CAP4453-Robot Vision Lecture 2-Basics of Images. Ulas Bagci

Transcription

1 CAP4453-Robot Vision Lecture 2-Basics of Images Ulas Bagci 1

2 Outline Image as a function Extracting useful information from Images Pixels / Histogram Convolution/Correlation Read Szeliski, Chapter 3. Read Shah, Chapter 2. Read/Program CV with Python, Chapters 1 and 2. 2

3 Vectors A vector is an array of numbers. The numbers are arranged in order. We can identify each individual number by its index in that ordering. 3

4 Vectors A vector is an array of numbers. The numbers are arranged in order. We can identify each individual number by its index in that ordering. The elements of the vector are identified by writing its name italic Typically, lower case and boldface R n 4

5 Matrices A matrix is a 2-D array of numbers, so each element is identified by two indices instead of one! 5

6 Transpose of a matrix The transpose of the matrix can be thought of as a mirror image across the main diagonal. 6

7 Transpose of a matrix The transpose of the matrix can be thought of as a mirror image across the main diagonal. Main diagonal 7

8 Multiplying Matrices and Vectors The matrix product of matrices A and B is a third matrix C. In order One of the most important operations involving matrices is multiplication of two for this product to be defined, A must have the same number of columns as B has matrices. If A is of shape m x n, and B is of shape n x p, then C is of shape m x p C=AB 8

9 Example 9

10 Example 10

11 Norms Sometimes we need to measure the size of a vector. Norm is a function to measure a size of a vector. 11

12 L1 Norm Commonly used in ML when the difference between zero and nonzero elements is very Important! 12

13 Max Norm This norm simplifies to the absolute value of the element with the largest magnitude In the vector. 13

14 PICTURES Computers use discrete form of the pictures The process transforming continuous space into discrete space is called digitization Picture Sampling+ Quantization Digital Picture 14

15 Lecture 2 - Filtering 8/27/ 15 15

16 Digitization/Sampling of 3D Image 16

17 Digitization of an arc 17

18 Definition A (2D) picture P is a function defined on a (finite) rectangular subset G of a regular planar orthogonal array. G is called (2D) grid, and an element of G is called pixel. P assigns a value of P(p) to each p 2 G 18

19 Definition A (2D) picture P is a function defined on a (finite) rectangular subset G of a regular planar orthogonal array. G is called (2D) grid, and an element of G is called pixel. P assigns a value of P(p) to each p 2 G 19

20 Definition Pictures are not only sampled, they are also quantized: they may have only a finite number of possible values (i.e., 0 to 255, 0-1, ) 20

21 2D vs 3D 21

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23 Resolution is a display parameter, defined in dots per inch (DPI) or equivalent measures of spatial pixel density, and its standard value for recent screen technologies is 72 dpi. Recent printer resolutions are in 300 dpi and/or 600 dpi. 23

24 Resolution Differences (Example) 24

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26 Source: F.F. Li 26

27 range domain 27

28 RGB Channels 28

29 Intensity profiles for selected (two) rows 29

30 Image Histogram? 30

31 Image Histogram 31

32 Example Reading of Histogram 32

33 Histogram Example Use ImageJ and/or FIJI Credit: Klette

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35 Credit: TechRadar 35

36 Pixel Transformation a function that takes an image (or images) an input and produces an output image (or scalar) 36

37 X 37

38 Mean (intensity) of an Image 38

39 Mean (intensity) of an Image 39

40 Standard Deviation (Intensity) of an Image Variation Square root of this is STD 40

41 Linear Operators Suppose D is an operator, f1 and f2 are images, and alpha/beta are scalars: Then, D is called linear operator 41

42 Linear Operators Suppose D is an operator, f1 and f2 are images, and alpha/beta are scalars: Then, D is called linear operator What is the most common linear operator? 42

43 Question Consider the image operator D as D=a.f + b, where f is image Is D linear operator or not? 43

44 Question Consider the image operator D as D=a.f + b, where f is image Is D linear operator or not? D(w1.f1 + w2.f2) = a(w1.f1 + w2.f2) + b =? w1.d(f1) + w2.d(f2) 44

45 Question Consider the image operator D as D=a.f + b, where f is image Is D linear operator or not? D(w1.f1 + w2.f2) = a(w1.f1 + w2.f2) + b =? w1.d(f1) + w2.d(f2) w1(a.f1+b) + w2(a.f2+b) NOT LINEAR! 45

46 Example Pixel Transformation compositing 46

47 Correlation and Convolution Convolution is a filtering operation, expresses the amount of overlap of one function as it is shifted over another function Correlation compares the similarity of two sets of data. (relatedness of the signals!) 47

48 Correlation (linear relationship) 48 ( ) ( ) = åå Ä k l l k h l k f h f,, Kernel Image = = h f h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 f 9 Ä h f h f h f h f h f h f h f h f h f h f = Ä f

49 Convolution ( k, l) h( - k l) åå f - f * h =, f = Image h 7 h 8 h 9 h = Kernel h 4 h 5 h 6 k l h 1 h 2 h 3 X - flip h h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 f Y - flip f 1 f 2 f 3 f 4 f 5 f 6 * f 7 f 8 f 9 h 9 h 8 h 7 h 6 h 5 h 4 h 3 h 2 h 1 f * h = + + f f f h 9 h h f f f h 8 h h f f f h 7 h h

50 Correlation and Convolution Convolution is associative F *( G * I) = ( F * G) * I 50

51 Convolution Example 51

52 Questions? 52