Mathematics in Machine Learning for students.pptx

Mathematics is very crucial in Machine Learning

1. Unveiling the Code in Medical Imaging

2. Over decades of research have made the image processing more accurate. Early picture obtained through submarine cables were not reconstructed properly. They were some alteration due to many errors that occurred during transportation of the image.

3. Application of linear Algebra to image processing Calvin White Department of Mathematics Georgia College

4. When we zoom in continuously, we observe little “squares” Those squared are pixels and their number depend on the type of device use to realize the picture. Application of linear Algebra to image processing Calvin White Department of Mathematics Georgia College

5. Array versus Matrix Operations Consider two 2 x 2 images 𝑎 11 𝑎 21 𝑎 12 𝑎 22 and 𝑏 11 𝑏 21 𝑏 12 𝑏 22 Array Product is: 𝑎 11 𝑏 11 𝑎 21 𝑏 21 𝑎 12 𝑏 12 𝑎 22 𝑏 22 Matrix Product is: 𝑎 11 𝑏 11 + 𝑎 12 𝑏 21 𝑎 21 𝑏 11 + 𝑎 22 𝑏 21 𝑎 11 𝑏 12 + 𝑎 12 𝑏 22 𝑎 21 𝑏 12 + 𝑎 22 𝑏 22

6. Spatial sub-sampling of 1024 X 1024, 8 bit image. The gray level quantization is the same for all images. Every image is produced from the previous by deleting every other column and row. (Digital Image Processing, 2nd E, by Gonzalez, Richard).

7. Spatial re-sampling of the images from Fig.2, 8 bit image. The size of each image is enlarged to the size of the 1024x1024 image. The gray level quantization is the same for all images. (Digital Image Processing, 2nd E, by Gonzalez, Richard).

8. We keep the spatial sampling but decrease k=8 to k=1 (gray levels from 256 to 2). (Digital Image Processing, 2nd E, by Gonzalez, Richard).

9. The image in c) is received as “and” of a) and b). The original image is a courtesy of Dr. Val Runge , Medical Center, Temple Texas

10. Array operation involving one or more images is carried out on a pixel-by-pixel basis. Ex. i) Raising an image to a power. Individual pixel is raised to that power. ii) Dividing an image by another: Division is between corresponding pixel pairs.

11. Linear versus Nonlinear Operations General operator, H , that produces an output image, g ( x , y ), for a given input image, f ( x , y ): H[f(x, y)] = g(x, y) H is said to be a linear operator if H[a i f i (x,y) + a j f j (x,y)] = a i H[f i (x,y)] + a j H[f j (x,y)] = a i g i (x,y) + a j g j (x,y)—Eq.(i) where, a i , a j – arbitrary constants f i (x,y), f j (x,y) – images of same size.

12. Suppose H is the sum operator, ∑ ∑ [𝑎 𝑖 𝑓𝑖 𝑥 , 𝑦 + 𝑎𝑗𝑓 𝑗 ( 𝑥 , 𝑦 ) ] = ∑ 𝑎 𝑖 𝑓 𝑖 ( 𝑥 , 𝑦 ) + ∑ 𝑎 𝑗 𝑓 𝑗 ( 𝑥 , 𝑦 ) = 𝑎 𝑖 ∑ 𝑓 𝑖 𝑥 , 𝑦 + 𝑎 𝑗 ∑ 𝑓 𝑗 ( 𝑥 , 𝑦 ) = 𝑎 𝑖 𝑔 𝑖 𝑥 , 𝑦 + 𝑎𝑗𝑔 𝑗 ( 𝑥 , 𝑦 ) Thus, ∑ operator is linear.

13. Consider max operation, Let f1 = 0 2 , f2 = 6 2 3 4 7 5 , a1 = 1, a2 = -1. T o T e s t Line arit y , 2 + (−1) 6 5 4 7 = −2 2 3 0 2 2 3 + −1 𝑚𝑎𝑥 6 5 4 7 = -4 LHS of eq(i): 𝑚𝑎𝑥 1 RHS of eq(i): 1 𝑚𝑎𝑥 LHS ≠ RHS So, max is non-linear operation.

