Probabilities, Greyscales, and Histograms: Chapter 3a G&WIntensity transformation exampleIntensity transformationGamma correctionGamma transformationsGamma transformationsPiecewise linear intensity transformationMore intensity transformationsHistogram of Image IntensitiesHistograms and NoiseSample SpacesConditional ProbabilitiesIndependenceRandom Variable (RV)Cumulative Distribution Function (cdf)Continuous Random VariablesProbability Density FunctionsExpected Value of a RVMean of a PDFSample MeanSample MeanApplication: Noisy ImagesApplication: Noisy ImagesGaussian DistributionGaussian PropertiesWhat is image segmentation?ThresholdingChoosing a threshold Role of noiseLow signal-to-noise ratioEffect of noise on image histogramEffect of illumination on histogramHistogram of Pixel Intensity DistributionClassification by ThresholdingImportant!Is the histogram suggesting the right threshold?Histogram as Superposition of PDF’s (probability density functions)Gaussian Mixture ModelExample: MRIExample: MRIFit with 3 weighted GaussiansSegmentation: Learning pdf’sSegmentation: Learning pdf’sHistogram Processing and EqualizationHistogramsHistogram equalizationHistogram EqualizationNext ClassUniv of Utah, CS6640 2010 1Probabilities, Greyscales, and Histograms:Chapter 3a G&WRoss Whitaker(slightly modified by Guido Gerig)SCI Institute, School of ComputingUniversity of UtahUniv of Utah, CS6640 2010 2Intensity transformation exampleg(x,y) = log(f(x,y))f(x1,y1) g(x1,y1)g(x1,y1) = log ( f(x1,y1) )f(x2,y2)g(x2,y2)g(x2,y2) = log ( f(x2,y2) )•We can drop the (x,y) and represent this kind of filter as an intensity transformation s=T(r). In this case s=log(r)-s: output intensity -r: input intensityUniv of Utah, CS6640 2010 3Intensity transformations = T(r)© 1992–2008 R. C. Gonzalez & R. E. WoodsUniv of Utah, CS6640 2010 4Gamma corrections = crγ© 1992–2008 R. C. Gonzalez & R. E. WoodsUniv of Utah, CS6640 2010 5Gamma transformations© 1992–2008 R. C. Gonzalez & R. E. WoodsUniv of Utah, CS6640 2010 6Gamma transformations© 1992–2008 R. C. Gonzalez & R. E. WoodsUniv of Utah, CS6640 2010 7Piecewise linear intensity transformation© 1992–2008 R. C. Gonzalez & R. E. Woods•More control•But also more parameters for user to specify•Graphical user interface can be usefulUniv of Utah, CS6640 2010 8More intensity transformations© 1992–2008 R. C. Gonzalez & R. E. WoodsUniv of Utah, CS6640 2010 9Histogram of Image Intensities• Create bins of intensities and count number of pixels at each level– Normalize or not (absolution vs % frequency)Grey level valueFrequencyUniv of Utah, CS6640 2010 11Histograms and Noise• What happens to the histogram if we add noise? – g(x, y) = f(x, y) + n(x, y)Grey level valueFrequencyUniv of Utah, CS6640 2010 12• S = Set of possible outcomes of a random event• Toy examples– Dice– Urn– Cards• ProbabilitiesSample SpacesUniv of Utah, CS6640 2010 13Conditional Probabilities• Multiple events– S2 = SxS Cartesian produce - sets– Dice - (2, 4)– Urn - (black, black)• P(A|B) - probability of A in second experiment knowledge of outcome of first experiment– This quantifies the effect of the first experiment on the second• P(A,B) - probability of A in second experiment and B in first experiment•P(A,B) = P(A|B)P(B)Univ of Utah, CS6640 2010 14Independence• P(A|B) = P(A)– The outcome of one experiment does not affect the other• Independence -> P(A,B) = P(A)P(B)• Dice– Each roll is unaffected by the previous (or history)• Urn– Independence -> put the stone back after each experiment• Cards– Put each card back after it is pickedUniv of Utah, CS6640 2010 15Random Variable (RV)• Variable (number) associated with the outcome of an random experiment• Dice– E.g. Assign 1-6 to the faces of dice• Urn– Assign 0 to black and 1 to white (or vise versa)• Cards– Lots of different schemes - depends on application• A function of a random variable is also a random variableUniv of Utah, CS6640 2010 16Cumulative Distribution Function (cdf)• F(x), where x is a RV• F(-infty) = 0, F(infty) = 1•F(x) non decreasingUniv of Utah, CS6640 2010 17Continuous Random Variables• f(x) is pdf (normalized to 1)• F(x) – cdf continuous– –> x is a continuous RV01)(xf)(xFUniv of Utah, CS6640 2010 18Probability Density Functions• f(x) is called a probability density function (pdf)• A probability density is not the same as a probability• The probability of a specific value as an outcome of continuous experiment is (generally) zero– To get meaningful numbers you must specify a rangeUniv of Utah, CS6640 2010 19Expected Value of a RV• Expectation is linear– E[ax] = aE[x] for a scalar (not random)– E[x + y] = E[x] + E[y]• Other properties– E[z] = z –––––– if z is not randomUniv of Utah, CS6640 2010 20Mean of a PDF• Mean: E[x] = m – also called “µ”– The mean is not a random variable–it is a fixed value for any PDF• Variance: E[(x - m)2] = E[x2] - 2E[mx] + E[m2] = E[x2] - m2 = E[x2] - E[x]2 – also called “σ2”– Standard deviation is σ– If a distribution has zero mean then: E[x2] = σ2Univ of Utah, CS6640 2010 21Sample Mean• Run an experiments– Take N samples from a pdf (RV)– Sum them up and divide by N• Let M be the result of that experiment– M is a random variableUniv of Utah, CS6640 2010 22Sample Mean• How close can we expect to be with a sample mean to the true mean?• Define a new random variable: D = (M - m)2.– Assume independence of sampling processRoot mean squared difference between true mean and sample mean is stdev/sqrt(N).As number of samples –> infty, sample mean –> true mean.Independence –> E[xy] = E[x]E[y]Number of terms off diagonalUniv of Utah, CS6640 2010 23Application: Noisy Images• Imagine N images of the same scene with random, independent, zero-mean noise added to each one– Nuclear medicine–radioactive events are random– Noise in sensors/electronics• Pixel value is s+nTrue pixel valueRandom noise:•Independent from one image to the next•Variance = σUniv of Utah, CS6640 2010 24Application: Noisy Images• If you take multiple images of the same scene you have– si= s + ni– S = (1/N) Σsi = s + (1/N) Σni – E[(S - s)2] = (1/N) E[ni 2] = (1/N) E[ni 2] - (1/N) E[ni]2= (1/N)σ2– Expected root mean squared error is σ/sqrt(N)• Application:– Digital cameras with large gain (high ISO, light sensitivity)– Not