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Image processingReadingSlide 3Pixel movementNoiseIdeal noise reductionSlide 7Practical noise reductionDiscrete convolutionDiscrete convolution in 2DConvolution representationMean filtersEffect of mean filtersGaussian filtersEffect of Gaussian filtersMedian filtersEffect of median filtersComparison: Gaussian noiseComparison: salt and pepper noiseEdge detectionWhat is an edge?GradientsLess than ideal edgesSteps in edge detectionEdge enhancementResults of Sobel edge detectionSecond derivative operatorsLocalization with the LaplacianSlide 29Marching squaresSharpening with the LaplacianSpectral impact of sharpeningSummaryNext class: Orthogonal functions Fourier seriesUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellImage processingUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 2ReadingJain, Kasturi, Schunck, Machine Vision. McGraw-Hill, 1995. Sections 4.2-4.4, 4.5(intro), 4.5.5, 4.5.6, 5.1-5.4.University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 3Image processingAn image processing operation typically defines a new image g in terms of an existing image f.The simplest operations are those that transform each pixel in isolation. These pixel-to-pixel operations can be written:Examples: threshold, RGB  grayscaleNote: a typical choice for mapping to grayscale is to apply the YIQ television matrix and keep the Y.€ g(x, y) = t( f (x, y))€ YIQ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥=0.299 0.587 0.1140.596 −0.275 −0.3210.212 −0.523 0.311 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥RGB ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 4Pixel movementSome operations preserve intensities, but move pixels around in the imageExamples: many amusing warps of images[Show image sequence.]% %( , ) ( ( , ), ( , ))g x y f x x y y x y=University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 5Noise Image processing is also useful for noise reduction and edge enhancement. We will focus on these applications for the remainder of the lecture… Common types of noise: Salt and pepper noise: contains random occurrences of black and white pixels Impulse noise: contains random occurrences of white pixels Gaussian noise: variations in intensity drawn from a Gaussian normal distributionUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 6Ideal noise reductionUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 7Ideal noise reductionUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 8Practical noise reductionHow can we “smooth” away noise in a single image?Is there a more abstract way to represent this sort of operation? Of course there is!University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 9Discrete convolutionFor a digital signal, we define discrete convolution as:where € g[i] = f [i]∗ h[i]= f [ ′i ]h[i − ′i ] ′i ∑= f [ ′i ]) h [ ′i − i] ′i ∑ € ) h [i] = h[−i]University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 10Discrete convolution in 2D Similarly, discrete convolution in 2D becomes:where € g[i, j] = f [i, j]∗ h[i, j]= f [ ′i , ′j ]h[i − ′i , j − ′j ] ′j ∑ ′i ∑= f [ ′i , ′j ]) h [ ′i − i, ′j − j] ′j ∑ ′i ∑ € ) h [i, j] = h[−i,− j]University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 11Convolution representationSince f and h are defined over finite regions, we can write them out in two-dimensional arrays:Note: This is not matrix multiplication!Q: What happens at the edges?University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 12Mean filtersHow can we represent our noise-reducing averaging filter as a convolution diagram (know as a mean filter)?University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 13Effect of mean filtersUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 14Gaussian filtersGaussian filters weigh pixels based on their distance from the center of the convolution filter. In particular:This does a decent job of blurring noise while preserving features of the image.What parameter controls the width of the Gaussian? What happens to the image as the Gaussian filter kernel gets wider?What is the constant C? What should we set it to?2 2 2( )/(2 )[ , ]i jeh i jCσ− +=University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 15Effect of Gaussian filtersUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 16Median filtersA median filter operates over an mxm region by selecting the median intensity in the region.What advantage does a median filter have over a mean filter?Is a median filter a kind of convolution?University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 17Effect of median filtersUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 18Comparison: Gaussian noiseUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 19Comparison: salt and pepper noiseUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 20Edge detectionOne of the most important uses of image processing is edge detection:Really easy for humansReally difficult for computersFundamental in computer visionImportant in many graphics applicationsUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 21What is an edge?Q: How might you detect an edge in 1D?University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 22GradientsThe gradient is the 2D equivalent of the derivative:Properties of the gradientIt’s a vectorPoints in the direction of maximum increase of fMagnitude is rate of increaseHow can we approximate the gradient in a discrete image?( , ) ,f ff x yx y⎛ ⎞∂ ∂∇ =⎜ ⎟⎝∂ ∂ ⎠University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 23Less than ideal edgesUniversity of Texas at Austin CS384G - Computer Graphics


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