14. Arithmetic Operations Arithmetic operations are array operations that are carried out between corresponding pixel pairs. Four arithmetic operations: s(x, y) = f(x, y) + g(x, y) d(x, y) = f(x, y) - g(x, y) p(x, y) = f(x, y) * g(x, y) v(x, y) = f(x, y) / g(x, y) Where, x = 0,1,2,…,M-1, y = 0,1,2,….N-1. All images are of size M (rows) x N (columns).

15. Set and Logical Operations Basic Set operation Let A - set composed of ordered pairs of real numbers. If pixel a = (x,y), is an element of A If a is not an element of A Set with no elements is called the null or empty set

16. If every element of a set A is also an element of a set B , then A is said to be a subset of B Union of two sets A and B Intersection of two sets A and B Two sets A and B are disjoint or mutually exclusiv e if they have no common elements

17. The set universe, U , is the set of all elements in a given application. complement of a set A is the set of elements that are not in A difference of two sets A and B ,

18. Illustration of Set Concept

19. Set operations on gray- scale images a) Original image b) Image Negative c) Union of (a) & constant image

20. Illustration of Logical Operators

21. Spatial Operations Spatial operations are performed directly on the pixels of a given image. (1) single-pixel operations, (2) neighborhood operations, & (3) geometric spatial transformations. Single-pixel operations s = T(z) z - intensity of a pixel in the original image s - (mapped) intensity of the corresponding pixel in the processed image.

22. Neighborhood operations

23. We can express the operation in equation form as where r and c are the row and column coordinates of the pixels whose coordinates are members of the set S xy .

24. Geometric spatial transformations They modify the spatial relationship between pixels in an image. a.k.a. rubber-sheet transformations. They consists of two basic operations: (1) spatial transformation of coordinates and (2) intensity interpolation that assigns intensity values to the spatially transformed pixels. The transformation of coordinates may be expressed as (v, w) - pixel coordinates in the original input image ( x , y ) - the corresponding pixel coordinates in the transformed output image.

25. One of the most commonly used spatial coordinate transformations is the affine transform Its General Form This transformation can scale, rotate, translate, or sheer a set of coordinate points, depending on the value chosen for the elements of matrix T .

26. a ) An image; b) The image after negative transformation. (Digital Image Processing, 2nd E, by Gonzalez, Richard).

27. An urban image and the results after applying power transformation with different power. (Digital Image Processing, 2nd E, by Gonzalez, Richard).

28. Examples R o t a tion

29. Vector and Matrix Operations Color images are formed in RGB color space by using red , green , and blue component images. Each pixel of an RGB image has 3 components, which can be organized in the form of a column vector.

30. forward mapping , consists of scanning the pixels of the input image and, at each location, (v, w), computing the spatial location, (x, y), of the corresponding pixel in the output image. inverse mapping , scans the output pixel locations and, at each location, ( x , y ), computes the corresponding location in the input image using (v, w, 1) = T -1 (x, y, 1) It then interpolates using either of nearest neighbor, bilinear, and bicubic interpolation techniques.

31. Euclidean distance , D, between a pixel vector z and an arbitrary point a in n -dimensional space is defined as the vector product This is a generalization of the 2-D Euclidean distance Sometimes is referred to as a vector norm, denoted by 𝑧 − 𝑎 .

32. An image of size M X N can be represented as a vector of dimension MN X 1. A broad range of linear processes can be applied to such images by using notation g = Hf + n f – MN X 1 vector representing Input image n - MN X 1 vector representing M X N noise pattern g - MN X 1 vector representing affected image H – MN X MN matrix representing linear process applied to input image

33. Image Registration Transform an image to align its pixels with those in another image of same scene. T r a n s f ormat i on is unknown. Transformation can be linear or nonlinear.

