1. The Imaging Chain 1. What energy is used to create the image? 2. How does the energy interact with matter? 3. How is the energy collected after the interaction? 4. How is the collected energy captured? 5. How is the collected signal manipulated? 6. How is the information displayed? 7. How is the information perceived?

2. The Imaging Chain: Light sources 1. What energy is used to create the image?

3. The Imaging Chain: Object interactions 2. How does the energy interact with matter?

4. The Imaging Chain: Object interactions 2. How does the energy interact with matter?

5. The Imaging Chain: Collection (optics) 3. How is the energy collected after the interaction?

6. The Imaging Chain: Capture (sensors) 4. How is the collected energy captured?

7. The Imaging Chain: Image processing 5. How is the collected signal manipulated?

8. The Imaging Chain: Display 6. How is the information displayed?

9. The Imaging Chain: Perception 7. How is the information perceived?

10. Topics: Statistics & Experimental Design The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

11. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

12. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

13. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

14. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

15. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

16. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

17. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

18. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

19. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

20. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

21. Topics: Statistics The Human Visual System Color Science Light Sources: Radiometry/Photometry Geometric Optics Tone-transfer Function Image Sensors Image Processing Displays & Output Colorimetry & Color Measurement Image Evaluation Psychophysics

22. July 7 - 11: The Imaging Chain, Statistics, Visual System, Color Science Overview July 14 - 18: Radiometry/Photometry, Geometric Optics, Tone-transfer Function July 21 - 25: Sensors, Image Processing, Displays & Output July 28 - Aug 1: Color Measurement, Image Evaluation, & Psychophysics Aug 4 - 8: Experimental Design & Capstone Projects Schedule:

23. MondayTuesdayWednesdayThursdayFriday Imaging Chain (Introduction) Jeff Pelz Statistics introduction María Helguera Human Visual System Jeff Pelz Human Visual System Jeff Pelz Color Science Overview Susan Farnand Lunch Imaging Chain (Introduction) Jeff Pelz Parametric Statistics María Helguera Human Visual System Jeff Pelz Color Science Overview Susan Farnand July 7-11

24. MondayTuesdayWednesdayThursdayFriday Sources: Radiometry/ Photometry Emmett Ientilucci Sources: Radiometry/ Photometry Emmett Ientilucci Geometric Optics Jeff Pelz Geometric Optics Jeff Pelz Tone-transfer Function Jon Arney Lunch Sources: Radiometry/ Photometry Emmett Ientilucci Sources: Radiometry/ Photometry Emmett Ientilucci Geometric Optics Jeff Pelz Tone-transfer Function Jon Arney July 14 -18

25. MondayTuesdayWednesdayThursdayFriday Sensors Robert Kremens Sensors Robert Kremens Image Processing Carl Salvaggio Image Processing Carl Salvaggio Displays & Output Mitchell Rosen Lunch Sensors Robert Kremens Image Processing Carl Salvaggio Image Processing Carl Salvaggio Displays & Output Mitchell Rosen July 21 - 25

26. MondayTuesdayWednesdayThursdayFriday Colorimetry Overview David Wyble Image Evaluation Larry Scarff/ Don Williams Image Evaluation Larry Scarff/ Don Williams Image Evaluation Larry Scarff/ Don Williams Psychophysics Susan Farnand Lunch Color Measurement David Wyble Image Evaluation Larry Scarff/ Don Williams Image Evaluation Larry Scarff/ Don Williams Image Evaluation Larry Scarff/ Don Williams July 28 - Aug 1

27. MondayTuesdayWednesdayThursdayFriday Experimental Design María Helguera/ Susan Farnand Experimental Design María Helguera/ Susan Farnand Experimental Design & Project María Helguera/ Susan Farnand / Larry Scarff Capstone Project Larry Scarff/ Susan Farnand Wrap up Susan Farnand Lunch Experimental Design María Helguera/ Susan Farnand Experimental Design María Helguera/ Susan Farnand Experimental Design & Project María Helguera/ Susan Farnand / Larry Scarff Capstone Project Larry Scarff/ Susan Farnand Aug 4 - Aug 8

28. Design of experiments Why is it important? We wish to draw meaningful conclusions from data collected Statistical methodology is the only objective approach to analysis

29. Design of experiments Recognize the problem Select factor to be varied, levels and ranges over which factors will be varied Select the response variable Choose experimental design: Sample size? Blocking? Randomization? Perform the experiment Statistical analysis Conclusions and recommendations

30. Let’s start easy We would like to compare the output of two systems. Design a testing protocol and run it several times RunSystem A System B 1y 1A y 1B 2y 2A y 2B 3y 3A y 3B ………

