Detecting Edges in Images. by Dr. Niels Lobo UCF EXCEL Applications of Calculus

Transcription

1 Detecting Edges in Images by Dr. Niels Lobo UCF EXCEL Applications of Calculus

2 Overview of talk We will first do three Examples of problems from the book, and then address the application to Computer Vision.

3 Calculus Problem A from the Book Example A: Textbook s Section 3.1 Example 6 Let D (t) be the U.S. National debt at time t. The table gives approximate values of this function by providing end of year estimates, in billions of dollars, from 1980 to Interpret and estimate the value of D' ( t) t D(t)

4 Calculus Problem A from the Book The derivative means the rate of change of D with respect to t when t = 1990, i.e, the rate of increase of national debt in To compute this, refer to Eqn. 3 in section 3.1, f ( a) = So, according to this eqn., we have D (1990) = D (1990) lim x a f lim t 1990 ( x) x f a D( t) t ( a) D(1990) 1990

5 Calculus Problem A from the Book Eqn. 3 in section 3.1, f ( a) = lim x a f ( x) x f ( a) a According to this eqn., we have D (1990) = lim D( t) D(1990) t 1990 t 1990 So we compute and tabulate values of the difference quotient as follows (the difference quotient records the average rate of change).

6 Calculus Problem A from the Book So we compute and tabulate values of the difference quotient as follows (the difference quotient records the average rate of change) t D( t) t D(1990)

7 Calculus Problem A from the Book Let us see how the table gets its values. t D( t) t D(1990)

8 Calculus Problem A from the Book Remember that the For t = 1980, compute original data was D( t) D(1990) t D(t) t which is divided by 10 (years) which is

9 Calculus Problem A from the Book So, that is the first entry. t D( t) t D(1990)

10 Calculus Problem A from the Book Similarly imilarly, get the rest. t D( t) t D(1990)

11 Calculus Problem A from the Book From the table we see that D (1990) lies somewhere between and billion dollars per year. (Here we are making the reasonable assumption that the debt didn t fluctuate wildly between 1980 and 2000.)

12 Calculus Problem A from the Book We estimate that the rate of increase of the national debt of the United States in 1990 was the average of these two numbers, namely D (1990) 303 billion dollars per year. (because 303 is the average of and )

13 Calculus Problem B from the Book Example B: Textbook s Section 3.1 Problem 31 Let T be the temperature (in ) in Dallas t hours after midnight on June 2, The table shows values of this function recorded every two hours. What is the meaning of? Estimate its value. T '(10) F t T

14 Calculus Problem B from the Book Solution: T '(10) in Dallas at 10 am. means the rate of change of temperature To estimate its value, first we calculate the difference quotient values. Need table of T ( t) t T (10) 10

15 Calculus Problem B from the Book Note the use of the word DIFFERENCE. Due to the fact that we do not have continuous data, (meaning we do not have the function values at all possible times), the best we can do is approximate the derivative by a DIFFERENCE. We would like to have been able to set the gap to being infinitesimally small. But we cannot. So, we settle for a larger gap.

16 Calculus Problem B from the Book Input t T gives a Diff Quotient table t T ( t) t T (10) / /2 -- 7/2 5/2

17 Calculus Problem B from the Book T '(10) lim To estimate, given by, t 1990 we take the average of the difference T ( t) t T (10) 10 quotient values at t=8, and t=12, to get the result ( )/2, i.e., 4.

18 Calculus Problem C from the Book Example C: Textbook s Section 3.1 Problem 32 Life expectancy improved dramatically in the 20th century. The table gives values of E(t), the life expectancy at birth (in years) of a male born in the year t in the United States. Interpret and estimate E '(1910) E'(1950) the values of and.

19 Calculus Problem C from the Book. t E(t) t E(t)

20 Calculus Problem C from the Book Solution: E '(1910) E'(1950) and refer to the rate of change (in this case, increase) of life expectancy at birth for males born in 1910 and 1950 respectively. To estimate these quantities, we build the difference quotient table for each of the required years.