34. Input 1 Input 2 Example : Align two images and combine them to produce a larger one Output

35. Image Transforms Approaches discussed till now work directly on spatial domain . Some tasks are best formulated by transforming the input images, carrying the specified task in a transform domain , and applying the inverse transform to return to the spatial domain . General form of 2-D linear transforms is given by: …..eq.(1)….eq.(1) f(x, y) - is the input image r(x, y, u, v) - is called the forward transformation kernel o u – 0,1,2,…..,M-1 o v – 0,1,2,…..,N-1

36. General approach for operating in the linear transform domain. x, y - spatial variables u, v – transform variables M, N - row and column dimensions of f . T(u, v) - is called the forward transform of f(x, y).

37. Given T(u, v) , we can recover f(x, y) using the inverse transform of T(u, v), ….eq.(2) o x – 0,1,2,…..,M-1 o y – 0,1,2,…..,N-1 o s(x, y, u, v) - is called the inverse transformation kernel .

38. (a) Image corrupted by sinusoidal interference. (b) Magnitude of the Fourier transform. (a) b) ( (c) d) ( (c) Mask (Filter) used to eliminate the energy bursts. (d) Result of computing the inverse of the modified Fourier transform.

39. The forward transformation kernel is said to be separable if Also the kernel is said to be symmetric if r 1 (x, y) is functionally equal to r 2 (x, y). Identical comments apply to the inverse kernel by replacing r with s in the preceding equations. Thus, forward & inverse kernels are given by:

40. Substituting these kernels into the general transform formulations, we get the D iscrete Fourier transform pair :

41. Fourier (forward and inverse) kernels are separable and symmetric allow 2-D transforms to be computed using 1-D transforms & f(x, y) is a square image of size M x M Then, eq.(1) & eq.(2) can be expressed in matrix form as F – M x M matrix with elements of input image f(x, y). A – M x M matrix with elements a ij = r 1 (x, y) T – resulting M x M matrix with values T(u, v), u, v = 0,1,2….., M-1

42. To obtain the inverse transform, we pre- and post-multiply above equation by an inverse transformation matrix B . If B = A -1 F can be recovered completely from its forward transform. If B ≠ A -1 Approximation is

43. Intensity Transformation Very first step in Digital Image Processing. It is purely subjective. It is a cosmetic procedure. It improves subjective qualities of images. It has two domains: Spatial domain Frequency domain

44. Spatial Domain Spatial means working in space i.e. (given image). It means working with pixel values or raw data. Let g( x, y) be original image where g is gray level values & ( x, y) is co-ordinates For 8-bit image, g can take values from – 255 where – black , 255 – white & others - shades of gray

45. Spatial Domain In an image with size 256 x 256, (x, y) can assume any value from (0 , 0) to (255 , 255). (0 , 0) Y X (255, 255)

46. Spatial Domain Applying transform modifies the image f(x,y) = T g(x,y) where, g(x,y) is original image T is transformation applied on g(x,y) f(x,y) is new modified image In spatial domain techniques simply T changes. Spatial domain enhancement is carried out in two ways: Point processing Neighborhood processing

47. Point Processing Here, we work on singe pixel i.e. T is 1 x 1 operator. New image depends on transform T and original image. Some important examples of point processing are: Digital Negative Contrast Stretching Thresholding Gray level slicing Bit plane slicing Dynamic range compression

48. Point Processing Modified Gray Identity Transformation: 2 5 5 T 125 s 10 2 5 5 10 125 Ori ginal G r a y L e v el r It does not modify the input image at all. In general, s = r

49. Point Processing 1) Digital Image Negative: Useful in large applications e.g. X-ray images. Negative means inverting gray levels. 255 Output Gray level 200 s 55 0 55 200 255 Input image intensity r

50. Point Processing Digital Negative can be obtained by: s = 255 – r (where, rmax = 255) when, r = 0; s = 255 & if r = 255; s = Generally, s = (L-1) –r where, L – total number of gray levels (e.g. 256 for 8-bit image)

51. Point Processing 2) Contrast Stretching: 255 w (r2, s2) n m (r1, s1) v l a b 255 Output Image Intensity s level r Input Image Intensity level

