PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 31Slide 32Slide 33Slide 34Slide 35MSU CSE 803Linear Operations Using MasksMasks are patterns used to define the weights used in averaging the neighbors of a pixel to compute some result at that pixelMSU CSE 803Expressing linear operations on neighborhoodsMSU CSE 803Images as functionsMSU CSE 803Neighborhood operationsAverage neighborhood to remove noise or high frequency patternsDetect boundaries at points of contrast using gradient computationCan use median filtering to smooth while keeping boundaries sharpMSU CSE 803Image processing examplesHistogram equalization; gamma correction; median filteringMSU CSE 803Histogram equalizationLeft image does not use all available gray levels. Image is recoded so that all gray levels are used and such that each gray level occurs in roughly the same number of pixels of the recoded image. (See algorithm in text, xv.)MSU CSE 803Histogram equalization can darken a bright image, perhaps improving contrastMSU CSE 803Histogram equalizationhttp://en.wikipedia.org/wiki/Histogram_equalizationMSU CSE 803Can define mapping of input gray level to output level (xv)Gamma correction: boost all gray levelsBoost low levels and reduce highMSU CSE 803Smoothing an image by averaging neighbors (boxcar)MSU CSE 803Properties of smoothing masksMSU CSE 803Types of ideal edges (in 1D) These types are also present in 2D and 3D images and are complicated by orientation variations.MSU CSE 803Boxcar smoothing filter exampleSo, reducing noise will also degrade the signal.MSU CSE 803Linear smoothing smoothes noise and blurs signalBlur: step is now rampInput image Row after 5x5 mean filterMSU CSE 803Gaussian smoothingMSU CSE 803Median filter replaces center with neighborhood median, not meanNoisy row of checkers imageMean filtering smoothes signal and ramps the boundaryMedian filter smoothes signal and preserves sharp boundaryMSU CSE 803Median filter is not linearAlgorithm requires comparisons and is more expensive than using maskCan sort all NxN pixel values and pick middleDo not need totally sorted data: O(N) algorithm existsX(n)=aX(n1)+bX(n2) -> Y(n)=aY(n1)+bY(n2)MSU CSE 803Scratches removed by using a median filterThin artifact removed, sharp boundaries preserved.MSU CSE 803Finding boundary pixelsComputing derivatives or gradients to locate region change.MSU CSE 8032 rows of intensity vs differenceMSU CSE 803Differencing used to estimate 1st and 2nd derivativesFirst differences2nd differencesMasks represent the first and 2nd differencesMSU CSE 803Step edges X mask [-1, 0, +1]Step edge is detected well, but edge location imprecise.MSU CSE 803Ramp and impulse X mask [-1, 0, +1]Ramp edge now yields a broad weak response. Impulse response is a “whip”, first up and then down.MSU CSE 8032nd derivative using mask [-1, 2, -1]Response is zero on constant region and a “double whip” amplifies and locates the step edge.MSU CSE 8032nd derivative using mask [-1, 2, -1]Weak response brackets the ramp edge. Bright impulse yields a double whip with gain of 3X original contrast.MSU CSE 803Estimating 2D image gradientMSU CSE 803Gradient from 3x3 neighborhoodEstimate both magnitude and direction of the edge.MSU CSE 803Alternative masks for gradientMSU CSE 803Prewitt versus Sobel masksSobel mask uses weights of 1,2,1 and -1,-2,-1 in order to give more weight to center estimate. The scaling factor is thus 1/8 and not 1/6.MSU CSE 803Computational short cutsMSU CSE 803Properties of derivative masksMSU CSE 803Next set of related slidesInterpret the properties of masks using the theory of vectors and dot