21 Calculus Problem C from the Book. t E( t) E(1910) t 1910 t E( t) E(1910) t

22 Calculus Problem C from the Book. t E( t) E(1910) t

23 Calculus Problem C from the Book Original table is: So, first entry is t E(t) ( )/ (-10) t E( t) E(1910) t

24 Calculus Problem C from the Book. t E( t) E(1910) t 1910 t E( t) E(1910) t Do we need whole table?

25 Calculus Problem C from the Book. t E( t) E(1910) t 1910 t E( t) E(1910) t ?? ?? ?? ?? 1940?? 2000?? 1950??

26 Calculus Problem C from the Book and we proceed to use the approach that E'(1910) is the average of 0.28 and 0.41, for an answer of Similarly, for E'(1950), the difference quotient table is (on the next slide)

27 Calculus Problem C from the Book. t E( t) E(1950) t 1950 t E( t) E(1950) t ?? ?? 1970?? 1920?? 1980?? 1930?? 1990?? ??

28 Calculus Problem C from the Book and E'(1910) is taken to be the average of 0.31 and 0.1, which is 0.205, indicating that the rate of life expectancy gain was slower in 1950 than in 1910.

29 Images and their Edges Computer Vision Good for substituting machine in place of eye - Can assist with recognition - Can assist with navigation - Can assist with manipulation

30 Computer Vision: Helpful Mirror.

31 Computer Vision: Driverless Cars Automated Driver Console Driverless Taxis Battlefield Urban Driving

32 Surveillance for Safety Camera Network Crime Watch Airport Security Your Title Text here Speaker Crowd Monitor U.S. assets abroad Border Control

33 Computer Vision Crime Watch

34 Computer Vision Airport Security

35 Computer Vision Monitor U.S. Assets Abroad

36 Computer Vision Border Security

37 Medical Imaging 3D Models Image Guided Surgery Revolutionizing Medical Science Automated Cancer Scans Computerized Fracture Estimation

38 Computer Vision A Basic Task: Detect Edges of Regions

39 Detecting Edges in an image This is an example of a picture you might see on a computer screen. 20 X 20 pixel image of black box on square white background. Pixel Values for image

40 Black Rectangle on White Background A

41 Computer Vision To

42 Plot values from a row To Pixel Values

43 Find jumps in the plot Denote the plot by, then we Compute I( x) x ( x I( x 1) 1) I(x) We can think of two values, A and B, moving along the row. So we get the calculation being merely (B-A)/1

44 Again, the plot of a row Pixel Values

45 Plot of difference of pairs B-A Pixel Values

46 Absolute Value of B-A Pixel Values

47 How to find strong edges To find an edge from this derivative plot, use a threshold.

48 Effect of Thresholding Threshold Bar 0 Pixel Values

49 Back to the other example

50 Back to the other example

51 This one has a drop and then a rise Pixel Values

52 Difference of pairs, B-A Pixel Values

53 Absolute Value of (B-A) Pixel Values

54 Thresholding Pixel Values

55 Back to Complete Image A Basic Task: Detect Edges of Regions

56 . Consider a typical image

57 For this typical image We know we can find the edges as we proceed along the horizontal direction i.e., along a row i.e., the x-direction

58 For this typical image What about the vertical direction??? i.e., along a column? i.e., along the y-direction?

59 The y-direction.

60 Compute the Vertical Edges So, just as for the x-direction, we can compute the difference quotient for the y-direction: I( y) I ( y 1) y ( y 1) which means we are to compute the difference between two neighboring points that are vertical.

61 Computer Vision So, at all points on the image, we have 2 answers (one from x- direction and one from y-direction.) How to give a unified answer at all the points on the image?

62 Computer Vision So, denote image by I ( x, y) Then, the two Difference quotients are: I y and I x

63 What to call these two? So, get the notion of the Gradient. The two quantities combine to give the Gradient Vector. See Page 1095 of text.

64 The Gradient.

65 Back to the complete image The Gradient Vector is a physical descriptor of two dimensional functions; The symbol for the gradient vector is, and to repeat, it has two parts, the partial derivatives I x and I y r I

66 Back to the complete image The magnitude of the gradient vector can be obtained by squaring the individual components, adding them, and taking the square root, to get one scalar number. This concept is introduced in your Calculus textbook on page 1095, Chapter 17.

67 Back to the complete image The magnitude of the gradient vector can be obtained by squaring the individual components, adding them, and r I taking the square root, r I s magn = I x 2 + I y 2

68 Computer Vision Apply the gradient magnitude computation to this image

69 Gradient Magnitude.

70 . Use a Threshold

71 Use a LOWER Threshold Edges too Thick!!

72 Thickness of edges Next week, will see how to get thin edges. In the meantime, look at some cool stuff.

73 Computer Vision A

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84 Computer Vision.