31. Visualize data For small data sets: Scatter plot

32. Visualize data For larger data sets: Histogram Divide horizontal axis into intervals (bins) Construct rectangle over interval with area proportional to number (frequency) of observations frequency, n i

33. Statistical inference Draw conclusions about a population using a sample from that population. Imagine hypothetical population containing a large number N of observations. Denote measure of location of population as 

34. Statistical inference Denote spread of population as variance 

35. Statistical inference A small group of observations is known as a sample. A statistic like the average is calculated from a set of data considered to be a sample from a population RunSystem A System B 1y 1A y 1B 2y 2A y 2B 3y 3A y 3B ………

36. Statistical inference Sample variance supplies a measure of the spread of the sample

37. Probability distribution functions P(a  x  b)

38. Probability distribution functions P(x i ) xixi P(x = x i ) = p(x i )

39. Mean, variance of pdf Mean is a measure of central tendency or location Variance measures the spread or dispersion

40. Normal distribution  = standard deviation = √    mean

41. Normal distribution, From previous examples we can see that mean =  and variance =  2 completely characterize the distribution. Knowing the pdf of the population from which sample is draw  determine pdf of particular statistic.

42. Normal distribution Probability that a positive deviation from the mean exceeds one standard deviation  is 0.1587  1/6 = percentage of the total area under the curve. (Same as negative deviation) Probability that a deviation in either direction will exceed one standard deviation  is 2 x 0.1587 = 0.3174 Chance that a positive deviation from the mean will exceed two  = 0.02275  1/40

43. Normal distribution Sample runs differ as a result of experimental error Often can be described by normal distribution

44. Standard Normal distribution, N(0,1) Values for N(0,1) are found in tables.

45. Standard Normal distribution, N(0,1)

46. Example: Suppose the outcome of a given experiment is approximately normally distributed with a  = 4.0 and  = 0.3. What is the probability that the outcome may be 4.4? Look in table in previous page, to find that the probability is 9%.

47.    distribution Another sampling distribution that can be defined in terms of normal random variables. Suppose z 1, z 2, …, z k are normally and independently distributed random variables with mean  = 0 and variance  2 = 1 (NID(0,1)), then let’s define Where  follows the chi-square distribution with k degrees of freedom.

48.    distribution k = 1 k = 5 k = 10 k = 15

49. Student’s t Distribution In practice we don’t know the theoretical parameter  This means we can’t really use and refer to the result of the table of standard normal distribution Assume that experimental standard deviation s can be used as an estimate of 

50. Student’s t Distribution Define a new variable It turns out that t has a known distribution. It was deduced by Gosset in 1908

51. Student’s t Distribution k=1 k=10 k=100 Probability points are given in tables. The form depends on the degree of uncertainty in s 2, measured by the number of degrees of freedom, k.

53. Inferences about differences in means Statistical hypothesis: Statement about the parameters of a probability distribution. Let’s go back to the example we started with, i.e., comparison of two imaging systems. We may think that the performance measurement of the two systems are equal.

54. Hypothesis testing First statement is the Null hypothesis, second statement is the Alternative hypothesis. In this case it is a two-sided alternative hypothesis. How to test hypothesis? Take a random sample, compute an appropriate test statistic and reject, or fail to reject the null hypothesis H 0. We need to specify a set of values for the test statistic that leads to rejection of H 0. This is the critical region.

55. Hypothesis testing Two errors can be made: Type I error: Reject null hypothesis when it is true Type II error: Null hypothesis is not rejected when it is not true In terms of probabilities:

56. Hypothesis testing We need to specify a value of the probability of type I error . This is known as significance level of the test. The test statistic for comparing the two systems is: Where

57. Hypothesis testing To determine whether to reject H 0, we would compare t 0 to the t distribution with k A +k B -2 degrees of freedom. If we reject H 0 and conclude that means are different. We have: System ASystem B

58. Hypothesis testing We have k A + k B – 2 = 18 Choose  = 0.05 We would reject H 0 if

59. Hypothesis testing

60. Since t 0 = -9.13 < -t 0.025,18 = -2.101 then we reject H 0 and conclude that the means are different. Hypothesis testing doesn’t always tell the whole story. It’s better to provide an interval within which the value of the parameter is expected to lie. Confidence interval. In other words, it’s better to find a confidence interval on the difference  A -  B

61. Confidence interval Using data from previous example: So the 95 percent confidence interval estimate on the difference in means extends from -1.43 to -0.89. Note that since  A –  B = 0 is not included in this interval, the data do not support the hypothesis that  A =  B at the 5% level of significance.