1. Computers in Medicine Bone visualization and analyses - An introduction to Matlab - Presented by: Kathryn Stok ETH Zurich kas@ethz.ch Prepared by: Steven Boyd University of Calgary

2. Objectives (I)  Provide an overview of Matlab – What is Matlab? – Basic commands – Accessing online help – Writing M-files – Programming style – Create plots

3. Objectives (II)  Work through examples – Reading an image file – Probe the data – Filtering – Create a histogram – Thresholding – Counting voxels – Calculating 2nd moment of area

4. Why bone visualization?  Carries the body → it should be strong  But in many cases, bone strength is impaired. – congenital deformities – osteoporosis – fracture – fatigue (stress fracture)

5. Bone architecture

6. Bone structure – femoral head  Young Normal  Osteoporotic

7. Bone structure – lumbar spine  Young Normal  Osteoporotic

8. Why bone visualization?  Bone (structure) analyses and visualization are of great importance in – Diagnosis – Treatment – Prevention

9. Distal radius 36 yr male64 yr female

10. Why Matlab?  High-performance language for technical computing. – Matrix Laboratory  Advantages – Easy to use: –No dimensioning required. –No compiling –Good for prototyping, testing – Functionality is greatly expanded by toolboxes. – Many fields of application.  Disadvantages: – Uses much memory (slow for large applications) – Not good for final software design

11. What is matlab?  Matlab language – matrix/array language – control flow (if, for, while) – functions  Working environment – workspace for computing, importing, and exporting data – text editor for creating M-files

12. This Matlab course  It covers: Using the development environment. Syntax of the language. Basic matrix operations. Basic graphics features. Writing Matlab scripts and functions (m-files)  It does not cover: Any of the toolboxes including Simulink. c-mex and f-mex files, Matlab compilers.

13. Possibilities…  Develop tools for: – Image processing and basic analysis. – Visualization: 2D and 3D. – Simulation: Simulate bone loss.

14. Matlab Environment

15. Matrices Albrecht Dürer  Magic square

16. Matrices semicolon  Entering matrices – explicitly

17. Matrices  Sum, transpose

18. Matrices  Diagonal (diag)

19. Matrices  Subscripts: A(i,j) row column

20. Matrices  Colon operator ' : '

21. Matrices  Colon operator ' : ' Matlab is 1-based

22. Matrix multiplication ( * )

23. Matrix multiplication (.* )

24. Variables  Scalar, vector, matrix

25. Variables  Check workspace – command line – window

26. Numbers  Integer  Real  Exponential  Imaginary -3 0.0001 1.60210e-20 5 + 2i

27. Operators: Arithmetic  +Addition  -Subtraction  *Multiplication (or.* element-wise)  /Division  ^Power (or.^ element-wise)  ' Transpose  ()Brackets for evaluation order A = [ 4 1 3] A.^2 = [16 1 9] A.*2 = [ 8 2 6]

28. Operators: Relational  <Less than  <=Less than or equal to  >Greater than  >=Greater than or equal to  ==Equal to  ~=Not equal to

29. Operators: Logical  &AND  |OR  ~NOT Type help ops to see: the arithmetic,relational and logical functions.

30. Matrices  Entering matrices – explicitly – load from external file – your own M-files – built-in function Albrecht Dürer

31. Generating matrices  Load data file

32. Generating matrices  M-files

33. Generating matrices  Built-in functions

34. Help  Help Window – pop-up window  Type help

35. Online help

38. Useful commands   (up arrow) – recall previous command – p  recalls previous command starting with ‘p’  clear – clear workspace – clear A Z Q clears only A, Z, and Q.  size – size(A) returns size of matrix A

39. Multidimensional matrices page k, Z row i, X column j, Y

40. Multidimensional matrices  Generate a 3D array manually

41. Advanced indexing  Matrices are always stored as columns – subscripts (i,j,k) – array dimension [d 1 d 2 d 3 ] – if A is of dimension 3 x 3 x 3 then A(3,3,3) = A(27) = 8 3 2 7 5 1 4 6 8 6 2 6 1 4 9 3 6 7 5 6 7 9 5 6 1 2 8 3 5 4 2 1 6 7 1 8 3 5 4 2 1 6 7 1 8 6 1 3 2 4 6 6 9 7 6 1 3 2 4 6 6 9 7 5 9 1 6 5 2 7 6 8 5 9 1 6 5 2 7 6 8 i j k

42. Advanced indexing  Matrices are always stored as columns – subscripts (i,j,k) – array dimension [d 1 d 2 d 3 ] – if A is of dimension 3 x 3 x 3 then A(3,3,3) = A(27) = 8 offset = i + (j-1)(d 1 ) + (k-1)(d 2 )(d 1 ) 3 2 7 5 1 4 6 8 6 2 6 1 4 9 3 6 7 5 6 7 9 5 6 1 2 8 3 5 4 2 1 6 7 1 8 3 5 4 2 1 6 7 1 8 6 1 3 2 4 6 6 9 7 6 1 3 2 4 6 6 9 7 5 9 1 6 5 2 7 6 8 5 9 1 6 5 2 7 6 8 i j k

43. Characters and Text  s = ' myimage ' – 7 character array (string)  num2str() - convert a number to string

44. User input  Keyboard input a_number = input('Enter number'); a_string = input('Enter filename', 's'); [filename,path] = uigetfile('*.*');  Using GUI

45. Graphics: plot

46. Graphics: plot(s) ‘hold on’ allows more plots to be added

47. Graphics: contours

48. Graphics: subplots

49. Graphics: labels and titles

50. … or use the Figure window !

