the oise parameters are kow. The solutio is ot to separate oise from the sigal but istead separate their statistical characteristics. Basic priciple i empirical oise diagostics from oisy images is: select for estimatio the statistical characteristic of the oisy sigal i which the presece of oise exhibits itself i easily detectable characteristic aomalies Measuremet of of locatio ad eergy of moire oise spectral peaks. A immediate example is measurig itesity of moire oise compoets. The characteristic property of arrow bad moire oise is that its eergy is cocetrated i a few of very arrow bad compoets. Therefore its spectrum has a few cocetrated peaks that are very easily detectable o the backgroud of the useful sigal spectrum that is much wider ad homogeeous. A practical algorithm for determiatio of frequecy ad eergy of moire oise compoets i images is illustrated i Fig Measurig variace of zero-mea additive white oise i images. Followig the above priciple of separatig statistical characteristics of oise ad useful sigal, oe ca measure variace of additive white oise i images by detectig oise correlatio fuctio peak i correlatio fuctio of the oisy image. For additive sigal idepedet white oise, correlatio fuctio of oisy image is: CF CF * { }= ( sigal + oise) = AV ( s + )( s + ) * * ( s) + CF ( ) + ( s ) + AV { s } = CF( s) CF( ) AV (2.7) + Correlatio fuctio of white oise CF ( ) is a delta fuctio that appears, i empirical measuremets, as a arrow peak i the correlatio fuctio origi of coordiates. Image sigal correlatio fuctio is usually a fuctio that decays to zero much more slowly. Therefore, sigal correlatio fuctio value i the origi of coordiates ca be accurately eough predicted from oisy sigal correlatio fuctio values i adjacet poits outside the extet of oise correlatio fuctio. Differece betwee the value of oisy sigal correlatio fuctio i co-ordiate origi ad the predicted oe provides estimate for variace of additive white oise. Fig.2.3 illustrates the method. 4

5 2.4.4 Diagostics of impulse oise. Mai statistical characteristic of impulse oise is the probability of appearace of oise impulses. Impulse oise maifests itself as easily detectable cotrast aomalies directly i the oisy sigal. Therefore, probability of oise impulses ca be estimated by detectig oise impulses directly i the oisy sigal. I practice, this process is a part of oise cleaig algorithms that work i two steps: detectio of oise impulses ad replacig sigal samples detected as substituted by oise by values approximated from eighbourig ot distorted samples. Fig. 2.4 illustrates impulse oise i a image ad its ifluece o the predictio error for image samples from their eighbours. This error ca be used for the detectio of oise impulses Quatizatio oise Quatizatio oise is characterized by the size of sigal quatizatio iterval. Sigal statistical characteristic i which oe ca easily detect the presece of quatizatio oise ad evaluate quatizatio itervals is image histogram for quatizatio oise maifests itself via appearace i image histogram of isolated ad easily detectable peaks (Fig. 2.5) 5

oise free image Additive oise, stdev=2/256

filtered image(eergy spectrum thresholdig);

6 oise free image Additive oise, stdev=2/256 Impulse oise, Pe=.6, stdev=2/256 Moire oise, stdev=2/256 Quatizatio oise, Q=4, stdev=2/256 Low pass filtered image(eergy spectrum thresholdig); stdev=2/256 Fig. 2. Differet image distortios with the same stadard deviatio of additive oise 6

6 4 2 8 6 Row wise averaged power spectrim spectrumspectrum 5 5 2 25 oise spectrum; thr=.5.5 5 5 2 25 Frequecy Fig. 2.2 Diagostics of moire oise.

7 Row wise averaged power spectrim spectrumspectrum oise spectrum; thr= Frequecy Fig. 2.2 Diagostics of moire oise. oise spectral peaks are detected i row wise averaged power spectrum of oisy image by compariso of predictio error of spectrum samples predicted (from left to right)) with a threshold which is a algorithm free parameter. The threshold ad the predictio method of spectrum samples deped o a priori kowledge regardig images ad oise. 7

Iitial ad additively oised image (STDoise=2) 6 4 2 8 6 4 -D spectra of iitial (red) ad oisy (blue) images 2 5 5 2 25 -D correlatio fuctio of the iitial image -D correlatio fuctio of the oisy image.9.

8 Iitial ad additively oised image (STDoise=2) D spectra of iitial (red) ad oisy (blue) images D correlatio fuctio of the iitial image -D correlatio fuctio of the oisy image Fig. 2.3 Diagostics of additive white oise i images. Plots show -D spectra (top) ad -D correlatio fuctios of oise free (bottom left) ad oisy (bottom right) images. Oe ca easily see peak i the correlatio fuctio of oisy image due to oise preset. 8

oisy image, Per=.3.Histograms^.5 of 2-D predictio error for the iitial ad oisy.25.2 oise free image.5. oisy image.5 5 5 2 25 Fig. 2.4 Diagostics of impulse oise.

9 oisy image, Per=.3.Histograms^.5 of 2-D predictio error for the iitial ad oisy.25.2 oise free image.5. oisy image Fig. 2.4 Diagostics of impulse oise. Impulse oise maifests itself as cotrast isolated peaks i the image ad appearace of heavy tails i histogram of predictio error of pixel grey levels from those of adjacet pixels 9

Image histogram 4 35 3 25 2 5 5 5 5 2 25 Gray level Fig. 2.5 Quatizatio oise diagostics.

10 Image histogram Gray level Fig. 2.5 Quatizatio oise diagostics. Quatizatio oise maifests itself via appearece of isolated peaks i image histogram.

Lecture 2: Poisson Stascs Probability Density Funcons Expectaon and Variance Es*mators

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