Gaussian Filtering - Auckland

5/25/2010Gaussian FilteringGaussian filteringg is used to blur imagesg and remove noise and detail.In one dimension, the Gaussian function is:G ( x) 12πσ 2 ex22σ 2Where σ is the standard deviation of the distribution.distribution The distribution isassumed to have a mean of 0.Shown graphically, we see the familiar bell shaped Gaussiandistribution.Gaussian distribution with mean 0 and σ 1181

5/25/2010Gaussian filtering Significant valuesxσ * G ( x ) / 0.399G ( x ) / G (0)011ee1234 0.5/σ 2 2/σ 2 9/4σ 2 8/σ 2 0.5/σ 2ee 2/σ 2ee 9/4σ 2ee 8/σ 2For σ 1:x012G ( x)0.3990.2420.05G ( x ) / G (0)10.60.125192

5/25/2010Gaussian FilteringStandard DeviationTh StandardTheSt d d deviationd i ti off theth GaussianGi ffunctionti playslan iimportantt trole in its behaviour.The values located between /- σ account for 68% of the set, while twostandard deviations from the mean (blue and brown) account for95%, and three standard deviations (blue, brown and green)account for 99.7%.This is very important when designing a Gaussian kernel of fixedlength.Distribution of the Gaussian function values (Wikipedia)203

5/25/2010Gaussian FilteringThe Gaussian function is used in numerous research areas:– It defines a probability distribution for noise or data.– It is a smoothing operator.– It is used in mathematics.The Gaussian function has important properties which are verified withrespect to its integral: I exp ( x )dx 2π In probabilistic terms, it describes 100% of the possible values of anygiven space when varying from negative to positive valuesGauss function is never equal to zero.It is a symmetric function.214

5/25/2010Gaussian FilteringWhen workingWhki withith iimages we needd tto use ththe ttwo didimensionalilGaussian function.This is simply the product of two 1D Gaussian functions (one for eachdirection) and is given by:G ( x, y ) 12πσ 2 ex2 y 22σ 2A graphical representation of the 2DGaussian distribution with mean(0,0)and σ 1 is shown to the right.225

5/25/2010Gaussian FilteringThe GaussianThGi filterfilt worksk byb usingi ththe 2D didistributiont ib ti as a point-spreadi tdfunction.This is achieved by convolving the 2D Gaussian distribution functionwith the image.We need to produce a discrete approximation to the Gaussian function.Thi thThistheoreticallyti ll requiresian iinfinitelyfi it l llarge convolutionl ti kkernel,l as ththeGaussian distribution is non-zero everywhere.Fortunately the distribution has approached very close to zero at aboutthree standard deviations from the mean. 99% of the distributionfalls within 3 standard deviations.This means we can normally limit the kernel size to contain only valueswithin three standard deviations of the mean.236

5/25/2010Gaussian FilteringGaussianGi kkernell coefficientsffi i t are sampledl d ffrom ththe 2D GGaussianifunction.x2 y 2 12G ( x, y ) e 2σ22πσWhere σ is the standard deviation of the distribution.The distribution is assumed to have a mean of zero.We need to discretize the continuous Gaussian functions to store it asdiscrete pixels.An integer valued 5 by 5 convolutionkernel approximating a Gaussianwith a σ of 1 is shown to the right,1273147416 26 16 4726 41 26 7416 26 16 41474411247

5/25/2010Gaussian FilteringThe Gaussian filter is a non-uniform low pass filter.The kernel coefficients diminish with increasing distance from thekernel’s centre.Central pixels have a higher weighting than those on the periphery.Larger values of σ produce a wider peak (greater blurring).g σ to maintain the GaussianKernel size must increase with increasingnature of the filter.Gaussian kernel coefficients depend on the value of σ.At the edge of the mask, coefficients must be close to 0.The kernel is rotationally symmetric with no directional bias.Gaussian kernel is separableseparable, which allows fast computationcomputation.Gaussian filters might not preserve image brightness.258

5/25/2010Gaussian Filtering examples Is the kernel1 6 1a 1D Gaussian kernel? Give a suitable integer-value 5 by 5 convolution maskthat approximates a Gaussian function with a σ of 1.4. How many standard deviations from the mean arerequired for a Gaussian function to fall to 5%5%, or 1% ofits peak value? What is the value of σ for which the value of theGaussian function is halved at /-1 x. Compute the horizontal Gaussian kernel with mean 0and σ 1, σ 5.269

5/25/2010Gaussian Filtering examplesApply the Gaussian filter to the image:Borders: keep border values as they are152025251510¼* 1211520242316102025 36 33 21 152044 55 51 35 202029 44 35 22 302015503020152050556030201521 25 24 25 30201565301530202115203020253015 20 24 23 16 10202515201015Original imageOr:12124212119161415119 28 38 35 23 15¼* 220 35 48 43 28 21119 31 42 36 26 2818 23 28 25 22 21*1/162021191614152710

5/25/2010Gaussian Filtering examplesApply the Gaussian filter (μ 0, σ 1)to the 1530152030202530202515201015Original image2811

5/25/2010Gaussian Filtering examplesApply the Gaussian filter (μ 0, σ 0.2)t thetoth 01530152030202530202515201015O i i l iimageOriginal2912

5/25/2010Gaussian FilteringGaussian filtering is used to remove noise and detaildetail. It is notparticularly effective at removing salt and pepper noise.Compare the results below with those achieved by the median filter.3013

5/25/2010Gaussian FilteringGaussian filtering is more effective at smoothing images. It has its basisin the human visual perception systemsystem. It has been found thatneurons create a similar filter when processing visual images.The halftone image at left has been smoothed with a Gaussian filterand is displayed to the right.3114

5/25/2010Gaussian FilteringThis is a common first step in edge detectiondetection.The images below have been processed with a Sobel filter commonlyused in edge detection applications. The image to the right has hada Gaussian filter applied prior to processing.3215

Gaussian filters might not preserve image brightness. 5/25/2010 9 Gaussian Filtering examples Is the kernel a 1D Gaussian kernel?Is the kernel 1 6 1 a 1D Gaussian kernel? Give a suitable integer-value 5 by 5 convolution mask that approximates a Gaussian function with a σof 1.4. .