51. Graphics: mesh

52. Flow control  if, else, and elseif  switch  while  for

53. if, else, and elseif if logical_expression statements end if (rem(a,2) == 0) disp(‘a is even’) end if (n<0) disp(‘Input must be positive’) elseif (rem(n,2) == 0) disp(‘a is even’) else disp(‘a is odd’) end

54. switch switch expression case value1 statements case value2 statements otherwise statements end switch (input_num) case (-1) disp(‘negative one’) case (0) disp(‘zero’) case (1) disp(‘positive one’) otherwise disp(‘othervalue’) end

55. while, for while expression statements end n = 1; while (prod(1:n) < 1e10) n = n + 1; end for index = start:inc:end statements end for i = 2:6 x(i) = 2*x(i-1); end for i = 1:m for j = 1:n A(i,j) = 1/(i + j - 1); end

56. Vectorization  Efficiency! – Example: Table of logarithms –Find log10 of numbers between 0 and 10. x = 0; for k = 1:1001 y(k) = log10(x); x = x + 0.01; end x = 0:0.01:10; y = log10(x); Looped codeVectorized code AVOID LOOPS!

57. Scripts and functions  Scripts – No input arguments nor output arguments. – Operate on data in the workspace – Useful for automating a series of steps that will be performed many times  Functions – Can accept input arguments and return output arguments – Variables are local to the function – Useful for extending the MATLAB language beyond the built-in functions

58. Script example Comment lines Computations Graphics output  Create m-files: – type edit – use the interface

59. Script example Hit return to advance plot

60. Script Example

61. Function example Function definition Help comments Error check Calculations function [y] = average(x) keyword output argument function name input argument

62. Function example semi-colon

63. M-files  Tips: – use descriptive variable names –e.g. number_students = 20 – functions end with “_fcn” –e.g. an_example_fcn(number_students) – indent if, for, while statements 3-4 spaces – add comments using ‘%’ – develop with smaller portions of data – debug by pasting line-by-line from M-file

64. Online tutorial http://www.imrtweb.ethz.ch/matlab/ Or google “mathworks tutorial” Fundamentals Exercises Examples

65. Part II: Practical Programming  Image analysis on 2D section – Read image data (micro-CT) – Plot the image – Examine image – Filter the image – Create a histogram – Threshold (distinguish between bone and marrow) – Calculate porosity – Centroid, principal axes

66. Read 2D Image  Want to import the image from micro-CT  Write a program to: – clear workspace – set filename – load file – Image data to visualize: /Phalanx/hp2d_34-95_380.bmp

67. Set up main program (a script file) comments clear workspace close all figure windows file to load load it Note: You need to write the 2D image reading function.

68. Read in 2D image Read the file Convert data

69. Display input data

72. Examine image  min, max, mean

73. Examine image  min, max, mean

74. Image Processing  Filter  Thresholding – histogram  Porosity = 1 - (bone area / total area)  Moment of area – First moment of area – Centroid – Second moment of area

75. Filtering – example 1

76. Filtering – example 2 filtered original imagefiltered noisy image original imagenoisy image

77. Filtering the data  Apply Gaussian filtration – Shape of filter depends on: window size [n1 n2] and 

78. Filtering Window= 3  = 1.2 Filter

79. Filtering Window= 5  = 1.2 Filter

80. Filtering Window= 15  = 1.2 Filter

81. Filtering Window= 15  = 7 Filter

82. Filtering Window= 15  = 100 Filter

83. Filtering: Convolution

84. Filtering: Edge Effects edge data nulled Important to begin with image dimensions large enough to account for edge effects

85. Filtering  Use built-in functions: – fspecial – imfilter – (smooth3)

86. Thresholding Good to try different thresholds to see the effect.  Segmentation of bone – divides bone from other tissue

87. Thresholding threshold2D = zeros matrix Find all indices (i,j) where element(i,j) > threshold threshold2D(i,j) = 1

88. Thresholding

89. Threshold = 90Threshold = 120

90. Histogram – lookfor and help

91. Histogram  hist works on vectors, not 2D matrices – need reshape function to vectorize

92. Porosity  Porosity = 1 - (Bone volume divided by total volume)  Need total bone volume (area) – Create mask: –filter image to remove holes –threshold filtered image – Count total “mask” pixels – Count total “bone” pixels within mask

93. Porosity  Create mask: original image filtered image threshold filtered image Image 3 super-imposed on image 1 123

94. Porosity Porosity = 1 - Mask pixels Bone pixels within mask Thresholded image (bone) Mask image Image of bone within the mask

95. Image analysis – determine porosity Original Segmenting the bone FilteredThresholded Creating the mask FilteredThresholded Find bone within mask Porosity = 1 - Area bone Area mask

96. Erosion, dilation ErodeErode = 2 Erode = 4 DilateDilate = 2 Dilate = 4

97. Part of MAIN program

98. Stress in Pure Bending  Mechanical stiffness – First moment of area – Centroid of area – Second moment of area

99. First moment of an area  First moment of area with respect to the X axis:  First moment of area with respect to the Y axis: Can be positive, negative, or zero depending on the axis location (mm 3 )

100. Centroid of an area  Centroid of area defined as:  Knowing the first moment:

101. First moment of area

102. Second moment of area  Stress resulting from bending: σ = My/I  Resistance against bending. – Take I x, I y, I xy about centroid C – Or use the parallel axis theorem: I x = I xc + Ad 2

103. Second moment of area - rotation  Transformation of 2nd moments:

104. Second moment of area - principal axes  Two values of  for which I xy = 0.  Can calculate the max/min 2nd moments: maximum associated with I x minimum associated with I y

105. Principal axes & Transformation

106. 3D: Fracture healing rightleft

107. 2D: Pompe Disease RadiusTibia

108. Mouse knees NTRTCTRL