52. Point Processing Reasons: Poor Illumination Wrong setting of lens aperture Idea behind Contrast Stretching is to make dark portion darker and bright portion brighter. In above figure, dotted line indicated Identity Transformation & solid line indicates Contrast Stretching. Dark portion is being made darker by assigning slope of < 1. Bright portion is being made brighter by assigning slope of > 1. Any set of slopes cant be generalized for all kind of images. Formulation is given below: s = l.r ; for ≤ r ≤ a = m(r-a) + v ; for a ≤ r ≤ b = n(r-b) + w ; for b ≤ r ≤ L-1

53. Point Processing 3) Thresholding: s a 255 255 Output Image In t e n s ity level r Input Image Intensity level

54. Point Processing Extreme Contrast Stretching yields Thresholding. In Contrast Stretching figure, if l & n slope are made ZERO & if m slope is increased then we get Thresholding Transformation. If r1 = r2, s1 = 0 & s2 = L-1 Then we get Thresholding function. Expression goes as under: s = 0; if r ≤ a s = L – 1 ; if r >a where, L is number of Gray levels. Note: It is a subjective phenomenon. Thresholded image has maximum contrast as it has only BLACK & WHITE gray values.

55. Point Processing 4) Gray Level Slicing (Intensity Slicing): L -1 L -1 s s 0 a b L -1 r a b L -1 r fig. (1) Slicing w/o background fig. (2) Slicing with background

56. Point Processing Thresholding splits the image in 2 parts At times, we need to highlight a specific range of gray levels. eg. X-ray scan, CT scan It looks similar to thresholding except that we select a band of gray levels. Formulation of Gray level slicing w/o background (fig. 1): ; for a ≤ r ≤ b ; otherwise s = L-1 = No background at all. Sometimes we may need to retain the background. Formulation of Gray level slicing with background (fig. 2): s = L-1 ; for a ≤ r ≤ b = r ; otherwise

57. Point Processing 5) Bit Plane Slicing: Here, we find the contribution made by each bit to the final image. Consider a 256 x 256 image with 256 gray levels i.e. 8-bit representation for each pixel. E.g. BLACK is represented as 0000_0000 & WHITE by 1111_1111. Consider LSB value of each pixel & plot image. Continue till MSB is reached. All 8 images will be binary. Observing the images we conclude that Higher order images contain visually sufficient data. Lower order bits contain suitable details of image. Hence, BPS can be used in Image Compression. We can transmit only higher order bits & remove lower order bits. E.g. Steganography

58. Point Processing Ex. Plot bit planes of the given 3 x 3 image. 1 2 4 3 2 7 5 2 1 1 1 1 1 1 1 1 1 1 1 1 1 - 00000 001 2 - 00000 010 - 00000 000 4 - 00000 100 3 - 00000 011 2 - 00000 010 7 - 00000 111 Max. Intensity is 7 thus 3 – bits 5 - 00000 101 2 - 00000 010 1 1 1 1 1 1 1 1 1 1 1 1 MSB plane Middle Plane LSB Plane

59. Point Processing 6) Dynamic Range Compression (Log transformation): s r At times, dynamic range of image exceeds the capability of display device. Some pixel values are so large that the other low value pixel gets obscured. E.g. stars in day time are not visible though present due to large intensity of sun. Thus dynamic range needs to be compressed.

60. Point Processing Log operator is an excellent compression function. Thus, Dynamic range compression is achieved using log operator. Formulation: s = C*log n (1 + |r|) where, C – normalization constant r – input intensity

61. Point Processing 7) Power law Transform: γ < 1 s γ = 1 γ > 1 r f(x, y) = C *g(x, y) γ s = C. r γ where, C & γ are positive constants

62. Point Processing The Transformation is shown for different values of ‘γ’ which is also the gamma correction factor. By changing γ, we obtain the family of transformation curves. Nonlinearity encountered during image capturing, storing & displaying can be corrected using gamma correction. Power Law Transform can be used to increase dynamic range of image.